1,1,86,131,0.2876087,"\int \cos ^2(c+d x) (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","\frac{a \left(-160 (A+2 C) \sin ^3(c+d x)+480 (A+C) \sin (c+d x)+15 (4 (4 A+3 C) (c+d x)+8 (A+C) \sin (2 (c+d x))+C \sin (4 (c+d x)))+96 C \sin ^5(c+d x)\right)}{480 d}","-\frac{a (5 A+4 C) \sin ^3(c+d x)}{15 d}+\frac{a (5 A+4 C) \sin (c+d x)}{5 d}+\frac{a (4 A+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 A+3 C)+\frac{a C \sin (c+d x) \cos ^4(c+d x)}{5 d}+\frac{a C \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"(a*(480*(A + C)*Sin[c + d*x] - 160*(A + 2*C)*Sin[c + d*x]^3 + 96*C*Sin[c + d*x]^5 + 15*(4*(4*A + 3*C)*(c + d*x) + 8*(A + C)*Sin[2*(c + d*x)] + C*Sin[4*(c + d*x)])))/(480*d)","A",1
2,1,77,108,0.2295285,"\int \cos (c+d x) (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","\frac{a (24 (4 A+3 C) \sin (c+d x)+24 (A+C) \sin (2 (c+d x))+48 A c+48 A d x+8 C \sin (3 (c+d x))+3 C \sin (4 (c+d x))+36 c C+36 C d x)}{96 d}","\frac{a (3 A+2 C) \sin (c+d x)}{3 d}+\frac{a (4 A+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 A+3 C)+\frac{a C \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{a C \sin (c+d x) \cos ^2(c+d x)}{3 d}",1,"(a*(48*A*c + 36*c*C + 48*A*d*x + 36*C*d*x + 24*(4*A + 3*C)*Sin[c + d*x] + 24*(A + C)*Sin[2*(c + d*x)] + 8*C*Sin[3*(c + d*x)] + 3*C*Sin[4*(c + d*x)]))/(96*d)","A",1
3,1,59,81,0.1336235,"\int (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","\frac{a (3 (4 A+3 C) \sin (c+d x)+12 A d x+3 C \sin (2 (c+d x))+C \sin (3 (c+d x))+6 c C+6 C d x)}{12 d}","\frac{a (3 A+C) \sin (c+d x)}{3 d}+\frac{1}{2} a x (2 A+C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 a d}-\frac{a C \sin (c+d x) \cos (c+d x)}{6 d}",1,"(a*(6*c*C + 12*A*d*x + 6*C*d*x + 3*(4*A + 3*C)*Sin[c + d*x] + 3*C*Sin[2*(c + d*x)] + C*Sin[3*(c + d*x)]))/(12*d)","A",1
4,1,52,58,0.4086106,"\int (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{a \left(4 A \tanh ^{-1}(\sin (c+d x))+4 A d x+4 C \sin (c+d x)+C \sin (2 (c+d x))+2 c C+2 C d x\right)}{4 d}","\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a x (2 A+C)+\frac{a C \sin (c+d x)}{d}+\frac{a C \sin (c+d x) \cos (c+d x)}{2 d}",1,"(a*(2*c*C + 4*A*d*x + 2*C*d*x + 4*A*ArcTanh[Sin[c + d*x]] + 4*C*Sin[c + d*x] + C*Sin[2*(c + d*x)]))/(4*d)","A",1
5,1,54,42,0.0248388,"\int (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{a A \tan (c+d x)}{d}+\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a C \sin (c) \cos (d x)}{d}+\frac{a C \cos (c) \sin (d x)}{d}+a C x","\frac{a A \tan (c+d x)}{d}+\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a C \sin (c+d x)}{d}+a C x",1,"a*C*x + (a*A*ArcTanh[Sin[c + d*x]])/d + (a*C*Cos[d*x]*Sin[c])/d + (a*C*Cos[c]*Sin[d*x])/d + (a*A*Tan[c + d*x])/d","A",1
6,1,67,58,0.0259267,"\int (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{a A \tan (c+d x)}{d}+\frac{a A \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d}+a C x","\frac{a (A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a A \tan (c+d x)}{d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}+a C x",1,"a*C*x + (a*A*ArcTanh[Sin[c + d*x]])/(2*d) + (a*C*ArcTanh[Sin[c + d*x]])/d + (a*A*Tan[c + d*x])/d + (a*A*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
7,1,56,86,0.2708159,"\int (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a \left(3 (A+2 C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(2 A \tan ^2(c+d x)+3 A \sec (c+d x)+6 (A+C)\right)\right)}{6 d}","\frac{a (2 A+3 C) \tan (c+d x)}{3 d}+\frac{a (A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}",1,"(a*(3*(A + 2*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(6*(A + C) + 3*A*Sec[c + d*x] + 2*A*Tan[c + d*x]^2)))/(6*d)","A",1
8,1,75,117,0.4085908,"\int (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{a \left(3 (3 A+4 C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(3 (3 A+4 C) \sec (c+d x)+8 A \tan ^2(c+d x)+6 A \sec ^3(c+d x)+24 (A+C)\right)\right)}{24 d}","\frac{a (2 A+3 C) \tan (c+d x)}{3 d}+\frac{a (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (3 A+4 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"(a*(3*(3*A + 4*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(24*(A + C) + 3*(3*A + 4*C)*Sec[c + d*x] + 6*A*Sec[c + d*x]^3 + 8*A*Tan[c + d*x]^2)))/(24*d)","A",1
9,1,123,194,0.4900444,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2),x]","\frac{a^2 (240 (6 A+5 C) \sin (c+d x)+15 (32 A+31 C) \sin (2 (c+d x))+160 A \sin (3 (c+d x))+30 A \sin (4 (c+d x))+840 A d x+200 C \sin (3 (c+d x))+75 C \sin (4 (c+d x))+24 C \sin (5 (c+d x))+5 C \sin (6 (c+d x))+420 c C+660 C d x)}{960 d}","-\frac{2 a^2 (5 A+4 C) \sin ^3(c+d x)}{15 d}+\frac{2 a^2 (5 A+4 C) \sin (c+d x)}{5 d}+\frac{a^2 (10 A+9 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{a^2 (14 A+11 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^2 x (14 A+11 C)+\frac{C \sin (c+d x) \cos ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{15 d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}",1,"(a^2*(420*c*C + 840*A*d*x + 660*C*d*x + 240*(6*A + 5*C)*Sin[c + d*x] + 15*(32*A + 31*C)*Sin[2*(c + d*x)] + 160*A*Sin[3*(c + d*x)] + 200*C*Sin[3*(c + d*x)] + 30*A*Sin[4*(c + d*x)] + 75*C*Sin[4*(c + d*x)] + 24*C*Sin[5*(c + d*x)] + 5*C*Sin[6*(c + d*x)]))/(960*d)","A",1
10,1,97,163,0.3825264,"\int \cos (c+d x) (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2),x]","\frac{a^2 (30 (14 A+11 C) \sin (c+d x)+120 (A+C) \sin (2 (c+d x))+20 A \sin (3 (c+d x))+240 A d x+45 C \sin (3 (c+d x))+15 C \sin (4 (c+d x))+3 C \sin (5 (c+d x))+120 c C+180 C d x)}{240 d}","\frac{a^2 (4 A+3 C) \sin (c+d x)}{3 d}+\frac{a^2 (4 A+3 C) \sin (c+d x) \cos (c+d x)}{12 d}+\frac{1}{4} a^2 x (4 A+3 C)+\frac{(10 A+3 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{30 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^2}{5 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{10 a d}",1,"(a^2*(120*c*C + 240*A*d*x + 180*C*d*x + 30*(14*A + 11*C)*Sin[c + d*x] + 120*(A + C)*Sin[2*(c + d*x)] + 20*A*Sin[3*(c + d*x)] + 45*C*Sin[3*(c + d*x)] + 15*C*Sin[4*(c + d*x)] + 3*C*Sin[5*(c + d*x)]))/(240*d)","A",1
11,1,73,123,0.2685685,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2),x]","\frac{a^2 (48 (4 A+3 C) \sin (c+d x)+24 (A+2 C) \sin (2 (c+d x))+144 A d x+16 C \sin (3 (c+d x))+3 C \sin (4 (c+d x))+84 C d x)}{96 d}","\frac{a^2 (12 A+7 C) \sin (c+d x)}{6 d}+\frac{a^2 (12 A+7 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} a^2 x (12 A+7 C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d}-\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{12 d}",1,"(a^2*(144*A*d*x + 84*C*d*x + 48*(4*A + 3*C)*Sin[c + d*x] + 24*(A + 2*C)*Sin[2*(c + d*x)] + 16*C*Sin[3*(c + d*x)] + 3*C*Sin[4*(c + d*x)]))/(96*d)","A",1
12,1,109,96,0.23132,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{a^2 \left(3 (4 A+7 C) \sin (c+d x)-12 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+24 A d x+6 C \sin (2 (c+d x))+C \sin (3 (c+d x))+12 C d x\right)}{12 d}","\frac{a^2 (A+C) \sin (c+d x)}{d}+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+a^2 x (2 A+C)+\frac{C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{3 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}",1,"(a^2*(24*A*d*x + 12*C*d*x - 12*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 3*(4*A + 7*C)*Sin[c + d*x] + 6*C*Sin[2*(c + d*x)] + C*Sin[3*(c + d*x)]))/(12*d)","A",1
13,1,109,112,0.417909,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{a^2 \left(4 A \tan (c+d x)-8 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+8 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+4 A c+4 A d x+8 C \sin (c+d x)+C \sin (2 (c+d x))+6 c C+6 C d x\right)}{4 d}","-\frac{a^2 (2 A-3 C) \sin (c+d x)}{2 d}-\frac{(2 A-C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{2 d}+\frac{2 a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^2 x (2 A+3 C)+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^2}{d}",1,"(a^2*(4*A*c + 6*c*C + 4*A*d*x + 6*C*d*x - 8*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 8*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 8*C*Sin[c + d*x] + C*Sin[2*(c + d*x)] + 4*A*Tan[c + d*x]))/(4*d)","A",1
14,1,293,112,2.2000995,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{1}{16} a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(-\frac{2 (3 A+2 C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{2 (3 A+2 C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{8 A \sin \left(\frac{d x}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{8 A \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{A}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{A}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{4 C \sin (c) \cos (d x)}{d}+\frac{4 C \cos (c) \sin (d x)}{d}+8 C x\right)","-\frac{a^2 (3 A-2 C) \sin (c+d x)}{2 d}+\frac{a^2 (3 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{d}+2 a^2 C x+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}",1,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(8*C*x - (2*(3*A + 2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (2*(3*A + 2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (4*C*Cos[d*x]*Sin[c])/d + (4*C*Cos[c]*Sin[d*x])/d + A/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (8*A*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - A/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (8*A*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/16","B",1
15,1,748,110,6.4284049,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^2 \left(5 A \sin \left(\frac{d x}{2}\right)+3 C \sin \left(\frac{d x}{2}\right)\right)}{12 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^2 \left(5 A \sin \left(\frac{d x}{2}\right)+3 C \sin \left(\frac{d x}{2}\right)\right)}{12 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{(-A-2 C) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^2 \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{4 d}+\frac{(A+2 C) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^2 \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{4 d}+\frac{\left(7 A \cos \left(\frac{c}{2}\right)-5 A \sin \left(\frac{c}{2}\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^2}{48 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\left(-5 A \sin \left(\frac{c}{2}\right)-7 A \cos \left(\frac{c}{2}\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^2}{48 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{A \sin \left(\frac{d x}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^2}{24 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{A \sin \left(\frac{d x}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^2}{24 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{1}{4} C x \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^2","\frac{a^2 (A+C) \tan (c+d x)}{d}+\frac{a^2 (A+2 C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A \tan (c+d x) \sec (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{3 d}+a^2 C x+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^2}{3 d}",1,"(C*x*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4)/4 + ((-A - 2*C)*(a + a*Cos[c + d*x])^2*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^4)/(4*d) + ((A + 2*C)*(a + a*Cos[c + d*x])^2*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^4)/(4*d) + (A*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*Sin[(d*x)/2])/(24*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + ((a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(7*A*Cos[c/2] - 5*A*Sin[c/2]))/(48*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + ((a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(5*A*Sin[(d*x)/2] + 3*C*Sin[(d*x)/2]))/(12*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (A*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*Sin[(d*x)/2])/(24*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + ((a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(-7*A*Cos[c/2] - 5*A*Sin[c/2]))/(48*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + ((a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(5*A*Sin[(d*x)/2] + 3*C*Sin[(d*x)/2]))/(12*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",1
16,1,262,147,1.1873193,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","-\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \left(24 (7 A+12 C) \cos ^4(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-48 (2 A+3 C) \sin (c)+45 A \sin (2 c+d x)+128 A \sin (c+2 d x)+21 A \sin (2 c+3 d x)+21 A \sin (4 c+3 d x)+32 A \sin (3 c+4 d x)+3 (15 A+4 C) \sin (d x)+12 C \sin (2 c+d x)+144 C \sin (c+2 d x)-48 C \sin (3 c+2 d x)+12 C \sin (2 c+3 d x)+12 C \sin (4 c+3 d x)+48 C \sin (3 c+4 d x))\right)}{768 d}","\frac{2 a^2 (2 A+3 C) \tan (c+d x)}{3 d}+\frac{a^2 (7 A+12 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (11 A+12 C) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{6 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}",1,"-1/768*(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*Sec[c + d*x]^4*(24*(7*A + 12*C)*Cos[c + d*x]^4*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(-48*(2*A + 3*C)*Sin[c] + 3*(15*A + 4*C)*Sin[d*x] + 45*A*Sin[2*c + d*x] + 12*C*Sin[2*c + d*x] + 128*A*Sin[c + 2*d*x] + 144*C*Sin[c + 2*d*x] - 48*C*Sin[3*c + 2*d*x] + 21*A*Sin[2*c + 3*d*x] + 12*C*Sin[2*c + 3*d*x] + 21*A*Sin[4*c + 3*d*x] + 12*C*Sin[4*c + 3*d*x] + 32*A*Sin[3*c + 4*d*x] + 48*C*Sin[3*c + 4*d*x])))/d","A",1
17,1,292,178,1.4699946,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","-\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \left(240 (3 A+4 C) \cos ^5(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-120 (A+3 C) \sin (2 c+d x)+210 A \sin (c+2 d x)+210 A \sin (3 c+2 d x)+360 A \sin (2 c+3 d x)+45 A \sin (3 c+4 d x)+45 A \sin (5 c+4 d x)+72 A \sin (4 c+5 d x)+40 (15 A+16 C) \sin (d x)+120 C \sin (c+2 d x)+120 C \sin (3 c+2 d x)+440 C \sin (2 c+3 d x)-60 C \sin (4 c+3 d x)+60 C \sin (3 c+4 d x)+60 C \sin (5 c+4 d x)+100 C \sin (4 c+5 d x))\right)}{3840 d}","\frac{a^2 (18 A+25 C) \tan (c+d x)}{15 d}+\frac{a^2 (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a^2 (9 A+10 C) \tan (c+d x) \sec ^2(c+d x)}{30 d}+\frac{a^2 (3 A+4 C) \tan (c+d x) \sec (c+d x)}{4 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{10 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}",1,"-1/3840*(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*Sec[c + d*x]^5*(240*(3*A + 4*C)*Cos[c + d*x]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(40*(15*A + 16*C)*Sin[d*x] - 120*(A + 3*C)*Sin[2*c + d*x] + 210*A*Sin[c + 2*d*x] + 120*C*Sin[c + 2*d*x] + 210*A*Sin[3*c + 2*d*x] + 120*C*Sin[3*c + 2*d*x] + 360*A*Sin[2*c + 3*d*x] + 440*C*Sin[2*c + 3*d*x] - 60*C*Sin[4*c + 3*d*x] + 45*A*Sin[3*c + 4*d*x] + 60*C*Sin[3*c + 4*d*x] + 45*A*Sin[5*c + 4*d*x] + 60*C*Sin[5*c + 4*d*x] + 72*A*Sin[4*c + 5*d*x] + 100*C*Sin[4*c + 5*d*x])))/d","A",1
18,1,145,237,0.6875601,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2),x]","\frac{a^3 (105 (184 A+155 C) \sin (c+d x)+105 (64 A+61 C) \sin (2 (c+d x))+2380 A \sin (3 (c+d x))+630 A \sin (4 (c+d x))+84 A \sin (5 (c+d x))+10920 A d x+2835 C \sin (3 (c+d x))+1155 C \sin (4 (c+d x))+399 C \sin (5 (c+d x))+105 C \sin (6 (c+d x))+15 C \sin (7 (c+d x))+5460 c C+8820 C d x)}{6720 d}","-\frac{a^3 (133 A+108 C) \sin ^3(c+d x)}{105 d}+\frac{a^3 (133 A+108 C) \sin (c+d x)}{35 d}+\frac{a^3 (154 A+129 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{(A+C) \sin (c+d x) \cos ^3(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{5 d}+\frac{a^3 (26 A+21 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^3 x (26 A+21 C)+\frac{C \sin (c+d x) \cos ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{14 a d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}",1,"(a^3*(5460*c*C + 10920*A*d*x + 8820*C*d*x + 105*(184*A + 155*C)*Sin[c + d*x] + 105*(64*A + 61*C)*Sin[2*(c + d*x)] + 2380*A*Sin[3*(c + d*x)] + 2835*C*Sin[3*(c + d*x)] + 630*A*Sin[4*(c + d*x)] + 1155*C*Sin[4*(c + d*x)] + 84*A*Sin[5*(c + d*x)] + 399*C*Sin[5*(c + d*x)] + 105*C*Sin[6*(c + d*x)] + 15*C*Sin[7*(c + d*x)]))/(6720*d)","A",1
19,1,123,188,0.4140751,"\int \cos (c+d x) (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2),x]","\frac{a^3 (120 (26 A+21 C) \sin (c+d x)+15 (64 A+63 C) \sin (2 (c+d x))+240 A \sin (3 (c+d x))+30 A \sin (4 (c+d x))+1800 A d x+380 C \sin (3 (c+d x))+135 C \sin (4 (c+d x))+36 C \sin (5 (c+d x))+5 C \sin (6 (c+d x))+900 c C+1380 C d x)}{960 d}","-\frac{a^3 (30 A+23 C) \sin ^3(c+d x)}{120 d}+\frac{a^3 (30 A+23 C) \sin (c+d x)}{10 d}+\frac{3 a^3 (30 A+23 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{1}{16} a^3 x (30 A+23 C)+\frac{(30 A+7 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{120 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^3}{6 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{10 a d}",1,"(a^3*(900*c*C + 1800*A*d*x + 1380*C*d*x + 120*(26*A + 21*C)*Sin[c + d*x] + 15*(64*A + 63*C)*Sin[2*(c + d*x)] + 240*A*Sin[3*(c + d*x)] + 380*C*Sin[3*(c + d*x)] + 30*A*Sin[4*(c + d*x)] + 135*C*Sin[4*(c + d*x)] + 36*C*Sin[5*(c + d*x)] + 5*C*Sin[6*(c + d*x)]))/(960*d)","A",1
20,1,97,148,0.375749,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2),x]","\frac{a^3 (60 (30 A+23 C) \sin (c+d x)+120 (3 A+4 C) \sin (2 (c+d x))+40 A \sin (3 (c+d x))+1200 A d x+170 C \sin (3 (c+d x))+45 C \sin (4 (c+d x))+6 C \sin (5 (c+d x))+780 C d x)}{480 d}","-\frac{a^3 (20 A+13 C) \sin ^3(c+d x)}{60 d}+\frac{a^3 (20 A+13 C) \sin (c+d x)}{5 d}+\frac{3 a^3 (20 A+13 C) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{1}{8} a^3 x (20 A+13 C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}-\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{20 d}",1,"(a^3*(1200*A*d*x + 780*C*d*x + 60*(30*A + 23*C)*Sin[c + d*x] + 120*(3*A + 4*C)*Sin[2*(c + d*x)] + 40*A*Sin[3*(c + d*x)] + 170*C*Sin[3*(c + d*x)] + 45*C*Sin[4*(c + d*x)] + 6*C*Sin[5*(c + d*x)]))/(480*d)","A",1
21,1,124,147,0.3376518,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{a^3 \left(8 (12 A+13 C) \sin (c+d x)+8 (A+4 C) \sin (2 (c+d x))-32 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+32 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+112 A d x+8 C \sin (3 (c+d x))+C \sin (4 (c+d x))+60 C d x\right)}{32 d}","\frac{5 a^3 (4 A+3 C) \sin (c+d x)}{8 d}+\frac{(4 A+5 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{8 d}+\frac{a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{8} a^3 x (28 A+15 C)+\frac{C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{4 a d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}",1,"(a^3*(112*A*d*x + 60*C*d*x - 32*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 32*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 8*(12*A + 13*C)*Sin[c + d*x] + 8*(A + 4*C)*Sin[2*(c + d*x)] + 8*C*Sin[3*(c + d*x)] + C*Sin[4*(c + d*x)]))/(32*d)","A",1
22,1,298,145,1.9660633,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{1}{96} a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(\frac{3 (4 A+15 C) \sin (c) \cos (d x)}{d}+\frac{3 (4 A+15 C) \cos (c) \sin (d x)}{d}+\frac{12 A \sin \left(\frac{d x}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{12 A \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{36 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{36 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+6 x (6 A+5 C)+\frac{9 C \sin (2 c) \cos (2 d x)}{d}+\frac{C \sin (3 c) \cos (3 d x)}{d}+\frac{9 C \cos (2 c) \sin (2 d x)}{d}+\frac{C \cos (3 c) \sin (3 d x)}{d}\right)","-\frac{(6 A-5 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{6 d}+\frac{3 a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^3 x (6 A+5 C)+\frac{5 a^3 C \sin (c+d x)}{2 d}-\frac{(3 A-C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{3 a d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}",1,"(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(6*(6*A + 5*C)*x - (36*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (36*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (3*(4*A + 15*C)*Cos[d*x]*Sin[c])/d + (9*C*Cos[2*d*x]*Sin[2*c])/d + (C*Cos[3*d*x]*Sin[3*c])/d + (3*(4*A + 15*C)*Cos[c]*Sin[d*x])/d + (9*C*Cos[2*c]*Sin[2*d*x])/d + (C*Cos[3*c]*Sin[3*d*x])/d + (12*A*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (12*A*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/96","B",1
23,1,214,160,1.984031,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{a^3 \left(12 A \tan (c+d x)+\frac{A}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{A}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-14 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+14 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+4 A c+4 A d x+12 C \sin (c+d x)+C \sin (2 (c+d x))-4 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+14 c C+14 C d x\right)}{4 d}","-\frac{5 a^3 (A-C) \sin (c+d x)}{2 d}+\frac{a^3 (7 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(4 A-C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{2 d}+\frac{1}{2} a^3 x (2 A+7 C)+\frac{3 A \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{2 a d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^3}{2 d}",1,"(a^3*(4*A*c + 14*c*C + 4*A*d*x + 14*C*d*x - 14*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 4*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 14*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 4*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + A/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 - A/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + 12*C*Sin[c + d*x] + C*Sin[2*(c + d*x)] + 12*A*Tan[c + d*x]))/(4*d)","A",1
24,1,832,156,6.3697232,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{3}{8} C x (\cos (c+d x) a+a)^3 \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{(-5 A-6 C) (\cos (c+d x) a+a)^3 \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 d}+\frac{(5 A+6 C) (\cos (c+d x) a+a)^3 \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 d}+\frac{C \cos (d x) (\cos (c+d x) a+a)^3 \sin (c) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d}+\frac{C \cos (c) (\cos (c+d x) a+a)^3 \sin (d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d}+\frac{(\cos (c+d x) a+a)^3 \left(11 A \sin \left(\frac{d x}{2}\right)+3 C \sin \left(\frac{d x}{2}\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{(\cos (c+d x) a+a)^3 \left(11 A \sin \left(\frac{d x}{2}\right)+3 C \sin \left(\frac{d x}{2}\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{(\cos (c+d x) a+a)^3 \left(5 A \cos \left(\frac{c}{2}\right)-4 A \sin \left(\frac{c}{2}\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{48 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{(\cos (c+d x) a+a)^3 \left(-5 A \cos \left(\frac{c}{2}\right)-4 A \sin \left(\frac{c}{2}\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{48 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{A (\cos (c+d x) a+a)^3 \sin \left(\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{48 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{A (\cos (c+d x) a+a)^3 \sin \left(\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{48 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}","\frac{a^3 (5 A+6 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(5 A+3 C) \tan (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{3 d}-\frac{5 a^3 A \sin (c+d x)}{2 d}+3 a^3 C x+\frac{A \tan (c+d x) \sec (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{2 a d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}",1,"(3*C*x*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6)/8 + ((-5*A - 6*C)*(a + a*Cos[c + d*x])^3*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^6)/(16*d) + ((5*A + 6*C)*(a + a*Cos[c + d*x])^3*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^6)/(16*d) + (C*Cos[d*x]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*Sin[c])/(8*d) + (C*Cos[c]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*Sin[d*x])/(8*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*Sin[(d*x)/2])/(48*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + ((a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(5*A*Cos[c/2] - 4*A*Sin[c/2]))/(48*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + ((a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(11*A*Sin[(d*x)/2] + 3*C*Sin[(d*x)/2]))/(24*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (A*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*Sin[(d*x)/2])/(48*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + ((a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-5*A*Cos[c/2] - 4*A*Sin[c/2]))/(48*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + ((a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(11*A*Sin[(d*x)/2] + 3*C*Sin[(d*x)/2]))/(24*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",1
25,1,334,169,1.4636631,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \left(\sec (c) (23 A \sin (2 c+d x)+88 A \sin (c+2 d x)-8 A \sin (3 c+2 d x)+15 A \sin (2 c+3 d x)+15 A \sin (4 c+3 d x)+24 A \sin (3 c+4 d x)-72 A \sin (c)+23 A \sin (d x)+4 C \sin (2 c+d x)+72 C \sin (c+2 d x)-24 C \sin (3 c+2 d x)+4 C \sin (2 c+3 d x)+4 C \sin (4 c+3 d x)+24 C \sin (3 c+4 d x)+24 C d x \cos (c)+16 C d x \cos (c+2 d x)+16 C d x \cos (3 c+2 d x)+4 C d x \cos (3 c+4 d x)+4 C d x \cos (5 c+4 d x)-72 C \sin (c)+4 C \sin (d x))-8 (15 A+28 C) \cos ^4(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{512 d}","\frac{5 a^3 (3 A+4 C) \tan (c+d x)}{8 d}+\frac{a^3 (15 A+28 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(5 A+4 C) \tan (c+d x) \sec (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{8 d}+a^3 C x+\frac{A \tan (c+d x) \sec ^2(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{4 a d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^3}{4 d}",1,"(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*Sec[c + d*x]^4*(-8*(15*A + 28*C)*Cos[c + d*x]^4*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c]*(24*C*d*x*Cos[c] + 16*C*d*x*Cos[c + 2*d*x] + 16*C*d*x*Cos[3*c + 2*d*x] + 4*C*d*x*Cos[3*c + 4*d*x] + 4*C*d*x*Cos[5*c + 4*d*x] - 72*A*Sin[c] - 72*C*Sin[c] + 23*A*Sin[d*x] + 4*C*Sin[d*x] + 23*A*Sin[2*c + d*x] + 4*C*Sin[2*c + d*x] + 88*A*Sin[c + 2*d*x] + 72*C*Sin[c + 2*d*x] - 8*A*Sin[3*c + 2*d*x] - 24*C*Sin[3*c + 2*d*x] + 15*A*Sin[2*c + 3*d*x] + 4*C*Sin[2*c + 3*d*x] + 15*A*Sin[4*c + 3*d*x] + 4*C*Sin[4*c + 3*d*x] + 24*A*Sin[3*c + 4*d*x] + 24*C*Sin[3*c + 4*d*x])))/(512*d)","A",1
26,1,294,194,1.4860601,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","-\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \left(240 (13 A+20 C) \cos ^5(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-240 (3 A+7 C) \sin (2 c+d x)+750 A \sin (c+2 d x)+750 A \sin (3 c+2 d x)+1520 A \sin (2 c+3 d x)+195 A \sin (3 c+4 d x)+195 A \sin (5 c+4 d x)+304 A \sin (4 c+5 d x)+80 (29 A+34 C) \sin (d x)+360 C \sin (c+2 d x)+360 C \sin (3 c+2 d x)+1840 C \sin (2 c+3 d x)-360 C \sin (4 c+3 d x)+180 C \sin (3 c+4 d x)+180 C \sin (5 c+4 d x)+440 C \sin (4 c+5 d x))\right)}{15360 d}","\frac{a^3 (38 A+55 C) \tan (c+d x)}{15 d}+\frac{a^3 (13 A+20 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 (109 A+140 C) \tan (c+d x) \sec (c+d x)}{120 d}+\frac{(11 A+10 C) \tan (c+d x) \sec ^2(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{30 d}+\frac{3 A \tan (c+d x) \sec ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{20 a d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}",1,"-1/15360*(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*Sec[c + d*x]^5*(240*(13*A + 20*C)*Cos[c + d*x]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(80*(29*A + 34*C)*Sin[d*x] - 240*(3*A + 7*C)*Sin[2*c + d*x] + 750*A*Sin[c + 2*d*x] + 360*C*Sin[c + 2*d*x] + 750*A*Sin[3*c + 2*d*x] + 360*C*Sin[3*c + 2*d*x] + 1520*A*Sin[2*c + 3*d*x] + 1840*C*Sin[2*c + 3*d*x] - 360*C*Sin[4*c + 3*d*x] + 195*A*Sin[3*c + 4*d*x] + 180*C*Sin[3*c + 4*d*x] + 195*A*Sin[5*c + 4*d*x] + 180*C*Sin[5*c + 4*d*x] + 304*A*Sin[4*c + 5*d*x] + 440*C*Sin[4*c + 5*d*x])))/d","A",1
27,1,358,225,2.0266221,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","-\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \sec ^6(c+d x) \left(480 (23 A+30 C) \cos ^6(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-160 (34 A+45 C) \sin (c)+2250 A \sin (2 c+d x)+7680 A \sin (c+2 d x)-480 A \sin (3 c+2 d x)+1955 A \sin (2 c+3 d x)+1955 A \sin (4 c+3 d x)+3264 A \sin (3 c+4 d x)+345 A \sin (4 c+5 d x)+345 A \sin (6 c+5 d x)+544 A \sin (5 c+6 d x)+30 (75 A+38 C) \sin (d x)+1140 C \sin (2 c+d x)+8160 C \sin (c+2 d x)-2640 C \sin (3 c+2 d x)+1590 C \sin (2 c+3 d x)+1590 C \sin (4 c+3 d x)+4080 C \sin (3 c+4 d x)-240 C \sin (5 c+4 d x)+450 C \sin (4 c+5 d x)+450 C \sin (6 c+5 d x)+720 C \sin (5 c+6 d x))\right)}{61440 d}","\frac{a^3 (34 A+45 C) \tan (c+d x)}{15 d}+\frac{a^3 (23 A+30 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 (73 A+90 C) \tan (c+d x) \sec ^2(c+d x)}{120 d}+\frac{a^3 (23 A+30 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(31 A+30 C) \tan (c+d x) \sec ^3(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{120 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{10 a d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}",1,"-1/61440*(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*Sec[c + d*x]^6*(480*(23*A + 30*C)*Cos[c + d*x]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(-160*(34*A + 45*C)*Sin[c] + 30*(75*A + 38*C)*Sin[d*x] + 2250*A*Sin[2*c + d*x] + 1140*C*Sin[2*c + d*x] + 7680*A*Sin[c + 2*d*x] + 8160*C*Sin[c + 2*d*x] - 480*A*Sin[3*c + 2*d*x] - 2640*C*Sin[3*c + 2*d*x] + 1955*A*Sin[2*c + 3*d*x] + 1590*C*Sin[2*c + 3*d*x] + 1955*A*Sin[4*c + 3*d*x] + 1590*C*Sin[4*c + 3*d*x] + 3264*A*Sin[3*c + 4*d*x] + 4080*C*Sin[3*c + 4*d*x] - 240*C*Sin[5*c + 4*d*x] + 345*A*Sin[4*c + 5*d*x] + 450*C*Sin[4*c + 5*d*x] + 345*A*Sin[6*c + 5*d*x] + 450*C*Sin[6*c + 5*d*x] + 544*A*Sin[5*c + 6*d*x] + 720*C*Sin[5*c + 6*d*x])))/d","A",1
28,1,167,279,0.9322629,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2),x]","\frac{a^4 (6720 (88 A+75 C) \sin (c+d x)+1680 (127 A+120 C) \sin (2 (c+d x))+80640 A \sin (3 (c+d x))+25200 A \sin (4 (c+d x))+5376 A \sin (5 (c+d x))+560 A \sin (6 (c+d x))+329280 A d x+91840 C \sin (3 (c+d x))+39480 C \sin (4 (c+d x))+14784 C \sin (5 (c+d x))+4480 C \sin (6 (c+d x))+960 C \sin (7 (c+d x))+105 C \sin (8 (c+d x))+164640 c C+271320 C d x)}{107520 d}","-\frac{4 a^4 (63 A+52 C) \sin ^3(c+d x)}{105 d}+\frac{4 a^4 (63 A+52 C) \sin (c+d x)}{35 d}+\frac{a^4 (2408 A+2007 C) \sin (c+d x) \cos ^3(c+d x)}{2240 d}+\frac{7 (8 A+7 C) \sin (c+d x) \cos ^3(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{120 d}+\frac{a^4 (392 A+323 C) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{1}{128} a^4 x (392 A+323 C)+\frac{(56 A+61 C) \sin (c+d x) \cos ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{336 d}+\frac{a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{14 d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}",1,"(a^4*(164640*c*C + 329280*A*d*x + 271320*C*d*x + 6720*(88*A + 75*C)*Sin[c + d*x] + 1680*(127*A + 120*C)*Sin[2*(c + d*x)] + 80640*A*Sin[3*(c + d*x)] + 91840*C*Sin[3*(c + d*x)] + 25200*A*Sin[4*(c + d*x)] + 39480*C*Sin[4*(c + d*x)] + 5376*A*Sin[5*(c + d*x)] + 14784*C*Sin[5*(c + d*x)] + 560*A*Sin[6*(c + d*x)] + 4480*C*Sin[6*(c + d*x)] + 960*C*Sin[7*(c + d*x)] + 105*C*Sin[8*(c + d*x)]))/(107520*d)","A",1
29,1,145,219,0.5862197,"\int \cos (c+d x) (a+a \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2),x]","\frac{a^4 (105 (392 A+323 C) \sin (c+d x)+420 (32 A+31 C) \sin (2 (c+d x))+4060 A \sin (3 (c+d x))+840 A \sin (4 (c+d x))+84 A \sin (5 (c+d x))+23520 A d x+5495 C \sin (3 (c+d x))+2100 C \sin (4 (c+d x))+651 C \sin (5 (c+d x))+140 C \sin (6 (c+d x))+15 C \sin (7 (c+d x))+11760 c C+18480 C d x)}{6720 d}","-\frac{8 a^4 (14 A+11 C) \sin ^3(c+d x)}{105 d}+\frac{16 a^4 (14 A+11 C) \sin (c+d x)}{35 d}+\frac{a^4 (14 A+11 C) \sin (c+d x) \cos ^3(c+d x)}{70 d}+\frac{27 a^4 (14 A+11 C) \sin (c+d x) \cos (c+d x)}{140 d}+\frac{1}{4} a^4 x (14 A+11 C)+\frac{(21 A+4 C) \sin (c+d x) (a \cos (c+d x)+a)^4}{105 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^5}{21 a d}",1,"(a^4*(11760*c*C + 23520*A*d*x + 18480*C*d*x + 105*(392*A + 323*C)*Sin[c + d*x] + 420*(32*A + 31*C)*Sin[2*(c + d*x)] + 4060*A*Sin[3*(c + d*x)] + 5495*C*Sin[3*(c + d*x)] + 840*A*Sin[4*(c + d*x)] + 2100*C*Sin[4*(c + d*x)] + 84*A*Sin[5*(c + d*x)] + 651*C*Sin[5*(c + d*x)] + 140*C*Sin[6*(c + d*x)] + 15*C*Sin[7*(c + d*x)]))/(6720*d)","A",1
30,1,119,179,0.3854282,"\int (a+a \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2),x]","\frac{a^4 (480 (14 A+11 C) \sin (c+d x)+15 (112 A+127 C) \sin (2 (c+d x))+320 A \sin (3 (c+d x))+30 A \sin (4 (c+d x))+4200 A d x+720 C \sin (3 (c+d x))+225 C \sin (4 (c+d x))+48 C \sin (5 (c+d x))+5 C \sin (6 (c+d x))+2940 C d x)}{960 d}","-\frac{2 a^4 (10 A+7 C) \sin ^3(c+d x)}{15 d}+\frac{4 a^4 (10 A+7 C) \sin (c+d x)}{5 d}+\frac{a^4 (10 A+7 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{27 a^4 (10 A+7 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{7}{16} a^4 x (10 A+7 C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^5}{6 a d}-\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{30 d}",1,"(a^4*(4200*A*d*x + 2940*C*d*x + 480*(14*A + 11*C)*Sin[c + d*x] + 15*(112*A + 127*C)*Sin[2*(c + d*x)] + 320*A*Sin[3*(c + d*x)] + 720*C*Sin[3*(c + d*x)] + 30*A*Sin[4*(c + d*x)] + 225*C*Sin[4*(c + d*x)] + 48*C*Sin[5*(c + d*x)] + 5*C*Sin[6*(c + d*x)]))/(960*d)","A",1
31,1,147,177,0.4859283,"\int (a+a \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{a^4 \left(30 (54 A+49 C) \sin (c+d x)+240 (A+2 C) \sin (2 (c+d x))+20 A \sin (3 (c+d x))-240 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+240 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+1440 A d x+145 C \sin (3 (c+d x))+30 C \sin (4 (c+d x))+3 C \sin (5 (c+d x))+840 C d x\right)}{240 d}","\frac{a^4 (10 A+7 C) \sin (c+d x)}{2 d}+\frac{(8 A+7 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{6 d}+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^4 x (12 A+7 C)+\frac{(5 A+7 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{15 d}+\frac{a C \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}",1,"(a^4*(1440*A*d*x + 840*C*d*x - 240*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 240*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 30*(54*A + 49*C)*Sin[c + d*x] + 240*(A + 2*C)*Sin[2*(c + d*x)] + 20*A*Sin[3*(c + d*x)] + 145*C*Sin[3*(c + d*x)] + 30*C*Sin[4*(c + d*x)] + 3*C*Sin[5*(c + d*x)]))/(240*d)","A",1
32,1,338,181,2.1663011,"\int (a+a \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \left(\frac{96 (4 A+7 C) \sin (c) \cos (d x)}{d}+\frac{24 (A+7 C) \sin (2 c) \cos (2 d x)}{d}+\frac{96 (4 A+7 C) \cos (c) \sin (d x)}{d}+\frac{24 (A+7 C) \cos (2 c) \sin (2 d x)}{d}+\frac{96 A \sin \left(\frac{d x}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{96 A \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{384 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{384 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+12 x (52 A+35 C)+\frac{32 C \sin (3 c) \cos (3 d x)}{d}+\frac{3 C \sin (4 c) \cos (4 d x)}{d}+\frac{32 C \cos (3 c) \sin (3 d x)}{d}+\frac{3 C \cos (4 c) \sin (4 d x)}{d}\right)}{1536}","\frac{5 a^4 (4 A+7 C) \sin (c+d x)}{8 d}-\frac{(12 A-35 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{24 d}+\frac{4 a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{8} a^4 x (52 A+35 C)-\frac{(12 A-7 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{12 d}-\frac{a (4 A-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^4}{d}",1,"(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*(12*(52*A + 35*C)*x - (384*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (384*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (96*(4*A + 7*C)*Cos[d*x]*Sin[c])/d + (24*(A + 7*C)*Cos[2*d*x]*Sin[2*c])/d + (32*C*Cos[3*d*x]*Sin[3*c])/d + (3*C*Cos[4*d*x]*Sin[4*c])/d + (96*(4*A + 7*C)*Cos[c]*Sin[d*x])/d + (24*(A + 7*C)*Cos[2*c]*Sin[2*d*x])/d + (32*C*Cos[3*c]*Sin[3*d*x])/d + (3*C*Cos[4*c]*Sin[4*d*x])/d + (96*A*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (96*A*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/1536","A",1
33,1,756,186,6.2380878,"\int (a+a \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{1}{8} x (2 A+3 C) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^4+\frac{(4 A+27 C) \sin (c) \cos (d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^4}{64 d}+\frac{(4 A+27 C) \cos (c) \sin (d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^4}{64 d}+\frac{(-13 A-2 C) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^4 \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{32 d}+\frac{(13 A+2 C) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^4 \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{32 d}+\frac{A \sin \left(\frac{d x}{2}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^4}{4 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{A \sin \left(\frac{d x}{2}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^4}{4 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{A \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^4}{64 d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}-\frac{A \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^4}{64 d \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{C \sin (2 c) \cos (2 d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^4}{16 d}+\frac{C \sin (3 c) \cos (3 d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^4}{192 d}+\frac{C \cos (2 c) \sin (2 d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^4}{16 d}+\frac{C \cos (3 c) \sin (3 d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^4}{192 d}","-\frac{5 a^4 (A-2 C) \sin (c+d x)}{2 d}+\frac{a^4 (13 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(9 A-4 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{3 d}+2 a^4 x (2 A+3 C)-\frac{(15 A-2 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{6 d}+\frac{2 a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d}",1,"((2*A + 3*C)*x*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8)/8 + ((-13*A - 2*C)*(a + a*Cos[c + d*x])^4*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^8)/(32*d) + ((13*A + 2*C)*(a + a*Cos[c + d*x])^4*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^8)/(32*d) + ((4*A + 27*C)*Cos[d*x]*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*Sin[c])/(64*d) + (C*Cos[2*d*x]*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*Sin[2*c])/(16*d) + (C*Cos[3*d*x]*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*Sin[3*c])/(192*d) + ((4*A + 27*C)*Cos[c]*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*Sin[d*x])/(64*d) + (C*Cos[2*c]*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*Sin[2*d*x])/(16*d) + (C*Cos[3*c]*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*Sin[3*d*x])/(192*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8)/(64*d*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (A*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*Sin[(d*x)/2])/(4*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) - (A*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8)/(64*d*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (A*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*Sin[(d*x)/2])/(4*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",1
34,1,386,198,6.2206789,"\int (a+a \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","a^4 \left(\frac{(2 A+13 C) (c+d x)}{2 d}+\frac{20 A \sin \left(\frac{1}{2} (c+d x)\right)+3 C \sin \left(\frac{1}{2} (c+d x)\right)}{3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{20 A \sin \left(\frac{1}{2} (c+d x)\right)+3 C \sin \left(\frac{1}{2} (c+d x)\right)}{3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{2 (3 A+2 C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{2 (3 A+2 C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{A \sin \left(\frac{1}{2} (c+d x)\right)}{6 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{13 A}{12 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{13 A}{12 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{A \sin \left(\frac{1}{2} (c+d x)\right)}{6 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{4 C \sin (c+d x)}{d}+\frac{C \sin (2 (c+d x))}{4 d}\right)","-\frac{5 a^4 (2 A-C) \sin (c+d x)}{2 d}+\frac{2 a^4 (3 A+2 C) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{(22 A+3 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{6 d}+\frac{1}{2} a^4 x (2 A+13 C)+\frac{(8 A+3 C) \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{3 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^4}{3 d}+\frac{2 a A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^3}{3 d}",1,"a^4*(((2*A + 13*C)*(c + d*x))/(2*d) - (2*(3*A + 2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (2*(3*A + 2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (13*A)/(12*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (A*Sin[(c + d*x)/2])/(6*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + (A*Sin[(c + d*x)/2])/(6*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (13*A)/(12*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (20*A*Sin[(c + d*x)/2] + 3*C*Sin[(c + d*x)/2])/(3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (20*A*Sin[(c + d*x)/2] + 3*C*Sin[(c + d*x)/2])/(3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (4*C*Sin[c + d*x])/d + (C*Sin[2*(c + d*x)])/(4*d))","A",1
35,1,350,200,2.1625677,"\int (a+a \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{a^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^4 \left(\sec (c) (105 A \sin (2 c+d x)+544 A \sin (c+2 d x)-96 A \sin (3 c+2 d x)+81 A \sin (2 c+3 d x)+81 A \sin (4 c+3 d x)+160 A \sin (3 c+4 d x)-480 A \sin (c)+105 A \sin (d x)+24 C \sin (2 c+d x)+288 C \sin (c+2 d x)-96 C \sin (3 c+2 d x)+30 C \sin (2 c+3 d x)+30 C \sin (4 c+3 d x)+96 C \sin (3 c+4 d x)+6 C \sin (4 c+5 d x)+6 C \sin (6 c+5 d x)+288 C d x \cos (c)+192 C d x \cos (c+2 d x)+192 C d x \cos (3 c+2 d x)+48 C d x \cos (3 c+4 d x)+48 C d x \cos (5 c+4 d x)-288 C \sin (c)+24 C \sin (d x))-24 (35 A+52 C) \cos ^4(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{3072 d}","-\frac{5 a^4 (7 A+4 C) \sin (c+d x)}{8 d}+\frac{a^4 (35 A+52 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(35 A+36 C) \tan (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{12 d}+4 a^4 C x+\frac{(7 A+4 C) \tan (c+d x) \sec (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{8 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}",1,"(a^4*Sec[(c + d*x)/2]^8*(1 + Sec[c + d*x])^4*(-24*(35*A + 52*C)*Cos[c + d*x]^4*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c]*(288*C*d*x*Cos[c] + 192*C*d*x*Cos[c + 2*d*x] + 192*C*d*x*Cos[3*c + 2*d*x] + 48*C*d*x*Cos[3*c + 4*d*x] + 48*C*d*x*Cos[5*c + 4*d*x] - 480*A*Sin[c] - 288*C*Sin[c] + 105*A*Sin[d*x] + 24*C*Sin[d*x] + 105*A*Sin[2*c + d*x] + 24*C*Sin[2*c + d*x] + 544*A*Sin[c + 2*d*x] + 288*C*Sin[c + 2*d*x] - 96*A*Sin[3*c + 2*d*x] - 96*C*Sin[3*c + 2*d*x] + 81*A*Sin[2*c + 3*d*x] + 30*C*Sin[2*c + 3*d*x] + 81*A*Sin[4*c + 3*d*x] + 30*C*Sin[4*c + 3*d*x] + 160*A*Sin[3*c + 4*d*x] + 96*C*Sin[3*c + 4*d*x] + 6*C*Sin[4*c + 5*d*x] + 6*C*Sin[6*c + 5*d*x])))/(3072*d)","A",1
36,1,389,207,1.8328897,"\int (a+a \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \left(\sec (c) (-480 A \sin (2 c+d x)+330 A \sin (c+2 d x)+330 A \sin (3 c+2 d x)+800 A \sin (2 c+3 d x)-30 A \sin (4 c+3 d x)+105 A \sin (3 c+4 d x)+105 A \sin (5 c+4 d x)+166 A \sin (4 c+5 d x)+1180 A \sin (d x)-780 C \sin (2 c+d x)+120 C \sin (c+2 d x)+120 C \sin (3 c+2 d x)+820 C \sin (2 c+3 d x)-180 C \sin (4 c+3 d x)+60 C \sin (3 c+4 d x)+60 C \sin (5 c+4 d x)+200 C \sin (4 c+5 d x)+150 C d x \cos (2 c+d x)+75 C d x \cos (2 c+3 d x)+75 C d x \cos (4 c+3 d x)+15 C d x \cos (4 c+5 d x)+15 C d x \cos (6 c+5 d x)+1220 C \sin (d x)+150 C d x \cos (d x))-240 (7 A+12 C) \cos ^5(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{7680 d}","\frac{a^4 (7 A+10 C) \tan (c+d x)}{2 d}+\frac{a^4 (7 A+12 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(7 A+8 C) \tan (c+d x) \sec (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{6 d}+a^4 C x+\frac{(7 A+5 C) \tan (c+d x) \sec ^2(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{15 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^4}{5 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^3}{5 d}",1,"(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*Sec[c + d*x]^5*(-240*(7*A + 12*C)*Cos[c + d*x]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c]*(150*C*d*x*Cos[d*x] + 150*C*d*x*Cos[2*c + d*x] + 75*C*d*x*Cos[2*c + 3*d*x] + 75*C*d*x*Cos[4*c + 3*d*x] + 15*C*d*x*Cos[4*c + 5*d*x] + 15*C*d*x*Cos[6*c + 5*d*x] + 1180*A*Sin[d*x] + 1220*C*Sin[d*x] - 480*A*Sin[2*c + d*x] - 780*C*Sin[2*c + d*x] + 330*A*Sin[c + 2*d*x] + 120*C*Sin[c + 2*d*x] + 330*A*Sin[3*c + 2*d*x] + 120*C*Sin[3*c + 2*d*x] + 800*A*Sin[2*c + 3*d*x] + 820*C*Sin[2*c + 3*d*x] - 30*A*Sin[4*c + 3*d*x] - 180*C*Sin[4*c + 3*d*x] + 105*A*Sin[3*c + 4*d*x] + 60*C*Sin[3*c + 4*d*x] + 105*A*Sin[5*c + 4*d*x] + 60*C*Sin[5*c + 4*d*x] + 166*A*Sin[4*c + 5*d*x] + 200*C*Sin[4*c + 5*d*x])))/(7680*d)","A",1
37,1,358,232,2.1590445,"\int (a+a \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","-\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \sec ^6(c+d x) \left(3360 (7 A+10 C) \cos ^6(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-640 (18 A+25 C) \sin (c)+3750 A \sin (2 c+d x)+15360 A \sin (c+2 d x)-1920 A \sin (3 c+2 d x)+3845 A \sin (2 c+3 d x)+3845 A \sin (4 c+3 d x)+6912 A \sin (3 c+4 d x)+735 A \sin (4 c+5 d x)+735 A \sin (6 c+5 d x)+1152 A \sin (5 c+6 d x)+30 (125 A+62 C) \sin (d x)+1860 C \sin (2 c+d x)+17280 C \sin (c+2 d x)-6720 C \sin (3 c+2 d x)+2670 C \sin (2 c+3 d x)+2670 C \sin (4 c+3 d x)+8640 C \sin (3 c+4 d x)-960 C \sin (5 c+4 d x)+810 C \sin (4 c+5 d x)+810 C \sin (6 c+5 d x)+1600 C \sin (5 c+6 d x))\right)}{122880 d}","\frac{4 a^4 (18 A+25 C) \tan (c+d x)}{15 d}+\frac{7 a^4 (7 A+10 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 (417 A+550 C) \tan (c+d x) \sec (c+d x)}{240 d}+\frac{(43 A+50 C) \tan (c+d x) \sec ^2(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{60 d}+\frac{(37 A+30 C) \tan (c+d x) \sec ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{120 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}+\frac{2 a A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{15 d}",1,"-1/122880*(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*Sec[c + d*x]^6*(3360*(7*A + 10*C)*Cos[c + d*x]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(-640*(18*A + 25*C)*Sin[c] + 30*(125*A + 62*C)*Sin[d*x] + 3750*A*Sin[2*c + d*x] + 1860*C*Sin[2*c + d*x] + 15360*A*Sin[c + 2*d*x] + 17280*C*Sin[c + 2*d*x] - 1920*A*Sin[3*c + 2*d*x] - 6720*C*Sin[3*c + 2*d*x] + 3845*A*Sin[2*c + 3*d*x] + 2670*C*Sin[2*c + 3*d*x] + 3845*A*Sin[4*c + 3*d*x] + 2670*C*Sin[4*c + 3*d*x] + 6912*A*Sin[3*c + 4*d*x] + 8640*C*Sin[3*c + 4*d*x] - 960*C*Sin[5*c + 4*d*x] + 735*A*Sin[4*c + 5*d*x] + 810*C*Sin[4*c + 5*d*x] + 735*A*Sin[6*c + 5*d*x] + 810*C*Sin[6*c + 5*d*x] + 1152*A*Sin[5*c + 6*d*x] + 1600*C*Sin[5*c + 6*d*x])))/d","A",1
38,1,390,263,3.1659061,"\int (a+a \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^8(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^8,x]","-\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \sec ^7(c+d x) \left(6720 (11 A+14 C) \cos ^7(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-140 (122 A+217 C) \sin (2 c+d x)+16415 A \sin (c+2 d x)+16415 A \sin (3 c+2 d x)+37296 A \sin (2 c+3 d x)-840 A \sin (4 c+3 d x)+7700 A \sin (3 c+4 d x)+7700 A \sin (5 c+4 d x)+12712 A \sin (4 c+5 d x)+1155 A \sin (5 c+6 d x)+1155 A \sin (7 c+6 d x)+1816 A \sin (6 c+7 d x)+560 (83 A+91 C) \sin (d x)+10710 C \sin (c+2 d x)+10710 C \sin (3 c+2 d x)+41244 C \sin (2 c+3 d x)-7560 C \sin (4 c+3 d x)+7560 C \sin (3 c+4 d x)+7560 C \sin (5 c+4 d x)+15848 C \sin (4 c+5 d x)-420 C \sin (6 c+5 d x)+1470 C \sin (5 c+6 d x)+1470 C \sin (7 c+6 d x)+2324 C \sin (6 c+7 d x))\right)}{430080 d}","\frac{a^4 (454 A+581 C) \tan (c+d x)}{105 d}+\frac{a^4 (11 A+14 C) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a^4 (247 A+308 C) \tan (c+d x) \sec ^2(c+d x)}{210 d}+\frac{a^4 (11 A+14 C) \tan (c+d x) \sec (c+d x)}{4 d}+\frac{(109 A+126 C) \tan (c+d x) \sec ^3(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{210 d}+\frac{(8 A+7 C) \tan (c+d x) \sec ^4(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{35 d}+\frac{A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}+\frac{2 a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{21 d}",1,"-1/430080*(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*Sec[c + d*x]^7*(6720*(11*A + 14*C)*Cos[c + d*x]^7*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(560*(83*A + 91*C)*Sin[d*x] - 140*(122*A + 217*C)*Sin[2*c + d*x] + 16415*A*Sin[c + 2*d*x] + 10710*C*Sin[c + 2*d*x] + 16415*A*Sin[3*c + 2*d*x] + 10710*C*Sin[3*c + 2*d*x] + 37296*A*Sin[2*c + 3*d*x] + 41244*C*Sin[2*c + 3*d*x] - 840*A*Sin[4*c + 3*d*x] - 7560*C*Sin[4*c + 3*d*x] + 7700*A*Sin[3*c + 4*d*x] + 7560*C*Sin[3*c + 4*d*x] + 7700*A*Sin[5*c + 4*d*x] + 7560*C*Sin[5*c + 4*d*x] + 12712*A*Sin[4*c + 5*d*x] + 15848*C*Sin[4*c + 5*d*x] - 420*C*Sin[6*c + 5*d*x] + 1155*A*Sin[5*c + 6*d*x] + 1470*C*Sin[5*c + 6*d*x] + 1155*A*Sin[7*c + 6*d*x] + 1470*C*Sin[7*c + 6*d*x] + 1816*A*Sin[6*c + 7*d*x] + 2324*C*Sin[6*c + 7*d*x])))/d","A",1
39,1,283,156,0.5615806,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(72 d x (4 A+5 C) \cos \left(c+\frac{d x}{2}\right)-96 A \sin \left(c+\frac{d x}{2}\right)-72 A \sin \left(c+\frac{3 d x}{2}\right)-72 A \sin \left(2 c+\frac{3 d x}{2}\right)+24 A \sin \left(2 c+\frac{5 d x}{2}\right)+24 A \sin \left(3 c+\frac{5 d x}{2}\right)+72 d x (4 A+5 C) \cos \left(\frac{d x}{2}\right)-480 A \sin \left(\frac{d x}{2}\right)-168 C \sin \left(c+\frac{d x}{2}\right)-120 C \sin \left(c+\frac{3 d x}{2}\right)-120 C \sin \left(2 c+\frac{3 d x}{2}\right)+40 C \sin \left(2 c+\frac{5 d x}{2}\right)+40 C \sin \left(3 c+\frac{5 d x}{2}\right)-5 C \sin \left(3 c+\frac{7 d x}{2}\right)-5 C \sin \left(4 c+\frac{7 d x}{2}\right)+3 C \sin \left(4 c+\frac{9 d x}{2}\right)+3 C \sin \left(5 c+\frac{9 d x}{2}\right)-552 C \sin \left(\frac{d x}{2}\right)\right)}{192 a d (\cos (c+d x)+1)}","\frac{(3 A+4 C) \sin ^3(c+d x)}{3 a d}-\frac{(3 A+4 C) \sin (c+d x)}{a d}-\frac{(A+C) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(4 A+5 C) \sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac{3 (4 A+5 C) \sin (c+d x) \cos (c+d x)}{8 a d}+\frac{3 x (4 A+5 C)}{8 a}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(72*(4*A + 5*C)*d*x*Cos[(d*x)/2] + 72*(4*A + 5*C)*d*x*Cos[c + (d*x)/2] - 480*A*Sin[(d*x)/2] - 552*C*Sin[(d*x)/2] - 96*A*Sin[c + (d*x)/2] - 168*C*Sin[c + (d*x)/2] - 72*A*Sin[c + (3*d*x)/2] - 120*C*Sin[c + (3*d*x)/2] - 72*A*Sin[2*c + (3*d*x)/2] - 120*C*Sin[2*c + (3*d*x)/2] + 24*A*Sin[2*c + (5*d*x)/2] + 40*C*Sin[2*c + (5*d*x)/2] + 24*A*Sin[3*c + (5*d*x)/2] + 40*C*Sin[3*c + (5*d*x)/2] - 5*C*Sin[3*c + (7*d*x)/2] - 5*C*Sin[4*c + (7*d*x)/2] + 3*C*Sin[4*c + (9*d*x)/2] + 3*C*Sin[5*c + (9*d*x)/2]))/(192*a*d*(1 + Cos[c + d*x]))","A",1
40,1,225,124,0.6162567,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-12 d x (2 A+3 C) \cos \left(c+\frac{d x}{2}\right)+12 A \sin \left(c+\frac{d x}{2}\right)+12 A \sin \left(c+\frac{3 d x}{2}\right)+12 A \sin \left(2 c+\frac{3 d x}{2}\right)-12 d x (2 A+3 C) \cos \left(\frac{d x}{2}\right)+60 A \sin \left(\frac{d x}{2}\right)+21 C \sin \left(c+\frac{d x}{2}\right)+18 C \sin \left(c+\frac{3 d x}{2}\right)+18 C \sin \left(2 c+\frac{3 d x}{2}\right)-2 C \sin \left(2 c+\frac{5 d x}{2}\right)-2 C \sin \left(3 c+\frac{5 d x}{2}\right)+C \sin \left(3 c+\frac{7 d x}{2}\right)+C \sin \left(4 c+\frac{7 d x}{2}\right)+69 C \sin \left(\frac{d x}{2}\right)\right)}{24 a d (\cos (c+d x)+1)}","-\frac{(3 A+4 C) \sin ^3(c+d x)}{3 a d}+\frac{(3 A+4 C) \sin (c+d x)}{a d}-\frac{(A+C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(2 A+3 C) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{x (2 A+3 C)}{2 a}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(-12*(2*A + 3*C)*d*x*Cos[(d*x)/2] - 12*(2*A + 3*C)*d*x*Cos[c + (d*x)/2] + 60*A*Sin[(d*x)/2] + 69*C*Sin[(d*x)/2] + 12*A*Sin[c + (d*x)/2] + 21*C*Sin[c + (d*x)/2] + 12*A*Sin[c + (3*d*x)/2] + 18*C*Sin[c + (3*d*x)/2] + 12*A*Sin[2*c + (3*d*x)/2] + 18*C*Sin[2*c + (3*d*x)/2] - 2*C*Sin[2*c + (5*d*x)/2] - 2*C*Sin[3*c + (5*d*x)/2] + C*Sin[3*c + (7*d*x)/2] + C*Sin[4*c + (7*d*x)/2]))/(24*a*d*(1 + Cos[c + d*x]))","A",1
41,1,159,98,0.3483481,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(4 d x (2 A+3 C) \cos \left(c+\frac{d x}{2}\right)+4 d x (2 A+3 C) \cos \left(\frac{d x}{2}\right)-16 A \sin \left(\frac{d x}{2}\right)-4 C \sin \left(c+\frac{d x}{2}\right)-3 C \sin \left(c+\frac{3 d x}{2}\right)-3 C \sin \left(2 c+\frac{3 d x}{2}\right)+C \sin \left(2 c+\frac{5 d x}{2}\right)+C \sin \left(3 c+\frac{5 d x}{2}\right)-20 C \sin \left(\frac{d x}{2}\right)\right)}{8 a d (\cos (c+d x)+1)}","-\frac{(A+2 C) \sin (c+d x)}{a d}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(2 A+3 C) \sin (c+d x) \cos (c+d x)}{2 a d}+\frac{x (2 A+3 C)}{2 a}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(4*(2*A + 3*C)*d*x*Cos[(d*x)/2] + 4*(2*A + 3*C)*d*x*Cos[c + (d*x)/2] - 16*A*Sin[(d*x)/2] - 20*C*Sin[(d*x)/2] - 4*C*Sin[c + (d*x)/2] - 3*C*Sin[c + (3*d*x)/2] - 3*C*Sin[2*c + (3*d*x)/2] + C*Sin[2*c + (5*d*x)/2] + C*Sin[3*c + (5*d*x)/2]))/(8*a*d*(1 + Cos[c + d*x]))","A",1
42,1,108,48,0.2555207,"\int \frac{A+C \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(4 A \sin \left(\frac{d x}{2}\right)+C \sin \left(c+\frac{d x}{2}\right)+C \sin \left(c+\frac{3 d x}{2}\right)+C \sin \left(2 c+\frac{3 d x}{2}\right)-2 C d x \cos \left(c+\frac{d x}{2}\right)+5 C \sin \left(\frac{d x}{2}\right)-2 C d x \cos \left(\frac{d x}{2}\right)\right)}{4 a d}","\frac{(A+C) \sin (c+d x)}{a d (\cos (c+d x)+1)}+\frac{C \sin (c+d x)}{a d}-\frac{C x}{a}",1,"(Sec[c/2]*Sec[(c + d*x)/2]*(-2*C*d*x*Cos[(d*x)/2] - 2*C*d*x*Cos[c + (d*x)/2] + 4*A*Sin[(d*x)/2] + 5*C*Sin[(d*x)/2] + C*Sin[c + (d*x)/2] + C*Sin[c + (3*d*x)/2] + C*Sin[2*c + (3*d*x)/2]))/(4*a*d)","B",1
43,1,114,48,0.3296381,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x]),x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right) \left(-A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+C d x\right)-(A+C) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)\right)}{a d (\cos (c+d x)+1)}","-\frac{(A+C) \sin (c+d x)}{d (a \cos (c+d x)+a)}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a d}+\frac{C x}{a}",1,"(2*Cos[(c + d*x)/2]*(Cos[(c + d*x)/2]*(C*d*x - A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - (A + C)*Sec[c/2]*Sin[(d*x)/2]))/(a*d*(1 + Cos[c + d*x]))","B",1
44,1,229,61,1.860113,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x]),x]","\frac{4 \cos \left(\frac{1}{2} (c+d x)\right) \cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right) \left((A+C) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+A \cos \left(\frac{1}{2} (c+d x)\right) \left(\frac{\sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{a d (\cos (c+d x)+1) (2 A+C \cos (2 (c+d x))+C)}","\frac{(2 A+C) \tan (c+d x)}{a d}-\frac{(A+C) \tan (c+d x)}{d (a \cos (c+d x)+a)}-\frac{A \tanh ^{-1}(\sin (c+d x))}{a d}",1,"(4*Cos[(c + d*x)/2]*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*((A + C)*Sec[c/2]*Sin[(d*x)/2] + A*Cos[(c + d*x)/2]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Sin[d*x]/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))))/(a*d*(1 + Cos[c + d*x])*(2*A + C + C*Cos[2*(c + d*x)]))","B",1
45,1,284,105,2.8022116,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x]),x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right) \left(-2 (3 A+2 C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\frac{4 A \sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{A}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{A}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+6 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+4 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-4 (A+C) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)\right)}{2 a d (\cos (c+d x)+1)}","-\frac{(2 A+C) \tan (c+d x)}{a d}+\frac{(3 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{(3 A+2 C) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(A+C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}",1,"(Cos[(c + d*x)/2]*(-4*(A + C)*Sec[c/2]*Sin[(d*x)/2] + Cos[(c + d*x)/2]*(-2*(3*A + 2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 4*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + A/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 - A/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - (4*A*Sin[d*x])/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))))/(2*a*d*(1 + Cos[c + d*x]))","B",1
46,1,765,133,6.4872072,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x]),x]","\frac{2 \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \left(5 A \sin \left(\frac{d x}{2}\right)+3 C \sin \left(\frac{d x}{2}\right)\right)}{3 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) (a \cos (c+d x)+a) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{2 \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \left(5 A \sin \left(\frac{d x}{2}\right)+3 C \sin \left(\frac{d x}{2}\right)\right)}{3 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) (a \cos (c+d x)+a) \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{(3 A+2 C) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d (a \cos (c+d x)+a)}+\frac{(-3 A-2 C) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d (a \cos (c+d x)+a)}+\frac{2 \sec \left(\frac{c}{2}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d (a \cos (c+d x)+a)}+\frac{\left(2 A \sin \left(\frac{c}{2}\right)-A \cos \left(\frac{c}{2}\right)\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) (a \cos (c+d x)+a) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\left(2 A \sin \left(\frac{c}{2}\right)+A \cos \left(\frac{c}{2}\right)\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) (a \cos (c+d x)+a) \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{A \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) (a \cos (c+d x)+a) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{A \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) (a \cos (c+d x)+a) \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}","\frac{(4 A+3 C) \tan ^3(c+d x)}{3 a d}+\frac{(4 A+3 C) \tan (c+d x)}{a d}-\frac{(3 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{(3 A+2 C) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(A+C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}",1,"((3*A + 2*C)*Cos[c/2 + (d*x)/2]^2*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]])/(d*(a + a*Cos[c + d*x])) + ((-3*A - 2*C)*Cos[c/2 + (d*x)/2]^2*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]])/(d*(a + a*Cos[c + d*x])) + (2*Cos[c/2 + (d*x)/2]*Sec[c/2]*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(d*(a + a*Cos[c + d*x])) + (A*Cos[c/2 + (d*x)/2]^2*Sin[(d*x)/2])/(3*d*(a + a*Cos[c + d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + (Cos[c/2 + (d*x)/2]^2*(-(A*Cos[c/2]) + 2*A*Sin[c/2]))/(3*d*(a + a*Cos[c + d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (2*Cos[c/2 + (d*x)/2]^2*(5*A*Sin[(d*x)/2] + 3*C*Sin[(d*x)/2]))/(3*d*(a + a*Cos[c + d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (A*Cos[c/2 + (d*x)/2]^2*Sin[(d*x)/2])/(3*d*(a + a*Cos[c + d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (Cos[c/2 + (d*x)/2]^2*(A*Cos[c/2] + 2*A*Sin[c/2]))/(3*d*(a + a*Cos[c + d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (2*Cos[c/2 + (d*x)/2]^2*(5*A*Sin[(d*x)/2] + 3*C*Sin[(d*x)/2]))/(3*d*(a + a*Cos[c + d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",0
47,1,399,191,0.8847838,"\int \frac{\cos ^4(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(72 d x (28 A+55 C) \cos \left(c+\frac{d x}{2}\right)+1176 A \sin \left(c+\frac{d x}{2}\right)-1912 A \sin \left(c+\frac{3 d x}{2}\right)-504 A \sin \left(2 c+\frac{3 d x}{2}\right)-120 A \sin \left(2 c+\frac{5 d x}{2}\right)-120 A \sin \left(3 c+\frac{5 d x}{2}\right)+24 A \sin \left(3 c+\frac{7 d x}{2}\right)+24 A \sin \left(4 c+\frac{7 d x}{2}\right)+672 A d x \cos \left(c+\frac{3 d x}{2}\right)+672 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+72 d x (28 A+55 C) \cos \left(\frac{d x}{2}\right)-3048 A \sin \left(\frac{d x}{2}\right)+1344 C \sin \left(c+\frac{d x}{2}\right)-3488 C \sin \left(c+\frac{3 d x}{2}\right)-1312 C \sin \left(2 c+\frac{3 d x}{2}\right)-285 C \sin \left(2 c+\frac{5 d x}{2}\right)-285 C \sin \left(3 c+\frac{5 d x}{2}\right)+57 C \sin \left(3 c+\frac{7 d x}{2}\right)+57 C \sin \left(4 c+\frac{7 d x}{2}\right)-7 C \sin \left(4 c+\frac{9 d x}{2}\right)-7 C \sin \left(5 c+\frac{9 d x}{2}\right)+3 C \sin \left(5 c+\frac{11 d x}{2}\right)+3 C \sin \left(6 c+\frac{11 d x}{2}\right)+1320 C d x \cos \left(c+\frac{3 d x}{2}\right)+1320 C d x \cos \left(2 c+\frac{3 d x}{2}\right)-5184 C \sin \left(\frac{d x}{2}\right)\right)}{384 a^2 d (\cos (c+d x)+1)^2}","\frac{8 (A+2 C) \sin ^3(c+d x)}{3 a^2 d}-\frac{8 (A+2 C) \sin (c+d x)}{a^2 d}-\frac{2 (A+2 C) \sin (c+d x) \cos ^4(c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{(28 A+55 C) \sin (c+d x) \cos ^3(c+d x)}{12 a^2 d}+\frac{(28 A+55 C) \sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac{x (28 A+55 C)}{8 a^2}-\frac{(A+C) \sin (c+d x) \cos ^5(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(72*(28*A + 55*C)*d*x*Cos[(d*x)/2] + 72*(28*A + 55*C)*d*x*Cos[c + (d*x)/2] + 672*A*d*x*Cos[c + (3*d*x)/2] + 1320*C*d*x*Cos[c + (3*d*x)/2] + 672*A*d*x*Cos[2*c + (3*d*x)/2] + 1320*C*d*x*Cos[2*c + (3*d*x)/2] - 3048*A*Sin[(d*x)/2] - 5184*C*Sin[(d*x)/2] + 1176*A*Sin[c + (d*x)/2] + 1344*C*Sin[c + (d*x)/2] - 1912*A*Sin[c + (3*d*x)/2] - 3488*C*Sin[c + (3*d*x)/2] - 504*A*Sin[2*c + (3*d*x)/2] - 1312*C*Sin[2*c + (3*d*x)/2] - 120*A*Sin[2*c + (5*d*x)/2] - 285*C*Sin[2*c + (5*d*x)/2] - 120*A*Sin[3*c + (5*d*x)/2] - 285*C*Sin[3*c + (5*d*x)/2] + 24*A*Sin[3*c + (7*d*x)/2] + 57*C*Sin[3*c + (7*d*x)/2] + 24*A*Sin[4*c + (7*d*x)/2] + 57*C*Sin[4*c + (7*d*x)/2] - 7*C*Sin[4*c + (9*d*x)/2] - 7*C*Sin[5*c + (9*d*x)/2] + 3*C*Sin[5*c + (11*d*x)/2] + 3*C*Sin[6*c + (11*d*x)/2]))/(384*a^2*d*(1 + Cos[c + d*x])^2)","B",1
48,1,341,163,0.6798651,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-72 d x (2 A+5 C) \cos \left(c+\frac{d x}{2}\right)-120 A \sin \left(c+\frac{d x}{2}\right)+164 A \sin \left(c+\frac{3 d x}{2}\right)+36 A \sin \left(2 c+\frac{3 d x}{2}\right)+12 A \sin \left(2 c+\frac{5 d x}{2}\right)+12 A \sin \left(3 c+\frac{5 d x}{2}\right)-48 A d x \cos \left(c+\frac{3 d x}{2}\right)-48 A d x \cos \left(2 c+\frac{3 d x}{2}\right)-72 d x (2 A+5 C) \cos \left(\frac{d x}{2}\right)+264 A \sin \left(\frac{d x}{2}\right)-156 C \sin \left(c+\frac{d x}{2}\right)+342 C \sin \left(c+\frac{3 d x}{2}\right)+118 C \sin \left(2 c+\frac{3 d x}{2}\right)+30 C \sin \left(2 c+\frac{5 d x}{2}\right)+30 C \sin \left(3 c+\frac{5 d x}{2}\right)-3 C \sin \left(3 c+\frac{7 d x}{2}\right)-3 C \sin \left(4 c+\frac{7 d x}{2}\right)+C \sin \left(4 c+\frac{9 d x}{2}\right)+C \sin \left(5 c+\frac{9 d x}{2}\right)-120 C d x \cos \left(c+\frac{3 d x}{2}\right)-120 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+516 C \sin \left(\frac{d x}{2}\right)\right)}{48 a^2 d (\cos (c+d x)+1)^2}","-\frac{(5 A+12 C) \sin ^3(c+d x)}{3 a^2 d}+\frac{(5 A+12 C) \sin (c+d x)}{a^2 d}-\frac{2 (2 A+5 C) \sin (c+d x) \cos ^3(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(2 A+5 C) \sin (c+d x) \cos (c+d x)}{a^2 d}-\frac{x (2 A+5 C)}{a^2}-\frac{(A+C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(-72*(2*A + 5*C)*d*x*Cos[(d*x)/2] - 72*(2*A + 5*C)*d*x*Cos[c + (d*x)/2] - 48*A*d*x*Cos[c + (3*d*x)/2] - 120*C*d*x*Cos[c + (3*d*x)/2] - 48*A*d*x*Cos[2*c + (3*d*x)/2] - 120*C*d*x*Cos[2*c + (3*d*x)/2] + 264*A*Sin[(d*x)/2] + 516*C*Sin[(d*x)/2] - 120*A*Sin[c + (d*x)/2] - 156*C*Sin[c + (d*x)/2] + 164*A*Sin[c + (3*d*x)/2] + 342*C*Sin[c + (3*d*x)/2] + 36*A*Sin[2*c + (3*d*x)/2] + 118*C*Sin[2*c + (3*d*x)/2] + 12*A*Sin[2*c + (5*d*x)/2] + 30*C*Sin[2*c + (5*d*x)/2] + 12*A*Sin[3*c + (5*d*x)/2] + 30*C*Sin[3*c + (5*d*x)/2] - 3*C*Sin[3*c + (7*d*x)/2] - 3*C*Sin[4*c + (7*d*x)/2] + C*Sin[4*c + (9*d*x)/2] + C*Sin[5*c + (9*d*x)/2]))/(48*a^2*d*(1 + Cos[c + d*x])^2)","B",1
49,1,273,141,0.6527371,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(36 d x (2 A+7 C) \cos \left(c+\frac{d x}{2}\right)+96 A \sin \left(c+\frac{d x}{2}\right)-80 A \sin \left(c+\frac{3 d x}{2}\right)+24 A d x \cos \left(c+\frac{3 d x}{2}\right)+24 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+36 d x (2 A+7 C) \cos \left(\frac{d x}{2}\right)-144 A \sin \left(\frac{d x}{2}\right)+147 C \sin \left(c+\frac{d x}{2}\right)-239 C \sin \left(c+\frac{3 d x}{2}\right)-63 C \sin \left(2 c+\frac{3 d x}{2}\right)-15 C \sin \left(2 c+\frac{5 d x}{2}\right)-15 C \sin \left(3 c+\frac{5 d x}{2}\right)+3 C \sin \left(3 c+\frac{7 d x}{2}\right)+3 C \sin \left(4 c+\frac{7 d x}{2}\right)+84 C d x \cos \left(c+\frac{3 d x}{2}\right)+84 C d x \cos \left(2 c+\frac{3 d x}{2}\right)-381 C \sin \left(\frac{d x}{2}\right)\right)}{48 a^2 d (\cos (c+d x)+1)^2}","-\frac{4 (A+4 C) \sin (c+d x)}{3 a^2 d}-\frac{2 (A+4 C) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{(2 A+7 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{x (2 A+7 C)}{2 a^2}-\frac{(A+C) \sin (c+d x) \cos ^3(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(36*(2*A + 7*C)*d*x*Cos[(d*x)/2] + 36*(2*A + 7*C)*d*x*Cos[c + (d*x)/2] + 24*A*d*x*Cos[c + (3*d*x)/2] + 84*C*d*x*Cos[c + (3*d*x)/2] + 24*A*d*x*Cos[2*c + (3*d*x)/2] + 84*C*d*x*Cos[2*c + (3*d*x)/2] - 144*A*Sin[(d*x)/2] - 381*C*Sin[(d*x)/2] + 96*A*Sin[c + (d*x)/2] + 147*C*Sin[c + (d*x)/2] - 80*A*Sin[c + (3*d*x)/2] - 239*C*Sin[c + (3*d*x)/2] - 63*C*Sin[2*c + (3*d*x)/2] - 15*C*Sin[2*c + (5*d*x)/2] - 15*C*Sin[3*c + (5*d*x)/2] + 3*C*Sin[3*c + (7*d*x)/2] + 3*C*Sin[4*c + (7*d*x)/2]))/(48*a^2*d*(1 + Cos[c + d*x])^2)","A",1
50,1,195,90,0.550694,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \left(-12 A \sin \left(c+\frac{d x}{2}\right)+8 A \sin \left(c+\frac{3 d x}{2}\right)+12 A \sin \left(\frac{d x}{2}\right)-30 C \sin \left(c+\frac{d x}{2}\right)+41 C \sin \left(c+\frac{3 d x}{2}\right)+9 C \sin \left(2 c+\frac{3 d x}{2}\right)+3 C \sin \left(2 c+\frac{5 d x}{2}\right)+3 C \sin \left(3 c+\frac{5 d x}{2}\right)-36 C d x \cos \left(c+\frac{d x}{2}\right)-12 C d x \cos \left(c+\frac{3 d x}{2}\right)-12 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+66 C \sin \left(\frac{d x}{2}\right)-36 C d x \cos \left(\frac{d x}{2}\right)\right)}{48 a^2 d}","\frac{(A+4 C) \sin (c+d x)}{3 a^2 d}+\frac{2 C \sin (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{2 C x}{a^2}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^3*(-36*C*d*x*Cos[(d*x)/2] - 36*C*d*x*Cos[c + (d*x)/2] - 12*C*d*x*Cos[c + (3*d*x)/2] - 12*C*d*x*Cos[2*c + (3*d*x)/2] + 12*A*Sin[(d*x)/2] + 66*C*Sin[(d*x)/2] - 12*A*Sin[c + (d*x)/2] - 30*C*Sin[c + (d*x)/2] + 8*A*Sin[c + (3*d*x)/2] + 41*C*Sin[c + (3*d*x)/2] + 9*C*Sin[2*c + (3*d*x)/2] + 3*C*Sin[2*c + (5*d*x)/2] + 3*C*Sin[3*c + (5*d*x)/2]))/(48*a^2*d)","B",1
51,1,141,66,0.3419131,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \left(2 A \sin \left(c+\frac{3 d x}{2}\right)+6 A \sin \left(\frac{d x}{2}\right)+12 C \sin \left(c+\frac{d x}{2}\right)-10 C \sin \left(c+\frac{3 d x}{2}\right)+9 C d x \cos \left(c+\frac{d x}{2}\right)+3 C d x \cos \left(c+\frac{3 d x}{2}\right)+3 C d x \cos \left(2 c+\frac{3 d x}{2}\right)-18 C \sin \left(\frac{d x}{2}\right)+9 C d x \cos \left(\frac{d x}{2}\right)\right)}{24 a^2 d}","\frac{(A-5 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{C x}{a^2}+\frac{(A+C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^3*(9*C*d*x*Cos[(d*x)/2] + 9*C*d*x*Cos[c + (d*x)/2] + 3*C*d*x*Cos[c + (3*d*x)/2] + 3*C*d*x*Cos[2*c + (3*d*x)/2] + 6*A*Sin[(d*x)/2] - 18*C*Sin[(d*x)/2] + 12*C*Sin[c + (d*x)/2] + 2*A*Sin[c + (3*d*x)/2] - 10*C*Sin[c + (3*d*x)/2]))/(24*a^2*d)","B",1
52,1,166,77,0.5769638,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^2,x]","-\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left((A+C) \tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)+(A+C) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+4 (2 A-C) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)+6 A \cos ^3\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{3 a^2 d (\cos (c+d x)+1)^2}","-\frac{2 (2 A-C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(A+C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(-2*Cos[(c + d*x)/2]*(6*A*Cos[(c + d*x)/2]^3*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (A + C)*Sec[c/2]*Sin[(d*x)/2] + 4*(2*A - C)*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + (A + C)*Cos[(c + d*x)/2]*Tan[c/2]))/(3*a^2*d*(1 + Cos[c + d*x])^2)","B",1
53,1,288,91,1.551478,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^2,x]","\frac{4 \cos \left(\frac{1}{2} (c+d x)\right) \cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right) \left((A+C) \tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)+(A+C) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+2 (7 A+C) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)+6 A \cos ^3\left(\frac{1}{2} (c+d x)\right) \left(\frac{\sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{3 a^2 d (\cos (c+d x)+1)^2 (2 A+C \cos (2 (c+d x))+C)}","\frac{(10 A+C) \tan (c+d x)}{3 a^2 d}-\frac{2 A \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{2 A \tan (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{(A+C) \tan (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(4*Cos[(c + d*x)/2]*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*((A + C)*Sec[c/2]*Sin[(d*x)/2] + 2*(7*A + C)*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + 6*A*Cos[(c + d*x)/2]^3*(2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Sin[d*x]/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))) + (A + C)*Cos[(c + d*x)/2]*Tan[c/2]))/(3*a^2*d*(1 + Cos[c + d*x])^2*(2*A + C + C*Cos[2*(c + d*x)]))","B",1
54,1,484,146,3.2453172,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^2,x]","-\frac{96 (7 A+2 C) \cos ^4\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+\sec \left(\frac{c}{2}\right) \sec (c) \cos \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(-126 A \sin \left(c-\frac{d x}{2}\right)+42 A \sin \left(c+\frac{d x}{2}\right)-98 A \sin \left(2 c+\frac{d x}{2}\right)-3 A \sin \left(c+\frac{3 d x}{2}\right)+37 A \sin \left(2 c+\frac{3 d x}{2}\right)-63 A \sin \left(3 c+\frac{3 d x}{2}\right)+75 A \sin \left(c+\frac{5 d x}{2}\right)+15 A \sin \left(2 c+\frac{5 d x}{2}\right)+39 A \sin \left(3 c+\frac{5 d x}{2}\right)-21 A \sin \left(4 c+\frac{5 d x}{2}\right)+32 A \sin \left(2 c+\frac{7 d x}{2}\right)+12 A \sin \left(3 c+\frac{7 d x}{2}\right)+20 A \sin \left(4 c+\frac{7 d x}{2}\right)-2 (7 A+10 C) \sin \left(\frac{d x}{2}\right)+(97 A+22 C) \sin \left(\frac{3 d x}{2}\right)-36 C \sin \left(c-\frac{d x}{2}\right)+36 C \sin \left(c+\frac{d x}{2}\right)-20 C \sin \left(2 c+\frac{d x}{2}\right)-18 C \sin \left(c+\frac{3 d x}{2}\right)+22 C \sin \left(2 c+\frac{3 d x}{2}\right)-18 C \sin \left(3 c+\frac{3 d x}{2}\right)+18 C \sin \left(c+\frac{5 d x}{2}\right)-6 C \sin \left(2 c+\frac{5 d x}{2}\right)+18 C \sin \left(3 c+\frac{5 d x}{2}\right)-6 C \sin \left(4 c+\frac{5 d x}{2}\right)+8 C \sin \left(2 c+\frac{7 d x}{2}\right)+8 C \sin \left(4 c+\frac{7 d x}{2}\right)\right)}{48 a^2 d (\cos (c+d x)+1)^2}","-\frac{4 (4 A+C) \tan (c+d x)}{3 a^2 d}+\frac{(7 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac{(7 A+2 C) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{2 (4 A+C) \tan (c+d x) \sec (c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(A+C) \tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"-1/48*(96*(7*A + 2*C)*Cos[(c + d*x)/2]^4*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Cos[(c + d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]^2*(-2*(7*A + 10*C)*Sin[(d*x)/2] + (97*A + 22*C)*Sin[(3*d*x)/2] - 126*A*Sin[c - (d*x)/2] - 36*C*Sin[c - (d*x)/2] + 42*A*Sin[c + (d*x)/2] + 36*C*Sin[c + (d*x)/2] - 98*A*Sin[2*c + (d*x)/2] - 20*C*Sin[2*c + (d*x)/2] - 3*A*Sin[c + (3*d*x)/2] - 18*C*Sin[c + (3*d*x)/2] + 37*A*Sin[2*c + (3*d*x)/2] + 22*C*Sin[2*c + (3*d*x)/2] - 63*A*Sin[3*c + (3*d*x)/2] - 18*C*Sin[3*c + (3*d*x)/2] + 75*A*Sin[c + (5*d*x)/2] + 18*C*Sin[c + (5*d*x)/2] + 15*A*Sin[2*c + (5*d*x)/2] - 6*C*Sin[2*c + (5*d*x)/2] + 39*A*Sin[3*c + (5*d*x)/2] + 18*C*Sin[3*c + (5*d*x)/2] - 21*A*Sin[4*c + (5*d*x)/2] - 6*C*Sin[4*c + (5*d*x)/2] + 32*A*Sin[2*c + (7*d*x)/2] + 8*C*Sin[2*c + (7*d*x)/2] + 12*A*Sin[3*c + (7*d*x)/2] + 20*A*Sin[4*c + (7*d*x)/2] + 8*C*Sin[4*c + (7*d*x)/2]))/(a^2*d*(1 + Cos[c + d*x])^2)","B",1
55,1,594,172,4.7823216,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^2,x]","\frac{192 (5 A+2 C) \cos ^4\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+\sec \left(\frac{c}{2}\right) \sec (c) \cos \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(-153 A \sin \left(c-\frac{d x}{2}\right)+21 A \sin \left(c+\frac{d x}{2}\right)-135 A \sin \left(2 c+\frac{d x}{2}\right)+25 A \sin \left(c+\frac{3 d x}{2}\right)+45 A \sin \left(2 c+\frac{3 d x}{2}\right)-85 A \sin \left(3 c+\frac{3 d x}{2}\right)+99 A \sin \left(c+\frac{5 d x}{2}\right)+21 A \sin \left(2 c+\frac{5 d x}{2}\right)+33 A \sin \left(3 c+\frac{5 d x}{2}\right)-45 A \sin \left(4 c+\frac{5 d x}{2}\right)+57 A \sin \left(2 c+\frac{7 d x}{2}\right)+18 A \sin \left(3 c+\frac{7 d x}{2}\right)+24 A \sin \left(4 c+\frac{7 d x}{2}\right)-15 A \sin \left(5 c+\frac{7 d x}{2}\right)+24 A \sin \left(3 c+\frac{9 d x}{2}\right)+11 A \sin \left(4 c+\frac{9 d x}{2}\right)+13 A \sin \left(5 c+\frac{9 d x}{2}\right)-3 (A+8 C) \sin \left(\frac{d x}{2}\right)+(155 A+66 C) \sin \left(\frac{3 d x}{2}\right)-60 C \sin \left(c-\frac{d x}{2}\right)+24 C \sin \left(c+\frac{d x}{2}\right)-60 C \sin \left(2 c+\frac{d x}{2}\right)-4 C \sin \left(c+\frac{3 d x}{2}\right)+36 C \sin \left(2 c+\frac{3 d x}{2}\right)-34 C \sin \left(3 c+\frac{3 d x}{2}\right)+42 C \sin \left(c+\frac{5 d x}{2}\right)+24 C \sin \left(3 c+\frac{5 d x}{2}\right)-18 C \sin \left(4 c+\frac{5 d x}{2}\right)+24 C \sin \left(2 c+\frac{7 d x}{2}\right)+3 C \sin \left(3 c+\frac{7 d x}{2}\right)+15 C \sin \left(4 c+\frac{7 d x}{2}\right)-6 C \sin \left(5 c+\frac{7 d x}{2}\right)+10 C \sin \left(3 c+\frac{9 d x}{2}\right)+3 C \sin \left(4 c+\frac{9 d x}{2}\right)+7 C \sin \left(5 c+\frac{9 d x}{2}\right)\right)}{48 a^2 d (\cos (c+d x)+1)^2}","\frac{(12 A+5 C) \tan ^3(c+d x)}{3 a^2 d}+\frac{(12 A+5 C) \tan (c+d x)}{a^2 d}-\frac{(5 A+2 C) \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(5 A+2 C) \tan (c+d x) \sec (c+d x)}{a^2 d}-\frac{2 (5 A+2 C) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(A+C) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(192*(5*A + 2*C)*Cos[(c + d*x)/2]^4*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Cos[(c + d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]^3*(-3*(A + 8*C)*Sin[(d*x)/2] + (155*A + 66*C)*Sin[(3*d*x)/2] - 153*A*Sin[c - (d*x)/2] - 60*C*Sin[c - (d*x)/2] + 21*A*Sin[c + (d*x)/2] + 24*C*Sin[c + (d*x)/2] - 135*A*Sin[2*c + (d*x)/2] - 60*C*Sin[2*c + (d*x)/2] + 25*A*Sin[c + (3*d*x)/2] - 4*C*Sin[c + (3*d*x)/2] + 45*A*Sin[2*c + (3*d*x)/2] + 36*C*Sin[2*c + (3*d*x)/2] - 85*A*Sin[3*c + (3*d*x)/2] - 34*C*Sin[3*c + (3*d*x)/2] + 99*A*Sin[c + (5*d*x)/2] + 42*C*Sin[c + (5*d*x)/2] + 21*A*Sin[2*c + (5*d*x)/2] + 33*A*Sin[3*c + (5*d*x)/2] + 24*C*Sin[3*c + (5*d*x)/2] - 45*A*Sin[4*c + (5*d*x)/2] - 18*C*Sin[4*c + (5*d*x)/2] + 57*A*Sin[2*c + (7*d*x)/2] + 24*C*Sin[2*c + (7*d*x)/2] + 18*A*Sin[3*c + (7*d*x)/2] + 3*C*Sin[3*c + (7*d*x)/2] + 24*A*Sin[4*c + (7*d*x)/2] + 15*C*Sin[4*c + (7*d*x)/2] - 15*A*Sin[5*c + (7*d*x)/2] - 6*C*Sin[5*c + (7*d*x)/2] + 24*A*Sin[3*c + (9*d*x)/2] + 10*C*Sin[3*c + (9*d*x)/2] + 11*A*Sin[4*c + (9*d*x)/2] + 3*C*Sin[4*c + (9*d*x)/2] + 13*A*Sin[5*c + (9*d*x)/2] + 7*C*Sin[5*c + (9*d*x)/2]))/(48*a^2*d*(1 + Cos[c + d*x])^2)","B",1
56,1,463,216,0.9156887,"\int \frac{\cos ^4(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","-\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(600 d x (6 A+23 C) \cos \left(c+\frac{d x}{2}\right)+4500 A \sin \left(c+\frac{d x}{2}\right)-4860 A \sin \left(c+\frac{3 d x}{2}\right)+900 A \sin \left(2 c+\frac{3 d x}{2}\right)-1452 A \sin \left(2 c+\frac{5 d x}{2}\right)-300 A \sin \left(3 c+\frac{5 d x}{2}\right)-60 A \sin \left(3 c+\frac{7 d x}{2}\right)-60 A \sin \left(4 c+\frac{7 d x}{2}\right)+1800 A d x \cos \left(c+\frac{3 d x}{2}\right)+1800 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+360 A d x \cos \left(2 c+\frac{5 d x}{2}\right)+360 A d x \cos \left(3 c+\frac{5 d x}{2}\right)+600 d x (6 A+23 C) \cos \left(\frac{d x}{2}\right)-7020 A \sin \left(\frac{d x}{2}\right)+11110 C \sin \left(c+\frac{d x}{2}\right)-15380 C \sin \left(c+\frac{3 d x}{2}\right)+380 C \sin \left(2 c+\frac{3 d x}{2}\right)-4777 C \sin \left(2 c+\frac{5 d x}{2}\right)-1625 C \sin \left(3 c+\frac{5 d x}{2}\right)-230 C \sin \left(3 c+\frac{7 d x}{2}\right)-230 C \sin \left(4 c+\frac{7 d x}{2}\right)+20 C \sin \left(4 c+\frac{9 d x}{2}\right)+20 C \sin \left(5 c+\frac{9 d x}{2}\right)-5 C \sin \left(5 c+\frac{11 d x}{2}\right)-5 C \sin \left(6 c+\frac{11 d x}{2}\right)+6900 C d x \cos \left(c+\frac{3 d x}{2}\right)+6900 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+1380 C d x \cos \left(2 c+\frac{5 d x}{2}\right)+1380 C d x \cos \left(3 c+\frac{5 d x}{2}\right)-20410 C \sin \left(\frac{d x}{2}\right)\right)}{480 a^3 d (\cos (c+d x)+1)^3}","-\frac{4 (9 A+34 C) \sin ^3(c+d x)}{15 a^3 d}+\frac{4 (9 A+34 C) \sin (c+d x)}{5 a^3 d}-\frac{(6 A+23 C) \sin (c+d x) \cos ^3(c+d x)}{3 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(6 A+23 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{x (6 A+23 C)}{2 a^3}-\frac{(A+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(3 A+13 C) \sin (c+d x) \cos ^4(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"-1/480*(Cos[(c + d*x)/2]*Sec[c/2]*(600*(6*A + 23*C)*d*x*Cos[(d*x)/2] + 600*(6*A + 23*C)*d*x*Cos[c + (d*x)/2] + 1800*A*d*x*Cos[c + (3*d*x)/2] + 6900*C*d*x*Cos[c + (3*d*x)/2] + 1800*A*d*x*Cos[2*c + (3*d*x)/2] + 6900*C*d*x*Cos[2*c + (3*d*x)/2] + 360*A*d*x*Cos[2*c + (5*d*x)/2] + 1380*C*d*x*Cos[2*c + (5*d*x)/2] + 360*A*d*x*Cos[3*c + (5*d*x)/2] + 1380*C*d*x*Cos[3*c + (5*d*x)/2] - 7020*A*Sin[(d*x)/2] - 20410*C*Sin[(d*x)/2] + 4500*A*Sin[c + (d*x)/2] + 11110*C*Sin[c + (d*x)/2] - 4860*A*Sin[c + (3*d*x)/2] - 15380*C*Sin[c + (3*d*x)/2] + 900*A*Sin[2*c + (3*d*x)/2] + 380*C*Sin[2*c + (3*d*x)/2] - 1452*A*Sin[2*c + (5*d*x)/2] - 4777*C*Sin[2*c + (5*d*x)/2] - 300*A*Sin[3*c + (5*d*x)/2] - 1625*C*Sin[3*c + (5*d*x)/2] - 60*A*Sin[3*c + (7*d*x)/2] - 230*C*Sin[3*c + (7*d*x)/2] - 60*A*Sin[4*c + (7*d*x)/2] - 230*C*Sin[4*c + (7*d*x)/2] + 20*C*Sin[4*c + (9*d*x)/2] + 20*C*Sin[5*c + (9*d*x)/2] - 5*C*Sin[5*c + (11*d*x)/2] - 5*C*Sin[6*c + (11*d*x)/2]))/(a^3*d*(1 + Cos[c + d*x])^3)","B",1
57,1,393,189,0.6845997,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(600 d x (2 A+13 C) \cos \left(c+\frac{d x}{2}\right)+2160 A \sin \left(c+\frac{d x}{2}\right)-1840 A \sin \left(c+\frac{3 d x}{2}\right)+720 A \sin \left(2 c+\frac{3 d x}{2}\right)-512 A \sin \left(2 c+\frac{5 d x}{2}\right)+600 A d x \cos \left(c+\frac{3 d x}{2}\right)+600 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+120 A d x \cos \left(2 c+\frac{5 d x}{2}\right)+120 A d x \cos \left(3 c+\frac{5 d x}{2}\right)+600 d x (2 A+13 C) \cos \left(\frac{d x}{2}\right)-2960 A \sin \left(\frac{d x}{2}\right)+7560 C \sin \left(c+\frac{d x}{2}\right)-9230 C \sin \left(c+\frac{3 d x}{2}\right)+930 C \sin \left(2 c+\frac{3 d x}{2}\right)-2782 C \sin \left(2 c+\frac{5 d x}{2}\right)-750 C \sin \left(3 c+\frac{5 d x}{2}\right)-105 C \sin \left(3 c+\frac{7 d x}{2}\right)-105 C \sin \left(4 c+\frac{7 d x}{2}\right)+15 C \sin \left(4 c+\frac{9 d x}{2}\right)+15 C \sin \left(5 c+\frac{9 d x}{2}\right)+3900 C d x \cos \left(c+\frac{3 d x}{2}\right)+3900 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+780 C d x \cos \left(2 c+\frac{5 d x}{2}\right)+780 C d x \cos \left(3 c+\frac{5 d x}{2}\right)-12760 C \sin \left(\frac{d x}{2}\right)\right)}{480 a^3 d (\cos (c+d x)+1)^3}","-\frac{2 (11 A+76 C) \sin (c+d x)}{15 a^3 d}-\frac{(11 A+76 C) \sin (c+d x) \cos ^2(c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(2 A+13 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac{x (2 A+13 C)}{2 a^3}-\frac{(A+C) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(A+11 C) \sin (c+d x) \cos ^3(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(600*(2*A + 13*C)*d*x*Cos[(d*x)/2] + 600*(2*A + 13*C)*d*x*Cos[c + (d*x)/2] + 600*A*d*x*Cos[c + (3*d*x)/2] + 3900*C*d*x*Cos[c + (3*d*x)/2] + 600*A*d*x*Cos[2*c + (3*d*x)/2] + 3900*C*d*x*Cos[2*c + (3*d*x)/2] + 120*A*d*x*Cos[2*c + (5*d*x)/2] + 780*C*d*x*Cos[2*c + (5*d*x)/2] + 120*A*d*x*Cos[3*c + (5*d*x)/2] + 780*C*d*x*Cos[3*c + (5*d*x)/2] - 2960*A*Sin[(d*x)/2] - 12760*C*Sin[(d*x)/2] + 2160*A*Sin[c + (d*x)/2] + 7560*C*Sin[c + (d*x)/2] - 1840*A*Sin[c + (3*d*x)/2] - 9230*C*Sin[c + (3*d*x)/2] + 720*A*Sin[2*c + (3*d*x)/2] + 930*C*Sin[2*c + (3*d*x)/2] - 512*A*Sin[2*c + (5*d*x)/2] - 2782*C*Sin[2*c + (5*d*x)/2] - 750*C*Sin[3*c + (5*d*x)/2] - 105*C*Sin[3*c + (7*d*x)/2] - 105*C*Sin[4*c + (7*d*x)/2] + 15*C*Sin[4*c + (9*d*x)/2] + 15*C*Sin[5*c + (9*d*x)/2]))/(480*a^3*d*(1 + Cos[c + d*x])^3)","B",1
58,1,283,136,0.940124,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","-\frac{\sec \left(\frac{c}{2}\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \left(120 A \sin \left(c+\frac{d x}{2}\right)-80 A \sin \left(c+\frac{3 d x}{2}\right)+60 A \sin \left(2 c+\frac{3 d x}{2}\right)-28 A \sin \left(2 c+\frac{5 d x}{2}\right)-160 A \sin \left(\frac{d x}{2}\right)+1125 C \sin \left(c+\frac{d x}{2}\right)-1215 C \sin \left(c+\frac{3 d x}{2}\right)+225 C \sin \left(2 c+\frac{3 d x}{2}\right)-363 C \sin \left(2 c+\frac{5 d x}{2}\right)-75 C \sin \left(3 c+\frac{5 d x}{2}\right)-15 C \sin \left(3 c+\frac{7 d x}{2}\right)-15 C \sin \left(4 c+\frac{7 d x}{2}\right)+900 C d x \cos \left(c+\frac{d x}{2}\right)+450 C d x \cos \left(c+\frac{3 d x}{2}\right)+450 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+90 C d x \cos \left(2 c+\frac{5 d x}{2}\right)+90 C d x \cos \left(3 c+\frac{5 d x}{2}\right)-1755 C \sin \left(\frac{d x}{2}\right)+900 C d x \cos \left(\frac{d x}{2}\right)\right)}{960 a^3 d}","\frac{(2 A+27 C) \sin (c+d x)}{15 a^3 d}+\frac{3 C \sin (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{3 C x}{a^3}-\frac{(A+C) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(A-9 C) \sin (c+d x) \cos ^2(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"-1/960*(Sec[c/2]*Sec[(c + d*x)/2]^5*(900*C*d*x*Cos[(d*x)/2] + 900*C*d*x*Cos[c + (d*x)/2] + 450*C*d*x*Cos[c + (3*d*x)/2] + 450*C*d*x*Cos[2*c + (3*d*x)/2] + 90*C*d*x*Cos[2*c + (5*d*x)/2] + 90*C*d*x*Cos[3*c + (5*d*x)/2] - 160*A*Sin[(d*x)/2] - 1755*C*Sin[(d*x)/2] + 120*A*Sin[c + (d*x)/2] + 1125*C*Sin[c + (d*x)/2] - 80*A*Sin[c + (3*d*x)/2] - 1215*C*Sin[c + (3*d*x)/2] + 60*A*Sin[2*c + (3*d*x)/2] + 225*C*Sin[2*c + (3*d*x)/2] - 28*A*Sin[2*c + (5*d*x)/2] - 363*C*Sin[2*c + (5*d*x)/2] - 75*C*Sin[3*c + (5*d*x)/2] - 15*C*Sin[3*c + (7*d*x)/2] - 15*C*Sin[4*c + (7*d*x)/2]))/(a^3*d)","B",1
59,1,227,114,0.5490774,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \left(-30 A \sin \left(c+\frac{d x}{2}\right)+30 A \sin \left(c+\frac{3 d x}{2}\right)+6 A \sin \left(2 c+\frac{5 d x}{2}\right)+30 A \sin \left(\frac{d x}{2}\right)+270 C \sin \left(c+\frac{d x}{2}\right)-230 C \sin \left(c+\frac{3 d x}{2}\right)+90 C \sin \left(2 c+\frac{3 d x}{2}\right)-64 C \sin \left(2 c+\frac{5 d x}{2}\right)+150 C d x \cos \left(c+\frac{d x}{2}\right)+75 C d x \cos \left(c+\frac{3 d x}{2}\right)+75 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+15 C d x \cos \left(2 c+\frac{5 d x}{2}\right)+15 C d x \cos \left(3 c+\frac{5 d x}{2}\right)-370 C \sin \left(\frac{d x}{2}\right)+150 C d x \cos \left(\frac{d x}{2}\right)\right)}{480 a^3 d}","\frac{(6 A-29 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{C x}{a^3}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(3 A-7 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^5*(150*C*d*x*Cos[(d*x)/2] + 150*C*d*x*Cos[c + (d*x)/2] + 75*C*d*x*Cos[c + (3*d*x)/2] + 75*C*d*x*Cos[2*c + (3*d*x)/2] + 15*C*d*x*Cos[2*c + (5*d*x)/2] + 15*C*d*x*Cos[3*c + (5*d*x)/2] + 30*A*Sin[(d*x)/2] - 370*C*Sin[(d*x)/2] - 30*A*Sin[c + (d*x)/2] + 270*C*Sin[c + (d*x)/2] + 30*A*Sin[c + (3*d*x)/2] - 230*C*Sin[c + (3*d*x)/2] + 90*C*Sin[2*c + (3*d*x)/2] + 6*A*Sin[2*c + (5*d*x)/2] - 64*C*Sin[2*c + (5*d*x)/2]))/(480*a^3*d)","A",1
60,1,129,98,0.3027199,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(10 A \sin \left(c+\frac{3 d x}{2}\right)+2 A \sin \left(2 c+\frac{5 d x}{2}\right)+20 (A+2 C) \sin \left(\frac{d x}{2}\right)-30 C \sin \left(c+\frac{d x}{2}\right)+20 C \sin \left(c+\frac{3 d x}{2}\right)-15 C \sin \left(2 c+\frac{3 d x}{2}\right)+7 C \sin \left(2 c+\frac{5 d x}{2}\right)\right)}{30 a^3 d (\cos (c+d x)+1)^3}","\frac{(2 A+7 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{2 (A-4 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}+\frac{(A+C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(20*(A + 2*C)*Sin[(d*x)/2] - 30*C*Sin[c + (d*x)/2] + 10*A*Sin[c + (3*d*x)/2] + 20*C*Sin[c + (3*d*x)/2] - 15*C*Sin[2*c + (3*d*x)/2] + 2*A*Sin[2*c + (5*d*x)/2] + 7*C*Sin[2*c + (5*d*x)/2]))/(30*a^3*d*(1 + Cos[c + d*x])^3)","A",1
61,1,203,115,1.0940187,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(15 (5 A-C) \sin \left(c+\frac{d x}{2}\right)-95 A \sin \left(c+\frac{3 d x}{2}\right)+15 A \sin \left(2 c+\frac{3 d x}{2}\right)-22 A \sin \left(2 c+\frac{5 d x}{2}\right)-5 (29 A-3 C) \sin \left(\frac{d x}{2}\right)+15 C \sin \left(c+\frac{3 d x}{2}\right)+3 C \sin \left(2 c+\frac{5 d x}{2}\right)\right)-240 A \cos ^6\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{30 a^3 d (\cos (c+d x)+1)^3}","-\frac{(22 A-3 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{(7 A-3 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"(-240*A*Cos[(c + d*x)/2]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Cos[(c + d*x)/2]*Sec[c/2]*(-5*(29*A - 3*C)*Sin[(d*x)/2] + 15*(5*A - C)*Sin[c + (d*x)/2] - 95*A*Sin[c + (3*d*x)/2] + 15*C*Sin[c + (3*d*x)/2] + 15*A*Sin[2*c + (3*d*x)/2] - 22*A*Sin[2*c + (5*d*x)/2] + 3*C*Sin[2*c + (5*d*x)/2]))/(30*a^3*d*(1 + Cos[c + d*x])^3)","A",1
62,1,596,129,6.3210657,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^3,x]","\frac{\frac{\sec \left(\frac{c}{2}\right) \sec (c) \cos (c+d x) \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \left(-600 A \sin \left(c-\frac{d x}{2}\right)+375 A \sin \left(c+\frac{d x}{2}\right)-480 A \sin \left(2 c+\frac{d x}{2}\right)-60 A \sin \left(c+\frac{3 d x}{2}\right)+402 A \sin \left(2 c+\frac{3 d x}{2}\right)-225 A \sin \left(3 c+\frac{3 d x}{2}\right)+315 A \sin \left(c+\frac{5 d x}{2}\right)+30 A \sin \left(2 c+\frac{5 d x}{2}\right)+240 A \sin \left(3 c+\frac{5 d x}{2}\right)-45 A \sin \left(4 c+\frac{5 d x}{2}\right)+72 A \sin \left(2 c+\frac{7 d x}{2}\right)+15 A \sin \left(3 c+\frac{7 d x}{2}\right)+57 A \sin \left(4 c+\frac{7 d x}{2}\right)-255 A \sin \left(\frac{d x}{2}\right)+567 A \sin \left(\frac{3 d x}{2}\right)-10 C \sin \left(c-\frac{d x}{2}\right)+10 C \sin \left(c+\frac{d x}{2}\right)-20 C \sin \left(2 c+\frac{d x}{2}\right)+22 C \sin \left(2 c+\frac{3 d x}{2}\right)+10 C \sin \left(c+\frac{5 d x}{2}\right)+10 C \sin \left(3 c+\frac{5 d x}{2}\right)+2 C \sin \left(2 c+\frac{7 d x}{2}\right)+2 C \sin \left(4 c+\frac{7 d x}{2}\right)-20 C \sin \left(\frac{d x}{2}\right)+22 C \sin \left(\frac{3 d x}{2}\right)\right) \left(A \sec ^2(c+d x)+C\right)}{60 d (\cos (c+d x)+1)^3 (2 A+C \cos (2 c+2 d x)+C)}+\frac{48 A \cos ^2(c+d x) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sec ^2(c+d x)+C\right) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d (\cos (c+d x)+1)^3 (2 A+C \cos (2 c+2 d x)+C)}-\frac{48 A \cos ^2(c+d x) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sec ^2(c+d x)+C\right) \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d (\cos (c+d x)+1)^3 (2 A+C \cos (2 c+2 d x)+C)}}{a^3}","\frac{2 (36 A+C) \tan (c+d x)}{15 a^3 d}-\frac{3 A \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{3 A \tan (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(9 A-C) \tan (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A+C) \tan (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"((48*A*Cos[c/2 + (d*x)/2]^6*Cos[c + d*x]^2*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*(C + A*Sec[c + d*x]^2))/(d*(1 + Cos[c + d*x])^3*(2*A + C + C*Cos[2*c + 2*d*x])) - (48*A*Cos[c/2 + (d*x)/2]^6*Cos[c + d*x]^2*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*(C + A*Sec[c + d*x]^2))/(d*(1 + Cos[c + d*x])^3*(2*A + C + C*Cos[2*c + 2*d*x])) + (Cos[c/2 + (d*x)/2]*Cos[c + d*x]*Sec[c/2]*Sec[c]*(C + A*Sec[c + d*x]^2)*(-255*A*Sin[(d*x)/2] - 20*C*Sin[(d*x)/2] + 567*A*Sin[(3*d*x)/2] + 22*C*Sin[(3*d*x)/2] - 600*A*Sin[c - (d*x)/2] - 10*C*Sin[c - (d*x)/2] + 375*A*Sin[c + (d*x)/2] + 10*C*Sin[c + (d*x)/2] - 480*A*Sin[2*c + (d*x)/2] - 20*C*Sin[2*c + (d*x)/2] - 60*A*Sin[c + (3*d*x)/2] + 402*A*Sin[2*c + (3*d*x)/2] + 22*C*Sin[2*c + (3*d*x)/2] - 225*A*Sin[3*c + (3*d*x)/2] + 315*A*Sin[c + (5*d*x)/2] + 10*C*Sin[c + (5*d*x)/2] + 30*A*Sin[2*c + (5*d*x)/2] + 240*A*Sin[3*c + (5*d*x)/2] + 10*C*Sin[3*c + (5*d*x)/2] - 45*A*Sin[4*c + (5*d*x)/2] + 72*A*Sin[2*c + (7*d*x)/2] + 2*C*Sin[2*c + (7*d*x)/2] + 15*A*Sin[3*c + (7*d*x)/2] + 57*A*Sin[4*c + (7*d*x)/2] + 2*C*Sin[4*c + (7*d*x)/2]))/(60*d*(1 + Cos[c + d*x])^3*(2*A + C + C*Cos[2*c + 2*d*x])))/a^3","B",1
63,1,597,192,4.76269,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^3,x]","-\frac{1920 (13 A+2 C) \cos ^6\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+\sec \left(\frac{c}{2}\right) \sec (c) \cos \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(-4329 A \sin \left(c-\frac{d x}{2}\right)+1989 A \sin \left(c+\frac{d x}{2}\right)-3575 A \sin \left(2 c+\frac{d x}{2}\right)-475 A \sin \left(c+\frac{3 d x}{2}\right)+2005 A \sin \left(2 c+\frac{3 d x}{2}\right)-2275 A \sin \left(3 c+\frac{3 d x}{2}\right)+2673 A \sin \left(c+\frac{5 d x}{2}\right)+105 A \sin \left(2 c+\frac{5 d x}{2}\right)+1593 A \sin \left(3 c+\frac{5 d x}{2}\right)-975 A \sin \left(4 c+\frac{5 d x}{2}\right)+1325 A \sin \left(2 c+\frac{7 d x}{2}\right)+255 A \sin \left(3 c+\frac{7 d x}{2}\right)+875 A \sin \left(4 c+\frac{7 d x}{2}\right)-195 A \sin \left(5 c+\frac{7 d x}{2}\right)+304 A \sin \left(3 c+\frac{9 d x}{2}\right)+90 A \sin \left(4 c+\frac{9 d x}{2}\right)+214 A \sin \left(5 c+\frac{9 d x}{2}\right)-5 (247 A+98 C) \sin \left(\frac{d x}{2}\right)+5 (761 A+106 C) \sin \left(\frac{3 d x}{2}\right)-654 C \sin \left(c-\frac{d x}{2}\right)+654 C \sin \left(c+\frac{d x}{2}\right)-490 C \sin \left(2 c+\frac{d x}{2}\right)-350 C \sin \left(c+\frac{3 d x}{2}\right)+530 C \sin \left(2 c+\frac{3 d x}{2}\right)-350 C \sin \left(3 c+\frac{3 d x}{2}\right)+378 C \sin \left(c+\frac{5 d x}{2}\right)-150 C \sin \left(2 c+\frac{5 d x}{2}\right)+378 C \sin \left(3 c+\frac{5 d x}{2}\right)-150 C \sin \left(4 c+\frac{5 d x}{2}\right)+190 C \sin \left(2 c+\frac{7 d x}{2}\right)-30 C \sin \left(3 c+\frac{7 d x}{2}\right)+190 C \sin \left(4 c+\frac{7 d x}{2}\right)-30 C \sin \left(5 c+\frac{7 d x}{2}\right)+44 C \sin \left(3 c+\frac{9 d x}{2}\right)+44 C \sin \left(5 c+\frac{9 d x}{2}\right)\right)}{480 a^3 d (\cos (c+d x)+1)^3}","-\frac{2 (76 A+11 C) \tan (c+d x)}{15 a^3 d}+\frac{(13 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{(13 A+2 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{(76 A+11 C) \tan (c+d x) \sec (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(11 A+C) \tan (c+d x) \sec (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A+C) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"-1/480*(1920*(13*A + 2*C)*Cos[(c + d*x)/2]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Cos[(c + d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]^2*(-5*(247*A + 98*C)*Sin[(d*x)/2] + 5*(761*A + 106*C)*Sin[(3*d*x)/2] - 4329*A*Sin[c - (d*x)/2] - 654*C*Sin[c - (d*x)/2] + 1989*A*Sin[c + (d*x)/2] + 654*C*Sin[c + (d*x)/2] - 3575*A*Sin[2*c + (d*x)/2] - 490*C*Sin[2*c + (d*x)/2] - 475*A*Sin[c + (3*d*x)/2] - 350*C*Sin[c + (3*d*x)/2] + 2005*A*Sin[2*c + (3*d*x)/2] + 530*C*Sin[2*c + (3*d*x)/2] - 2275*A*Sin[3*c + (3*d*x)/2] - 350*C*Sin[3*c + (3*d*x)/2] + 2673*A*Sin[c + (5*d*x)/2] + 378*C*Sin[c + (5*d*x)/2] + 105*A*Sin[2*c + (5*d*x)/2] - 150*C*Sin[2*c + (5*d*x)/2] + 1593*A*Sin[3*c + (5*d*x)/2] + 378*C*Sin[3*c + (5*d*x)/2] - 975*A*Sin[4*c + (5*d*x)/2] - 150*C*Sin[4*c + (5*d*x)/2] + 1325*A*Sin[2*c + (7*d*x)/2] + 190*C*Sin[2*c + (7*d*x)/2] + 255*A*Sin[3*c + (7*d*x)/2] - 30*C*Sin[3*c + (7*d*x)/2] + 875*A*Sin[4*c + (7*d*x)/2] + 190*C*Sin[4*c + (7*d*x)/2] - 195*A*Sin[5*c + (7*d*x)/2] - 30*C*Sin[5*c + (7*d*x)/2] + 304*A*Sin[3*c + (9*d*x)/2] + 44*C*Sin[3*c + (9*d*x)/2] + 90*A*Sin[4*c + (9*d*x)/2] + 214*A*Sin[5*c + (9*d*x)/2] + 44*C*Sin[5*c + (9*d*x)/2]))/(a^3*d*(1 + Cos[c + d*x])^3)","B",1
64,1,798,225,6.4753478,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^3,x]","\frac{4 (23 A+6 C) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^3}-\frac{4 (23 A+6 C) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^3}+\frac{\sec \left(\frac{c}{2}\right) \sec (c) \sec ^3(c+d x) \left(-2484 A \sin \left(\frac{d x}{2}\right)-1764 C \sin \left(\frac{d x}{2}\right)+12622 A \sin \left(\frac{3 d x}{2}\right)+3372 C \sin \left(\frac{3 d x}{2}\right)-13340 A \sin \left(c-\frac{d x}{2}\right)-3480 C \sin \left(c-\frac{d x}{2}\right)+4140 A \sin \left(c+\frac{d x}{2}\right)+2100 C \sin \left(c+\frac{d x}{2}\right)-11684 A \sin \left(2 c+\frac{d x}{2}\right)-3144 C \sin \left(2 c+\frac{d x}{2}\right)-450 A \sin \left(c+\frac{3 d x}{2}\right)-960 C \sin \left(c+\frac{3 d x}{2}\right)+5022 A \sin \left(2 c+\frac{3 d x}{2}\right)+2232 C \sin \left(2 c+\frac{3 d x}{2}\right)-8050 A \sin \left(3 c+\frac{3 d x}{2}\right)-2100 C \sin \left(3 c+\frac{3 d x}{2}\right)+9230 A \sin \left(c+\frac{5 d x}{2}\right)+2460 C \sin \left(c+\frac{5 d x}{2}\right)+630 A \sin \left(2 c+\frac{5 d x}{2}\right)-390 C \sin \left(2 c+\frac{5 d x}{2}\right)+4230 A \sin \left(3 c+\frac{5 d x}{2}\right)+1710 C \sin \left(3 c+\frac{5 d x}{2}\right)-4370 A \sin \left(4 c+\frac{5 d x}{2}\right)-1140 C \sin \left(4 c+\frac{5 d x}{2}\right)+5347 A \sin \left(2 c+\frac{7 d x}{2}\right)+1422 C \sin \left(2 c+\frac{7 d x}{2}\right)+875 A \sin \left(3 c+\frac{7 d x}{2}\right)-60 C \sin \left(3 c+\frac{7 d x}{2}\right)+2747 A \sin \left(4 c+\frac{7 d x}{2}\right)+1032 C \sin \left(4 c+\frac{7 d x}{2}\right)-1725 A \sin \left(5 c+\frac{7 d x}{2}\right)-450 C \sin \left(5 c+\frac{7 d x}{2}\right)+2375 A \sin \left(3 c+\frac{9 d x}{2}\right)+630 C \sin \left(3 c+\frac{9 d x}{2}\right)+655 A \sin \left(4 c+\frac{9 d x}{2}\right)+60 C \sin \left(4 c+\frac{9 d x}{2}\right)+1375 A \sin \left(5 c+\frac{9 d x}{2}\right)+480 C \sin \left(5 c+\frac{9 d x}{2}\right)-345 A \sin \left(6 c+\frac{9 d x}{2}\right)-90 C \sin \left(6 c+\frac{9 d x}{2}\right)+544 A \sin \left(4 c+\frac{11 d x}{2}\right)+144 C \sin \left(4 c+\frac{11 d x}{2}\right)+200 A \sin \left(5 c+\frac{11 d x}{2}\right)+30 C \sin \left(5 c+\frac{11 d x}{2}\right)+344 A \sin \left(6 c+\frac{11 d x}{2}\right)+114 C \sin \left(6 c+\frac{11 d x}{2}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{960 d (\cos (c+d x) a+a)^3}","\frac{4 (34 A+9 C) \tan ^3(c+d x)}{15 a^3 d}+\frac{4 (34 A+9 C) \tan (c+d x)}{5 a^3 d}-\frac{(23 A+6 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{(23 A+6 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{(23 A+6 C) \tan (c+d x) \sec ^2(c+d x)}{3 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(13 A+3 C) \tan (c+d x) \sec ^2(c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"(4*(23*A + 6*C)*Cos[c/2 + (d*x)/2]^6*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]])/(d*(a + a*Cos[c + d*x])^3) - (4*(23*A + 6*C)*Cos[c/2 + (d*x)/2]^6*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]])/(d*(a + a*Cos[c + d*x])^3) + (Cos[c/2 + (d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]^3*(-2484*A*Sin[(d*x)/2] - 1764*C*Sin[(d*x)/2] + 12622*A*Sin[(3*d*x)/2] + 3372*C*Sin[(3*d*x)/2] - 13340*A*Sin[c - (d*x)/2] - 3480*C*Sin[c - (d*x)/2] + 4140*A*Sin[c + (d*x)/2] + 2100*C*Sin[c + (d*x)/2] - 11684*A*Sin[2*c + (d*x)/2] - 3144*C*Sin[2*c + (d*x)/2] - 450*A*Sin[c + (3*d*x)/2] - 960*C*Sin[c + (3*d*x)/2] + 5022*A*Sin[2*c + (3*d*x)/2] + 2232*C*Sin[2*c + (3*d*x)/2] - 8050*A*Sin[3*c + (3*d*x)/2] - 2100*C*Sin[3*c + (3*d*x)/2] + 9230*A*Sin[c + (5*d*x)/2] + 2460*C*Sin[c + (5*d*x)/2] + 630*A*Sin[2*c + (5*d*x)/2] - 390*C*Sin[2*c + (5*d*x)/2] + 4230*A*Sin[3*c + (5*d*x)/2] + 1710*C*Sin[3*c + (5*d*x)/2] - 4370*A*Sin[4*c + (5*d*x)/2] - 1140*C*Sin[4*c + (5*d*x)/2] + 5347*A*Sin[2*c + (7*d*x)/2] + 1422*C*Sin[2*c + (7*d*x)/2] + 875*A*Sin[3*c + (7*d*x)/2] - 60*C*Sin[3*c + (7*d*x)/2] + 2747*A*Sin[4*c + (7*d*x)/2] + 1032*C*Sin[4*c + (7*d*x)/2] - 1725*A*Sin[5*c + (7*d*x)/2] - 450*C*Sin[5*c + (7*d*x)/2] + 2375*A*Sin[3*c + (9*d*x)/2] + 630*C*Sin[3*c + (9*d*x)/2] + 655*A*Sin[4*c + (9*d*x)/2] + 60*C*Sin[4*c + (9*d*x)/2] + 1375*A*Sin[5*c + (9*d*x)/2] + 480*C*Sin[5*c + (9*d*x)/2] - 345*A*Sin[6*c + (9*d*x)/2] - 90*C*Sin[6*c + (9*d*x)/2] + 544*A*Sin[4*c + (11*d*x)/2] + 144*C*Sin[4*c + (11*d*x)/2] + 200*A*Sin[5*c + (11*d*x)/2] + 30*C*Sin[5*c + (11*d*x)/2] + 344*A*Sin[6*c + (11*d*x)/2] + 114*C*Sin[6*c + (11*d*x)/2]))/(960*d*(a + a*Cos[c + d*x])^3)","B",1
65,1,513,223,1.0980971,"\int \frac{\cos ^4(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(14700 d x (2 A+21 C) \cos \left(c+\frac{d x}{2}\right)+66080 A \sin \left(c+\frac{d x}{2}\right)-57120 A \sin \left(c+\frac{3 d x}{2}\right)+30240 A \sin \left(2 c+\frac{3 d x}{2}\right)-22400 A \sin \left(2 c+\frac{5 d x}{2}\right)+6720 A \sin \left(3 c+\frac{5 d x}{2}\right)-4160 A \sin \left(3 c+\frac{7 d x}{2}\right)+17640 A d x \cos \left(c+\frac{3 d x}{2}\right)+17640 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+5880 A d x \cos \left(2 c+\frac{5 d x}{2}\right)+5880 A d x \cos \left(3 c+\frac{5 d x}{2}\right)+840 A d x \cos \left(3 c+\frac{7 d x}{2}\right)+840 A d x \cos \left(4 c+\frac{7 d x}{2}\right)+14700 d x (2 A+21 C) \cos \left(\frac{d x}{2}\right)-79520 A \sin \left(\frac{d x}{2}\right)+386190 C \sin \left(c+\frac{d x}{2}\right)-422478 C \sin \left(c+\frac{3 d x}{2}\right)+132930 C \sin \left(2 c+\frac{3 d x}{2}\right)-181461 C \sin \left(2 c+\frac{5 d x}{2}\right)+3675 C \sin \left(3 c+\frac{5 d x}{2}\right)-36003 C \sin \left(3 c+\frac{7 d x}{2}\right)-9555 C \sin \left(4 c+\frac{7 d x}{2}\right)-945 C \sin \left(4 c+\frac{9 d x}{2}\right)-945 C \sin \left(5 c+\frac{9 d x}{2}\right)+105 C \sin \left(5 c+\frac{11 d x}{2}\right)+105 C \sin \left(6 c+\frac{11 d x}{2}\right)+185220 C d x \cos \left(c+\frac{3 d x}{2}\right)+185220 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+61740 C d x \cos \left(2 c+\frac{5 d x}{2}\right)+61740 C d x \cos \left(3 c+\frac{5 d x}{2}\right)+8820 C d x \cos \left(3 c+\frac{7 d x}{2}\right)+8820 C d x \cos \left(4 c+\frac{7 d x}{2}\right)-539490 C \sin \left(\frac{d x}{2}\right)\right)}{6720 a^4 d (\cos (c+d x)+1)^4}","-\frac{32 (5 A+54 C) \sin (c+d x)}{105 a^4 d}-\frac{(10 A+129 C) \sin (c+d x) \cos ^3(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{16 (5 A+54 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)}+\frac{(2 A+21 C) \sin (c+d x) \cos (c+d x)}{2 a^4 d}+\frac{x (2 A+21 C)}{2 a^4}-\frac{(A+C) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{2 C \sin (c+d x) \cos ^4(c+d x)}{5 a d (a \cos (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(14700*(2*A + 21*C)*d*x*Cos[(d*x)/2] + 14700*(2*A + 21*C)*d*x*Cos[c + (d*x)/2] + 17640*A*d*x*Cos[c + (3*d*x)/2] + 185220*C*d*x*Cos[c + (3*d*x)/2] + 17640*A*d*x*Cos[2*c + (3*d*x)/2] + 185220*C*d*x*Cos[2*c + (3*d*x)/2] + 5880*A*d*x*Cos[2*c + (5*d*x)/2] + 61740*C*d*x*Cos[2*c + (5*d*x)/2] + 5880*A*d*x*Cos[3*c + (5*d*x)/2] + 61740*C*d*x*Cos[3*c + (5*d*x)/2] + 840*A*d*x*Cos[3*c + (7*d*x)/2] + 8820*C*d*x*Cos[3*c + (7*d*x)/2] + 840*A*d*x*Cos[4*c + (7*d*x)/2] + 8820*C*d*x*Cos[4*c + (7*d*x)/2] - 79520*A*Sin[(d*x)/2] - 539490*C*Sin[(d*x)/2] + 66080*A*Sin[c + (d*x)/2] + 386190*C*Sin[c + (d*x)/2] - 57120*A*Sin[c + (3*d*x)/2] - 422478*C*Sin[c + (3*d*x)/2] + 30240*A*Sin[2*c + (3*d*x)/2] + 132930*C*Sin[2*c + (3*d*x)/2] - 22400*A*Sin[2*c + (5*d*x)/2] - 181461*C*Sin[2*c + (5*d*x)/2] + 6720*A*Sin[3*c + (5*d*x)/2] + 3675*C*Sin[3*c + (5*d*x)/2] - 4160*A*Sin[3*c + (7*d*x)/2] - 36003*C*Sin[3*c + (7*d*x)/2] - 9555*C*Sin[4*c + (7*d*x)/2] - 945*C*Sin[4*c + (9*d*x)/2] - 945*C*Sin[5*c + (9*d*x)/2] + 105*C*Sin[5*c + (11*d*x)/2] + 105*C*Sin[6*c + (11*d*x)/2]))/(6720*a^4*d*(1 + Cos[c + d*x])^4)","B",1
66,1,371,174,0.7618483,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^4,x]","-\frac{\sec \left(\frac{c}{2}\right) \sec ^7\left(\frac{1}{2} (c+d x)\right) \left(2520 A \sin \left(c+\frac{d x}{2}\right)-1764 A \sin \left(c+\frac{3 d x}{2}\right)+1260 A \sin \left(2 c+\frac{3 d x}{2}\right)-588 A \sin \left(2 c+\frac{5 d x}{2}\right)+420 A \sin \left(3 c+\frac{5 d x}{2}\right)-144 A \sin \left(3 c+\frac{7 d x}{2}\right)-2520 A \sin \left(\frac{d x}{2}\right)+46130 C \sin \left(c+\frac{d x}{2}\right)-46116 C \sin \left(c+\frac{3 d x}{2}\right)+18060 C \sin \left(2 c+\frac{3 d x}{2}\right)-19292 C \sin \left(2 c+\frac{5 d x}{2}\right)+2100 C \sin \left(3 c+\frac{5 d x}{2}\right)-3791 C \sin \left(3 c+\frac{7 d x}{2}\right)-735 C \sin \left(4 c+\frac{7 d x}{2}\right)-105 C \sin \left(4 c+\frac{9 d x}{2}\right)-105 C \sin \left(5 c+\frac{9 d x}{2}\right)+29400 C d x \cos \left(c+\frac{d x}{2}\right)+17640 C d x \cos \left(c+\frac{3 d x}{2}\right)+17640 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+5880 C d x \cos \left(2 c+\frac{5 d x}{2}\right)+5880 C d x \cos \left(3 c+\frac{5 d x}{2}\right)+840 C d x \cos \left(3 c+\frac{7 d x}{2}\right)+840 C d x \cos \left(4 c+\frac{7 d x}{2}\right)-60830 C \sin \left(\frac{d x}{2}\right)+29400 C d x \cos \left(\frac{d x}{2}\right)\right)}{26880 a^4 d}","\frac{2 (3 A+122 C) \sin (c+d x)}{105 a^4 d}+\frac{(3 A-88 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{4 C \sin (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac{4 C x}{a^4}-\frac{(A+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{2 (A-6 C) \sin (c+d x) \cos ^3(c+d x)}{35 a d (a \cos (c+d x)+a)^3}",1,"-1/26880*(Sec[c/2]*Sec[(c + d*x)/2]^7*(29400*C*d*x*Cos[(d*x)/2] + 29400*C*d*x*Cos[c + (d*x)/2] + 17640*C*d*x*Cos[c + (3*d*x)/2] + 17640*C*d*x*Cos[2*c + (3*d*x)/2] + 5880*C*d*x*Cos[2*c + (5*d*x)/2] + 5880*C*d*x*Cos[3*c + (5*d*x)/2] + 840*C*d*x*Cos[3*c + (7*d*x)/2] + 840*C*d*x*Cos[4*c + (7*d*x)/2] - 2520*A*Sin[(d*x)/2] - 60830*C*Sin[(d*x)/2] + 2520*A*Sin[c + (d*x)/2] + 46130*C*Sin[c + (d*x)/2] - 1764*A*Sin[c + (3*d*x)/2] - 46116*C*Sin[c + (3*d*x)/2] + 1260*A*Sin[2*c + (3*d*x)/2] + 18060*C*Sin[2*c + (3*d*x)/2] - 588*A*Sin[2*c + (5*d*x)/2] - 19292*C*Sin[2*c + (5*d*x)/2] + 420*A*Sin[3*c + (5*d*x)/2] + 2100*C*Sin[3*c + (5*d*x)/2] - 144*A*Sin[3*c + (7*d*x)/2] - 3791*C*Sin[3*c + (7*d*x)/2] - 735*C*Sin[4*c + (7*d*x)/2] - 105*C*Sin[4*c + (9*d*x)/2] - 105*C*Sin[5*c + (9*d*x)/2]))/(a^4*d)","B",1
67,1,315,152,0.7413177,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^7\left(\frac{1}{2} (c+d x)\right) \left(-350 A \sin \left(c+\frac{d x}{2}\right)+336 A \sin \left(c+\frac{3 d x}{2}\right)-210 A \sin \left(2 c+\frac{3 d x}{2}\right)+182 A \sin \left(2 c+\frac{5 d x}{2}\right)+26 A \sin \left(3 c+\frac{7 d x}{2}\right)+560 A \sin \left(\frac{d x}{2}\right)+8260 C \sin \left(c+\frac{d x}{2}\right)-7140 C \sin \left(c+\frac{3 d x}{2}\right)+3780 C \sin \left(2 c+\frac{3 d x}{2}\right)-2800 C \sin \left(2 c+\frac{5 d x}{2}\right)+840 C \sin \left(3 c+\frac{5 d x}{2}\right)-520 C \sin \left(3 c+\frac{7 d x}{2}\right)+3675 C d x \cos \left(c+\frac{d x}{2}\right)+2205 C d x \cos \left(c+\frac{3 d x}{2}\right)+2205 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+735 C d x \cos \left(2 c+\frac{5 d x}{2}\right)+735 C d x \cos \left(3 c+\frac{5 d x}{2}\right)+105 C d x \cos \left(3 c+\frac{7 d x}{2}\right)+105 C d x \cos \left(4 c+\frac{7 d x}{2}\right)-9940 C \sin \left(\frac{d x}{2}\right)+3675 C d x \cos \left(\frac{d x}{2}\right)\right)}{13440 a^4 d}","\frac{(16 A-215 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(8 A-55 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{C x}{a^4}-\frac{(A+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{2 (2 A-5 C) \sin (c+d x) \cos ^2(c+d x)}{35 a d (a \cos (c+d x)+a)^3}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^7*(3675*C*d*x*Cos[(d*x)/2] + 3675*C*d*x*Cos[c + (d*x)/2] + 2205*C*d*x*Cos[c + (3*d*x)/2] + 2205*C*d*x*Cos[2*c + (3*d*x)/2] + 735*C*d*x*Cos[2*c + (5*d*x)/2] + 735*C*d*x*Cos[3*c + (5*d*x)/2] + 105*C*d*x*Cos[3*c + (7*d*x)/2] + 105*C*d*x*Cos[4*c + (7*d*x)/2] + 560*A*Sin[(d*x)/2] - 9940*C*Sin[(d*x)/2] - 350*A*Sin[c + (d*x)/2] + 8260*C*Sin[c + (d*x)/2] + 336*A*Sin[c + (3*d*x)/2] - 7140*C*Sin[c + (3*d*x)/2] - 210*A*Sin[2*c + (3*d*x)/2] + 3780*C*Sin[2*c + (3*d*x)/2] + 182*A*Sin[2*c + (5*d*x)/2] - 2800*C*Sin[2*c + (5*d*x)/2] + 840*C*Sin[3*c + (5*d*x)/2] + 26*A*Sin[3*c + (7*d*x)/2] - 520*C*Sin[3*c + (7*d*x)/2]))/(13440*a^4*d)","B",1
68,1,179,138,0.409707,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-70 (2 A+9 C) \sin \left(c+\frac{d x}{2}\right)+168 A \sin \left(c+\frac{3 d x}{2}\right)+56 A \sin \left(2 c+\frac{5 d x}{2}\right)+8 A \sin \left(3 c+\frac{7 d x}{2}\right)+70 (2 A+9 C) \sin \left(\frac{d x}{2}\right)+441 C \sin \left(c+\frac{3 d x}{2}\right)-315 C \sin \left(2 c+\frac{3 d x}{2}\right)+147 C \sin \left(2 c+\frac{5 d x}{2}\right)-105 C \sin \left(3 c+\frac{5 d x}{2}\right)+36 C \sin \left(3 c+\frac{7 d x}{2}\right)\right)}{420 a^4 d (\cos (c+d x)+1)^4}","\frac{4 (2 A+9 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}+\frac{(23 A-54 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{2 (3 A-4 C) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(70*(2*A + 9*C)*Sin[(d*x)/2] - 70*(2*A + 9*C)*Sin[c + (d*x)/2] + 168*A*Sin[c + (3*d*x)/2] + 441*C*Sin[c + (3*d*x)/2] - 315*C*Sin[2*c + (3*d*x)/2] + 56*A*Sin[2*c + (5*d*x)/2] + 147*C*Sin[2*c + (5*d*x)/2] - 105*C*Sin[3*c + (5*d*x)/2] + 8*A*Sin[3*c + (7*d*x)/2] + 36*C*Sin[3*c + (7*d*x)/2]))/(420*a^4*d*(1 + Cos[c + d*x])^4)","A",1
69,1,159,136,0.376774,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(126 A \sin \left(c+\frac{3 d x}{2}\right)+42 A \sin \left(2 c+\frac{5 d x}{2}\right)+6 A \sin \left(3 c+\frac{7 d x}{2}\right)+70 (3 A+4 C) \sin \left(\frac{d x}{2}\right)-175 C \sin \left(c+\frac{d x}{2}\right)+168 C \sin \left(c+\frac{3 d x}{2}\right)-105 C \sin \left(2 c+\frac{3 d x}{2}\right)+91 C \sin \left(2 c+\frac{5 d x}{2}\right)+13 C \sin \left(3 c+\frac{7 d x}{2}\right)\right)}{420 a^4 d (\cos (c+d x)+1)^4}","\frac{(6 A+13 C) \sin (c+d x)}{105 d \left(a^4 \cos (c+d x)+a^4\right)}+\frac{(6 A+13 C) \sin (c+d x)}{105 d \left(a^2 \cos (c+d x)+a^2\right)^2}+\frac{(3 A-11 C) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac{(A+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(70*(3*A + 4*C)*Sin[(d*x)/2] - 175*C*Sin[c + (d*x)/2] + 126*A*Sin[c + (3*d*x)/2] + 168*C*Sin[c + (3*d*x)/2] - 105*C*Sin[2*c + (3*d*x)/2] + 42*A*Sin[2*c + (5*d*x)/2] + 91*C*Sin[2*c + (5*d*x)/2] + 6*A*Sin[3*c + (7*d*x)/2] + 13*C*Sin[3*c + (7*d*x)/2]))/(420*a^4*d*(1 + Cos[c + d*x])^4)","A",1
70,1,245,145,1.7605764,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+a \cos (c+d x))^4} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(70 (31 A-2 C) \sin \left(c+\frac{d x}{2}\right)-2625 A \sin \left(c+\frac{3 d x}{2}\right)+735 A \sin \left(2 c+\frac{3 d x}{2}\right)-1015 A \sin \left(2 c+\frac{5 d x}{2}\right)+105 A \sin \left(3 c+\frac{5 d x}{2}\right)-160 A \sin \left(3 c+\frac{7 d x}{2}\right)-70 (49 A-2 C) \sin \left(\frac{d x}{2}\right)+168 C \sin \left(c+\frac{3 d x}{2}\right)+56 C \sin \left(2 c+\frac{5 d x}{2}\right)+8 C \sin \left(3 c+\frac{7 d x}{2}\right)\right)-6720 A \cos ^8\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{420 a^4 d (\cos (c+d x)+1)^4}","-\frac{8 (20 A-C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(55 A-8 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{2 (5 A-2 C) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"(-6720*A*Cos[(c + d*x)/2]^8*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Cos[(c + d*x)/2]*Sec[c/2]*(-70*(49*A - 2*C)*Sin[(d*x)/2] + 70*(31*A - 2*C)*Sin[c + (d*x)/2] - 2625*A*Sin[c + (3*d*x)/2] + 168*C*Sin[c + (3*d*x)/2] + 735*A*Sin[2*c + (3*d*x)/2] - 1015*A*Sin[2*c + (5*d*x)/2] + 56*C*Sin[2*c + (5*d*x)/2] + 105*A*Sin[3*c + (5*d*x)/2] - 160*A*Sin[3*c + (7*d*x)/2] + 8*C*Sin[3*c + (7*d*x)/2]))/(420*a^4*d*(1 + Cos[c + d*x])^4)","A",1
71,1,680,161,6.4324788,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^4,x]","\frac{\frac{\sec \left(\frac{c}{2}\right) \sec (c) \cos (c+d x) \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \left(-20524 A \sin \left(c-\frac{d x}{2}\right)+14644 A \sin \left(c+\frac{d x}{2}\right)-16660 A \sin \left(2 c+\frac{d x}{2}\right)-4690 A \sin \left(c+\frac{3 d x}{2}\right)+14378 A \sin \left(2 c+\frac{3 d x}{2}\right)-9100 A \sin \left(3 c+\frac{3 d x}{2}\right)+11668 A \sin \left(c+\frac{5 d x}{2}\right)-630 A \sin \left(2 c+\frac{5 d x}{2}\right)+9358 A \sin \left(3 c+\frac{5 d x}{2}\right)-2940 A \sin \left(4 c+\frac{5 d x}{2}\right)+4228 A \sin \left(2 c+\frac{7 d x}{2}\right)+315 A \sin \left(3 c+\frac{7 d x}{2}\right)+3493 A \sin \left(4 c+\frac{7 d x}{2}\right)-420 A \sin \left(5 c+\frac{7 d x}{2}\right)+664 A \sin \left(3 c+\frac{9 d x}{2}\right)+105 A \sin \left(4 c+\frac{9 d x}{2}\right)+559 A \sin \left(5 c+\frac{9 d x}{2}\right)-10780 A \sin \left(\frac{d x}{2}\right)+18788 A \sin \left(\frac{3 d x}{2}\right)-126 C \sin \left(c-\frac{d x}{2}\right)+126 C \sin \left(c+\frac{d x}{2}\right)-210 C \sin \left(2 c+\frac{d x}{2}\right)+252 C \sin \left(2 c+\frac{3 d x}{2}\right)+132 C \sin \left(c+\frac{5 d x}{2}\right)+132 C \sin \left(3 c+\frac{5 d x}{2}\right)+42 C \sin \left(2 c+\frac{7 d x}{2}\right)+42 C \sin \left(4 c+\frac{7 d x}{2}\right)+6 C \sin \left(3 c+\frac{9 d x}{2}\right)+6 C \sin \left(5 c+\frac{9 d x}{2}\right)-210 C \sin \left(\frac{d x}{2}\right)+252 C \sin \left(\frac{3 d x}{2}\right)\right) \left(A \sec ^2(c+d x)+C\right)}{840 d (\cos (c+d x)+1)^4 (2 A+C \cos (2 c+2 d x)+C)}+\frac{128 A \cos ^2(c+d x) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sec ^2(c+d x)+C\right) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d (\cos (c+d x)+1)^4 (2 A+C \cos (2 c+2 d x)+C)}-\frac{128 A \cos ^2(c+d x) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sec ^2(c+d x)+C\right) \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d (\cos (c+d x)+1)^4 (2 A+C \cos (2 c+2 d x)+C)}}{a^4}","\frac{2 (332 A+3 C) \tan (c+d x)}{105 a^4 d}-\frac{(88 A-3 C) \tan (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{4 A \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{4 A \tan (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac{2 (6 A-C) \tan (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A+C) \tan (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"((128*A*Cos[c/2 + (d*x)/2]^8*Cos[c + d*x]^2*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*(C + A*Sec[c + d*x]^2))/(d*(1 + Cos[c + d*x])^4*(2*A + C + C*Cos[2*c + 2*d*x])) - (128*A*Cos[c/2 + (d*x)/2]^8*Cos[c + d*x]^2*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*(C + A*Sec[c + d*x]^2))/(d*(1 + Cos[c + d*x])^4*(2*A + C + C*Cos[2*c + 2*d*x])) + (Cos[c/2 + (d*x)/2]*Cos[c + d*x]*Sec[c/2]*Sec[c]*(C + A*Sec[c + d*x]^2)*(-10780*A*Sin[(d*x)/2] - 210*C*Sin[(d*x)/2] + 18788*A*Sin[(3*d*x)/2] + 252*C*Sin[(3*d*x)/2] - 20524*A*Sin[c - (d*x)/2] - 126*C*Sin[c - (d*x)/2] + 14644*A*Sin[c + (d*x)/2] + 126*C*Sin[c + (d*x)/2] - 16660*A*Sin[2*c + (d*x)/2] - 210*C*Sin[2*c + (d*x)/2] - 4690*A*Sin[c + (3*d*x)/2] + 14378*A*Sin[2*c + (3*d*x)/2] + 252*C*Sin[2*c + (3*d*x)/2] - 9100*A*Sin[3*c + (3*d*x)/2] + 11668*A*Sin[c + (5*d*x)/2] + 132*C*Sin[c + (5*d*x)/2] - 630*A*Sin[2*c + (5*d*x)/2] + 9358*A*Sin[3*c + (5*d*x)/2] + 132*C*Sin[3*c + (5*d*x)/2] - 2940*A*Sin[4*c + (5*d*x)/2] + 4228*A*Sin[2*c + (7*d*x)/2] + 42*C*Sin[2*c + (7*d*x)/2] + 315*A*Sin[3*c + (7*d*x)/2] + 3493*A*Sin[4*c + (7*d*x)/2] + 42*C*Sin[4*c + (7*d*x)/2] - 420*A*Sin[5*c + (7*d*x)/2] + 664*A*Sin[3*c + (9*d*x)/2] + 6*C*Sin[3*c + (9*d*x)/2] + 105*A*Sin[4*c + (9*d*x)/2] + 559*A*Sin[5*c + (9*d*x)/2] + 6*C*Sin[5*c + (9*d*x)/2]))/(840*d*(1 + Cos[c + d*x])^4*(2*A + C + C*Cos[2*c + 2*d*x])))/a^4","B",1
72,1,784,224,6.4933597,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^4,x]","-\frac{8 (21 A+2 C) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d (a \cos (c+d x)+a)^4}+\frac{8 (21 A+2 C) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right) \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d (a \cos (c+d x)+a)^4}+\frac{\sec \left(\frac{c}{2}\right) \sec (c) \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x) \left(183162 A \sin \left(c-\frac{d x}{2}\right)-100842 A \sin \left(c+\frac{d x}{2}\right)+155526 A \sin \left(2 c+\frac{d x}{2}\right)+37380 A \sin \left(c+\frac{3 d x}{2}\right)-101148 A \sin \left(2 c+\frac{3 d x}{2}\right)+102900 A \sin \left(3 c+\frac{3 d x}{2}\right)-119364 A \sin \left(c+\frac{5 d x}{2}\right)+8820 A \sin \left(2 c+\frac{5 d x}{2}\right)-78204 A \sin \left(3 c+\frac{5 d x}{2}\right)+49980 A \sin \left(4 c+\frac{5 d x}{2}\right)-64053 A \sin \left(2 c+\frac{7 d x}{2}\right)-3885 A \sin \left(3 c+\frac{7 d x}{2}\right)-44733 A \sin \left(4 c+\frac{7 d x}{2}\right)+15435 A \sin \left(5 c+\frac{7 d x}{2}\right)-21987 A \sin \left(3 c+\frac{9 d x}{2}\right)-3675 A \sin \left(4 c+\frac{9 d x}{2}\right)-16107 A \sin \left(5 c+\frac{9 d x}{2}\right)+2205 A \sin \left(6 c+\frac{9 d x}{2}\right)-3456 A \sin \left(4 c+\frac{11 d x}{2}\right)-840 A \sin \left(5 c+\frac{11 d x}{2}\right)-2616 A \sin \left(6 c+\frac{11 d x}{2}\right)+73206 A \sin \left(\frac{d x}{2}\right)-166668 A \sin \left(\frac{3 d x}{2}\right)+17220 C \sin \left(c-\frac{d x}{2}\right)-17220 C \sin \left(c+\frac{d x}{2}\right)+14140 C \sin \left(2 c+\frac{d x}{2}\right)+9800 C \sin \left(c+\frac{3 d x}{2}\right)-15160 C \sin \left(2 c+\frac{3 d x}{2}\right)+9800 C \sin \left(3 c+\frac{3 d x}{2}\right)-10920 C \sin \left(c+\frac{5 d x}{2}\right)+4760 C \sin \left(2 c+\frac{5 d x}{2}\right)-10920 C \sin \left(3 c+\frac{5 d x}{2}\right)+4760 C \sin \left(4 c+\frac{5 d x}{2}\right)-5890 C \sin \left(2 c+\frac{7 d x}{2}\right)+1470 C \sin \left(3 c+\frac{7 d x}{2}\right)-5890 C \sin \left(4 c+\frac{7 d x}{2}\right)+1470 C \sin \left(5 c+\frac{7 d x}{2}\right)-2030 C \sin \left(3 c+\frac{9 d x}{2}\right)+210 C \sin \left(4 c+\frac{9 d x}{2}\right)-2030 C \sin \left(5 c+\frac{9 d x}{2}\right)+210 C \sin \left(6 c+\frac{9 d x}{2}\right)-320 C \sin \left(4 c+\frac{11 d x}{2}\right)-320 C \sin \left(6 c+\frac{11 d x}{2}\right)+14140 C \sin \left(\frac{d x}{2}\right)-15160 C \sin \left(\frac{3 d x}{2}\right)\right)}{6720 d (a \cos (c+d x)+a)^4}","-\frac{32 (54 A+5 C) \tan (c+d x)}{105 a^4 d}+\frac{(21 A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{(21 A+2 C) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{16 (54 A+5 C) \tan (c+d x) \sec (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(129 A+10 C) \tan (c+d x) \sec (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(A+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{2 A \tan (c+d x) \sec (c+d x)}{5 a d (a \cos (c+d x)+a)^3}",1,"(-8*(21*A + 2*C)*Cos[c/2 + (d*x)/2]^8*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]])/(d*(a + a*Cos[c + d*x])^4) + (8*(21*A + 2*C)*Cos[c/2 + (d*x)/2]^8*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]])/(d*(a + a*Cos[c + d*x])^4) + (Cos[c/2 + (d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]^2*(73206*A*Sin[(d*x)/2] + 14140*C*Sin[(d*x)/2] - 166668*A*Sin[(3*d*x)/2] - 15160*C*Sin[(3*d*x)/2] + 183162*A*Sin[c - (d*x)/2] + 17220*C*Sin[c - (d*x)/2] - 100842*A*Sin[c + (d*x)/2] - 17220*C*Sin[c + (d*x)/2] + 155526*A*Sin[2*c + (d*x)/2] + 14140*C*Sin[2*c + (d*x)/2] + 37380*A*Sin[c + (3*d*x)/2] + 9800*C*Sin[c + (3*d*x)/2] - 101148*A*Sin[2*c + (3*d*x)/2] - 15160*C*Sin[2*c + (3*d*x)/2] + 102900*A*Sin[3*c + (3*d*x)/2] + 9800*C*Sin[3*c + (3*d*x)/2] - 119364*A*Sin[c + (5*d*x)/2] - 10920*C*Sin[c + (5*d*x)/2] + 8820*A*Sin[2*c + (5*d*x)/2] + 4760*C*Sin[2*c + (5*d*x)/2] - 78204*A*Sin[3*c + (5*d*x)/2] - 10920*C*Sin[3*c + (5*d*x)/2] + 49980*A*Sin[4*c + (5*d*x)/2] + 4760*C*Sin[4*c + (5*d*x)/2] - 64053*A*Sin[2*c + (7*d*x)/2] - 5890*C*Sin[2*c + (7*d*x)/2] - 3885*A*Sin[3*c + (7*d*x)/2] + 1470*C*Sin[3*c + (7*d*x)/2] - 44733*A*Sin[4*c + (7*d*x)/2] - 5890*C*Sin[4*c + (7*d*x)/2] + 15435*A*Sin[5*c + (7*d*x)/2] + 1470*C*Sin[5*c + (7*d*x)/2] - 21987*A*Sin[3*c + (9*d*x)/2] - 2030*C*Sin[3*c + (9*d*x)/2] - 3675*A*Sin[4*c + (9*d*x)/2] + 210*C*Sin[4*c + (9*d*x)/2] - 16107*A*Sin[5*c + (9*d*x)/2] - 2030*C*Sin[5*c + (9*d*x)/2] + 2205*A*Sin[6*c + (9*d*x)/2] + 210*C*Sin[6*c + (9*d*x)/2] - 3456*A*Sin[4*c + (11*d*x)/2] - 320*C*Sin[4*c + (11*d*x)/2] - 840*A*Sin[5*c + (11*d*x)/2] - 2616*A*Sin[6*c + (11*d*x)/2] - 320*C*Sin[6*c + (11*d*x)/2]))/(6720*d*(a + a*Cos[c + d*x])^4)","B",1
73,1,361,257,4.5857592,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^4,x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(4 (412 A+139 C) \tan \left(\frac{c}{2}\right) \cos ^5\left(\frac{1}{2} (c+d x)\right)+6 (31 A+17 C) \tan \left(\frac{c}{2}\right) \cos ^3\left(\frac{1}{2} (c+d x)\right)+15 (A+C) \tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)+15 (A+C) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+8 (2512 A+559 C) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^6\left(\frac{1}{2} (c+d x)\right)+4 (412 A+139 C) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^4\left(\frac{1}{2} (c+d x)\right)+6 (31 A+17 C) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)+280 \cos ^7\left(\frac{1}{2} (c+d x)\right) \left(\sec (c) \sin (d x) \sec (c+d x) \left(A \sec ^2(c+d x)-6 A \sec (c+d x)+32 A+3 C\right)+6 (11 A+2 C) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+A \tan (c) \sec ^2(c+d x)-6 A \tan (c) \sec (c+d x)\right)\right)}{105 a^4 d (\cos (c+d x)+1)^4}","\frac{4 (454 A+83 C) \tan ^3(c+d x)}{105 a^4 d}+\frac{4 (454 A+83 C) \tan (c+d x)}{35 a^4 d}-\frac{2 (11 A+2 C) \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{2 (11 A+2 C) \tan (c+d x) \sec (c+d x)}{a^4 d}-\frac{4 (11 A+2 C) \tan (c+d x) \sec ^2(c+d x)}{3 a^4 d (\cos (c+d x)+1)}-\frac{(178 A+31 C) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{2 (8 A+C) \tan (c+d x) \sec ^2(c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"(2*Cos[(c + d*x)/2]*(15*(A + C)*Sec[c/2]*Sin[(d*x)/2] + 6*(31*A + 17*C)*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + 4*(412*A + 139*C)*Cos[(c + d*x)/2]^4*Sec[c/2]*Sin[(d*x)/2] + 8*(2512*A + 559*C)*Cos[(c + d*x)/2]^6*Sec[c/2]*Sin[(d*x)/2] + 15*(A + C)*Cos[(c + d*x)/2]*Tan[c/2] + 6*(31*A + 17*C)*Cos[(c + d*x)/2]^3*Tan[c/2] + 4*(412*A + 139*C)*Cos[(c + d*x)/2]^5*Tan[c/2] + 280*Cos[(c + d*x)/2]^7*(6*(11*A + 2*C)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c]*Sec[c + d*x]*(32*A + 3*C - 6*A*Sec[c + d*x] + A*Sec[c + d*x]^2)*Sin[d*x] - 6*A*Sec[c + d*x]*Tan[c] + A*Sec[c + d*x]^2*Tan[c])))/(105*a^4*d*(1 + Cos[c + d*x])^4)","A",1
74,1,114,223,0.9677736,"\int \cos ^3(c+d x) \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (2 (9306 A+9095 C) \cos (c+d x)+16 (297 A+415 C) \cos (2 (c+d x))+1980 A \cos (3 (c+d x))+30096 A+3175 C \cos (3 (c+d x))+700 C \cos (4 (c+d x))+315 C \cos (5 (c+d x))+26420 C)}{27720 d}","\frac{2 a (99 A+80 C) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{4 (99 A+80 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 a d}-\frac{8 (99 A+80 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3465 d}+\frac{4 a (99 A+80 C) \sin (c+d x)}{495 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \cos ^4(c+d x) \sqrt{a \cos (c+d x)+a}}{11 d}+\frac{2 a C \sin (c+d x) \cos ^4(c+d x)}{99 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(30096*A + 26420*C + 2*(9306*A + 9095*C)*Cos[c + d*x] + 16*(297*A + 415*C)*Cos[2*(c + d*x)] + 1980*A*Cos[3*(c + d*x)] + 3175*C*Cos[3*(c + d*x)] + 700*C*Cos[4*(c + d*x)] + 315*C*Cos[5*(c + d*x)])*Tan[(c + d*x)/2])/(27720*d)","A",1
75,1,92,180,0.4963972,"\int \cos ^2(c+d x) \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (16 (42 A+47 C) \cos (c+d x)+4 (63 A+83 C) \cos (2 (c+d x))+1596 A+80 C \cos (3 (c+d x))+35 C \cos (4 (c+d x))+1321 C)}{1260 d}","\frac{2 (21 A+16 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 a d}-\frac{4 (21 A+16 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 a (21 A+16 C) \sin (c+d x)}{45 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{9 d}+\frac{2 a C \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(1596*A + 1321*C + 16*(42*A + 47*C)*Cos[c + d*x] + 4*(63*A + 83*C)*Cos[2*(c + d*x)] + 80*C*Cos[3*(c + d*x)] + 35*C*Cos[4*(c + d*x)])*Tan[(c + d*x)/2])/(1260*d)","A",1
76,1,74,137,0.2801081,"\int \cos (c+d x) \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((140 A+141 C) \cos (c+d x)+280 A+36 C \cos (2 (c+d x))+15 C \cos (3 (c+d x))+228 C)}{210 d}","\frac{2 (35 A+18 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 a (35 A+27 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{7 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 a d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(280*A + 228*C + (140*A + 141*C)*Cos[c + d*x] + 36*C*Cos[2*(c + d*x)] + 15*C*Cos[3*(c + d*x)])*Tan[(c + d*x)/2])/(210*d)","A",1
77,1,58,95,0.1162117,"\int \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (30 A+8 C \cos (c+d x)+3 C \cos (2 (c+d x))+19 C)}{15 d}","\frac{2 a (15 A+7 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}-\frac{4 C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(30*A + 19*C + 8*C*Cos[c + d*x] + 3*C*Cos[2*(c + d*x)])*Tan[(c + d*x)/2])/(15*d)","A",1
78,1,82,96,0.1421825,"\int \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} A \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+C \left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{3}{2} (c+d x)\right)\right)\right)}{3 d}","\frac{2 \sqrt{a} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{2 a C \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(3*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + C*(3*Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(3*d)","A",1
79,1,91,94,0.2284789,"\int \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) (A+2 C \cos (c+d x))+\sqrt{2} A \cos (c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{2 d}","-\frac{a (A-2 C) \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{d}+\frac{\sqrt{a} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]*(Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x] + 2*(A + 2*C*Cos[c + d*x])*Sin[(c + d*x)/2]))/(2*d)","A",1
80,1,103,110,0.3822761,"\int \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(\sqrt{2} (3 A+8 C) \cos ^2(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+A \left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \sin \left(\frac{3}{2} (c+d x)\right)\right)\right)}{8 d}","\frac{\sqrt{a} (3 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a A \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^2*(Sqrt[2]*(3*A + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^2 + A*(Sin[(c + d*x)/2] + 3*Sin[(3*(c + d*x))/2])))/(8*d)","A",1
81,1,115,153,1.0975007,"\int \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\sqrt{a (\cos (c+d x)+1)} \left(\tan \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) (3 (5 A+8 C) \cos (2 (c+d x))+20 A \cos (c+d x)+31 A+24 C)+3 \sqrt{2} (5 A+8 C) \sec \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{48 d}","\frac{a (5 A+8 C) \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (5 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{a A \tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(3*Sqrt[2]*(5*A + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Sec[(c + d*x)/2] + (31*A + 24*C + 20*A*Cos[c + d*x] + 3*(5*A + 8*C)*Cos[2*(c + d*x)])*Sec[c + d*x]^3*Tan[(c + d*x)/2]))/(48*d)","A",1
82,1,145,196,1.8823567,"\int \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{\sqrt{a (\cos (c+d x)+1)} \left(\frac{1}{2} \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) ((539 A+432 C) \cos (c+d x)+4 (35 A+48 C) \cos (2 (c+d x))+105 A \cos (3 (c+d x))+332 A+144 C \cos (3 (c+d x))+192 C)+3 \sqrt{2} (35 A+48 C) \sec \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{384 d}","\frac{a (35 A+48 C) \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (35 A+48 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a (35 A+48 C) \tan (c+d x) \sec (c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(3*Sqrt[2]*(35*A + 48*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Sec[(c + d*x)/2] + ((332*A + 192*C + (539*A + 432*C)*Cos[c + d*x] + 4*(35*A + 48*C)*Cos[2*(c + d*x)] + 105*A*Cos[3*(c + d*x)] + 144*C*Cos[3*(c + d*x)])*Sec[c + d*x]^4*Tan[(c + d*x)/2])/2))/(384*d)","A",1
83,1,115,225,0.9563028,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (2 (5566 A+5789 C) \cos (c+d x)+8 (429 A+581 C) \cos (2 (c+d x))+660 A \cos (3 (c+d x))+21736 A+1645 C \cos (3 (c+d x))+490 C \cos (4 (c+d x))+105 C \cos (5 (c+d x))+18494 C)}{9240 d}","\frac{2 a^2 (33 A+28 C) \sin (c+d x) \cos ^3(c+d x)}{231 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (143 A+112 C) \sin (c+d x)}{165 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (143 A+112 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{385 d}-\frac{4 a (143 A+112 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{1155 d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}+\frac{2 a C \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{33 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(21736*A + 18494*C + 2*(5566*A + 5789*C)*Cos[c + d*x] + 8*(429*A + 581*C)*Cos[2*(c + d*x)] + 660*A*Cos[3*(c + d*x)] + 1645*C*Cos[3*(c + d*x)] + 490*C*Cos[4*(c + d*x)] + 105*C*Cos[5*(c + d*x)])*Tan[(c + d*x)/2])/(9240*d)","A",1
84,1,93,174,0.5447629,"\int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (2 (756 A+799 C) \cos (c+d x)+4 (63 A+137 C) \cos (2 (c+d x))+3276 A+170 C \cos (3 (c+d x))+35 C \cos (4 (c+d x))+2689 C)}{1260 d}","\frac{8 a^2 (63 A+47 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (63 A+22 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{315 d}+\frac{2 a (63 A+47 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{21 a d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(3276*A + 2689*C + 2*(756*A + 799*C)*Cos[c + d*x] + 4*(63*A + 137*C)*Cos[2*(c + d*x)] + 170*C*Cos[3*(c + d*x)] + 35*C*Cos[4*(c + d*x)])*Tan[(c + d*x)/2])/(1260*d)","A",1
85,1,75,132,0.2549177,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((140 A+253 C) \cos (c+d x)+700 A+78 C \cos (2 (c+d x))+15 C \cos (3 (c+d x))+494 C)}{210 d}","\frac{8 a^2 (35 A+19 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (35 A+19 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 a d}-\frac{4 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(700*A + 494*C + (140*A + 253*C)*Cos[c + d*x] + 78*C*Cos[2*(c + d*x)] + 15*C*Cos[3*(c + d*x)])*Tan[(c + d*x)/2])/(210*d)","A",1
86,1,95,133,0.3572675,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (10 A+6 C \cos (c+d x)+C \cos (2 (c+d x))+13 C)+5 \sqrt{2} A \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{5 d}","\frac{2 a^{3/2} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a^2 (5 A+4 C) \sin (c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{5 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(5*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + (10*A + 13*C + 6*C*Cos[c + d*x] + C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(5*d)","A",1
87,1,106,136,0.3386419,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) (3 A+10 C \cos (c+d x)+C \cos (2 (c+d x))+C)+9 \sqrt{2} A \cos (c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{6 d}","\frac{3 a^{3/2} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a^2 (3 A-8 C) \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{a (3 A-2 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^{3/2}}{d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]*(9*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x] + 2*(3*A + C + 10*C*Cos[c + d*x] + C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(6*d)","A",1
88,1,118,147,0.5771633,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) (7 A \cos (c+d x)+2 A+4 C \cos (2 (c+d x))+4 C)+\sqrt{2} (7 A+8 C) \cos ^2(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{8 d}","\frac{a^{3/2} (7 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^2 (5 A-8 C) \sin (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{3 a A \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^{3/2}}{2 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^2*(Sqrt[2]*(7*A + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^2 + 2*(2*A + 4*C + 7*A*Cos[c + d*x] + 4*C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(8*d)","A",1
89,1,124,155,0.8864706,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (3 (11 A+8 C) \cos (2 (c+d x))+44 A \cos (c+d x)+49 A+24 C)+3 \sqrt{2} (11 A+24 C) \cos ^3(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{48 d}","\frac{a^{3/2} (11 A+24 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 (19 A+24 C) \tan (c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d}+\frac{a A \tan (c+d x) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^3*(3*Sqrt[2]*(11*A + 24*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^3 + (49*A + 24*C + 44*A*Cos[c + d*x] + 3*(11*A + 8*C)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
90,1,152,200,1.4818665,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (7 (55 A+48 C) \cos (c+d x)+4 (25 A+16 C) \cos (2 (c+d x))+75 A \cos (3 (c+d x))+164 A+112 C \cos (3 (c+d x))+64 C)+2 \sqrt{2} (75 A+112 C) \cos ^4(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{256 d}","\frac{a^{3/2} (75 A+112 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (75 A+112 C) \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (13 A+16 C) \tan (c+d x) \sec (c+d x)}{32 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{8 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^4*(2*Sqrt[2]*(75*A + 112*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^4 + (164*A + 64*C + 7*(55*A + 48*C)*Cos[c + d*x] + 4*(25*A + 16*C)*Cos[2*(c + d*x)] + 75*A*Cos[3*(c + d*x)] + 112*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(256*d)","A",1
91,1,174,245,2.2665021,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (12 (1273 A+880 C) \cos (c+d x)+4 (3059 A+3280 C) \cos (2 (c+d x))+2660 A \cos (3 (c+d x))+1995 A \cos (4 (c+d x))+13313 A+3520 C \cos (3 (c+d x))+2640 C \cos (4 (c+d x))+10480 C)+60 \sqrt{2} (133 A+176 C) \cos ^5(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{15360 d}","\frac{a^{3/2} (133 A+176 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^2 (133 A+176 C) \tan (c+d x)}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (67 A+80 C) \tan (c+d x) \sec ^2(c+d x)}{240 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (133 A+176 C) \tan (c+d x) \sec (c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}+\frac{3 a A \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{40 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^5*(60*Sqrt[2]*(133*A + 176*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^5 + (13313*A + 10480*C + 12*(1273*A + 880*C)*Cos[c + d*x] + 4*(3059*A + 3280*C)*Cos[2*(c + d*x)] + 2660*A*Cos[3*(c + d*x)] + 3520*C*Cos[3*(c + d*x)] + 1995*A*Cos[4*(c + d*x)] + 2640*C*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]))/(15360*d)","A",1
92,1,138,273,1.2877803,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (8 (222794 A+226573 C) \cos (c+d x)+(581152 A+746519 C) \cos (2 (c+d x))+148720 A \cos (3 (c+d x))+20020 A \cos (4 (c+d x))+3233516 A+287060 C \cos (3 (c+d x))+94010 C \cos (4 (c+d x))+23940 C \cos (5 (c+d x))+3465 C \cos (6 (c+d x))+2798182 C)}{720720 d}","\frac{2 a^3 (2717 A+2224 C) \sin (c+d x) \cos ^3(c+d x)}{9009 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (10439 A+8368 C) \sin (c+d x)}{6435 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (143 A+136 C) \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{1287 d}-\frac{4 a^2 (10439 A+8368 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{45045 d}+\frac{2 a (10439 A+8368 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{15015 d}+\frac{10 a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(3233516*A + 2798182*C + 8*(222794*A + 226573*C)*Cos[c + d*x] + (581152*A + 746519*C)*Cos[2*(c + d*x)] + 148720*A*Cos[3*(c + d*x)] + 287060*C*Cos[3*(c + d*x)] + 20020*A*Cos[4*(c + d*x)] + 94010*C*Cos[4*(c + d*x)] + 23940*C*Cos[5*(c + d*x)] + 3465*C*Cos[6*(c + d*x)])*Tan[(c + d*x)/2])/(720720*d)","A",1
93,1,117,211,0.8793004,"\int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (2 (6666 A+6989 C) \cos (c+d x)+16 (198 A+325 C) \cos (2 (c+d x))+396 A \cos (3 (c+d x))+27456 A+1735 C \cos (3 (c+d x))+448 C \cos (4 (c+d x))+63 C \cos (5 (c+d x))+22928 C)}{5544 d}","\frac{64 a^3 (33 A+25 C) \sin (c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (33 A+25 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{693 d}+\frac{2 (99 A+26 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{693 d}+\frac{2 a (33 A+25 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{231 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d}+\frac{10 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{99 a d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(27456*A + 22928*C + 2*(6666*A + 6989*C)*Cos[c + d*x] + 16*(198*A + 325*C)*Cos[2*(c + d*x)] + 396*A*Cos[3*(c + d*x)] + 1735*C*Cos[3*(c + d*x)] + 448*C*Cos[4*(c + d*x)] + 63*C*Cos[5*(c + d*x)])*Tan[(c + d*x)/2])/(5544*d)","A",1
94,1,95,169,0.4617018,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (4 (588 A+779 C) \cos (c+d x)+4 (63 A+254 C) \cos (2 (c+d x))+7476 A+260 C \cos (3 (c+d x))+35 C \cos (4 (c+d x))+5653 C)}{1260 d}","\frac{64 a^3 (21 A+13 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (21 A+13 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 a (21 A+13 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}-\frac{4 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{63 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(7476*A + 5653*C + 4*(588*A + 779*C)*Cos[c + d*x] + 4*(63*A + 254*C)*Cos[2*(c + d*x)] + 260*C*Cos[3*(c + d*x)] + 35*C*Cos[4*(c + d*x)])*Tan[(c + d*x)/2])/(1260*d)","A",1
95,1,115,170,0.5314122,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) ((28 A+101 C) \cos (c+d x)+224 A+24 C \cos (2 (c+d x))+3 C \cos (3 (c+d x))+208 C)+84 \sqrt{2} A \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{84 d}","\frac{2 a^{5/2} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a^3 (49 A+32 C) \sin (c+d x)}{21 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (7 A+8 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{21 d}+\frac{2 a C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(84*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*(224*A + 208*C + (28*A + 101*C)*Cos[c + d*x] + 24*C*Cos[2*(c + d*x)] + 3*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(84*d)","A",1
96,1,127,173,0.5965499,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) ((60 A+181 C) \cos (c+d x)+30 A+28 C \cos (2 (c+d x))+3 C \cos (3 (c+d x))+28 C)+150 \sqrt{2} A \cos (c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{60 d}","\frac{5 a^{5/2} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{a^3 (15 A+64 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (15 A-16 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}-\frac{a (5 A-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^{5/2}}{d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]*(150*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x] + 2*(30*A + 28*C + (60*A + 181*C)*Cos[c + d*x] + 28*C*Cos[2*(c + d*x)] + 3*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(60*d)","A",1
97,1,137,184,0.7732196,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(4 \sin \left(\frac{1}{2} (c+d x)\right) ((33 A+6 C) \cos (c+d x)+6 A+32 C \cos (2 (c+d x))+2 C \cos (3 (c+d x))+32 C)+6 \sqrt{2} (19 A+8 C) \cos ^2(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{48 d}","\frac{a^{5/2} (19 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^3 (27 A-56 C) \sin (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (21 A-8 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{12 d}+\frac{5 a A \tan (c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^{5/2}}{2 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^2*(6*Sqrt[2]*(19*A + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^2 + 4*(6*A + 32*C + (33*A + 6*C)*Cos[c + d*x] + 32*C*Cos[2*(c + d*x)] + 2*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
98,1,142,192,1.1259517,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) ((68 A+72 C) \cos (c+d x)+3 (25 A+8 C) \cos (2 (c+d x))+91 A+24 C \cos (3 (c+d x))+24 C)+15 \sqrt{2} (5 A+8 C) \cos ^3(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{48 d}","\frac{5 a^{5/2} (5 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}-\frac{a^3 (49 A-24 C) \sin (c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (31 A+24 C) \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{24 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^{5/2}}{3 d}+\frac{5 a A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^{3/2}}{12 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^3*(15*Sqrt[2]*(5*A + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^3 + (91*A + 24*C + (68*A + 72*C)*Cos[c + d*x] + 3*(25*A + 8*C)*Cos[2*(c + d*x)] + 24*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
99,1,153,200,1.7128759,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) ((2203 A+1584 C) \cos (c+d x)+4 (163 A+48 C) \cos (2 (c+d x))+489 A \cos (3 (c+d x))+844 A+528 C \cos (3 (c+d x))+192 C)+6 \sqrt{2} (163 A+304 C) \cos ^4(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{768 d}","\frac{a^{5/2} (163 A+304 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^3 (299 A+432 C) \tan (c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (17 A+16 C) \tan (c+d x) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{32 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{4 d}+\frac{5 a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{24 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^4*(6*Sqrt[2]*(163*A + 304*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^4 + (844*A + 192*C + (2203*A + 1584*C)*Cos[c + d*x] + 4*(163*A + 48*C)*Cos[2*(c + d*x)] + 489*A*Cos[3*(c + d*x)] + 528*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(768*d)","A",1
100,1,176,245,2.0807587,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (12 (2343 A+1360 C) \cos (c+d x)+4 (6509 A+6640 C) \cos (2 (c+d x))+5660 A \cos (3 (c+d x))+4245 A \cos (4 (c+d x))+24863 A+5440 C \cos (3 (c+d x))+6000 C \cos (4 (c+d x))+20560 C)+60 \sqrt{2} (283 A+400 C) \cos ^5(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{15360 d}","\frac{a^{5/2} (283 A+400 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^3 (283 A+400 C) \tan (c+d x)}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (787 A+1040 C) \tan (c+d x) \sec (c+d x)}{960 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (79 A+80 C) \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{240 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{8 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^5*(60*Sqrt[2]*(283*A + 400*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^5 + (24863*A + 20560*C + 12*(2343*A + 1360*C)*Cos[c + d*x] + 4*(6509*A + 6640*C)*Cos[2*(c + d*x)] + 5660*A*Cos[3*(c + d*x)] + 5440*C*Cos[3*(c + d*x)] + 4245*A*Cos[4*(c + d*x)] + 6000*C*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]))/(15360*d)","A",1
101,1,198,290,2.7244726,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^6(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (14 (4591 A+4056 C) \cos (c+d x)+16 (1711 A+1496 C) \cos (2 (c+d x))+21721 A \cos (3 (c+d x))+4060 A \cos (4 (c+d x))+3045 A \cos (5 (c+d x))+27412 A+25448 C \cos (3 (c+d x))+5216 C \cos (4 (c+d x))+3912 C \cos (5 (c+d x))+18720 C)+24 \sqrt{2} (1015 A+1304 C) \cos ^6(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{24576 d}","\frac{a^{5/2} (1015 A+1304 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{512 d}+\frac{a^3 (1015 A+1304 C) \tan (c+d x)}{512 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (109 A+136 C) \tan (c+d x) \sec ^2(c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (1015 A+1304 C) \tan (c+d x) \sec (c+d x)}{768 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (23 A+24 C) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{96 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}+\frac{a A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{3/2}}{12 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^6*(24*Sqrt[2]*(1015*A + 1304*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^6 + (27412*A + 18720*C + 14*(4591*A + 4056*C)*Cos[c + d*x] + 16*(1711*A + 1496*C)*Cos[2*(c + d*x)] + 21721*A*Cos[3*(c + d*x)] + 25448*C*Cos[3*(c + d*x)] + 4060*A*Cos[4*(c + d*x)] + 5216*C*Cos[4*(c + d*x)] + 3045*A*Cos[5*(c + d*x)] + 3912*C*Cos[5*(c + d*x)])*Sin[(c + d*x)/2]))/(24576*d)","A",1
102,1,121,236,0.6240233,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(2 \sin \left(\frac{1}{2} (c+d x)\right) (-2 (84 A+131 C) \cos (c+d x)+4 (63 A+92 C) \cos (2 (c+d x))+2436 A-10 C \cos (3 (c+d x))+35 C \cos (4 (c+d x))+2389 C)-2520 (A+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{1260 d \sqrt{a (\cos (c+d x)+1)}}","\frac{2 (21 A+19 C) \sin (c+d x) \cos ^2(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (21 A+29 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 a d}+\frac{4 (147 A+143 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 C \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}}-\frac{2 C \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*(-2520*(A + C)*ArcTanh[Sin[(c + d*x)/2]] + 2*(2436*A + 2389*C - 2*(84*A + 131*C)*Cos[c + d*x] + 4*(63*A + 92*C)*Cos[2*(c + d*x)] - 10*C*Cos[3*(c + d*x)] + 35*C*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]))/(1260*d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
103,1,89,193,0.3269388,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(105 (A+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 \sin ^3\left(\frac{1}{2} (c+d x)\right) (70 A+24 C \cos (c+d x)+15 C \cos (2 (c+d x))+101 C)\right)}{105 d \sqrt{a (\cos (c+d x)+1)}}","\frac{2 (35 A+31 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 a d}-\frac{4 (35 A+37 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}-\frac{2 C \sin (c+d x) \cos ^2(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}",1,"(2*Cos[(c + d*x)/2]*(105*(A + C)*ArcTanh[Sin[(c + d*x)/2]] - 2*(70*A + 101*C + 24*C*Cos[c + d*x] + 15*C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]^3))/(105*d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
104,1,87,152,0.2349492,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(2 \sin \left(\frac{1}{2} (c+d x)\right) (30 A-2 C \cos (c+d x)+3 C \cos (2 (c+d x))+29 C)-30 (A+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{15 d \sqrt{a (\cos (c+d x)+1)}}","\frac{2 (15 A+14 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}-\frac{2 C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 a d}",1,"(Cos[(c + d*x)/2]*(-30*(A + C)*ArcTanh[Sin[(c + d*x)/2]] + 2*(30*A + 29*C - 2*C*Cos[c + d*x] + 3*C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(15*d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
105,1,63,109,0.1058148,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(3 (A+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-4 C \sin ^3\left(\frac{1}{2} (c+d x)\right)\right)}{3 d \sqrt{a (\cos (c+d x)+1)}}","\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 a d}-\frac{4 C \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}",1,"(2*Cos[(c + d*x)/2]*(3*(A + C)*ArcTanh[Sin[(c + d*x)/2]] - 4*C*Sin[(c + d*x)/2]^3))/(3*d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
106,1,83,115,0.2587191,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(-\left((A+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)+\sqrt{2} A \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 C \sin \left(\frac{1}{2} (c+d x)\right)\right)}{d \sqrt{a (\cos (c+d x)+1)}}","-\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 C \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}",1,"(2*Cos[(c + d*x)/2]*(-((A + C)*ArcTanh[Sin[(c + d*x)/2]]) + Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*C*Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
107,1,89,113,0.3354923,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(2 (A+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\sqrt{2} A \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 A \sin \left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)}{d \sqrt{a (\cos (c+d x)+1)}}","\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{A \tan (c+d x)}{d \sqrt{a \cos (c+d x)+a}}-\frac{A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(Cos[(c + d*x)/2]*(2*(A + C)*ArcTanh[Sin[(c + d*x)/2]] - Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*A*Sec[c + d*x]*Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
108,1,113,159,0.7553696,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(-8 (A+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\sqrt{2} (7 A+8 C) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+A \left(5 \sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)\right) \sec ^2(c+d x)\right)}{4 d \sqrt{a (\cos (c+d x)+1)}}","\frac{(7 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{A \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec (c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*(-8*(A + C)*ArcTanh[Sin[(c + d*x)/2]] + Sqrt[2]*(7*A + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + A*Sec[c + d*x]^2*(5*Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2])))/(4*d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
109,1,131,200,1.3528562,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(48 (A+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-3 \sqrt{2} (9 A+8 C) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+\sin \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) (3 (7 A+8 C) \cos (2 (c+d x))-4 A \cos (c+d x)+37 A+24 C)\right)}{24 d \sqrt{a (\cos (c+d x)+1)}}","\frac{(7 A+8 C) \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}-\frac{(9 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{A \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{A \tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*(48*(A + C)*ArcTanh[Sin[(c + d*x)/2]] - 3*Sqrt[2]*(9*A + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + (37*A + 24*C - 4*A*Cos[c + d*x] + 3*(7*A + 8*C)*Cos[2*(c + d*x)])*Sec[c + d*x]^3*Sin[(c + d*x)/2]))/(24*d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
110,1,174,243,1.9844522,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5)/Sqrt[a + a*Cos[c + d*x]],x]","-\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \left(\sin \left(\frac{1}{2} (c+d x)\right) ((221 A+144 C) \cos (c+d x)-4 (43 A+48 C) \cos (2 (c+d x))+63 A \cos (3 (c+d x))-364 A+48 C \cos (3 (c+d x))-192 C)+768 (A+C) \cos ^4(c+d x) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-6 \sqrt{2} (107 A+112 C) \cos ^4(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{384 d \sqrt{a (\cos (c+d x)+1)}}","-\frac{(21 A+16 C) \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{(107 A+112 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 \sqrt{a} d}-\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(43 A+48 C) \tan (c+d x) \sec (c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}-\frac{A \tan (c+d x) \sec ^2(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}",1,"-1/384*(Cos[(c + d*x)/2]*Sec[c + d*x]^4*(768*(A + C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[c + d*x]^4 - 6*Sqrt[2]*(107*A + 112*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^4 + (-364*A - 192*C + (221*A + 144*C)*Cos[c + d*x] - 4*(43*A + 48*C)*Cos[2*(c + d*x)] + 63*A*Cos[3*(c + d*x)] + 48*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
111,1,157,259,1.0136298,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{\frac{1}{2} \sin \left(\frac{1}{2} (c+d x)\right) \cos ^3\left(\frac{1}{2} (c+d x)\right) (6 (140 A+277 C) \cos (c+d x)-4 (35 A+64 C) \cos (2 (c+d x))+1190 A+18 C \cos (3 (c+d x))-15 C \cos (4 (c+d x))+2161 C)-105 (11 A+19 C) \cos ^5\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{105 d \left(\sin ^2\left(\frac{1}{2} (c+d x)\right)-1\right) (a (\cos (c+d x)+1))^{3/2}}","\frac{(11 A+19 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(245 A+397 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{210 a^2 d}-\frac{(A+C) \sin (c+d x) \cos ^4(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(7 A+11 C) \sin (c+d x) \cos ^3(c+d x)}{14 a d \sqrt{a \cos (c+d x)+a}}-\frac{(35 A+67 C) \sin (c+d x) \cos ^2(c+d x)}{70 a d \sqrt{a \cos (c+d x)+a}}-\frac{(455 A+799 C) \sin (c+d x)}{105 a d \sqrt{a \cos (c+d x)+a}}",1,"(-105*(11*A + 19*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^5 + (Cos[(c + d*x)/2]^3*(1190*A + 2161*C + 6*(140*A + 277*C)*Cos[c + d*x] - 4*(35*A + 64*C)*Cos[2*(c + d*x)] + 18*C*Cos[3*(c + d*x)] - 15*C*Cos[4*(c + d*x)])*Sin[(c + d*x)/2])/2)/(105*d*(a*(1 + Cos[c + d*x]))^(3/2)*(-1 + Sin[(c + d*x)/2]^2))","A",1
112,1,136,214,0.6561921,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{5 (7 A+15 C) \cos ^5\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\sin \left(\frac{1}{2} (c+d x)\right) \cos ^3\left(\frac{1}{2} (c+d x)\right) ((20 A+39 C) \cos (c+d x)+25 A-2 C \cos (2 (c+d x))+C \cos (3 (c+d x))+47 C)}{5 d \left(\sin ^2\left(\frac{1}{2} (c+d x)\right)-1\right) (a (\cos (c+d x)+1))^{3/2}}","-\frac{(7 A+15 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(5 A+13 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{10 a^2 d}-\frac{(A+C) \sin (c+d x) \cos ^3(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(5 A+9 C) \sin (c+d x) \cos ^2(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}+\frac{(15 A+31 C) \sin (c+d x)}{5 a d \sqrt{a \cos (c+d x)+a}}",1,"(5*(7*A + 15*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^5 - Cos[(c + d*x)/2]^3*(25*A + 47*C + (20*A + 39*C)*Cos[c + d*x] - 2*C*Cos[2*(c + d*x)] + C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2])/(5*d*(a*(1 + Cos[c + d*x]))^(3/2)*(-1 + Sin[(c + d*x)/2]^2))","A",1
113,1,94,169,0.6142878,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{3 (3 A+11 C) \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\tan \left(\frac{1}{2} (c+d x)\right) (3 A+12 C \cos (c+d x)-2 C \cos (2 (c+d x))+17 C)}{6 a d \sqrt{a (\cos (c+d x)+1)}}","\frac{(3 A+11 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(3 A+7 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{6 a^2 d}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(3 A+13 C) \sin (c+d x)}{3 a d \sqrt{a \cos (c+d x)+a}}",1,"(3*(3*A + 11*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2] - (3*A + 17*C + 12*C*Cos[c + d*x] - 2*C*Cos[2*(c + d*x)])*Tan[(c + d*x)/2])/(6*a*d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
114,1,77,114,0.4567743,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) (A+4 C \cos (c+d x)+5 C)+(A-7 C) \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a d \sqrt{a (\cos (c+d x)+1)}}","\frac{(A-7 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(A+C) \sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{2 C \sin (c+d x)}{a d \sqrt{a \cos (c+d x)+a}}",1,"((A - 7*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2] + (A + 5*C + 4*C*Cos[c + d*x])*Tan[(c + d*x)/2])/(2*a*d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
115,1,129,125,0.7589932,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(A+C) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^3\left(\frac{1}{2} (c+d x)\right)+(5 A-3 C) \cos ^5\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-4 \sqrt{2} A \cos ^5\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)}{d \left(\sin ^2\left(\frac{1}{2} (c+d x)\right)-1\right) (a (\cos (c+d x)+1))^{3/2}}","-\frac{(5 A-3 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A+C) \sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"((5*A - 3*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^5 - 4*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^5 + (A + C)*Cos[(c + d*x)/2]^3*Sin[(c + d*x)/2])/(d*(a*(1 + Cos[c + d*x]))^(3/2)*(-1 + Sin[(c + d*x)/2]^2))","A",1
116,1,167,158,2.2731895,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2),x]","\frac{\cos ^3\left(\frac{1}{2} (c+d x)\right) \cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right) \left(2 (9 A+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\frac{12 \sqrt{2} A \cos ^2\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right) (2 A \sec (c+d x)+3 A+C)}{\sin ^2\left(\frac{1}{2} (c+d x)\right)-1}\right)}{d (a (\cos (c+d x)+1))^{3/2} (2 A+C \cos (2 (c+d x))+C)}","\frac{(9 A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{3 A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}+\frac{(3 A+C) \tan (c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \tan (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]^3*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*(2*(9*A + C)*ArcTanh[Sin[(c + d*x)/2]] + (12*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^2 - 2*(3*A + C + 2*A*Sec[c + d*x])*Sin[(c + d*x)/2])/(-1 + Sin[(c + d*x)/2]^2)))/(d*(a*(1 + Cos[c + d*x]))^(3/2)*(2*A + C + C*Cos[2*(c + d*x)]))","A",1
117,1,211,217,3.0702885,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^(3/2),x]","-\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(A+C \cos ^2(c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right) ((7 A+2 C) \cos (2 (c+d x))+6 A \cos (c+d x)+3 A+2 C)+(13 A+5 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{3}{2} (c+d x)\right)\right)^2 \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\frac{(19 A+8 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{3}{2} (c+d x)\right)\right)^2 \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{2}}\right)}{4 a d \sqrt{a (\cos (c+d x)+1)} (2 A+C \cos (2 (c+d x))+C)}","\frac{(19 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{(13 A+5 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(7 A+2 C) \tan (c+d x)}{4 a d \sqrt{a \cos (c+d x)+a}}+\frac{(2 A+C) \tan (c+d x) \sec (c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \tan (c+d x) \sec (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"-1/4*((A + C*Cos[c + d*x]^2)*Sec[(c + d*x)/2]*Sec[c + d*x]^2*((13*A + 5*C)*ArcTanh[Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Cos[(3*(c + d*x))/2])^2 - ((19*A + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Cos[(3*(c + d*x))/2])^2)/Sqrt[2] + (3*A + 2*C + 6*A*Cos[c + d*x] + (7*A + 2*C)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(a*d*Sqrt[a*(1 + Cos[c + d*x])]*(2*A + C + C*Cos[2*(c + d*x)]))","A",1
118,1,205,266,3.9000216,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^(3/2),x]","\frac{-48 (17 A+9 C) \cos ^5\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 \sqrt{2} (47 A+24 C) \cos ^5\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)-\sin \left(\frac{1}{2} (c+d x)\right) \cos ^3\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) (3 (55 A+36 C) \cos (c+d x)+(74 A+48 C) \cos (2 (c+d x))+63 A \cos (3 (c+d x))+106 A+36 C \cos (3 (c+d x))+48 C)}{48 d \left(\sin ^2\left(\frac{1}{2} (c+d x)\right)-1\right) (a (\cos (c+d x)+1))^{3/2}}","-\frac{(47 A+24 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 a^{3/2} d}+\frac{(17 A+9 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{3 (7 A+4 C) \tan (c+d x)}{8 a d \sqrt{a \cos (c+d x)+a}}+\frac{(5 A+3 C) \tan (c+d x) \sec ^2(c+d x)}{6 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \tan (c+d x) \sec ^2(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(13 A+6 C) \tan (c+d x) \sec (c+d x)}{12 a d \sqrt{a \cos (c+d x)+a}}",1,"(-48*(17*A + 9*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^5 + 12*Sqrt[2]*(47*A + 24*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^5 - Cos[(c + d*x)/2]^3*(106*A + 48*C + 3*(55*A + 36*C)*Cos[c + d*x] + (74*A + 48*C)*Cos[2*(c + d*x)] + 63*A*Cos[3*(c + d*x)] + 36*C*Cos[3*(c + d*x)])*Sec[c + d*x]^3*Sin[(c + d*x)/2])/(48*d*(a*(1 + Cos[c + d*x]))^(3/2)*(-1 + Sin[(c + d*x)/2]^2))","A",1
119,1,129,259,1.3161544,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) (5 (255 A+887 C) \cos (c+d x)+16 (15 A+52 C) \cos (2 (c+d x))+975 A-40 C \cos (3 (c+d x))+12 C \cos (4 (c+d x))+3491 C)-30 (75 A+283 C) \cos ^3\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{240 a d (a (\cos (c+d x)+1))^{3/2}}","-\frac{(75 A+283 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(195 A+787 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{240 a^3 d}+\frac{(45 A+157 C) \sin (c+d x) \cos ^2(c+d x)}{80 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(465 A+1729 C) \sin (c+d x)}{120 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x) \cos ^4(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{(5 A+21 C) \sin (c+d x) \cos ^3(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"(-30*(75*A + 283*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 + (975*A + 3491*C + 5*(255*A + 887*C)*Cos[c + d*x] + 16*(15*A + 52*C)*Cos[2*(c + d*x)] - 40*C*Cos[3*(c + d*x)] + 12*C*Cos[4*(c + d*x)])*Tan[(c + d*x)/2])/(240*a*d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
120,1,112,212,0.8618216,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{6 (19 A+163 C) \cos ^3\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\tan \left(\frac{1}{2} (c+d x)\right) ((39 A+479 C) \cos (c+d x)+27 A+80 C \cos (2 (c+d x))-8 C \cos (3 (c+d x))+379 C)}{48 a d (a (\cos (c+d x)+1))^{3/2}}","\frac{(19 A+163 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{5 (3 A+19 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{48 a^3 d}-\frac{(21 A+197 C) \sin (c+d x)}{24 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x) \cos ^3(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{(A+17 C) \sin (c+d x) \cos ^2(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"(6*(19*A + 163*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 - (27*A + 379*C + (39*A + 479*C)*Cos[c + d*x] + 80*C*Cos[2*(c + d*x)] - 8*C*Cos[3*(c + d*x)])*Tan[(c + d*x)/2])/(48*a*d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
121,1,95,165,0.7179108,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) (5 (A+17 C) \cos (c+d x)+A+16 C \cos (2 (c+d x))+65 C)+10 (A-15 C) \cos ^3\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{16 a d (a (\cos (c+d x)+1))^{3/2}}","\frac{5 (A-15 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(A+9 C) \sin (c+d x)}{4 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{(3 A-13 C) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"(10*(A - 15*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 + (A + 65*C + 5*(A + 17*C)*Cos[c + d*x] + 16*C*Cos[2*(c + d*x)])*Tan[(c + d*x)/2])/(16*a*d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
122,1,89,124,0.4589633,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) ((3 A-13 C) \cos (c+d x)+7 A-9 C)+2 (3 A+19 C) \cos ^3\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{16 a d (a (\cos (c+d x)+1))^{3/2}}","\frac{(3 A+19 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(3 A-13 C) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}+\frac{(A+C) \sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(2*(3*A + 19*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 + (7*A - 9*C + (3*A - 13*C)*Cos[c + d*x])*Tan[(c + d*x)/2])/(16*a*d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
123,1,124,162,1.7109748,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) ((5 C-11 A) \cos (c+d x)-15 A+C)-2 (43 A-5 C) \cos ^3\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+64 \sqrt{2} A \cos ^3\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)}{16 a d (a (\cos (c+d x)+1))^{3/2}}","-\frac{(43 A-5 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(11 A-5 C) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(-2*(43*A - 5*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 + 64*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 + (-15*A + C + (-11*A + 5*C)*Cos[c + d*x])*Tan[(c + d*x)/2])/(16*a*d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
124,1,185,199,4.5527067,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\cos ^5\left(\frac{1}{2} (c+d x)\right) \cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right) \left((230 A+6 C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\frac{1}{2} \tan \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sec ^3\left(\frac{1}{2} (c+d x)\right) (2 (55 A+7 C) \cos (c+d x)+(35 A+3 C) \cos (2 (c+d x))+67 A+3 C)-160 \sqrt{2} A \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{4 d (a (\cos (c+d x)+1))^{5/2} (2 A+C \cos (2 (c+d x))+C)}","\frac{(115 A+3 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{5 A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}+\frac{(35 A+3 C) \tan (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(15 A-C) \tan (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \tan (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(Cos[(c + d*x)/2]^5*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*((230*A + 6*C)*ArcTanh[Sin[(c + d*x)/2]] - 160*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + ((67*A + 3*C + 2*(55*A + 7*C)*Cos[c + d*x] + (35*A + 3*C)*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^3*Sec[c + d*x]*Tan[(c + d*x)/2])/2))/(4*d*(a*(1 + Cos[c + d*x]))^(5/2)*(2*A + C + C*Cos[2*(c + d*x)]))","A",1
125,1,408,262,6.1740247,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^(5/2),x]","\frac{4 \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right) \cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right) \left(-\frac{A+C}{16 \left(1-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{A+C}{16 \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1\right)^2}-\frac{27 A+11 C}{16 \left(1-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{27 A+11 C}{16 \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1\right)}-\frac{1}{8} (219 A+43 C) \tanh ^{-1}\left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)+4 \sqrt{2} (6 A+C) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)-\frac{12 A \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}+\frac{2 A \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{3}{2} A \left(\frac{2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}+\sqrt{2} \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)\right)-6 \sqrt{2} A \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)\right)}{d (a (\cos (c+d x)+1))^{5/2} (2 A+C \cos (2 c+2 d x)+C)}","\frac{(39 A+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{5/2} d}-\frac{(219 A+43 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(63 A+11 C) \tan (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(31 A+7 C) \tan (c+d x) \sec (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(19 A+3 C) \tan (c+d x) \sec (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \tan (c+d x) \sec (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(4*Cos[c/2 + (d*x)/2]^5*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*(-1/8*((219*A + 43*C)*ArcTanh[Sin[c/2 + (d*x)/2]]) - 6*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[c/2 + (d*x)/2]] + 4*Sqrt[2]*(6*A + C)*ArcTanh[Sqrt[2]*Sin[c/2 + (d*x)/2]] - (A + C)/(16*(1 - Sin[c/2 + (d*x)/2])^2) - (27*A + 11*C)/(16*(1 - Sin[c/2 + (d*x)/2])) + (A + C)/(16*(1 + Sin[c/2 + (d*x)/2])^2) + (27*A + 11*C)/(16*(1 + Sin[c/2 + (d*x)/2])) + (2*A*Sin[c/2 + (d*x)/2])/(1 - 2*Sin[c/2 + (d*x)/2]^2)^2 - (12*A*Sin[c/2 + (d*x)/2])/(1 - 2*Sin[c/2 + (d*x)/2]^2) + (3*A*(Sqrt[2]*ArcTanh[Sqrt[2]*Sin[c/2 + (d*x)/2]] + (2*Sin[c/2 + (d*x)/2])/(1 - 2*Sin[c/2 + (d*x)/2]^2)))/2))/(d*(a*(1 + Cos[c + d*x]))^(5/2)*(2*A + C + C*Cos[2*c + 2*d*x]))","A",1
126,1,964,196,6.3655839,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(-\frac{(9 A+7 C) \cot (c)}{15 d}+\frac{(506 A+435 C) \cos (d x) \sin (c)}{1848 d}+\frac{(18 A+19 C) \cos (2 d x) \sin (2 c)}{180 d}+\frac{(44 A+57 C) \cos (3 d x) \sin (3 c)}{1232 d}+\frac{C \cos (4 d x) \sin (4 c)}{72 d}+\frac{C \cos (5 d x) \sin (5 c)}{176 d}+\frac{(506 A+435 C) \cos (c) \sin (d x)}{1848 d}+\frac{(18 A+19 C) \cos (2 c) \sin (2 d x)}{180 d}+\frac{(44 A+57 C) \cos (3 c) \sin (3 d x)}{1232 d}+\frac{C \cos (4 c) \sin (4 d x)}{72 d}+\frac{C \cos (5 c) \sin (5 d x)}{176 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{3 A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{7 C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{30 d}-\frac{5 A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d \sqrt{\cot ^2(c)+1}}-\frac{15 C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{77 d \sqrt{\cot ^2(c)+1}}\right)","\frac{10 a (11 A+9 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 a (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a (11 A+9 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{2 a (9 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 a (11 A+9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{11 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-1/15*((9*A + 7*C)*Cot[c])/d + ((506*A + 435*C)*Cos[d*x]*Sin[c])/(1848*d) + ((18*A + 19*C)*Cos[2*d*x]*Sin[2*c])/(180*d) + ((44*A + 57*C)*Cos[3*d*x]*Sin[3*c])/(1232*d) + (C*Cos[4*d*x]*Sin[4*c])/(72*d) + (C*Cos[5*d*x]*Sin[5*c])/(176*d) + ((506*A + 435*C)*Cos[c]*Sin[d*x])/(1848*d) + ((18*A + 19*C)*Cos[2*c]*Sin[2*d*x])/(180*d) + ((44*A + 57*C)*Cos[3*c]*Sin[3*d*x])/(1232*d) + (C*Cos[4*c]*Sin[4*d*x])/(72*d) + (C*Cos[5*c]*Sin[5*d*x])/(176*d)) - (5*A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (15*C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(77*d*Sqrt[1 + Cot[c]^2]) - (3*A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) - (7*C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(30*d))","C",0
127,1,918,165,6.2901222,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(-\frac{(9 A+7 C) \cot (c)}{15 d}+\frac{(28 A+23 C) \cos (d x) \sin (c)}{84 d}+\frac{(18 A+19 C) \cos (2 d x) \sin (2 c)}{180 d}+\frac{C \cos (3 d x) \sin (3 c)}{28 d}+\frac{C \cos (4 d x) \sin (4 c)}{72 d}+\frac{(28 A+23 C) \cos (c) \sin (d x)}{84 d}+\frac{(18 A+19 C) \cos (2 c) \sin (2 d x)}{180 d}+\frac{C \cos (3 c) \sin (3 d x)}{28 d}+\frac{C \cos (4 c) \sin (4 d x)}{72 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{3 A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{7 C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{30 d}-\frac{A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{5 C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a (9 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 a (7 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-1/15*((9*A + 7*C)*Cot[c])/d + ((28*A + 23*C)*Cos[d*x]*Sin[c])/(84*d) + ((18*A + 19*C)*Cos[2*d*x]*Sin[2*c])/(180*d) + (C*Cos[3*d*x]*Sin[3*c])/(28*d) + (C*Cos[4*d*x]*Sin[4*c])/(72*d) + ((28*A + 23*C)*Cos[c]*Sin[d*x])/(84*d) + ((18*A + 19*C)*Cos[2*c]*Sin[2*d*x])/(180*d) + (C*Cos[3*c]*Sin[3*d*x])/(28*d) + (C*Cos[4*c]*Sin[4*d*x])/(72*d)) - (A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (5*C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (3*A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) - (7*C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(30*d))","C",0
128,1,872,134,6.2633492,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(-\frac{(5 A+3 C) \cot (c)}{5 d}+\frac{(28 A+23 C) \cos (d x) \sin (c)}{84 d}+\frac{C \cos (2 d x) \sin (2 c)}{10 d}+\frac{C \cos (3 d x) \sin (3 c)}{28 d}+\frac{(28 A+23 C) \cos (c) \sin (d x)}{84 d}+\frac{C \cos (2 c) \sin (2 d x)}{10 d}+\frac{C \cos (3 c) \sin (3 d x)}{28 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{3 C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{5 C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (7 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-1/5*((5*A + 3*C)*Cot[c])/d + ((28*A + 23*C)*Cos[d*x]*Sin[c])/(84*d) + (C*Cos[2*d*x]*Sin[2*c])/(10*d) + (C*Cos[3*d*x]*Sin[3*c])/(28*d) + ((28*A + 23*C)*Cos[c]*Sin[d*x])/(84*d) + (C*Cos[2*c]*Sin[2*d*x])/(10*d) + (C*Cos[3*c]*Sin[3*d*x])/(28*d)) - (A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (5*C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) - (3*C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d))","C",0
129,1,824,101,6.2838923,"\int \frac{(a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(-\frac{(5 A+3 C) \cot (c)}{5 d}+\frac{C \cos (d x) \sin (c)}{3 d}+\frac{C \cos (2 d x) \sin (2 c)}{10 d}+\frac{C \cos (c) \sin (d x)}{3 d}+\frac{C \cos (2 c) \sin (2 d x)}{10 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{3 C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-1/5*((5*A + 3*C)*Cot[c])/d + (C*Cos[d*x]*Sin[c])/(3*d) + (C*Cos[2*d*x]*Sin[2*c])/(10*d) + (C*Cos[c]*Sin[d*x])/(3*d) + (C*Cos[2*c]*Sin[2*d*x])/(10*d)) - (A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) - (C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) - (3*C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d))","C",0
130,1,813,95,6.3487191,"\int \frac{(a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(-\frac{(-2 A+C+C \cos (2 c)) \csc (c) \sec (c)}{2 d}+\frac{A \sec (c+d x) \sin (d x) \sec (c)}{d}+\frac{C \cos (d x) \sin (c)}{3 d}+\frac{C \cos (c) \sin (d x)}{3 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 a C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-1/2*((-2*A + C + C*Cos[2*c])*Csc[c]*Sec[c])/d + (C*Cos[d*x]*Sin[c])/(3*d) + (C*Cos[c]*Sin[d*x])/(3*d) + (A*Sec[c]*Sec[c + d*x]*Sin[d*x])/d) - (A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) - (C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) + (A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) - (C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d))","C",0
131,1,817,95,6.3570045,"\int \frac{(a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(\frac{A \sec (c) \sin (d x) \sec ^2(c+d x)}{3 d}+\frac{\sec (c) (A \sin (c)+3 A \sin (d x)) \sec (c+d x)}{3 d}-\frac{(-2 A+C+C \cos (2 c)) \csc (c) \sec (c)}{2 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-1/2*((-2*A + C + C*Cos[2*c])*Csc[c]*Sec[c])/d + (A*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (Sec[c]*Sec[c + d*x]*(A*Sin[c] + 3*A*Sin[d*x]))/(3*d)) - (A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) + (A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) - (C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d))","C",0
132,1,851,132,6.4597853,"\int \frac{(a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(\frac{A \sec (c) \sin (d x) \sec ^3(c+d x)}{5 d}+\frac{\sec (c) (3 A \sin (c)+5 A \sin (d x)) \sec ^2(c+d x)}{15 d}+\frac{\sec (c) (5 A \sin (c)+9 A \sin (d x)+15 C \sin (d x)) \sec (c+d x)}{15 d}+\frac{(3 A+5 C) \csc (c) \sec (c)}{5 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{3 A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}+\frac{C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (3 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(((3*A + 5*C)*Csc[c]*Sec[c])/(5*d) + (A*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(5*d) + (Sec[c]*Sec[c + d*x]^2*(3*A*Sin[c] + 5*A*Sin[d*x]))/(15*d) + (Sec[c]*Sec[c + d*x]*(5*A*Sin[c] + 9*A*Sin[d*x] + 15*C*Sin[d*x]))/(15*d)) - (A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) + (3*A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) + (C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d))","C",0
133,1,895,165,6.5417446,"\int \frac{(a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(\frac{A \sec (c) \sin (d x) \sec ^4(c+d x)}{7 d}+\frac{\sec (c) (5 A \sin (c)+7 A \sin (d x)) \sec ^3(c+d x)}{35 d}+\frac{\sec (c) (21 A \sin (c)+25 A \sin (d x)+35 C \sin (d x)) \sec ^2(c+d x)}{105 d}+\frac{\sec (c) (25 A \sin (c)+35 C \sin (c)+63 A \sin (d x)+105 C \sin (d x)) \sec (c+d x)}{105 d}+\frac{(3 A+5 C) \csc (c) \sec (c)}{5 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{3 A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}+\frac{C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{5 A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (5 A+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (5 A+7 C) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a (3 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(((3*A + 5*C)*Csc[c]*Sec[c])/(5*d) + (A*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(7*d) + (Sec[c]*Sec[c + d*x]^3*(5*A*Sin[c] + 7*A*Sin[d*x]))/(35*d) + (Sec[c]*Sec[c + d*x]^2*(21*A*Sin[c] + 25*A*Sin[d*x] + 35*C*Sin[d*x]))/(105*d) + (Sec[c]*Sec[c + d*x]*(25*A*Sin[c] + 35*C*Sin[c] + 63*A*Sin[d*x] + 105*C*Sin[d*x]))/(105*d)) - (5*A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) + (3*A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) + (C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d))","C",0
134,1,982,230,6.3099236,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^2 \left(-\frac{(9 A+7 C) \cot (c)}{15 d}+\frac{(1122 A+941 C) \cos (d x) \sin (c)}{3696 d}+\frac{(18 A+19 C) \cos (2 d x) \sin (2 c)}{180 d}+\frac{(44 A+101 C) \cos (3 d x) \sin (3 c)}{2464 d}+\frac{C \cos (4 d x) \sin (4 c)}{72 d}+\frac{C \cos (5 d x) \sin (5 c)}{352 d}+\frac{(1122 A+941 C) \cos (c) \sin (d x)}{3696 d}+\frac{(18 A+19 C) \cos (2 c) \sin (2 d x)}{180 d}+\frac{(44 A+101 C) \cos (3 c) \sin (3 d x)}{2464 d}+\frac{C \cos (4 c) \sin (4 d x)}{72 d}+\frac{C \cos (5 c) \sin (5 d x)}{352 d}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{3 A (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{7 C (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{30 d}-\frac{2 A (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d \sqrt{\cot ^2(c)+1}}-\frac{50 C (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{231 d \sqrt{\cot ^2(c)+1}}","\frac{8 a^2 (33 A+25 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^2 (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 (99 A+89 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{693 d}+\frac{4 a^2 (9 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{8 a^2 (33 A+25 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{8 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{99 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(-1/15*((9*A + 7*C)*Cot[c])/d + ((1122*A + 941*C)*Cos[d*x]*Sin[c])/(3696*d) + ((18*A + 19*C)*Cos[2*d*x]*Sin[2*c])/(180*d) + ((44*A + 101*C)*Cos[3*d*x]*Sin[3*c])/(2464*d) + (C*Cos[4*d*x]*Sin[4*c])/(72*d) + (C*Cos[5*d*x]*Sin[5*c])/(352*d) + ((1122*A + 941*C)*Cos[c]*Sin[d*x])/(3696*d) + ((18*A + 19*C)*Cos[2*c]*Sin[2*d*x])/(180*d) + ((44*A + 101*C)*Cos[3*c]*Sin[3*d*x])/(2464*d) + (C*Cos[4*c]*Sin[4*d*x])/(72*d) + (C*Cos[5*c]*Sin[5*d*x])/(352*d)) - (2*A*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*Sqrt[1 + Cot[c]^2]) - (50*C*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(231*d*Sqrt[1 + Cot[c]^2]) - (3*A*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) - (7*C*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(30*d)","C",0
135,1,936,197,6.2870732,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^2 \left(-\frac{4 (3 A+2 C) \cot (c)}{15 d}+\frac{(28 A+23 C) \cos (d x) \sin (c)}{84 d}+\frac{(18 A+37 C) \cos (2 d x) \sin (2 c)}{360 d}+\frac{C \cos (3 d x) \sin (3 c)}{28 d}+\frac{C \cos (4 d x) \sin (4 c)}{144 d}+\frac{(28 A+23 C) \cos (c) \sin (d x)}{84 d}+\frac{(18 A+37 C) \cos (2 c) \sin (2 d x)}{360 d}+\frac{C \cos (3 c) \sin (3 d x)}{28 d}+\frac{C \cos (4 c) \sin (4 d x)}{144 d}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{2 A (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 C (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{A (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{5 C (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^2 (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{16 a^2 (3 A+2 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 (21 A+19 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^2 (7 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{8 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*((-4*(3*A + 2*C)*Cot[c])/(15*d) + ((28*A + 23*C)*Cos[d*x]*Sin[c])/(84*d) + ((18*A + 37*C)*Cos[2*d*x]*Sin[2*c])/(360*d) + (C*Cos[3*d*x]*Sin[3*c])/(28*d) + (C*Cos[4*d*x]*Sin[4*c])/(144*d) + ((28*A + 23*C)*Cos[c]*Sin[d*x])/(84*d) + ((18*A + 37*C)*Cos[2*c]*Sin[2*d*x])/(360*d) + (C*Cos[3*c]*Sin[3*d*x])/(28*d) + (C*Cos[4*c]*Sin[4*d*x])/(144*d)) - (A*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (5*C*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (2*A*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d) - (4*C*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*d)","C",0
136,1,890,164,6.3445262,"\int \frac{(a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^2 \left(-\frac{(5 A+3 C) \cot (c)}{5 d}+\frac{(28 A+51 C) \cos (d x) \sin (c)}{168 d}+\frac{C \cos (2 d x) \sin (2 c)}{10 d}+\frac{C \cos (3 d x) \sin (3 c)}{56 d}+\frac{(28 A+51 C) \cos (c) \sin (d x)}{168 d}+\frac{C \cos (2 c) \sin (2 d x)}{10 d}+\frac{C \cos (3 c) \sin (3 d x)}{56 d}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{A (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{3 C (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{2 A (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{2 C (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d \sqrt{\cot ^2(c)+1}}","\frac{8 a^2 (7 A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (35 A+33 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{8 C \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)}{35 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(-1/5*((5*A + 3*C)*Cot[c])/d + ((28*A + 51*C)*Cos[d*x]*Sin[c])/(168*d) + (C*Cos[2*d*x]*Sin[2*c])/(10*d) + (C*Cos[3*d*x]*Sin[3*c])/(56*d) + ((28*A + 51*C)*Cos[c]*Sin[d*x])/(168*d) + (C*Cos[2*c]*Sin[2*d*x])/(10*d) + (C*Cos[3*c]*Sin[3*d*x])/(56*d)) - (2*A*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (2*C*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*Sqrt[1 + Cot[c]^2]) - (A*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) - (3*C*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d)","C",0
137,1,658,160,6.425043,"\int \frac{(a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","-\frac{A \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^2 \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{d \sqrt{\cot ^2(c)+1}}-\frac{2 C \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^2 \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{5 d}-\frac{C \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^2 \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{3 d \sqrt{\cot ^2(c)+1}}+\sqrt{\cos (c+d x)} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^2 \left(-\frac{\csc (c) \sec (c) (5 A \cos (2 c)-5 A+8 C \cos (2 c)+8 C)}{20 d}+\frac{A \sec (c) \sin (d x) \sec (c+d x)}{2 d}+\frac{C \sin (c) \cos (d x)}{3 d}+\frac{C \sin (2 c) \cos (2 d x)}{20 d}+\frac{C \cos (c) \sin (d x)}{3 d}+\frac{C \cos (2 c) \sin (2 d x)}{20 d}\right)","\frac{4 a^2 (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a^2 (15 A-7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}-\frac{2 (5 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)}{5 d}+\frac{16 a^2 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{d \sqrt{\cos (c+d x)}}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(-1/20*((-5*A + 8*C + 5*A*Cos[2*c] + 8*C*Cos[2*c])*Csc[c]*Sec[c])/d + (C*Cos[d*x]*Sin[c])/(3*d) + (C*Cos[2*d*x]*Sin[2*c])/(20*d) + (C*Cos[c]*Sin[d*x])/(3*d) + (A*Sec[c]*Sec[c + d*x]*Sin[d*x])/(2*d) + (C*Cos[2*c]*Sin[2*d*x])/(20*d)) - (A*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) - (C*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (2*C*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d)","C",0
138,1,865,156,6.4762202,"\int \frac{(a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^2 \left(\frac{A \sec (c) \sin (d x) \sec ^2(c+d x)}{6 d}+\frac{\sec (c) (A \sin (c)+6 A \sin (d x)) \sec (c+d x)}{6 d}-\frac{(-2 A+C+C \cos (2 c)) \csc (c) \sec (c)}{2 d}+\frac{C \cos (d x) \sin (c)}{6 d}+\frac{C \cos (c) \sin (d x)}{6 d}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{A (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{C (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{2 A (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{2 C (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}","\frac{8 a^2 (A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^2 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 a^2 (5 A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{8 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(-1/2*((-2*A + C + C*Cos[2*c])*Csc[c]*Sec[c])/d + (C*Cos[d*x]*Sin[c])/(6*d) + (C*Cos[c]*Sin[d*x])/(6*d) + (A*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(6*d) + (Sec[c]*Sec[c + d*x]*(A*Sin[c] + 6*A*Sin[d*x]))/(6*d)) - (2*A*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (2*C*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) + (A*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) - (C*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d)","C",0
139,1,656,156,6.5202439,"\int \frac{(a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{2 A \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^2 \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{5 d}-\frac{A \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^2 \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{C \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^2 \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{d \sqrt{\cot ^2(c)+1}}+\sqrt{\cos (c+d x)} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^2 \left(\frac{\sec (c) \sec (c+d x) (10 A \sin (c)+24 A \sin (d x)+15 C \sin (d x))}{30 d}-\frac{\csc (c) \sec (c) (-16 A+5 C \cos (2 c)-5 C)}{20 d}+\frac{A \sec (c) \sin (d x) \sec ^3(c+d x)}{10 d}+\frac{\sec (c) \sec ^2(c+d x) (3 A \sin (c)+10 A \sin (d x))}{30 d}\right)","\frac{4 a^2 (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^2 (17 A+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}-\frac{16 a^2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(-1/20*((-16*A - 5*C + 5*C*Cos[2*c])*Csc[c]*Sec[c])/d + (A*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(10*d) + (Sec[c]*Sec[c + d*x]^2*(3*A*Sin[c] + 10*A*Sin[d*x]))/(30*d) + (Sec[c]*Sec[c + d*x]*(10*A*Sin[c] + 24*A*Sin[d*x] + 15*C*Sin[d*x]))/(30*d)) - (A*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (C*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) + (2*A*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d)","C",0
140,1,913,197,6.6202781,"\int \frac{(a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^2 \left(\frac{A \sec (c) \sin (d x) \sec ^4(c+d x)}{14 d}+\frac{\sec (c) (5 A \sin (c)+14 A \sin (d x)) \sec ^3(c+d x)}{70 d}+\frac{\sec (c) (42 A \sin (c)+60 A \sin (d x)+35 C \sin (d x)) \sec ^2(c+d x)}{210 d}+\frac{\sec (c) (60 A \sin (c)+35 C \sin (c)+126 A \sin (d x)+210 C \sin (d x)) \sec (c+d x)}{210 d}+\frac{(3 A+5 C) \csc (c) \sec (c)}{5 d}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{3 A (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}+\frac{C (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{2 A (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d \sqrt{\cot ^2(c)+1}}-\frac{2 C (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}","\frac{8 a^2 (3 A+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (33 A+35 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (3 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{8 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(((3*A + 5*C)*Csc[c]*Sec[c])/(5*d) + (A*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(14*d) + (Sec[c]*Sec[c + d*x]^3*(5*A*Sin[c] + 14*A*Sin[d*x]))/(70*d) + (Sec[c]*Sec[c + d*x]^2*(42*A*Sin[c] + 60*A*Sin[d*x] + 35*C*Sin[d*x]))/(210*d) + (Sec[c]*Sec[c + d*x]*(60*A*Sin[c] + 35*C*Sin[c] + 126*A*Sin[d*x] + 210*C*Sin[d*x]))/(210*d)) - (2*A*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*Sqrt[1 + Cot[c]^2]) - (2*C*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) + (3*A*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) + (C*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d)","C",0
141,1,955,230,6.7125503,"\int \frac{(a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^2 \left(\frac{A \sec (c) \sin (d x) \sec ^5(c+d x)}{18 d}+\frac{\sec (c) (7 A \sin (c)+18 A \sin (d x)) \sec ^4(c+d x)}{126 d}+\frac{\sec (c) (90 A \sin (c)+112 A \sin (d x)+63 C \sin (d x)) \sec ^3(c+d x)}{630 d}+\frac{\sec (c) (112 A \sin (c)+63 C \sin (c)+150 A \sin (d x)+210 C \sin (d x)) \sec ^2(c+d x)}{630 d}+\frac{\sec (c) (25 A \sin (c)+35 C \sin (c)+56 A \sin (d x)+84 C \sin (d x)) \sec (c+d x)}{105 d}+\frac{4 (2 A+3 C) \csc (c) \sec (c)}{15 d}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{4 A (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{2 C (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{5 A (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^2 (5 A+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{16 a^2 (2 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^2 (5 A+7 C) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (19 A+21 C) \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{16 a^2 (2 A+3 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{8 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*((4*(2*A + 3*C)*Csc[c]*Sec[c])/(15*d) + (A*Sec[c]*Sec[c + d*x]^5*Sin[d*x])/(18*d) + (Sec[c]*Sec[c + d*x]^4*(7*A*Sin[c] + 18*A*Sin[d*x]))/(126*d) + (Sec[c]*Sec[c + d*x]^3*(90*A*Sin[c] + 112*A*Sin[d*x] + 63*C*Sin[d*x]))/(630*d) + (Sec[c]*Sec[c + d*x]*(25*A*Sin[c] + 35*C*Sin[c] + 56*A*Sin[d*x] + 84*C*Sin[d*x]))/(105*d) + (Sec[c]*Sec[c + d*x]^2*(112*A*Sin[c] + 63*C*Sin[c] + 150*A*Sin[d*x] + 210*C*Sin[d*x]))/(630*d)) - (5*A*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (C*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) + (4*A*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*d) + (2*C*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d)","C",0
142,1,1028,279,6.355716,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^3 \left(-\frac{(221 A+175 C) \cot (c)}{390 d}+\frac{(2134 A+1811 C) \cos (d x) \sin (c)}{7392 d}+\frac{(7592 A+7825 C) \cos (2 d x) \sin (2 c)}{74880 d}+\frac{(132 A+215 C) \cos (3 d x) \sin (3 c)}{4928 d}+\frac{(13 A+59 C) \cos (4 d x) \sin (4 c)}{3744 d}+\frac{3 C \cos (5 d x) \sin (5 c)}{704 d}+\frac{C \cos (6 d x) \sin (6 c)}{1664 d}+\frac{(2134 A+1811 C) \cos (c) \sin (d x)}{7392 d}+\frac{(7592 A+7825 C) \cos (2 c) \sin (2 d x)}{74880 d}+\frac{(132 A+215 C) \cos (3 c) \sin (3 d x)}{4928 d}+\frac{(13 A+59 C) \cos (4 c) \sin (4 d x)}{3744 d}+\frac{3 C \cos (5 c) \sin (5 d x)}{704 d}+\frac{C \cos (6 c) \sin (6 d x)}{1664 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{17 A (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{60 d}-\frac{35 C (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{156 d}-\frac{11 A (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{42 d \sqrt{\cot ^2(c)+1}}-\frac{95 C (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{462 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (121 A+95 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (221 A+175 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{195 d}+\frac{40 a^3 (143 A+118 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{9009 d}+\frac{4 a^3 (221 A+175 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{585 d}+\frac{2 (143 A+145 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{1287 d}+\frac{4 a^3 (121 A+95 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{12 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{143 a d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-1/390*((221*A + 175*C)*Cot[c])/d + ((2134*A + 1811*C)*Cos[d*x]*Sin[c])/(7392*d) + ((7592*A + 7825*C)*Cos[2*d*x]*Sin[2*c])/(74880*d) + ((132*A + 215*C)*Cos[3*d*x]*Sin[3*c])/(4928*d) + ((13*A + 59*C)*Cos[4*d*x]*Sin[4*c])/(3744*d) + (3*C*Cos[5*d*x]*Sin[5*c])/(704*d) + (C*Cos[6*d*x]*Sin[6*c])/(1664*d) + ((2134*A + 1811*C)*Cos[c]*Sin[d*x])/(7392*d) + ((7592*A + 7825*C)*Cos[2*c]*Sin[2*d*x])/(74880*d) + ((132*A + 215*C)*Cos[3*c]*Sin[3*d*x])/(4928*d) + ((13*A + 59*C)*Cos[4*c]*Sin[4*d*x])/(3744*d) + (3*C*Cos[5*c]*Sin[5*d*x])/(704*d) + (C*Cos[6*c]*Sin[6*d*x])/(1664*d)) - (11*A*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (95*C*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(462*d*Sqrt[1 + Cot[c]^2]) - (17*A*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(60*d) - (35*C*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(156*d)","C",0
143,1,982,246,6.3197128,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^3 \left(-\frac{(7 A+5 C) \cot (c)}{10 d}+\frac{(2354 A+1953 C) \cos (d x) \sin (c)}{7392 d}+\frac{(18 A+25 C) \cos (2 d x) \sin (2 c)}{240 d}+\frac{(44 A+189 C) \cos (3 d x) \sin (3 c)}{4928 d}+\frac{C \cos (4 d x) \sin (4 c)}{96 d}+\frac{C \cos (5 d x) \sin (5 c)}{704 d}+\frac{(2354 A+1953 C) \cos (c) \sin (d x)}{7392 d}+\frac{(18 A+25 C) \cos (2 c) \sin (2 d x)}{240 d}+\frac{(44 A+189 C) \cos (3 c) \sin (3 d x)}{4928 d}+\frac{C \cos (4 c) \sin (4 d x)}{96 d}+\frac{C \cos (5 c) \sin (5 d x)}{704 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{7 A (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 d}-\frac{C (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}-\frac{13 A (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{42 d \sqrt{\cot ^2(c)+1}}-\frac{5 C (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{22 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (143 A+105 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (7 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 a^3 (44 A+35 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{385 d}+\frac{2 (33 A+35 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{231 d}+\frac{4 a^3 (143 A+105 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{4 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{33 a d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-1/10*((7*A + 5*C)*Cot[c])/d + ((2354*A + 1953*C)*Cos[d*x]*Sin[c])/(7392*d) + ((18*A + 25*C)*Cos[2*d*x]*Sin[2*c])/(240*d) + ((44*A + 189*C)*Cos[3*d*x]*Sin[3*c])/(4928*d) + (C*Cos[4*d*x]*Sin[4*c])/(96*d) + (C*Cos[5*d*x]*Sin[5*c])/(704*d) + ((2354*A + 1953*C)*Cos[c]*Sin[d*x])/(7392*d) + ((18*A + 25*C)*Cos[2*c]*Sin[2*d*x])/(240*d) + ((44*A + 189*C)*Cos[3*c]*Sin[3*d*x])/(4928*d) + (C*Cos[4*c]*Sin[4*d*x])/(96*d) + (C*Cos[5*c]*Sin[5*d*x])/(704*d)) - (13*A*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (5*C*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(22*d*Sqrt[1 + Cot[c]^2]) - (7*A*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d) - (C*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(4*d)","C",0
144,1,936,213,6.4033733,"\int \frac{(a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^3 \left(-\frac{(27 A+17 C) \cot (c)}{30 d}+\frac{(84 A+97 C) \cos (d x) \sin (c)}{336 d}+\frac{(18 A+73 C) \cos (2 d x) \sin (2 c)}{720 d}+\frac{3 C \cos (3 d x) \sin (3 c)}{112 d}+\frac{C \cos (4 d x) \sin (4 c)}{288 d}+\frac{(84 A+97 C) \cos (c) \sin (d x)}{336 d}+\frac{(18 A+73 C) \cos (2 c) \sin (2 d x)}{720 d}+\frac{3 C \cos (3 c) \sin (3 d x)}{112 d}+\frac{C \cos (4 c) \sin (4 d x)}{288 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{9 A (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 d}-\frac{17 C (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{60 d}-\frac{A (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d \sqrt{\cot ^2(c)+1}}-\frac{11 C (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{42 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (21 A+11 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (27 A+17 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{8 a^3 (21 A+16 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{2 (63 A+73 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{315 d}+\frac{4 C \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-1/30*((27*A + 17*C)*Cot[c])/d + ((84*A + 97*C)*Cos[d*x]*Sin[c])/(336*d) + ((18*A + 73*C)*Cos[2*d*x]*Sin[2*c])/(720*d) + (3*C*Cos[3*d*x]*Sin[3*c])/(112*d) + (C*Cos[4*d*x]*Sin[4*c])/(288*d) + ((84*A + 97*C)*Cos[c]*Sin[d*x])/(336*d) + ((18*A + 73*C)*Cos[2*c]*Sin[2*d*x])/(720*d) + (3*C*Cos[3*c]*Sin[3*d*x])/(112*d) + (C*Cos[4*c]*Sin[4*d*x])/(288*d)) - (A*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(2*d*Sqrt[1 + Cot[c]^2]) - (11*C*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (9*A*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d) - (17*C*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(60*d)","C",0
145,1,926,217,6.5258018,"\int \frac{(a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^3 \left(-\frac{(15 \cos (2 c) A+5 A+14 C+14 C \cos (2 c)) \csc (c) \sec (c)}{40 d}+\frac{A \sec (c+d x) \sin (d x) \sec (c)}{4 d}+\frac{(28 A+107 C) \cos (d x) \sin (c)}{336 d}+\frac{3 C \cos (2 d x) \sin (2 c)}{40 d}+\frac{C \cos (3 d x) \sin (3 c)}{112 d}+\frac{(28 A+107 C) \cos (c) \sin (d x)}{336 d}+\frac{3 C \cos (2 c) \sin (2 d x)}{40 d}+\frac{C \cos (3 c) \sin (3 d x)}{112 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{A (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}-\frac{7 C (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 d}-\frac{5 A (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{6 d \sqrt{\cot ^2(c)+1}}-\frac{13 C (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{42 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (35 A+13 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (5 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{4 a^3 (35 A-41 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}-\frac{2 (35 A-11 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{35 d}-\frac{2 (7 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)^2}{7 a d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt{\cos (c+d x)}}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-1/40*((5*A + 14*C + 15*A*Cos[2*c] + 14*C*Cos[2*c])*Csc[c]*Sec[c])/d + ((28*A + 107*C)*Cos[d*x]*Sin[c])/(336*d) + (3*C*Cos[2*d*x]*Sin[2*c])/(40*d) + (C*Cos[3*d*x]*Sin[3*c])/(112*d) + ((28*A + 107*C)*Cos[c]*Sin[d*x])/(336*d) + (A*Sec[c]*Sec[c + d*x]*Sin[d*x])/(4*d) + (3*C*Cos[2*c]*Sin[2*d*x])/(40*d) + (C*Cos[3*c]*Sin[3*d*x])/(112*d)) - (5*A*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(6*d*Sqrt[1 + Cot[c]^2]) - (13*C*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (A*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(4*d) - (7*C*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d)","C",0
146,1,909,211,6.5864373,"\int \frac{(a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^3 \left(\frac{A \sec (c) \sin (d x) \sec ^2(c+d x)}{12 d}+\frac{\sec (c) (A \sin (c)+9 A \sin (d x)) \sec (c+d x)}{12 d}-\frac{(5 \cos (2 c) A-25 A+18 C+18 C \cos (2 c)) \csc (c) \sec (c)}{40 d}+\frac{C \cos (d x) \sin (c)}{4 d}+\frac{C \cos (2 d x) \sin (2 c)}{40 d}+\frac{C \cos (c) \sin (d x)}{4 d}+\frac{C \cos (2 c) \sin (2 d x)}{40 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{A (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}-\frac{9 C (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 d}-\frac{5 A (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{6 d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (5 A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (5 A-9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{8 a^3 (10 A-3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}-\frac{2 (35 A-3 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{15 d}+\frac{4 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{a d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-1/40*((-25*A + 18*C + 5*A*Cos[2*c] + 18*C*Cos[2*c])*Csc[c]*Sec[c])/d + (C*Cos[d*x]*Sin[c])/(4*d) + (C*Cos[2*d*x]*Sin[2*c])/(40*d) + (C*Cos[c]*Sin[d*x])/(4*d) + (A*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(12*d) + (Sec[c]*Sec[c + d*x]*(A*Sin[c] + 9*A*Sin[d*x]))/(12*d) + (C*Cos[2*c]*Sin[2*d*x])/(40*d)) - (5*A*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(6*d*Sqrt[1 + Cot[c]^2]) - (C*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(2*d*Sqrt[1 + Cot[c]^2]) + (A*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(4*d) - (9*C*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d)","C",0
147,1,905,213,6.6189174,"\int \frac{(a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^3 \left(\frac{A \sec (c) \sin (d x) \sec ^3(c+d x)}{20 d}+\frac{\sec (c) (A \sin (c)+5 A \sin (d x)) \sec ^2(c+d x)}{20 d}+\frac{\sec (c) (5 A \sin (c)+18 A \sin (d x)+5 C \sin (d x)) \sec (c+d x)}{20 d}-\frac{(-36 A+5 C+15 C \cos (2 c)) \csc (c) \sec (c)}{40 d}+\frac{C \cos (d x) \sin (c)}{12 d}+\frac{C \cos (c) \sin (d x)}{12 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{9 A (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 d}-\frac{C (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}-\frac{A (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d \sqrt{\cot ^2(c)+1}}-\frac{5 C (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{6 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (3 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (9 A-5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{4 a^3 (21 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 (11 A+5 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{5 d \sqrt{\cos (c+d x)}}+\frac{4 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{5 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-1/40*((-36*A + 5*C + 15*C*Cos[2*c])*Csc[c]*Sec[c])/d + (C*Cos[d*x]*Sin[c])/(12*d) + (C*Cos[c]*Sin[d*x])/(12*d) + (A*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(20*d) + (Sec[c]*Sec[c + d*x]^2*(A*Sin[c] + 5*A*Sin[d*x]))/(20*d) + (Sec[c]*Sec[c + d*x]*(5*A*Sin[c] + 18*A*Sin[d*x] + 5*C*Sin[d*x]))/(20*d)) - (A*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(2*d*Sqrt[1 + Cot[c]^2]) - (5*C*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(6*d*Sqrt[1 + Cot[c]^2]) + (9*A*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d) - (C*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(4*d)","C",0
148,1,920,213,6.7330066,"\int \frac{(a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^3 \left(\frac{A \sec (c) \sin (d x) \sec ^4(c+d x)}{28 d}+\frac{\sec (c) (5 A \sin (c)+21 A \sin (d x)) \sec ^3(c+d x)}{140 d}+\frac{\sec (c) (63 A \sin (c)+130 A \sin (d x)+35 C \sin (d x)) \sec ^2(c+d x)}{420 d}+\frac{\sec (c) (130 A \sin (c)+35 C \sin (c)+294 A \sin (d x)+315 C \sin (d x)) \sec (c+d x)}{420 d}-\frac{(-28 A-25 C+5 C \cos (2 c)) \csc (c) \sec (c)}{40 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{7 A (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 d}+\frac{C (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}-\frac{13 A (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{42 d \sqrt{\cot ^2(c)+1}}-\frac{5 C (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{6 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (13 A+35 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (7 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (7 A+5 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{8 a^3 (53 A+70 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)}}+\frac{12 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{35 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-1/40*((-28*A - 25*C + 5*C*Cos[2*c])*Csc[c]*Sec[c])/d + (A*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(28*d) + (Sec[c]*Sec[c + d*x]^3*(5*A*Sin[c] + 21*A*Sin[d*x]))/(140*d) + (Sec[c]*Sec[c + d*x]^2*(63*A*Sin[c] + 130*A*Sin[d*x] + 35*C*Sin[d*x]))/(420*d) + (Sec[c]*Sec[c + d*x]*(130*A*Sin[c] + 35*C*Sin[c] + 294*A*Sin[d*x] + 315*C*Sin[d*x]))/(420*d)) - (13*A*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (5*C*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(6*d*Sqrt[1 + Cot[c]^2]) + (7*A*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d) + (C*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(4*d)","C",0
149,1,955,246,6.7851031,"\int \frac{(a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^3 \left(\frac{A \sec (c) \sin (d x) \sec ^5(c+d x)}{36 d}+\frac{\sec (c) (7 A \sin (c)+27 A \sin (d x)) \sec ^4(c+d x)}{252 d}+\frac{\sec (c) (135 A \sin (c)+238 A \sin (d x)+63 C \sin (d x)) \sec ^3(c+d x)}{1260 d}+\frac{\sec (c) (238 A \sin (c)+63 C \sin (c)+330 A \sin (d x)+315 C \sin (d x)) \sec ^2(c+d x)}{1260 d}+\frac{\sec (c) (110 A \sin (c)+105 C \sin (c)+238 A \sin (d x)+378 C \sin (d x)) \sec (c+d x)}{420 d}+\frac{(17 A+27 C) \csc (c) \sec (c)}{30 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{17 A (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{60 d}+\frac{9 C (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 d}-\frac{11 A (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{42 d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (11 A+21 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (17 A+27 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{8 a^3 (16 A+21 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (73 A+63 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (17 A+27 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{4 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(((17*A + 27*C)*Csc[c]*Sec[c])/(30*d) + (A*Sec[c]*Sec[c + d*x]^5*Sin[d*x])/(36*d) + (Sec[c]*Sec[c + d*x]^4*(7*A*Sin[c] + 27*A*Sin[d*x]))/(252*d) + (Sec[c]*Sec[c + d*x]^3*(135*A*Sin[c] + 238*A*Sin[d*x] + 63*C*Sin[d*x]))/(1260*d) + (Sec[c]*Sec[c + d*x]^2*(238*A*Sin[c] + 63*C*Sin[c] + 330*A*Sin[d*x] + 315*C*Sin[d*x]))/(1260*d) + (Sec[c]*Sec[c + d*x]*(110*A*Sin[c] + 105*C*Sin[c] + 238*A*Sin[d*x] + 378*C*Sin[d*x]))/(420*d)) - (11*A*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (C*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(2*d*Sqrt[1 + Cot[c]^2]) + (17*A*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(60*d) + (9*C*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d)","C",0
150,1,997,279,6.8678634,"\int \frac{(a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^3 \left(\frac{A \sec (c) \sin (d x) \sec ^6(c+d x)}{44 d}+\frac{\sec (c) (3 A \sin (c)+11 A \sin (d x)) \sec ^5(c+d x)}{132 d}+\frac{\sec (c) (77 A \sin (c)+126 A \sin (d x)+33 C \sin (d x)) \sec ^4(c+d x)}{924 d}+\frac{\sec (c) (630 A \sin (c)+165 C \sin (c)+770 A \sin (d x)+693 C \sin (d x)) \sec ^3(c+d x)}{4620 d}+\frac{\sec (c) (770 A \sin (c)+693 C \sin (c)+1050 A \sin (d x)+1430 C \sin (d x)) \sec ^2(c+d x)}{4620 d}+\frac{\sec (c) (525 A \sin (c)+715 C \sin (c)+1155 A \sin (d x)+1617 C \sin (d x)) \sec (c+d x)}{2310 d}+\frac{(5 A+7 C) \csc (c) \sec (c)}{10 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{A (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}+\frac{7 C (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 d}-\frac{5 A (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{22 d \sqrt{\cot ^2(c)+1}}-\frac{13 C (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{42 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (105 A+143 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}-\frac{4 a^3 (5 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a^3 (105 A+143 C) \sin (c+d x)}{231 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{8 a^3 (35 A+44 C) \sin (c+d x)}{385 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (35 A+33 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (5 A+7 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{4 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{33 a d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac{11}{2}}(c+d x)}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(((5*A + 7*C)*Csc[c]*Sec[c])/(10*d) + (A*Sec[c]*Sec[c + d*x]^6*Sin[d*x])/(44*d) + (Sec[c]*Sec[c + d*x]^5*(3*A*Sin[c] + 11*A*Sin[d*x]))/(132*d) + (Sec[c]*Sec[c + d*x]^4*(77*A*Sin[c] + 126*A*Sin[d*x] + 33*C*Sin[d*x]))/(924*d) + (Sec[c]*Sec[c + d*x]^3*(630*A*Sin[c] + 165*C*Sin[c] + 770*A*Sin[d*x] + 693*C*Sin[d*x]))/(4620*d) + (Sec[c]*Sec[c + d*x]^2*(770*A*Sin[c] + 693*C*Sin[c] + 1050*A*Sin[d*x] + 1430*C*Sin[d*x]))/(4620*d) + (Sec[c]*Sec[c + d*x]*(525*A*Sin[c] + 715*C*Sin[c] + 1155*A*Sin[d*x] + 1617*C*Sin[d*x]))/(2310*d)) - (5*A*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(22*d*Sqrt[1 + Cot[c]^2]) - (13*C*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) + (A*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(4*d) + (7*C*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d)","C",0
151,1,1219,192,6.6745074,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","-\frac{3 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}-\frac{21 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 (\cos (c+d x) a+a)}+\frac{\sqrt{\cos (c+d x)} \left(\frac{2 (10 \cos (c) A+5 A+5 C+16 C \cos (c)) \csc (c)}{5 d}+\frac{(28 A+51 C) \cos (d x) \sin (c)}{21 d}-\frac{2 C \cos (2 d x) \sin (2 c)}{5 d}+\frac{C \cos (3 d x) \sin (3 c)}{7 d}+\frac{2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}+\frac{(28 A+51 C) \cos (c) \sin (d x)}{21 d}-\frac{2 C \cos (2 c) \sin (2 d x)}{5 d}+\frac{C \cos (3 c) \sin (3 d x)}{7 d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (c+d x) a+a}-\frac{5 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}-\frac{15 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}","\frac{5 (7 A+9 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a d}-\frac{3 (5 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(7 A+9 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 a d}-\frac{(5 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}+\frac{5 (7 A+9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 a d}",1,"(((-3*I)/4)*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) - (((21*I)/20)*C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*((2*(5*A + 5*C + 10*A*Cos[c] + 16*C*Cos[c])*Csc[c])/(5*d) + ((28*A + 51*C)*Cos[d*x]*Sin[c])/(21*d) - (2*C*Cos[2*d*x]*Sin[2*c])/(5*d) + (C*Cos[3*d*x]*Sin[3*c])/(7*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + ((28*A + 51*C)*Cos[c]*Sin[d*x])/(21*d) - (2*C*Cos[2*c]*Sin[2*d*x])/(5*d) + (C*Cos[3*c]*Sin[3*d*x])/(7*d)))/(a + a*Cos[c + d*x]) - (5*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) - (15*C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2])","C",0
152,1,1170,159,6.6144765,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","\frac{3 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}+\frac{21 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 (\cos (c+d x) a+a)}+\frac{\sqrt{\cos (c+d x)} \left(-\frac{2 (10 \cos (c) A+5 A+5 C+16 C \cos (c)) \csc (c)}{5 d}-\frac{4 C \cos (d x) \sin (c)}{3 d}+\frac{2 C \cos (2 d x) \sin (2 c)}{5 d}-\frac{2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}-\frac{4 C \cos (c) \sin (d x)}{3 d}+\frac{2 C \cos (2 c) \sin (2 d x)}{5 d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (c+d x) a+a}+\frac{A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}+\frac{5 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}","-\frac{(3 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (5 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(5 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}-\frac{(3 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"(((3*I)/4)*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) + (((21*I)/20)*C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*((-2*(5*A + 5*C + 10*A*Cos[c] + 16*C*Cos[c])*Csc[c])/(5*d) - (4*C*Cos[d*x]*Sin[c])/(3*d) + (2*C*Cos[2*d*x]*Sin[2*c])/(5*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d - (4*C*Cos[c]*Sin[d*x])/(3*d) + (2*C*Cos[2*c]*Sin[2*d*x])/(5*d)))/(a + a*Cos[c + d*x]) + (A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) + (5*C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2])","C",0
153,1,1126,122,6.49107,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","-\frac{i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}-\frac{3 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}+\frac{\sqrt{\cos (c+d x)} \left(\frac{2 (A+C+2 C \cos (c)) \csc (c)}{d}+\frac{4 C \cos (d x) \sin (c)}{3 d}+\frac{2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}+\frac{4 C \cos (c) \sin (d x)}{3 d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (c+d x) a+a}-\frac{A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}-\frac{5 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}","\frac{(3 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{(A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(3 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"((-1/4*I)*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) - (((3*I)/4)*C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*((2*(A + C + 2*C*Cos[c])*Csc[c])/d + (4*C*Cos[d*x]*Sin[c])/(3*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (4*C*Cos[c]*Sin[d*x])/(3*d)))/(a + a*Cos[c + d*x]) - (A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) - (5*C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2])","C",1
154,1,1095,83,6.4837754,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])),x]","\frac{i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}+\frac{3 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}+\frac{\sqrt{\cos (c+d x)} \left(-\frac{2 (A+C+2 C \cos (c)) \csc (c)}{d}-\frac{2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (c+d x) a+a}-\frac{A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}+\frac{C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}","\frac{(A-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \cos (c+d x)+a)}",1,"((I/4)*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) + (((3*I)/4)*C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*((-2*(A + C + 2*C*Cos[c])*Csc[c])/d - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d))/(a + a*Cos[c + d*x]) - (A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) + (C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2])","C",1
155,1,1128,113,6.6771121,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])),x]","-\frac{3 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}-\frac{i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}+\frac{\sqrt{\cos (c+d x)} \left(\frac{(\cos (c) A+2 A+C \cos (c)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c)}{d}+\frac{4 A \sec (c+d x) \sin (d x) \sec (c)}{d}+\frac{2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (c+d x) a+a}+\frac{A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}-\frac{C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}","-\frac{(A-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(3 A+C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(3 A+C) \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}-\frac{(A+C) \sin (c+d x)}{d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)}",1,"(((-3*I)/4)*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) - ((I/4)*C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*(((2*A + A*Cos[c] + C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/d + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (4*A*Sec[c]*Sec[c + d*x]*Sin[d*x])/d))/(a + a*Cos[c + d*x]) + (A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) - (C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2])","C",1
156,1,1163,150,7.0234554,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])),x]","\frac{3 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}+\frac{i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}+\frac{\sqrt{\cos (c+d x)} \left(\frac{4 A \sec (c) \sin (d x) \sec ^2(c+d x)}{3 d}+\frac{4 \sec (c) (A \sin (c)-3 A \sin (d x)) \sec (c+d x)}{3 d}-\frac{(\cos (c) A+2 A+C \cos (c)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c)}{d}-\frac{2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (c+d x) a+a}-\frac{5 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}-\frac{C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}","\frac{(5 A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{(3 A+C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A+C) \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{(5 A+3 C) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(3 A+C) \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}",1,"(((3*I)/4)*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) + ((I/4)*C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*(-(((2*A + A*Cos[c] + C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (4*A*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (4*Sec[c]*Sec[c + d*x]*(A*Sin[c] - 3*A*Sin[d*x]))/(3*d)))/(a + a*Cos[c + d*x]) - (5*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) - (C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2])","C",1
157,1,1207,192,7.2583701,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \cos (c+d x))} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Cos[c + d*x])),x]","-\frac{21 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 (\cos (c+d x) a+a)}-\frac{3 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}+\frac{\sqrt{\cos (c+d x)} \left(\frac{4 A \sec (c) \sin (d x) \sec ^3(c+d x)}{5 d}+\frac{4 \sec (c) (3 A \sin (c)-5 A \sin (d x)) \sec ^2(c+d x)}{15 d}-\frac{4 \sec (c) (5 A \sin (c)-24 A \sin (d x)-15 C \sin (d x)) \sec (c+d x)}{15 d}+\frac{(5 \cos (c) A+16 A+10 C+5 C \cos (c)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c)}{5 d}+\frac{2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (c+d x) a+a}+\frac{5 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}+\frac{C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}","-\frac{(5 A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 (7 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{(A+C) \sin (c+d x)}{d \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)}-\frac{(5 A+3 C) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{(7 A+5 C) \sin (c+d x)}{5 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{3 (7 A+5 C) \sin (c+d x)}{5 a d \sqrt{\cos (c+d x)}}",1,"(((-21*I)/20)*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) - (((3*I)/4)*C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*(((16*A + 10*C + 5*A*Cos[c] + 5*C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/(5*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (4*A*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(5*d) + (4*Sec[c]*Sec[c + d*x]^2*(3*A*Sin[c] - 5*A*Sin[d*x]))/(15*d) - (4*Sec[c]*Sec[c + d*x]*(5*A*Sin[c] - 24*A*Sin[d*x] - 15*C*Sin[d*x]))/(15*d)))/(a + a*Cos[c + d*x]) + (5*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) + (C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2])","C",0
158,1,1248,196,6.8345882,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","\frac{2 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^2}+\frac{28 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 (\cos (c+d x) a+a)^2}+\frac{\sqrt{\cos (c+d x)} \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{2 (A+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{8 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+2 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{8 (5 \cos (c) A+5 A+10 C+18 C \cos (c)) \csc (c)}{5 d}-\frac{16 C \cos (d x) \sin (c)}{3 d}+\frac{4 C \cos (2 d x) \sin (2 c)}{5 d}-\frac{16 C \cos (c) \sin (d x)}{3 d}+\frac{4 C \cos (2 c) \sin (2 d x)}{5 d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^2}+\frac{10 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}+\frac{10 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}","-\frac{5 (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{4 (5 A+14 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{(A+3 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{4 (5 A+14 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a^2 d}-\frac{5 (A+3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((2*I)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^2 + (((28*I)/5)*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^2 + (10*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) + (10*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*((-8*(5*A + 10*C + 5*A*Cos[c] + 18*C*Cos[c])*Csc[c])/(5*d) - (16*C*Cos[d*x]*Sin[c])/(3*d) + (4*C*Cos[2*d*x]*Sin[2*c])/(5*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + 2*C*Sin[(d*x)/2]))/d - (16*C*Cos[c]*Sin[d*x])/(3*d) + (4*C*Cos[2*c]*Sin[2*d*x])/(5*d) + (2*(A + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])^2","C",0
159,1,1209,161,6.7044763,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","-\frac{i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (c+d x) a+a)^2}-\frac{7 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (c+d x) a+a)^2}+\frac{\sqrt{\cos (c+d x)} \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (A+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+3 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{4 (A+3 C+4 C \cos (c)) \csc (c)}{d}+\frac{8 C \cos (d x) \sin (c)}{3 d}+\frac{8 C \cos (c) \sin (d x)}{3 d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^2}-\frac{4 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}-\frac{20 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}","\frac{2 (A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{2 (A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((-1/2*I)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^2 - (((7*I)/2)*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^2 - (4*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) - (20*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*((4*(A + 3*C + 4*C*Cos[c])*Csc[c])/d + (8*C*Cos[d*x]*Sin[c])/(3*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + 3*C*Sin[(d*x)/2]))/d + (8*C*Cos[c]*Sin[d*x])/(3*d) - (2*(A + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])^2","C",0
160,1,814,126,6.5670435,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","\frac{2 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^2}+\frac{\sqrt{\cos (c+d x)} \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{2 (A+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{8 C \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{8 C \cot \left(\frac{c}{2}\right)}{d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^2}-\frac{2 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}+\frac{10 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}","\frac{(A-5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(A-5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d (\cos (c+d x)+1)}+\frac{4 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((2*I)*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^2 - (2*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) + (10*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*((-8*C*Cot[c/2])/d - (8*C*Sec[c/2]*Sec[c/2 + (d*x)/2]*Sin[(d*x)/2])/d + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) + (2*(A + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])^2","C",1
161,1,1176,125,6.6436641,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2),x]","\frac{i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (c+d x) a+a)^2}-\frac{i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (c+d x) a+a)^2}+\frac{\sqrt{\cos (c+d x)} \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (A+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{4 (A-C) \csc (c)}{d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^2}-\frac{4 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}-\frac{4 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}","\frac{2 (A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d (\cos (c+d x)+1)}-\frac{(A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \cos (c+d x)+a)^2}",1,"((I/2)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^2 - ((I/2)*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^2 - (4*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) - (4*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*((-4*(A - C)*Csc[c])/d - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - C*Sin[(d*x)/2]))/d - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) - (2*(A + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])^2","C",1
162,1,834,155,6.6985303,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2),x]","-\frac{2 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^2}+\frac{\sqrt{\cos (c+d x)} \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{2 (A+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{8 A \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{8 A \cot \left(\frac{c}{2}\right) \sec (c)}{d}+\frac{8 A \sec (c) \sec (c+d x) \sin (d x)}{d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^2}+\frac{10 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}-\frac{2 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}","-\frac{(5 A-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(5 A-C) \sin (c+d x)}{3 a^2 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1)}-\frac{4 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{4 A \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(A+C) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}",1,"((-2*I)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^2 + (10*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) - (2*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*((8*A*Cot[c/2]*Sec[c])/d + (8*A*Sec[c/2]*Sec[c/2 + (d*x)/2]*Sin[(d*x)/2])/d + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) + (8*A*Sec[c]*Sec[c + d*x]*Sin[d*x])/d + (2*(A + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])^2","C",1
163,1,1245,189,7.4193673,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^2} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2),x]","\frac{7 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (c+d x) a+a)^2}+\frac{i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (c+d x) a+a)^2}+\frac{\sqrt{\cos (c+d x)} \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (A+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(3 A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{2 (3 \cos (c) A+4 A+C \cos (c)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c)}{d}+\frac{8 A \sec (c) \sec ^2(c+d x) \sin (d x)}{3 d}+\frac{8 \sec (c) \sec (c+d x) (A \sin (c)-6 A \sin (d x))}{3 d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^2}-\frac{20 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}-\frac{4 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}","\frac{2 (5 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(7 A+C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(7 A+C) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x) (\cos (c+d x)+1)}+\frac{2 (5 A+C) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(7 A+C) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(A+C) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"(((7*I)/2)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^2 + ((I/2)*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^2 - (20*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) - (4*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*((-2*(4*A + 3*A*Cos[c] + C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/d - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(3*A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (8*A*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (8*Sec[c]*Sec[c + d*x]*(A*Sin[c] - 6*A*Sin[d*x]))/(3*d) - (2*(A + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])^2","C",0
164,1,1333,250,7.1313308,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(7/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","\frac{49 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}+\frac{231 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}+\frac{\sqrt{\cos (c+d x)} \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{2 (A+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(7 A \sin \left(\frac{d x}{2}\right)+12 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{8 (7 A+12 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(29 A \sin \left(\frac{d x}{2}\right)+99 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (20 \cos (c) A+29 A+99 C+132 C \cos (c)) \csc (c)}{5 d}-\frac{16 C \cos (d x) \sin (c)}{d}+\frac{8 C \cos (2 d x) \sin (2 c)}{5 d}-\frac{16 C \cos (c) \sin (d x)}{d}+\frac{8 C \cos (2 c) \sin (2 d x)}{5 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^3}+\frac{26 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}+\frac{42 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}","-\frac{(13 A+63 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{7 (7 A+33 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(13 A+63 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{7 (7 A+33 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 a^3 d}-\frac{(13 A+63 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 a^3 d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{2 (A+6 C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(((49*I)/10)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 + (((231*I)/10)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 + (26*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) + (42*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*((-4*(29*A + 99*C + 20*A*Cos[c] + 132*C*Cos[c])*Csc[c])/(5*d) - (16*C*Cos[d*x]*Sin[c])/d + (8*C*Cos[2*d*x]*Sin[2*c])/(5*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(7*A*Sin[(d*x)/2] + 12*C*Sin[(d*x)/2]))/(15*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(29*A*Sin[(d*x)/2] + 99*C*Sin[(d*x)/2]))/(5*d) - (16*C*Cos[c]*Sin[d*x])/d + (8*C*Cos[2*c]*Sin[2*d*x])/(5*d) + (8*(7*A + 12*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (2*(A + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",0
165,1,1296,209,6.9676079,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","-\frac{9 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}-\frac{119 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}+\frac{\sqrt{\cos (c+d x)} \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{2 (A+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(9 A \sin \left(\frac{d x}{2}\right)+19 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 (9 A+19 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(9 A \sin \left(\frac{d x}{2}\right)+59 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 (9 A+59 C+60 C \cos (c)) \csc (c)}{5 d}+\frac{16 C \cos (d x) \sin (c)}{3 d}+\frac{16 C \cos (c) \sin (d x)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^3}-\frac{2 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}-\frac{22 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}","\frac{(A+11 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{(9 A+119 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(9 A+119 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A+11 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a^3 d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 a d (a \cos (c+d x)+a)^2}",1,"(((-9*I)/10)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 - (((119*I)/10)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 - (2*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) - (22*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*((4*(9*A + 59*C + 60*C*Cos[c])*Csc[c])/(5*d) + (16*C*Cos[d*x]*Sin[c])/(3*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(9*A*Sin[(d*x)/2] + 19*C*Sin[(d*x)/2]))/(15*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(9*A*Sin[(d*x)/2] + 59*C*Sin[(d*x)/2]))/(5*d) + (16*C*Cos[c]*Sin[d*x])/(3*d) - (4*(9*A + 19*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (2*(A + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",0
166,1,1271,178,6.8216824,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","-\frac{i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}+\frac{49 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}+\frac{\sqrt{\cos (c+d x)} \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{2 (A+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(2 A \sin \left(\frac{d x}{2}\right)+7 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{8 (2 A+7 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-29 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (-A+29 C+20 C \cos (c)) \csc (c)}{5 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^3}-\frac{2 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}+\frac{26 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}","\frac{(A-13 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(A-49 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(A-13 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{2 (A-4 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"((-1/10*I)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 + (((49*I)/10)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 - (2*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) + (26*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*((-4*(-A + 29*C + 20*C*Cos[c])*Csc[c])/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - 29*C*Sin[(d*x)/2]))/(5*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(2*A*Sin[(d*x)/2] + 7*C*Sin[(d*x)/2]))/(15*d) + (8*(2*A + 7*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (2*(A + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",1
167,1,1259,180,6.7763174,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","\frac{i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}-\frac{9 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}+\frac{\sqrt{\cos (c+d x)} \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{2 (A+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-9 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 (A-9 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-9 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (A-9 C) \csc (c)}{5 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^3}-\frac{2 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}-\frac{2 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}","\frac{(A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(A-9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{2 (2 A-3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}",1,"((I/10)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 - (((9*I)/10)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 - (2*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) - (2*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*((-4*(A - 9*C)*Csc[c])/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - 9*C*Sin[(d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] - 9*C*Sin[(d*x)/2]))/(15*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (4*(A - 9*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (2*(A + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",1
168,1,1265,184,6.7937109,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3),x]","\frac{9 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}-\frac{i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}+\frac{\sqrt{\cos (c+d x)} \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{2 (A+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{8 \sec \left(\frac{c}{2}\right) \left(3 A \sin \left(\frac{d x}{2}\right)-2 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{8 (3 A-2 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(9 A \sin \left(\frac{d x}{2}\right)-C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (9 A-C) \csc (c)}{5 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^3}-\frac{2 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}-\frac{2 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}","\frac{(3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(9 A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(9 A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{2 (3 A-2 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}",1,"(((9*I)/10)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 - ((I/10)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 - (2*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) - (2*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*((-4*(9*A - C)*Csc[c])/(5*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(3*A*Sin[(d*x)/2] - 2*C*Sin[(d*x)/2]))/(15*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(9*A*Sin[(d*x)/2] - C*Sin[(d*x)/2]))/(5*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) - (8*(3*A - 2*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (2*(A + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",1
169,1,1301,219,7.0408736,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^3} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3),x]","-\frac{49 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}+\frac{i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}+\frac{\sqrt{\cos (c+d x)} \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{2 (A+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(11 A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 (11 A+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(29 A \sin \left(\frac{d x}{2}\right)-C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{2 (29 \cos (c) A+20 A-C \cos (c)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c)}{5 d}+\frac{16 A \sec (c) \sec (c+d x) \sin (d x)}{d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^3}+\frac{26 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}-\frac{2 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}","-\frac{(13 A-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(49 A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(49 A-C) \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{(13 A-C) \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}-\frac{2 (4 A-C) \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}",1,"(((-49*I)/10)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 + ((I/10)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 + (26*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) - (2*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*((2*(20*A + 29*A*Cos[c] - C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(29*A*Sin[(d*x)/2] - C*Sin[(d*x)/2]))/(5*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(11*A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(15*d) + (16*A*Sec[c]*Sec[c + d*x]*Sin[d*x])/d + (4*(11*A + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (2*(A + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",1
170,1,1331,242,7.7202563,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^3} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^3),x]","\frac{119 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}+\frac{9 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}+\frac{\sqrt{\cos (c+d x)} \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{2 (A+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{8 \sec \left(\frac{c}{2}\right) \left(8 A \sin \left(\frac{d x}{2}\right)+3 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{8 (8 A+3 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(59 A \sin \left(\frac{d x}{2}\right)+9 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{2 (59 \cos (c) A+60 A+9 C \cos (c)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c)}{5 d}+\frac{16 A \sec (c) \sec ^2(c+d x) \sin (d x)}{3 d}+\frac{16 \sec (c) \sec (c+d x) (A \sin (c)-9 A \sin (d x))}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^3}-\frac{22 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}-\frac{2 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}","\frac{(11 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{(119 A+9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(119 A+9 C) \sin (c+d x)}{30 d \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(11 A+C) \sin (c+d x)}{2 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(119 A+9 C) \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{(A+C) \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}-\frac{2 A \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"(((119*I)/10)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 + (((9*I)/10)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 - (22*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) - (2*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*((-2*(60*A + 59*A*Cos[c] + 9*C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/(5*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(8*A*Sin[(d*x)/2] + 3*C*Sin[(d*x)/2]))/(15*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(59*A*Sin[(d*x)/2] + 9*C*Sin[(d*x)/2]))/(5*d) + (16*A*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (16*Sec[c]*Sec[c + d*x]*(A*Sin[c] - 9*A*Sin[d*x]))/(3*d) - (8*(8*A + 3*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (2*(A + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",0
171,1,129,214,0.8301656,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} (48 A+35 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} (2 (48 A+53 C) \cos (c+d x)+144 A+28 C \cos (2 (c+d x))+12 C \cos (3 (c+d x))+133 C)\right)}{384 d}","\frac{a (48 A+35 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (48 A+35 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a (48 A+35 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(3*Sqrt[2]*(48*A + 35*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(144*A + 133*C + 2*(48*A + 53*C)*Cos[c + d*x] + 28*C*Cos[2*(c + d*x)] + 12*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(384*d)","A",1
172,1,112,169,0.4950244,"\int \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} (8 A+5 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} (24 A+10 C \cos (c+d x)+4 C \cos (2 (c+d x))+19 C)\right)}{48 d}","\frac{\sqrt{a} (8 A+5 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a (8 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(3*Sqrt[2]*(8*A + 5*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(24*A + 19*C + 10*C*Cos[c + d*x] + 4*C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
173,1,98,124,0.2771054,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(\sqrt{2} (8 A+3 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 C \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{3}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}\right)}{8 d}","\frac{\sqrt{a} (8 A+3 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}+\frac{a C \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(Sqrt[2]*(8*A + 3*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*C*Sqrt[Cos[c + d*x]]*(2*Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(8*d)","A",1
174,1,100,117,0.2448214,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) (2 A+C \cos (c+d x))+\sqrt{2} C \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}\right)}{2 d \sqrt{\cos (c+d x)}}","-\frac{a (2 A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(Sqrt[2]*C*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*(2*A + C*Cos[c + d*x])*Sin[(c + d*x)/2]))/(2*d*Sqrt[Cos[c + d*x]])","A",1
175,1,90,116,0.2292329,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(2 A \sin \left(\frac{3}{2} (c+d x)\right)+3 \sqrt{2} C \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{3}{2}}(c+d x)\right)}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 \sqrt{a} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(3*Sqrt[2]*C*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(3/2) + 2*A*Sin[(3*(c + d*x))/2]))/(3*d*Cos[c + d*x]^(3/2))","A",1
176,1,73,123,0.2734982,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((8 A+15 C) \cos (2 (c+d x))+8 A \cos (c+d x)+14 A+15 C)}{15 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 a (8 A+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(14*A + 15*C + 8*A*Cos[c + d*x] + (8*A + 15*C)*Cos[2*(c + d*x)])*Tan[(c + d*x)/2])/(15*d*Cos[c + d*x]^(5/2))","A",1
177,1,101,168,0.446526,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (3 (36 A+35 C) \cos (c+d x)+(24 A+35 C) \cos (2 (c+d x))+24 A \cos (3 (c+d x))+54 A+35 C \cos (3 (c+d x))+35 C)}{105 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 a (24 A+35 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a (24 A+35 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(54*A + 35*C + 3*(36*A + 35*C)*Cos[c + d*x] + (24*A + 35*C)*Cos[2*(c + d*x)] + 24*A*Cos[3*(c + d*x)] + 35*C*Cos[3*(c + d*x)])*Tan[(c + d*x)/2])/(105*d*Cos[c + d*x]^(7/2))","A",1
178,1,124,213,0.6699171,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (2 (88 A+63 C) \cos (c+d x)+11 (16 A+21 C) \cos (2 (c+d x))+32 A \cos (3 (c+d x))+32 A \cos (4 (c+d x))+214 A+42 C \cos (3 (c+d x))+42 C \cos (4 (c+d x))+189 C)}{315 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{8 a (16 A+21 C) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a (16 A+21 C) \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a (16 A+21 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(214*A + 189*C + 2*(88*A + 63*C)*Cos[c + d*x] + 11*(16*A + 21*C)*Cos[2*(c + d*x)] + 32*A*Cos[3*(c + d*x)] + 42*C*Cos[3*(c + d*x)] + 32*A*Cos[4*(c + d*x)] + 42*C*Cos[4*(c + d*x)])*Tan[(c + d*x)/2])/(315*d*Cos[c + d*x]^(9/2))","A",1
179,1,147,265,1.5523451,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(15 \sqrt{2} (176 A+133 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} (2 (880 A+1007 C) \cos (c+d x)+4 (80 A+181 C) \cos (2 (c+d x))+2960 A+228 C \cos (3 (c+d x))+48 C \cos (4 (c+d x))+2671 C)\right)}{3840 d}","\frac{a^{3/2} (176 A+133 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^2 (80 A+67 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{240 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (176 A+133 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (176 A+133 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}+\frac{3 a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{40 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(15*Sqrt[2]*(176*A + 133*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(2960*A + 2671*C + 2*(880*A + 1007*C)*Cos[c + d*x] + 4*(80*A + 181*C)*Cos[2*(c + d*x)] + 228*C*Cos[3*(c + d*x)] + 48*C*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]))/(3840*d)","A",1
180,1,128,218,0.8732623,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(\sqrt{2} (112 A+75 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} ((32 A+62 C) \cos (c+d x)+112 A+20 C \cos (2 (c+d x))+4 C \cos (3 (c+d x))+95 C)\right)}{128 d}","\frac{a^{3/2} (112 A+75 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (16 A+13 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{32 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (112 A+75 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{8 d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(Sqrt[2]*(112*A + 75*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(112*A + 95*C + (32*A + 62*C)*Cos[c + d*x] + 20*C*Cos[2*(c + d*x)] + 4*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(128*d)","A",1
181,1,113,171,0.5721531,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} (24 A+11 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} (24 A+22 C \cos (c+d x)+4 C \cos (2 (c+d x))+37 C)\right)}{48 d}","\frac{a^{3/2} (24 A+11 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 (24 A+19 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}{3 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(3*Sqrt[2]*(24*A + 11*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(24*A + 37*C + 22*C*Cos[c + d*x] + 4*C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
182,1,119,175,0.5531603,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(\sqrt{2} (8 A+7 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+2 \sin \left(\frac{1}{2} (c+d x)\right) (8 A+7 C \cos (c+d x)+C \cos (2 (c+d x))+C)\right)}{8 d \sqrt{\cos (c+d x)}}","\frac{a^{3/2} (8 A+7 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^2 (8 A-5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}-\frac{a (4 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{d \sqrt{\cos (c+d x)}}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(Sqrt[2]*(8*A + 7*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*(8*A + C + 7*C*Cos[c + d*x] + C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(8*d*Sqrt[Cos[c + d*x]])","A",1
183,1,116,161,0.5187358,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (20 A \cos (c+d x)+4 A+3 C \cos (2 (c+d x))+3 C)+9 \sqrt{2} C \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{3}{2}}(c+d x)\right)}{6 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{3 a^{3/2} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a^2 (8 A-3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{d \sqrt{\cos (c+d x)}}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(9*Sqrt[2]*C*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(3/2) + (4*A + 3*C + 20*A*Cos[c + d*x] + 3*C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(6*d*Cos[c + d*x]^(3/2))","A",1
184,1,121,163,0.7066404,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) ((6 A+5 C) \cos (2 (c+d x))+6 A \cos (c+d x)+8 A+5 C)+5 \sqrt{2} C \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{5}{2}}(c+d x)\right)}{5 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 a^{3/2} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a^2 (4 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{5 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(5*Sqrt[2]*C*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(5/2) + (8*A + 5*C + 6*A*Cos[c + d*x] + (6*A + 5*C)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(5*d*Cos[c + d*x]^(5/2))","A",1
185,1,102,172,0.5412402,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((468 A+525 C) \cos (c+d x)+2 (52 A+35 C) \cos (2 (c+d x))+104 A \cos (3 (c+d x))+164 A+175 C \cos (3 (c+d x))+70 C)}{210 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 a^2 (4 A+5 C) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (104 A+175 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{6 a A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(164*A + 70*C + (468*A + 525*C)*Cos[c + d*x] + 2*(52*A + 35*C)*Cos[2*(c + d*x)] + 104*A*Cos[3*(c + d*x)] + 175*C*Cos[3*(c + d*x)])*Tan[(c + d*x)/2])/(210*d*Cos[c + d*x]^(7/2))","A",1
186,1,123,219,0.738313,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((748 A+567 C) \cos (c+d x)+(748 A+882 C) \cos (2 (c+d x))+136 A \cos (3 (c+d x))+136 A \cos (4 (c+d x))+752 A+189 C \cos (3 (c+d x))+189 C \cos (4 (c+d x))+693 C)}{630 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{2 a^2 (136 A+189 C) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (52 A+63 C) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (136 A+189 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(752*A + 693*C + (748*A + 567*C)*Cos[c + d*x] + (748*A + 882*C)*Cos[2*(c + d*x)] + 136*A*Cos[3*(c + d*x)] + 189*C*Cos[3*(c + d*x)] + 136*A*Cos[4*(c + d*x)] + 189*C*Cos[4*(c + d*x)])*Tan[(c + d*x)/2])/(630*d*Cos[c + d*x]^(9/2))","A",1
187,1,146,266,0.8162114,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((4228 A+4147 C) \cos (c+d x)+2 (728 A+737 C) \cos (2 (c+d x))+1456 A \cos (3 (c+d x))+224 A \cos (4 (c+d x))+224 A \cos (5 (c+d x))+1652 A+1859 C \cos (3 (c+d x))+286 C \cos (4 (c+d x))+286 C \cos (5 (c+d x))+1188 C)}{2310 d \cos ^{\frac{11}{2}}(c+d x)}","\frac{8 a^2 (112 A+143 C) \sin (c+d x)}{1155 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (112 A+143 C) \sin (c+d x)}{385 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (28 A+33 C) \sin (c+d x)}{231 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (112 A+143 C) \sin (c+d x)}{1155 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{33 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac{11}{2}}(c+d x)}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(1652*A + 1188*C + (4228*A + 4147*C)*Cos[c + d*x] + 2*(728*A + 737*C)*Cos[2*(c + d*x)] + 1456*A*Cos[3*(c + d*x)] + 1859*C*Cos[3*(c + d*x)] + 224*A*Cos[4*(c + d*x)] + 286*C*Cos[4*(c + d*x)] + 224*A*Cos[5*(c + d*x)] + 286*C*Cos[5*(c + d*x)])*Tan[(c + d*x)/2])/(2310*d*Cos[c + d*x]^(11/2))","A",1
188,1,170,312,2.4684558,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} (1304 A+1015 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} ((2896 A+3234 C) \cos (c+d x)+4 (184 A+315 C) \cos (2 (c+d x))+96 A \cos (3 (c+d x))+4648 A+428 C \cos (3 (c+d x))+112 C \cos (4 (c+d x))+16 C \cos (5 (c+d x))+4193 C)\right)}{3072 d}","\frac{a^{5/2} (1304 A+1015 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{512 d}+\frac{a^3 (136 A+109 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (1304 A+1015 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{768 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (1304 A+1015 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{512 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (24 A+23 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{96 d}+\frac{a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{12 d}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(3*Sqrt[2]*(1304*A + 1015*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(4648*A + 4193*C + (2896*A + 3234*C)*Cos[c + d*x] + 4*(184*A + 315*C)*Cos[2*(c + d*x)] + 96*A*Cos[3*(c + d*x)] + 428*C*Cos[3*(c + d*x)] + 112*C*Cos[4*(c + d*x)] + 16*C*Cos[5*(c + d*x)])*Sin[(c + d*x)/2]))/(3072*d)","A",1
189,1,148,265,1.6042739,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(15 \sqrt{2} (400 A+283 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} ((2720 A+3874 C) \cos (c+d x)+4 (80 A+331 C) \cos (2 (c+d x))+6320 A+348 C \cos (3 (c+d x))+48 C \cos (4 (c+d x))+5521 C)\right)}{3840 d}","\frac{a^{5/2} (400 A+283 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^3 (1040 A+787 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{960 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (400 A+283 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (80 A+79 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{240 d}+\frac{a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{8 d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(15*Sqrt[2]*(400*A + 283*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(6320*A + 5521*C + (2720*A + 3874*C)*Cos[c + d*x] + 4*(80*A + 331*C)*Cos[2*(c + d*x)] + 348*C*Cos[3*(c + d*x)] + 48*C*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]))/(3840*d)","A",1
190,1,131,218,0.9600471,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} (304 A+163 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} ((96 A+362 C) \cos (c+d x)+528 A+92 C \cos (2 (c+d x))+12 C \cos (3 (c+d x))+581 C)\right)}{384 d}","\frac{a^{5/2} (304 A+163 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^3 (432 A+299 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (16 A+17 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{32 d}+\frac{5 a C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}{24 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}{4 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(3*Sqrt[2]*(304*A + 163*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(528*A + 581*C + (96*A + 362*C)*Cos[c + d*x] + 92*C*Cos[2*(c + d*x)] + 12*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(384*d)","A",1
191,1,142,222,0.9195666,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(15 \sqrt{2} (8 A+5 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+2 \sin \left(\frac{1}{2} (c+d x)\right) (3 (8 A+27 C) \cos (c+d x)+48 A+17 C \cos (2 (c+d x))+2 C \cos (3 (c+d x))+17 C)\right)}{48 d \sqrt{\cos (c+d x)}}","\frac{5 a^{5/2} (8 A+5 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}-\frac{a^3 (24 A-49 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{24 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A-3 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{4 d}-\frac{a (6 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}{3 d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{d \sqrt{\cos (c+d x)}}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(15*Sqrt[2]*(8*A + 5*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*(48*A + 17*C + 3*(8*A + 27*C)*Cos[c + d*x] + 17*C*Cos[2*(c + d*x)] + 2*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d*Sqrt[Cos[c + d*x]])","A",1
192,1,141,218,0.7596589,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(6 \sqrt{2} (8 A+19 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{3}{2}}(c+d x)+2 \sin \left(\frac{1}{2} (c+d x)\right) ((128 A+9 C) \cos (c+d x)+16 A+33 C \cos (2 (c+d x))+3 C \cos (3 (c+d x))+33 C)\right)}{48 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{a^{5/2} (8 A+19 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^3 (56 A-27 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{12 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{10 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \sqrt{\cos (c+d x)}}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(6*Sqrt[2]*(8*A + 19*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(3/2) + 2*(16*A + 33*C + (128*A + 9*C)*Cos[c + d*x] + 33*C*Cos[2*(c + d*x)] + 3*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d*Cos[c + d*x]^(3/2))","A",1
193,1,141,210,0.8963812,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) ((112 A+45 C) \cos (c+d x)+4 (43 A+15 C) \cos (2 (c+d x))+196 A+15 C \cos (3 (c+d x))+60 C)+300 \sqrt{2} C \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{5}{2}}(c+d x)\right)}{120 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{5 a^{5/2} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a^3 (64 A+15 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (8 A+5 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(300*Sqrt[2]*C*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(5/2) + 2*(196*A + 60*C + (112*A + 45*C)*Cos[c + d*x] + 4*(43*A + 15*C)*Cos[2*(c + d*x)] + 15*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(120*d*Cos[c + d*x]^(5/2))","A",1
194,1,151,210,1.3298264,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(4 \sin \left(\frac{1}{2} (c+d x)\right) ((93 A+84 C) \cos (c+d x)+(23 A+7 C) \cos (2 (c+d x))+23 A \cos (3 (c+d x))+29 A+28 C \cos (3 (c+d x))+7 C)+84 \sqrt{2} C \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{7}{2}}(c+d x)\right)}{84 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 a^{5/2} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a^3 (32 A+49 C) \sin (c+d x)}{21 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (8 A+7 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(84*Sqrt[2]*C*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(7/2) + 4*(29*A + 7*C + (93*A + 84*C)*Cos[c + d*x] + (23*A + 7*C)*Cos[2*(c + d*x)] + 23*A*Cos[3*(c + d*x)] + 28*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(84*d*Cos[c + d*x]^(7/2))","A",1
195,1,127,219,0.9177132,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (4 (698 A+441 C) \cos (c+d x)+4 (803 A+966 C) \cos (2 (c+d x))+584 A \cos (3 (c+d x))+584 A \cos (4 (c+d x))+2908 A+588 C \cos (3 (c+d x))+903 C \cos (4 (c+d x))+2961 C)}{1260 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{2 a^3 (8 A+11 C) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (584 A+903 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (64 A+63 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{10 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(2908*A + 2961*C + 4*(698*A + 441*C)*Cos[c + d*x] + 4*(803*A + 966*C)*Cos[2*(c + d*x)] + 584*A*Cos[3*(c + d*x)] + 588*C*Cos[3*(c + d*x)] + 584*A*Cos[4*(c + d*x)] + 903*C*Cos[4*(c + d*x)])*Tan[(c + d*x)/2])/(1260*d*Cos[c + d*x]^(9/2))","A",1
196,1,149,266,0.9865219,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (2 (5014 A+4983 C) \cos (c+d x)+52 (71 A+66 C) \cos (2 (c+d x))+3692 A \cos (3 (c+d x))+568 A \cos (4 (c+d x))+568 A \cos (5 (c+d x))+3628 A+4587 C \cos (3 (c+d x))+759 C \cos (4 (c+d x))+759 C \cos (5 (c+d x))+2673 C)}{2772 d \cos ^{\frac{11}{2}}(c+d x)}","\frac{2 a^3 (568 A+759 C) \sin (c+d x)}{693 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (232 A+297 C) \sin (c+d x)}{693 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a^3 (568 A+759 C) \sin (c+d x)}{693 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (32 A+33 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{10 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac{11}{2}}(c+d x)}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(3628*A + 2673*C + 2*(5014*A + 4983*C)*Cos[c + d*x] + 52*(71*A + 66*C)*Cos[2*(c + d*x)] + 3692*A*Cos[3*(c + d*x)] + 4587*C*Cos[3*(c + d*x)] + 568*A*Cos[4*(c + d*x)] + 759*C*Cos[4*(c + d*x)] + 568*A*Cos[5*(c + d*x)] + 759*C*Cos[5*(c + d*x)])*Tan[(c + d*x)/2])/(2772*d*Cos[c + d*x]^(11/2))","A",1
197,1,171,313,0.9052344,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{15}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(15/2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (1120 (347 A+286 C) \cos (c+d x)+14 (30334 A+32747 C) \cos (2 (c+d x))+125520 A \cos (3 (c+d x))+125520 A \cos (4 (c+d x))+16736 A \cos (5 (c+d x))+16736 A \cos (6 (c+d x))+343612 A+141570 C \cos (3 (c+d x))+156585 C \cos (4 (c+d x))+20878 C \cos (5 (c+d x))+20878 C \cos (6 (c+d x))+322751 C)}{180180 d \cos ^{\frac{13}{2}}(c+d x)}","\frac{8 a^3 (8368 A+10439 C) \sin (c+d x)}{45045 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (8368 A+10439 C) \sin (c+d x)}{15015 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (2224 A+2717 C) \sin (c+d x)}{9009 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a^3 (8368 A+10439 C) \sin (c+d x)}{45045 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (136 A+143 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{1287 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{10 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac{13}{2}}(c+d x)}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(343612*A + 322751*C + 1120*(347*A + 286*C)*Cos[c + d*x] + 14*(30334*A + 32747*C)*Cos[2*(c + d*x)] + 125520*A*Cos[3*(c + d*x)] + 141570*C*Cos[3*(c + d*x)] + 125520*A*Cos[4*(c + d*x)] + 156585*C*Cos[4*(c + d*x)] + 16736*A*Cos[5*(c + d*x)] + 20878*C*Cos[5*(c + d*x)] + 16736*A*Cos[6*(c + d*x)] + 20878*C*Cos[6*(c + d*x)])*Tan[(c + d*x)/2])/(180180*d*Cos[c + d*x]^(13/2))","A",1
198,1,349,226,2.1796628,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(4 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} (24 A-2 C \cos (c+d x)+4 C \cos (2 (c+d x))+25 C)-\frac{3 i \sqrt{2} e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \left(16 \sqrt{2} (A+C) \log \left(1+e^{i (c+d x)}\right)-(8 A+9 C) \sinh ^{-1}\left(e^{i (c+d x)}\right)+8 A \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)-16 \sqrt{2} A \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)-8 i A d x+9 C \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)-16 \sqrt{2} C \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)-9 i C d x\right)}{\sqrt{1+e^{2 i (c+d x)}}}\right)}{48 d \sqrt{a (\cos (c+d x)+1)}}","-\frac{(8 A+9 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{(8 A+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*(((-3*I)*Sqrt[2]*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*((-8*I)*A*d*x - (9*I)*C*d*x - (8*A + 9*C)*ArcSinh[E^(I*(c + d*x))] + 16*Sqrt[2]*(A + C)*Log[1 + E^(I*(c + d*x))] + 8*A*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] + 9*C*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] - 16*Sqrt[2]*A*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]] - 16*Sqrt[2]*C*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]]))/Sqrt[1 + E^((2*I)*(c + d*x))] + 4*Sqrt[Cos[c + d*x]]*(24*A + 25*C - 2*C*Cos[c + d*x] + 4*C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d*Sqrt[a*(1 + Cos[c + d*x])])","C",1
199,1,344,183,1.6189148,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(\frac{4 C \left(\sin \left(\frac{3}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}}{d}+\frac{\sqrt{2} e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \left(8 i \sqrt{2} (A+C) \log \left(1+e^{i (c+d x)}\right)-i (8 A+7 C) \sinh ^{-1}\left(e^{i (c+d x)}\right)+8 i A \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)-8 i \sqrt{2} A \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)+8 A d x+7 i C \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)-8 i \sqrt{2} C \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)+7 C d x\right)}{d \sqrt{1+e^{2 i (c+d x)}}}\right)}{8 \sqrt{a (\cos (c+d x)+1)}}","\frac{(8 A+7 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}-\frac{C \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*((Sqrt[2]*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(8*A*d*x + 7*C*d*x - I*(8*A + 7*C)*ArcSinh[E^(I*(c + d*x))] + (8*I)*Sqrt[2]*(A + C)*Log[1 + E^(I*(c + d*x))] + (8*I)*A*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] + (7*I)*C*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] - (8*I)*Sqrt[2]*A*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]] - (8*I)*Sqrt[2]*C*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]]))/(d*Sqrt[1 + E^((2*I)*(c + d*x))]) + (4*C*Sqrt[Cos[c + d*x]]*(-2*Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2]))/d))/(8*Sqrt[a*(1 + Cos[c + d*x])])","C",1
200,1,104,133,0.2362546,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]),x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(2 (A+C) \tan ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)-\sqrt{2} C \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 C \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)}\right)}{d \sqrt{a (\cos (c+d x)+1)}}","\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*(-(Sqrt[2]*C*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]) + 2*(A + C)*ArcTan[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]] + 2*C*Sqrt[Cos[c + d*x]]*Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
201,1,235,135,3.4846474,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]),x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(\frac{(A+C) \csc ^3\left(\frac{1}{2} (c+d x)\right) \left(5 \cos ^2(c+d x) (\cos (c+d x)+2) \left(-\cos (c+d x)+\cos (c+d x) \sqrt{2-2 \sec (c+d x)} \tanh ^{-1}\left(\sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right) (-\sec (c+d x))}\right)+1\right)-\sin ^4\left(\frac{1}{2} (c+d x)\right) \sin ^2(c+d x) \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};-\sec (c+d x) \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}{2 \cos ^{\frac{5}{2}}(c+d x)}+5 C \left(\sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)-\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)\right)}{5 d \sqrt{a (\cos (c+d x)+1)}}","-\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(2*Cos[(c + d*x)/2]*(5*C*(Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] - (2*Sin[(c + d*x)/2])/Sqrt[Cos[c + d*x]]) + ((A + C)*Csc[(c + d*x)/2]^3*(5*Cos[c + d*x]^2*(2 + Cos[c + d*x])*(1 - Cos[c + d*x] + ArcTanh[Sqrt[-(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]]*Cos[c + d*x]*Sqrt[2 - 2*Sec[c + d*x]]) - Hypergeometric2F1[2, 5/2, 7/2, -(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^4*Sin[c + d*x]^2))/(2*Cos[c + d*x]^(5/2))))/(5*d*Sqrt[a*(1 + Cos[c + d*x])])","C",0
202,1,556,136,6.7490913,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]),x]","\frac{2 (A+C) \cot \left(\frac{c}{2}+\frac{d x}{2}\right) \csc ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \left(-12 \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right) \cos ^4\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(2,2,\frac{7}{2};1,\frac{9}{2};-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}\right)-12 \left(3 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-7 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+4\right) \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right) \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}\right)+7 \sqrt{-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}} \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3 \left(8 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-20 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+15\right) \left(\left(3-7 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sqrt{-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}-3 \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right) \tanh ^{-1}\left(\sqrt{-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}\right)\right)\right)}{63 d \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{7/2} \sqrt{a (\cos (c+d x)+1)}}-\frac{8 C \sin ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{3/2} \sqrt{a (\cos (c+d x)+1)}}","\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{2 A \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(-8*C*Cos[c/2 + (d*x)/2]*Sin[c/2 + (d*x)/2]^3)/(3*d*Sqrt[a*(1 + Cos[c + d*x])]*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) + (2*(A + C)*Cot[c/2 + (d*x)/2]*Csc[c/2 + (d*x)/2]^4*(-12*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 7/2}, {1, 9/2}, -(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]*Sin[c/2 + (d*x)/2]^8 - 12*Hypergeometric2F1[2, 7/2, 9/2, -(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]*Sin[c/2 + (d*x)/2]^8*(4 - 7*Sin[c/2 + (d*x)/2]^2 + 3*Sin[c/2 + (d*x)/2]^4) + 7*Sqrt[-(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]*(1 - 2*Sin[c/2 + (d*x)/2]^2)^3*(15 - 20*Sin[c/2 + (d*x)/2]^2 + 8*Sin[c/2 + (d*x)/2]^4)*((3 - 7*Sin[c/2 + (d*x)/2]^2)*Sqrt[-(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))] - 3*ArcTanh[Sqrt[-(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]]*(1 - 2*Sin[c/2 + (d*x)/2]^2))))/(63*d*Sqrt[a*(1 + Cos[c + d*x])]*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2))","C",0
203,1,1765,181,7.7052999,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]),x]","-\frac{2 (A+C) \cot \left(\frac{c}{2}+\frac{d x}{2}\right) \left(440 \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right) \sin ^{16}\left(\frac{c}{2}+\frac{d x}{2}\right)+69120 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^{16}\left(\frac{c}{2}+\frac{d x}{2}\right)-42048 \sin ^{16}\left(\frac{c}{2}+\frac{d x}{2}\right)-1500 \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right) \sin ^{14}\left(\frac{c}{2}+\frac{d x}{2}\right)-414720 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^{14}\left(\frac{c}{2}+\frac{d x}{2}\right)+226656 \sin ^{14}\left(\frac{c}{2}+\frac{d x}{2}\right)+1770 \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right) \sin ^{12}\left(\frac{c}{2}+\frac{d x}{2}\right)+1080000 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^{12}\left(\frac{c}{2}+\frac{d x}{2}\right)-518760 \sin ^{12}\left(\frac{c}{2}+\frac{d x}{2}\right)-710 \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right) \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)-40 \cos ^6\left(\frac{1}{2} (c+d x)\right) \, _4F_3\left(2,2,2,\frac{9}{2};1,1,\frac{11}{2};\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right) \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)+60 \cos ^4\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(2,2,\frac{9}{2};1,\frac{11}{2};\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right) \left(4 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-5\right) \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)-1598400 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)+655812 \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)+1458000 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)-486630 \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)-833760 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)+210105 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)+291060 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-48825 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-56700 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+4725 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+4725 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \csc ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{675 d \sqrt{a (\cos (c+d x)+1)} \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{7/2} \left(2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1\right)}+\frac{C \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{3 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{5/2}}+4 \left(\frac{2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}+\frac{\sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{3/2}}\right)\right)}{15 d \sqrt{a (\cos (c+d x)+1)}}-\frac{C \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{a (\cos (c+d x)+1)} \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{5/2}}","\frac{2 (13 A+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 A \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"-((C*Cos[c/2 + (d*x)/2]*Sin[c/2 + (d*x)/2])/(d*Sqrt[a*(1 + Cos[c + d*x])]*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2))) - (2*(A + C)*Cot[c/2 + (d*x)/2]*Csc[c/2 + (d*x)/2]^6*(4725*Sin[c/2 + (d*x)/2]^2 - 48825*Sin[c/2 + (d*x)/2]^4 + 210105*Sin[c/2 + (d*x)/2]^6 - 486630*Sin[c/2 + (d*x)/2]^8 + 655812*Sin[c/2 + (d*x)/2]^10 - 710*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^10 - 40*Cos[(c + d*x)/2]^6*HypergeometricPFQ[{2, 2, 2, 9/2}, {1, 1, 11/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^10 - 518760*Sin[c/2 + (d*x)/2]^12 + 1770*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12 + 226656*Sin[c/2 + (d*x)/2]^14 - 1500*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14 - 42048*Sin[c/2 + (d*x)/2]^16 + 440*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^16 + 4725*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 56700*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^2*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 291060*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^4*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 833760*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^6*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 1458000*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^8*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 1598400*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^10*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 1080000*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^12*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 414720*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^14*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 69120*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^16*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 60*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 9/2}, {1, 11/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^10*(-5 + 4*Sin[c/2 + (d*x)/2]^2)))/(675*d*Sqrt[a*(1 + Cos[c + d*x])]*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2)*(-1 + 2*Sin[c/2 + (d*x)/2]^2)) + (C*Cos[c/2 + (d*x)/2]*((3*Sin[c/2 + (d*x)/2])/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2) + 4*(Sin[c/2 + (d*x)/2]/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2) + (2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])))/(15*d*Sqrt[a*(1 + Cos[c + d*x])])","C",0
204,1,2490,224,9.8849622,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{9}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*Sqrt[a + a*Cos[c + d*x]]),x]","\text{Result too large to show}","\frac{2 (31 A+35 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{2 (43 A+35 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 A \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"(-2*C*Cos[c/2 + (d*x)/2]*Sin[c/2 + (d*x)/2])/(3*d*Sqrt[a*(1 + Cos[c + d*x])]*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2)) + (2*(A + C)*Cot[c/2 + (d*x)/2]*Csc[c/2 + (d*x)/2]^8*(363825*Sin[c/2 + (d*x)/2]^2 - 4729725*Sin[c/2 + (d*x)/2]^4 + 26785605*Sin[c/2 + (d*x)/2]^6 - 86790165*Sin[c/2 + (d*x)/2]^8 + 177677808*Sin[c/2 + (d*x)/2]^10 - 239283044*Sin[c/2 + (d*x)/2]^12 + 52080*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12 + 560*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12 + 213120160*Sin[c/2 + (d*x)/2]^14 - 168280*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14 - 2240*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14 - 121497024*Sin[c/2 + (d*x)/2]^16 + 212520*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^16 + 3360*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^16 + 40125184*Sin[c/2 + (d*x)/2]^18 - 124320*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^18 - 2240*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^18 - 5840384*Sin[c/2 + (d*x)/2]^20 + 28000*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^20 + 560*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^20 + 363825*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 5336100*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^2*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 34636140*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^4*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 131060160*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^6*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 320535600*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^8*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 530671680*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^10*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 604296000*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^12*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 468948480*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^14*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 237726720*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^16*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 70963200*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^18*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 9461760*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^20*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 1120*Cos[(c + d*x)/2]^6*HypergeometricPFQ[{2, 2, 2, 11/2}, {1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12*(-6 + 5*Sin[c/2 + (d*x)/2]^2) + 280*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 11/2}, {1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12*(103 - 164*Sin[c/2 + (d*x)/2]^2 + 70*Sin[c/2 + (d*x)/2]^4)))/(40425*d*Sqrt[a*(1 + Cos[c + d*x])]*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(9/2)*(-1 + 2*Sin[c/2 + (d*x)/2]^2)) + (2*C*Cos[c/2 + (d*x)/2]*((5*Sin[c/2 + (d*x)/2])/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2) + 2*((3*Sin[c/2 + (d*x)/2])/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2) + 4*(Sin[c/2 + (d*x)/2]/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2) + (2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]))))/(105*d*Sqrt[a*(1 + Cos[c + d*x])])","C",0
205,1,370,245,2.5057854,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{\cos ^3\left(\frac{1}{2} (c+d x)\right) \left(-2 \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) (2 A+3 C \cos (c+d x)-C \cos (2 (c+d x))+6 C)+\frac{\sqrt{2} e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \left(2 i \sqrt{2} (5 A+13 C) \log \left(1+e^{i (c+d x)}\right)-i (8 A+19 C) \sinh ^{-1}\left(e^{i (c+d x)}\right)+8 i A \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)-10 i \sqrt{2} A \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)+8 A d x+19 i C \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)-26 i \sqrt{2} C \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)+19 C d x\right)}{\sqrt{1+e^{2 i (c+d x)}}}\right)}{4 d (a (\cos (c+d x)+1))^{3/2}}","\frac{(8 A+19 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{(5 A+13 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(A+2 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(2 A+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 a d \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]^3*((Sqrt[2]*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(8*A*d*x + 19*C*d*x - I*(8*A + 19*C)*ArcSinh[E^(I*(c + d*x))] + (2*I)*Sqrt[2]*(5*A + 13*C)*Log[1 + E^(I*(c + d*x))] + (8*I)*A*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] + (19*I)*C*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] - (10*I)*Sqrt[2]*A*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]] - (26*I)*Sqrt[2]*C*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]]))/Sqrt[1 + E^((2*I)*(c + d*x))] - 2*Sqrt[Cos[c + d*x]]*(2*A + 6*C + 3*C*Cos[c + d*x] - C*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]*Tan[(c + d*x)/2]))/(4*d*(a*(1 + Cos[c + d*x]))^(3/2))","C",1
206,1,238,188,2.0960535,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{\cos ^3\left(\frac{1}{2} (c+d x)\right) \left(\frac{2 \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) (A+2 C \cos (c+d x)+3 C)}{d}+\frac{i \sqrt{2} e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \left(\sqrt{2} (A+9 C) \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+6 C \sinh ^{-1}\left(e^{i (c+d x)}\right)-6 C \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{d \sqrt{1+e^{2 i (c+d x)}}}\right)}{2 (a (\cos (c+d x)+1))^{3/2}}","\frac{(A+9 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{3 C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(A+3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a d \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]^3*((I*Sqrt[2]*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(6*C*ArcSinh[E^(I*(c + d*x))] + Sqrt[2]*(A + 9*C)*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - 6*C*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))/(d*Sqrt[1 + E^((2*I)*(c + d*x))]) + (2*Sqrt[Cos[c + d*x]]*(A + 3*C + 2*C*Cos[c + d*x])*Sec[(c + d*x)/2]*Tan[(c + d*x)/2])/d))/(2*(a*(1 + Cos[c + d*x]))^(3/2))","C",1
207,1,227,145,1.9227747,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)),x]","\frac{\cos ^3\left(\frac{1}{2} (c+d x)\right) \left(-\frac{(A+C) \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right)}{d}-\frac{i e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \left(-\sqrt{2} (3 A-5 C) \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+4 C \sinh ^{-1}\left(e^{i (c+d x)}\right)-4 C \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{\sqrt{2} d \sqrt{1+e^{2 i (c+d x)}}}\right)}{(a (\cos (c+d x)+1))^{3/2}}","\frac{(3 A-5 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]^3*(((-I)*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(4*C*ArcSinh[E^(I*(c + d*x))] - Sqrt[2]*(3*A - 5*C)*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - 4*C*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))/(Sqrt[2]*d*Sqrt[1 + E^((2*I)*(c + d*x))]) - ((A + C)*Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]*Tan[(c + d*x)/2])/d))/(a*(1 + Cos[c + d*x]))^(3/2)","C",1
208,1,434,152,3.9500364,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)),x]","\frac{\cos ^3\left(\frac{1}{2} (c+d x)\right) \left(\frac{(A-7 C) \csc ^3\left(\frac{1}{2} (c+d x)\right) \left(5 (4 \cos (c+d x)+\cos (2 (c+d x))+1) \left(-\cos (c+d x)+\cos (c+d x) \sqrt{2-2 \sec (c+d x)} \tanh ^{-1}\left(\sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right) (-\sec (c+d x))}\right)+1\right)-2 \sin ^4\left(\frac{1}{2} (c+d x)\right) \sin (c+d x) \tan (c+d x) \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};-\sec (c+d x) \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}{2 \cos ^{\frac{3}{2}}(c+d x)}+\frac{5 (A+C) \left(2 \sin \left(\frac{1}{2} (c+d x)\right)-1\right)}{\sqrt{\cos (c+d x)} \left(\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)\right)^2}-\frac{5 (A+C) \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+1\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)-1\right) \sqrt{\cos (c+d x)}}-\frac{20 (A+C) \sqrt{\cos (c+d x)}}{\sin \left(\frac{1}{2} (c+d x)\right)-1}-\frac{20 (A+C) \sqrt{\cos (c+d x)}}{\sin \left(\frac{1}{2} (c+d x)\right)+1}+30 (A+C) \tan ^{-1}\left(\frac{1-2 \sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)-30 (A+C) \tan ^{-1}\left(\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)+1}{\sqrt{\cos (c+d x)}}\right)+\frac{80 C \sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)}{10 d (a (\cos (c+d x)+1))^{3/2}}","-\frac{(7 A-C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A+C) \sin (c+d x)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]^3*(30*(A + C)*ArcTan[(1 - 2*Sin[(c + d*x)/2])/Sqrt[Cos[c + d*x]]] - 30*(A + C)*ArcTan[(1 + 2*Sin[(c + d*x)/2])/Sqrt[Cos[c + d*x]]] - (20*(A + C)*Sqrt[Cos[c + d*x]])/(-1 + Sin[(c + d*x)/2]) + (80*C*Sin[(c + d*x)/2])/Sqrt[Cos[c + d*x]] - (20*(A + C)*Sqrt[Cos[c + d*x]])/(1 + Sin[(c + d*x)/2]) + (5*(A + C)*(-1 + 2*Sin[(c + d*x)/2]))/(Sqrt[Cos[c + d*x]]*(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^2) - (5*(A + C)*(1 + 2*Sin[(c + d*x)/2]))/(Sqrt[Cos[c + d*x]]*(-1 + Sin[(c + d*x)/2])) + ((A - 7*C)*Csc[(c + d*x)/2]^3*(5*(1 + 4*Cos[c + d*x] + Cos[2*(c + d*x)])*(1 - Cos[c + d*x] + ArcTanh[Sqrt[-(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]]*Cos[c + d*x]*Sqrt[2 - 2*Sec[c + d*x]]) - 2*Hypergeometric2F1[2, 5/2, 7/2, -(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^4*Sin[c + d*x]*Tan[c + d*x]))/(2*Cos[c + d*x]^(3/2))))/(10*d*(a*(1 + Cos[c + d*x]))^(3/2))","C",0
209,1,1192,201,6.7876052,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)),x]","-\frac{(A+C) \left(5 \tan ^{-1}\left(\frac{1-2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}\right)+\frac{3 \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}{1-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)}+\frac{\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1}{\left(1-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{3/2}}+\frac{(A+C) \left(5 \tan ^{-1}\left(\frac{2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1}{\sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}\right)+\frac{3 \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}{\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1}+\frac{1-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1\right) \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{3/2}}+\frac{16 C \sin \left(\frac{c}{2}+\frac{d x}{2}\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (a (\cos (c+d x)+1))^{3/2} \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}+\frac{8 C \sin \left(\frac{c}{2}+\frac{d x}{2}\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (a (\cos (c+d x)+1))^{3/2} \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{3/2}}+\frac{(A+C) \left(2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{6 d (a (\cos (c+d x)+1))^{3/2} \left(1-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{3/2}}-\frac{(A+C) \left(1-2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{6 d (a (\cos (c+d x)+1))^{3/2} \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1\right) \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{3/2}}+\frac{(A-7 C) \cot ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \csc ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \left(-12 \cos ^4\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(2,2,\frac{7}{2};1,\frac{9}{2};-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}\right) \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)-12 \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}\right) \left(3 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-7 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+4\right) \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)+7 \sqrt{-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}} \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3 \left(8 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-20 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+15\right) \left(\left(3-7 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sqrt{-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}-3 \tanh ^{-1}\left(\sqrt{-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}\right) \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)\right)\right)}{63 d (a (\cos (c+d x)+1))^{3/2} \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{7/2}}","\frac{(11 A+3 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(7 A+3 C) \sin (c+d x)}{6 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(19 A+3 C) \sin (c+d x)}{6 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(8*C*Cos[c/2 + (d*x)/2]^3*Sin[c/2 + (d*x)/2])/(3*d*(a*(1 + Cos[c + d*x]))^(3/2)*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) - ((A + C)*Cos[c/2 + (d*x)/2]^3*(1 - 2*Sin[c/2 + (d*x)/2]))/(6*d*(a*(1 + Cos[c + d*x]))^(3/2)*(1 + Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) + ((A + C)*Cos[c/2 + (d*x)/2]^3*(1 + 2*Sin[c/2 + (d*x)/2]))/(6*d*(a*(1 + Cos[c + d*x]))^(3/2)*(1 - Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) + (16*C*Cos[c/2 + (d*x)/2]^3*Sin[c/2 + (d*x)/2])/(3*d*(a*(1 + Cos[c + d*x]))^(3/2)*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) - ((A + C)*Cos[c/2 + (d*x)/2]^3*(5*ArcTan[(1 - 2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]] + (1 + Sin[c/2 + (d*x)/2])/((1 - Sin[c/2 + (d*x)/2])*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) + (3*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])/(1 - Sin[c/2 + (d*x)/2])))/(d*(a*(1 + Cos[c + d*x]))^(3/2)) + ((A + C)*Cos[c/2 + (d*x)/2]^3*(5*ArcTan[(1 + 2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]] + (1 - Sin[c/2 + (d*x)/2])/((1 + Sin[c/2 + (d*x)/2])*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) + (3*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])/(1 + Sin[c/2 + (d*x)/2])))/(d*(a*(1 + Cos[c + d*x]))^(3/2)) + ((A - 7*C)*Cot[c/2 + (d*x)/2]^3*Csc[c/2 + (d*x)/2]^2*(-12*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 7/2}, {1, 9/2}, -(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]*Sin[c/2 + (d*x)/2]^8 - 12*Hypergeometric2F1[2, 7/2, 9/2, -(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]*Sin[c/2 + (d*x)/2]^8*(4 - 7*Sin[c/2 + (d*x)/2]^2 + 3*Sin[c/2 + (d*x)/2]^4) + 7*Sqrt[-(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]*(1 - 2*Sin[c/2 + (d*x)/2]^2)^3*(15 - 20*Sin[c/2 + (d*x)/2]^2 + 8*Sin[c/2 + (d*x)/2]^4)*((3 - 7*Sin[c/2 + (d*x)/2]^2)*Sqrt[-(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))] - 3*ArcTanh[Sqrt[-(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]]*(1 - 2*Sin[c/2 + (d*x)/2]^2))))/(63*d*(a*(1 + Cos[c + d*x]))^(3/2)*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2))","C",0
210,1,2422,248,7.8453477,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Cos[c + d*x])^(3/2)),x]","\text{Result too large to show}","-\frac{(15 A+7 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(13 A+5 C) \sin (c+d x)}{10 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{(9 A+5 C) \sin (c+d x)}{10 a d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}+\frac{(49 A+25 C) \sin (c+d x)}{10 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(8*C*Cos[c/2 + (d*x)/2]^3*Sin[c/2 + (d*x)/2])/(5*d*(a*(1 + Cos[c + d*x]))^(3/2)*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2)) - ((A + C)*Cos[c/2 + (d*x)/2]^3*(1 - 2*Sin[c/2 + (d*x)/2]))/(10*d*(a*(1 + Cos[c + d*x]))^(3/2)*(1 + Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2)) + ((A + C)*Cos[c/2 + (d*x)/2]^3*(1 + 2*Sin[c/2 + (d*x)/2]))/(10*d*(a*(1 + Cos[c + d*x]))^(3/2)*(1 - Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2)) + (32*C*Cos[c/2 + (d*x)/2]^3*(Sin[c/2 + (d*x)/2]/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2) + (2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]))/(15*d*(a*(1 + Cos[c + d*x]))^(3/2)) + ((A + C)*Cos[c/2 + (d*x)/2]^3*(105*ArcTan[(1 - 2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]] - (4 + 3*Sin[c/2 + (d*x)/2])/((1 - Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) + (19 + 29*Sin[c/2 + (d*x)/2])/((1 - Sin[c/2 + (d*x)/2])*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) + (67*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])/(1 - Sin[c/2 + (d*x)/2])))/(15*d*(a*(1 + Cos[c + d*x]))^(3/2)) - ((A + C)*Cos[c/2 + (d*x)/2]^3*(105*ArcTan[(1 + 2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]] - (4 - 3*Sin[c/2 + (d*x)/2])/((1 + Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) + (19 - 29*Sin[c/2 + (d*x)/2])/((1 + Sin[c/2 + (d*x)/2])*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) + (67*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])/(1 + Sin[c/2 + (d*x)/2])))/(15*d*(a*(1 + Cos[c + d*x]))^(3/2)) + ((-A + 7*C)*Cot[c/2 + (d*x)/2]^3*Csc[c/2 + (d*x)/2]^4*(4725*Sin[c/2 + (d*x)/2]^2 - 48825*Sin[c/2 + (d*x)/2]^4 + 210105*Sin[c/2 + (d*x)/2]^6 - 486630*Sin[c/2 + (d*x)/2]^8 + 655812*Sin[c/2 + (d*x)/2]^10 - 710*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^10 - 40*Cos[(c + d*x)/2]^6*HypergeometricPFQ[{2, 2, 2, 9/2}, {1, 1, 11/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^10 - 518760*Sin[c/2 + (d*x)/2]^12 + 1770*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12 + 226656*Sin[c/2 + (d*x)/2]^14 - 1500*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14 - 42048*Sin[c/2 + (d*x)/2]^16 + 440*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^16 + 4725*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 56700*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^2*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 291060*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^4*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 833760*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^6*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 1458000*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^8*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 1598400*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^10*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 1080000*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^12*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 414720*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^14*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 69120*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^16*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 60*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 9/2}, {1, 11/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^10*(-5 + 4*Sin[c/2 + (d*x)/2]^2)))/(675*d*(a*(1 + Cos[c + d*x]))^(3/2)*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2)*(-1 + 2*Sin[c/2 + (d*x)/2]^2))","C",0
211,1,256,237,2.2760947,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\cos ^5\left(\frac{1}{2} (c+d x)\right) \left(\sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) ((7 A+55 C) \cos (c+d x)+3 A+8 C \cos (2 (c+d x))+43 C)+\frac{i \sqrt{2} e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \left(\sqrt{2} (3 A+115 C) \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+80 C \sinh ^{-1}\left(e^{i (c+d x)}\right)-80 C \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{\sqrt{1+e^{2 i (c+d x)}}}\right)}{8 d (a (\cos (c+d x)+1))^{5/2}}","\frac{(3 A+115 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{5 C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}+\frac{(3 A+35 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(A-15 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]^5*((I*Sqrt[2]*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(80*C*ArcSinh[E^(I*(c + d*x))] + Sqrt[2]*(3*A + 115*C)*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - 80*C*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))/Sqrt[1 + E^((2*I)*(c + d*x))] + Sqrt[Cos[c + d*x]]*(3*A + 43*C + (7*A + 55*C)*Cos[c + d*x] + 8*C*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^3*Tan[(c + d*x)/2]))/(8*d*(a*(1 + Cos[c + d*x]))^(5/2))","C",1
212,1,244,192,1.9516246,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\cos ^5\left(\frac{1}{2} (c+d x)\right) \left(\sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) ((A-15 C) \cos (c+d x)+5 A-11 C)-\frac{i \sqrt{2} e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \left(-\sqrt{2} (5 A-43 C) \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+32 C \sinh ^{-1}\left(e^{i (c+d x)}\right)-32 C \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{\sqrt{1+e^{2 i (c+d x)}}}\right)}{8 d (a (\cos (c+d x)+1))^{5/2}}","\frac{(5 A-43 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(5 A-11 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]^5*(((-I)*Sqrt[2]*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(32*C*ArcSinh[E^(I*(c + d*x))] - Sqrt[2]*(5*A - 43*C)*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - 32*C*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))/Sqrt[1 + E^((2*I)*(c + d*x))] + Sqrt[Cos[c + d*x]]*(5*A - 11*C + (A - 15*C)*Cos[c + d*x])*Sec[(c + d*x)/2]^3*Tan[(c + d*x)/2]))/(8*d*(a*(1 + Cos[c + d*x]))^(5/2))","C",1
213,1,200,154,1.5565951,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)),x]","\frac{\cos ^5\left(\frac{1}{2} (c+d x)\right) \left(-\frac{1}{2} \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) ((9 A-7 C) \cos (c+d x)+13 A-3 C)+\frac{i (19 A+3 C) e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)}{\sqrt{1+e^{2 i (c+d x)}}}\right)}{4 d (a (\cos (c+d x)+1))^{5/2}}","\frac{(19 A+3 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(9 A-7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(Cos[(c + d*x)/2]^5*((I*(19*A + 3*C)*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/Sqrt[1 + E^((2*I)*(c + d*x))] - (Sqrt[Cos[c + d*x]]*(13*A - 3*C + (9*A - 7*C)*Cos[c + d*x])*Sec[(c + d*x)/2]^3*Tan[(c + d*x)/2])/2))/(4*d*(a*(1 + Cos[c + d*x]))^(5/2))","C",1
214,1,211,199,2.7563456,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)),x]","\frac{\cos ^5\left(\frac{1}{2} (c+d x)\right) \left(\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) (10 (17 A+C) \cos (c+d x)+(49 A+C) \cos (2 (c+d x))+113 A+C)}{4 \sqrt{\cos (c+d x)}}-\frac{5 i (15 A-C) e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)}{\sqrt{1+e^{2 i (c+d x)}}}\right)}{4 d (a (\cos (c+d x)+1))^{5/2}}","-\frac{5 (15 A-C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(49 A+C) \sin (c+d x)}{16 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(13 A-3 C) \sin (c+d x)}{16 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x)}{4 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}",1,"(Cos[(c + d*x)/2]^5*(((-5*I)*(15*A - C)*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/Sqrt[1 + E^((2*I)*(c + d*x))] + ((113*A + C + 10*(17*A + C)*Cos[c + d*x] + (49*A + C)*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^3*Tan[(c + d*x)/2])/(4*Sqrt[Cos[c + d*x]])))/(4*d*(a*(1 + Cos[c + d*x]))^(5/2))","C",1
215,1,239,246,3.5706422,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(5/2)),x]","\frac{\cos ^5\left(\frac{1}{2} (c+d x)\right) \left(-\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) ((1537 A+81 C) \cos (c+d x)+2 (503 A+39 C) \cos (2 (c+d x))+299 A \cos (3 (c+d x))+878 A+27 C \cos (3 (c+d x))+78 C)}{8 \cos ^{\frac{3}{2}}(c+d x)}+\frac{3 i (163 A+19 C) e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)}{\sqrt{1+e^{2 i (c+d x)}}}\right)}{12 d (a (\cos (c+d x)+1))^{5/2}}","\frac{(163 A+19 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{5 (19 A+3 C) \sin (c+d x)}{48 a^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(299 A+27 C) \sin (c+d x)}{48 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(17 A+C) \sin (c+d x)}{16 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}",1,"(Cos[(c + d*x)/2]^5*(((3*I)*(163*A + 19*C)*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/Sqrt[1 + E^((2*I)*(c + d*x))] - ((878*A + 78*C + (1537*A + 81*C)*Cos[c + d*x] + 2*(503*A + 39*C)*Cos[2*(c + d*x)] + 299*A*Cos[3*(c + d*x)] + 27*C*Cos[3*(c + d*x)])*Sec[(c + d*x)/2]^3*Tan[(c + d*x)/2])/(8*Cos[c + d*x]^(3/2))))/(12*d*(a*(1 + Cos[c + d*x]))^(5/2))","C",1
216,1,89,92,0.1391242,"\int \cos ^3(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{3 B (c+d x)}{8 d}+\frac{B \sin (2 (c+d x))}{4 d}+\frac{B \sin (4 (c+d x))}{32 d}+\frac{C \sin ^5(c+d x)}{5 d}-\frac{2 C \sin ^3(c+d x)}{3 d}+\frac{C \sin (c+d x)}{d}","\frac{B \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 B \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 B x}{8}+\frac{C \sin ^5(c+d x)}{5 d}-\frac{2 C \sin ^3(c+d x)}{3 d}+\frac{C \sin (c+d x)}{d}",1,"(3*B*(c + d*x))/(8*d) + (C*Sin[c + d*x])/d - (2*C*Sin[c + d*x]^3)/(3*d) + (C*Sin[c + d*x]^5)/(5*d) + (B*Sin[2*(c + d*x)])/(4*d) + (B*Sin[4*(c + d*x)])/(32*d)","A",1
217,1,73,76,0.0934241,"\int \cos ^2(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{B \sin ^3(c+d x)}{3 d}+\frac{B \sin (c+d x)}{d}+\frac{3 C (c+d x)}{8 d}+\frac{C \sin (2 (c+d x))}{4 d}+\frac{C \sin (4 (c+d x))}{32 d}","-\frac{B \sin ^3(c+d x)}{3 d}+\frac{B \sin (c+d x)}{d}+\frac{C \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 C \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 C x}{8}",1,"(3*C*(c + d*x))/(8*d) + (B*Sin[c + d*x])/d - (B*Sin[c + d*x]^3)/(3*d) + (C*Sin[2*(c + d*x)])/(4*d) + (C*Sin[4*(c + d*x)])/(32*d)","A",1
218,1,57,54,0.0638397,"\int \cos (c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{B (c+d x)}{2 d}+\frac{B \sin (2 (c+d x))}{4 d}-\frac{C \sin ^3(c+d x)}{3 d}+\frac{C \sin (c+d x)}{d}","\frac{B \sin (c+d x) \cos (c+d x)}{2 d}+\frac{B x}{2}-\frac{C \sin ^3(c+d x)}{3 d}+\frac{C \sin (c+d x)}{d}",1,"(B*(c + d*x))/(2*d) + (C*Sin[c + d*x])/d - (C*Sin[c + d*x]^3)/(3*d) + (B*Sin[2*(c + d*x)])/(4*d)","A",1
219,1,35,38,0.0630826,"\int \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[B*Cos[c + d*x] + C*Cos[c + d*x]^2,x]","\frac{4 B \sin (c+d x)+C (2 (c+d x)+\sin (2 (c+d x)))}{4 d}","\frac{B \sin (c+d x)}{d}+\frac{C \sin (c+d x) \cos (c+d x)}{2 d}+\frac{C x}{2}",1,"(4*B*Sin[c + d*x] + C*(2*(c + d*x) + Sin[2*(c + d*x)]))/(4*d)","A",1
220,1,26,15,0.0083202,"\int \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","B x+\frac{C \sin (c) \cos (d x)}{d}+\frac{C \cos (c) \sin (d x)}{d}","B x+\frac{C \sin (c+d x)}{d}",1,"B*x + (C*Cos[d*x]*Sin[c])/d + (C*Cos[c]*Sin[d*x])/d","A",1
221,1,16,16,0.0063679,"\int \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{B \tanh ^{-1}(\sin (c+d x))}{d}+C x","\frac{B \tanh ^{-1}(\sin (c+d x))}{d}+C x",1,"C*x + (B*ArcTanh[Sin[c + d*x]])/d","A",1
222,1,24,24,0.0075508,"\int \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{B \tan (c+d x)}{d}+\frac{C \tanh ^{-1}(\sin (c+d x))}{d}","\frac{B \tan (c+d x)}{d}+\frac{C \tanh ^{-1}(\sin (c+d x))}{d}",1,"(C*ArcTanh[Sin[c + d*x]])/d + (B*Tan[c + d*x])/d","A",1
223,1,47,47,0.0126638,"\int \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{B \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{B \tan (c+d x) \sec (c+d x)}{2 d}+\frac{C \tan (c+d x)}{d}","\frac{B \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{B \tan (c+d x) \sec (c+d x)}{2 d}+\frac{C \tan (c+d x)}{d}",1,"(B*ArcTanh[Sin[c + d*x]])/(2*d) + (C*Tan[c + d*x])/d + (B*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
224,1,60,63,0.1579305,"\int \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{B \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}+\frac{C \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{C \tan (c+d x) \sec (c+d x)}{2 d}","\frac{B \tan ^3(c+d x)}{3 d}+\frac{B \tan (c+d x)}{d}+\frac{C \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{C \tan (c+d x) \sec (c+d x)}{2 d}",1,"(C*ArcTanh[Sin[c + d*x]])/(2*d) + (C*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (B*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d","A",1
225,1,76,85,0.2180924,"\int \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{B \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 B \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)}{8 d}+\frac{C \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}","\frac{3 B \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{B \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 B \tan (c+d x) \sec (c+d x)}{8 d}+\frac{C \tan ^3(c+d x)}{3 d}+\frac{C \tan (c+d x)}{d}",1,"(B*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (3*B*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x]))/(8*d) + (C*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d","A",1
226,1,102,125,0.3409531,"\int \cos ^2(c+d x) (a+a \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a (60 (6 B+5 C) \sin (c+d x)+120 (B+C) \sin (2 (c+d x))+40 B \sin (3 (c+d x))+15 B \sin (4 (c+d x))+180 B d x+50 C \sin (3 (c+d x))+15 C \sin (4 (c+d x))+6 C \sin (5 (c+d x))+180 C d x)}{480 d}","-\frac{a (5 B+4 C) \sin ^3(c+d x)}{15 d}+\frac{a (5 B+4 C) \sin (c+d x)}{5 d}+\frac{a (B+C) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a (B+C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3}{8} a x (B+C)+\frac{a C \sin (c+d x) \cos ^4(c+d x)}{5 d}",1,"(a*(180*B*d*x + 180*C*d*x + 60*(6*B + 5*C)*Sin[c + d*x] + 120*(B + C)*Sin[2*(c + d*x)] + 40*B*Sin[3*(c + d*x)] + 50*C*Sin[3*(c + d*x)] + 15*B*Sin[4*(c + d*x)] + 15*C*Sin[4*(c + d*x)] + 6*C*Sin[5*(c + d*x)]))/(480*d)","A",1
227,1,76,97,0.2744134,"\int \cos (c+d x) (a+a \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a (72 (B+C) \sin (c+d x)+24 (B+C) \sin (2 (c+d x))+8 B \sin (3 (c+d x))+48 B d x+8 C \sin (3 (c+d x))+3 C \sin (4 (c+d x))+36 C d x)}{96 d}","-\frac{a (B+C) \sin ^3(c+d x)}{3 d}+\frac{a (B+C) \sin (c+d x)}{d}+\frac{a (4 B+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 B+3 C)+\frac{a C \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"(a*(48*B*d*x + 36*C*d*x + 72*(B + C)*Sin[c + d*x] + 24*(B + C)*Sin[2*(c + d*x)] + 8*B*Sin[3*(c + d*x)] + 8*C*Sin[3*(c + d*x)] + 3*C*Sin[4*(c + d*x)]))/(96*d)","A",1
228,1,65,85,0.1767399,"\int (a+a \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + a*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a (3 (4 B+3 C) \sin (c+d x)+3 (B+C) \sin (2 (c+d x))+6 B c+6 B d x+C \sin (3 (c+d x))+6 c C+6 C d x)}{12 d}","\frac{a (3 B+C) \sin (c+d x)}{3 d}+\frac{a (3 B-C) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{1}{2} a x (B+C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 a d}",1,"(a*(6*B*c + 6*c*C + 6*B*d*x + 6*C*d*x + 3*(4*B + 3*C)*Sin[c + d*x] + 3*(B + C)*Sin[2*(c + d*x)] + C*Sin[3*(c + d*x)]))/(12*d)","A",1
229,1,44,47,0.0978184,"\int (a+a \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{a (4 (B+C) \sin (c+d x)+4 B d x+C \sin (2 (c+d x))+2 c C+2 C d x)}{4 d}","\frac{a (B+C) \sin (c+d x)}{d}+\frac{1}{2} a x (2 B+C)+\frac{a C \sin (c+d x) \cos (c+d x)}{2 d}",1,"(a*(2*c*C + 4*B*d*x + 2*C*d*x + 4*(B + C)*Sin[c + d*x] + C*Sin[2*(c + d*x)]))/(4*d)","A",1
230,1,46,32,0.024232,"\int (a+a \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+a B x+\frac{a C \sin (c) \cos (d x)}{d}+\frac{a C \cos (c) \sin (d x)}{d}+a C x","\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+a x (B+C)+\frac{a C \sin (c+d x)}{d}",1,"a*B*x + a*C*x + (a*B*ArcTanh[Sin[c + d*x]])/d + (a*C*Cos[d*x]*Sin[c])/d + (a*C*Cos[c]*Sin[d*x])/d","A",1
231,1,43,32,0.0184731,"\int (a+a \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{a B \tan (c+d x)}{d}+\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d}+a C x","\frac{a (B+C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a B \tan (c+d x)}{d}+a C x",1,"a*C*x + (a*B*ArcTanh[Sin[c + d*x]])/d + (a*C*ArcTanh[Sin[c + d*x]])/d + (a*B*Tan[c + d*x])/d","A",1
232,1,75,56,0.0260464,"\int (a+a \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a B \tan (c+d x)}{d}+\frac{a B \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a B \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a C \tan (c+d x)}{d}+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d}","\frac{a (B+C) \tan (c+d x)}{d}+\frac{a (B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a B \tan (c+d x) \sec (c+d x)}{2 d}",1,"(a*B*ArcTanh[Sin[c + d*x]])/(2*d) + (a*C*ArcTanh[Sin[c + d*x]])/d + (a*B*Tan[c + d*x])/d + (a*C*Tan[c + d*x])/d + (a*B*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
233,1,56,86,0.3309806,"\int (a+a \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{a \left(3 (B+C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(3 (B+C) \sec (c+d x)+2 B \tan ^2(c+d x)+6 (B+C)\right)\right)}{6 d}","\frac{a (2 B+3 C) \tan (c+d x)}{3 d}+\frac{a (B+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (B+C) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a B \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"(a*(3*(B + C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(6*(B + C) + 3*(B + C)*Sec[c + d*x] + 2*B*Tan[c + d*x]^2)))/(6*d)","A",1
234,1,77,106,0.3736487,"\int (a+a \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{a \left(3 (3 B+4 C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x) \left(8 (B+C) (\cos (2 (c+d x))+2) \sec (c+d x)+6 B \sec ^2(c+d x)+9 B+12 C\right)\right)}{24 d}","\frac{a (B+C) \tan ^3(c+d x)}{3 d}+\frac{a (B+C) \tan (c+d x)}{d}+\frac{a (3 B+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (3 B+4 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a B \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"(a*(3*(3*B + 4*C)*ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*(9*B + 12*C + 8*(B + C)*(2 + Cos[2*(c + d*x)])*Sec[c + d*x] + 6*B*Sec[c + d*x]^2)*Tan[c + d*x]))/(24*d)","A",1
235,1,104,160,0.3264814,"\int \cos (c+d x) (a+a \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^2 (60 (12 B+11 C) \sin (c+d x)+240 (B+C) \sin (2 (c+d x))+80 B \sin (3 (c+d x))+15 B \sin (4 (c+d x))+420 B d x+90 C \sin (3 (c+d x))+30 C \sin (4 (c+d x))+6 C \sin (5 (c+d x))+360 C d x)}{480 d}","-\frac{a^2 (10 B+9 C) \sin ^3(c+d x)}{15 d}+\frac{a^2 (10 B+9 C) \sin (c+d x)}{5 d}+\frac{a^2 (5 B+6 C) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{a^2 (7 B+6 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a^2 x (7 B+6 C)+\frac{C \sin (c+d x) \cos ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{5 d}",1,"(a^2*(420*B*d*x + 360*C*d*x + 60*(12*B + 11*C)*Sin[c + d*x] + 240*(B + C)*Sin[2*(c + d*x)] + 80*B*Sin[3*(c + d*x)] + 90*C*Sin[3*(c + d*x)] + 15*B*Sin[4*(c + d*x)] + 30*C*Sin[4*(c + d*x)] + 6*C*Sin[5*(c + d*x)]))/(480*d)","A",1
236,1,86,129,0.334687,"\int (a+a \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^2 (24 (7 B+6 C) \sin (c+d x)+48 (B+C) \sin (2 (c+d x))+8 B \sin (3 (c+d x))+96 B d x+16 C \sin (3 (c+d x))+3 C \sin (4 (c+d x))+84 c C+84 C d x)}{96 d}","\frac{a^2 (8 B+7 C) \sin (c+d x)}{6 d}+\frac{a^2 (8 B+7 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} a^2 x (8 B+7 C)+\frac{(4 B-C) \sin (c+d x) (a \cos (c+d x)+a)^2}{12 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d}",1,"(a^2*(84*c*C + 96*B*d*x + 84*C*d*x + 24*(7*B + 6*C)*Sin[c + d*x] + 48*(B + C)*Sin[2*(c + d*x)] + 8*B*Sin[3*(c + d*x)] + 16*C*Sin[3*(c + d*x)] + 3*C*Sin[4*(c + d*x)]))/(96*d)","A",1
237,1,61,94,0.1941955,"\int (a+a \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{a^2 (3 (8 B+7 C) \sin (c+d x)+3 (B+2 C) \sin (2 (c+d x))+18 B d x+C \sin (3 (c+d x))+12 C d x)}{12 d}","\frac{2 a^2 (3 B+2 C) \sin (c+d x)}{3 d}+\frac{a^2 (3 B+2 C) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{1}{2} a^2 x (3 B+2 C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}",1,"(a^2*(18*B*d*x + 12*C*d*x + 3*(8*B + 7*C)*Sin[c + d*x] + 3*(B + 2*C)*Sin[2*(c + d*x)] + C*Sin[3*(c + d*x)]))/(12*d)","A",1
238,1,96,82,0.1815541,"\int (a+a \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{a^2 \left(4 (B+2 C) \sin (c+d x)-4 B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+8 B d x+C \sin (2 (c+d x))+6 C d x\right)}{4 d}","\frac{a^2 (2 B+3 C) \sin (c+d x)}{2 d}+\frac{a^2 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^2 x (4 B+3 C)+\frac{C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{2 d}",1,"(a^2*(8*B*d*x + 6*C*d*x - 4*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 4*(B + 2*C)*Sin[c + d*x] + C*Sin[2*(c + d*x)]))/(4*d)","A",1
239,1,143,74,0.3401441,"\int (a+a \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{a^2 \left(B \tan (c+d x)-2 B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+B c+B d x+C \sin (c+d x)-C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 c C+2 C d x\right)}{d}","-\frac{a^2 (B-C) \sin (c+d x)}{d}+\frac{a^2 (2 B+C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{B \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{d}+a^2 x (B+2 C)",1,"(a^2*(B*c + 2*c*C + B*d*x + 2*C*d*x - 2*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + C*Sin[c + d*x] + B*Tan[c + d*x]))/d","A",1
240,1,277,88,1.3629658,"\int (a+a \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{1}{16} a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(\frac{4 (2 B+C) \sin \left(\frac{d x}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{4 (2 B+C) \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{2 (3 B+4 C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{2 (3 B+4 C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{B}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{B}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+4 C x\right)","\frac{a^2 (3 B+2 C) \tan (c+d x)}{2 d}+\frac{a^2 (3 B+4 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{B \tan (c+d x) \sec (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{2 d}+a^2 C x",1,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(4*C*x - (2*(3*B + 4*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (2*(3*B + 4*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + B/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (4*(2*B + C)*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - B/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (4*(2*B + C)*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/16","B",1
241,1,451,113,5.8868823,"\int (a+a \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(\frac{4 (5 B+6 C) \sin \left(\frac{d x}{2}\right)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{4 (5 B+6 C) \sin \left(\frac{d x}{2}\right)}{\left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{(7 B+3 C) \cos \left(\frac{c}{2}\right)-(5 B+3 C) \sin \left(\frac{c}{2}\right)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{(5 B+3 C) \sin \left(\frac{c}{2}\right)+(7 B+3 C) \cos \left(\frac{c}{2}\right)}{\left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-6 (2 B+3 C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 (2 B+3 C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{2 B \sin \left(\frac{d x}{2}\right)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2 B \sin \left(\frac{d x}{2}\right)}{\left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}\right)}{48 d}","\frac{a^2 (5 B+6 C) \tan (c+d x)}{3 d}+\frac{a^2 (2 B+3 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 (4 B+3 C) \tan (c+d x) \sec (c+d x)}{6 d}+\frac{B \tan (c+d x) \sec ^2(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{3 d}",1,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(-6*(2*B + 3*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*(2*B + 3*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*B*Sin[(d*x)/2])/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + ((7*B + 3*C)*Cos[c/2] - (5*B + 3*C)*Sin[c/2])/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (4*(5*B + 6*C)*Sin[(d*x)/2])/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (2*B*Sin[(d*x)/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - ((7*B + 3*C)*Cos[c/2] + (5*B + 3*C)*Sin[c/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (4*(5*B + 6*C)*Sin[(d*x)/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(48*d)","B",1
242,1,262,144,1.145791,"\int (a+a \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","-\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \left(24 (7 B+8 C) \cos ^4(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-24 (4 B+5 C) \sin (c)+45 B \sin (2 c+d x)+128 B \sin (c+2 d x)+21 B \sin (2 c+3 d x)+21 B \sin (4 c+3 d x)+32 B \sin (3 c+4 d x)+3 (15 B+8 C) \sin (d x)+24 C \sin (2 c+d x)+136 C \sin (c+2 d x)-24 C \sin (3 c+2 d x)+24 C \sin (2 c+3 d x)+24 C \sin (4 c+3 d x)+40 C \sin (3 c+4 d x))\right)}{768 d}","\frac{a^2 (4 B+5 C) \tan (c+d x)}{3 d}+\frac{a^2 (7 B+8 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (5 B+4 C) \tan (c+d x) \sec ^2(c+d x)}{12 d}+\frac{a^2 (7 B+8 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{B \tan (c+d x) \sec ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{4 d}",1,"-1/768*(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*Sec[c + d*x]^4*(24*(7*B + 8*C)*Cos[c + d*x]^4*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(-24*(4*B + 5*C)*Sin[c] + 3*(15*B + 8*C)*Sin[d*x] + 45*B*Sin[2*c + d*x] + 24*C*Sin[2*c + d*x] + 128*B*Sin[c + 2*d*x] + 136*C*Sin[c + 2*d*x] - 24*C*Sin[3*c + 2*d*x] + 21*B*Sin[2*c + 3*d*x] + 24*C*Sin[2*c + 3*d*x] + 21*B*Sin[4*c + 3*d*x] + 24*C*Sin[4*c + 3*d*x] + 32*B*Sin[3*c + 4*d*x] + 40*C*Sin[3*c + 4*d*x])))/d","A",1
243,1,280,169,1.3320777,"\int (a+a \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","-\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \left(240 (6 B+7 C) \cos ^5(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-240 (B+2 C) \sin (2 c+d x)+420 B \sin (c+2 d x)+420 B \sin (3 c+2 d x)+720 B \sin (2 c+3 d x)+90 B \sin (3 c+4 d x)+90 B \sin (5 c+4 d x)+144 B \sin (4 c+5 d x)+80 (15 B+14 C) \sin (d x)+330 C \sin (c+2 d x)+330 C \sin (3 c+2 d x)+800 C \sin (2 c+3 d x)+105 C \sin (3 c+4 d x)+105 C \sin (5 c+4 d x)+160 C \sin (4 c+5 d x))\right)}{7680 d}","\frac{a^2 (9 B+10 C) \tan ^3(c+d x)}{15 d}+\frac{a^2 (9 B+10 C) \tan (c+d x)}{5 d}+\frac{a^2 (6 B+7 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (6 B+5 C) \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac{a^2 (6 B+7 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{B \tan (c+d x) \sec ^4(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{5 d}",1,"-1/7680*(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*Sec[c + d*x]^5*(240*(6*B + 7*C)*Cos[c + d*x]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(80*(15*B + 14*C)*Sin[d*x] - 240*(B + 2*C)*Sin[2*c + d*x] + 420*B*Sin[c + 2*d*x] + 330*C*Sin[c + 2*d*x] + 420*B*Sin[3*c + 2*d*x] + 330*C*Sin[3*c + 2*d*x] + 720*B*Sin[2*c + 3*d*x] + 800*C*Sin[2*c + 3*d*x] + 90*B*Sin[3*c + 4*d*x] + 105*C*Sin[3*c + 4*d*x] + 90*B*Sin[5*c + 4*d*x] + 105*C*Sin[5*c + 4*d*x] + 144*B*Sin[4*c + 5*d*x] + 160*C*Sin[4*c + 5*d*x])))/d","A",1
244,1,130,201,0.4157891,"\int \cos (c+d x) (a+a \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^3 (120 (23 B+21 C) \sin (c+d x)+15 (64 B+63 C) \sin (2 (c+d x))+340 B \sin (3 (c+d x))+90 B \sin (4 (c+d x))+12 B \sin (5 (c+d x))+1560 B d x+380 C \sin (3 (c+d x))+135 C \sin (4 (c+d x))+36 C \sin (5 (c+d x))+5 C \sin (6 (c+d x))+1380 C d x)}{960 d}","-\frac{a^3 (19 B+17 C) \sin ^3(c+d x)}{15 d}+\frac{a^3 (19 B+17 C) \sin (c+d x)}{5 d}+\frac{a^3 (22 B+21 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{(3 B+4 C) \sin (c+d x) \cos ^3(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d}+\frac{a^3 (26 B+23 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^3 x (26 B+23 C)+\frac{a C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}",1,"(a^3*(1560*B*d*x + 1380*C*d*x + 120*(23*B + 21*C)*Sin[c + d*x] + 15*(64*B + 63*C)*Sin[2*(c + d*x)] + 340*B*Sin[3*(c + d*x)] + 380*C*Sin[3*(c + d*x)] + 90*B*Sin[4*(c + d*x)] + 135*C*Sin[4*(c + d*x)] + 12*B*Sin[5*(c + d*x)] + 36*C*Sin[5*(c + d*x)] + 5*C*Sin[6*(c + d*x)]))/(960*d)","A",1
245,1,108,154,0.4291715,"\int (a+a \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^3 (60 (26 B+23 C) \sin (c+d x)+480 (B+C) \sin (2 (c+d x))+120 B \sin (3 (c+d x))+15 B \sin (4 (c+d x))+900 B d x+170 C \sin (3 (c+d x))+45 C \sin (4 (c+d x))+6 C \sin (5 (c+d x))+780 c C+780 C d x)}{480 d}","-\frac{a^3 (15 B+13 C) \sin ^3(c+d x)}{60 d}+\frac{a^3 (15 B+13 C) \sin (c+d x)}{5 d}+\frac{3 a^3 (15 B+13 C) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{1}{8} a^3 x (15 B+13 C)+\frac{(5 B-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}",1,"(a^3*(780*c*C + 900*B*d*x + 780*C*d*x + 60*(26*B + 23*C)*Sin[c + d*x] + 480*(B + C)*Sin[2*(c + d*x)] + 120*B*Sin[3*(c + d*x)] + 170*C*Sin[3*(c + d*x)] + 15*B*Sin[4*(c + d*x)] + 45*C*Sin[4*(c + d*x)] + 6*C*Sin[5*(c + d*x)]))/(480*d)","A",1
246,1,86,116,0.3091875,"\int (a+a \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{a^3 (24 (15 B+13 C) \sin (c+d x)+24 (3 B+4 C) \sin (2 (c+d x))+8 B \sin (3 (c+d x))+240 B d x+24 C \sin (3 (c+d x))+3 C \sin (4 (c+d x))+180 C d x)}{96 d}","-\frac{a^3 (4 B+3 C) \sin ^3(c+d x)}{12 d}+\frac{a^3 (4 B+3 C) \sin (c+d x)}{d}+\frac{3 a^3 (4 B+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5}{8} a^3 x (4 B+3 C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}",1,"(a^3*(240*B*d*x + 180*C*d*x + 24*(15*B + 13*C)*Sin[c + d*x] + 24*(3*B + 4*C)*Sin[2*(c + d*x)] + 8*B*Sin[3*(c + d*x)] + 24*C*Sin[3*(c + d*x)] + 3*C*Sin[4*(c + d*x)]))/(96*d)","A",1
247,1,113,111,0.2597766,"\int (a+a \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{a^3 \left(9 (4 B+5 C) \sin (c+d x)+3 (B+3 C) \sin (2 (c+d x))-12 B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+42 B d x+C \sin (3 (c+d x))+30 C d x\right)}{12 d}","\frac{5 a^3 (B+C) \sin (c+d x)}{2 d}+\frac{(3 B+5 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{6 d}+\frac{a^3 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^3 x (7 B+5 C)+\frac{a C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}",1,"(a^3*(42*B*d*x + 30*C*d*x - 12*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 9*(4*B + 5*C)*Sin[c + d*x] + 3*(B + 3*C)*Sin[2*(c + d*x)] + C*Sin[3*(c + d*x)]))/(12*d)","A",1
248,1,272,110,1.729897,"\int (a+a \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{1}{32} a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(\frac{4 (B+3 C) \sin (c) \cos (d x)}{d}+\frac{4 (B+3 C) \cos (c) \sin (d x)}{d}-\frac{4 (3 B+C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{4 (3 B+C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{4 B \sin \left(\frac{d x}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{4 B \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+2 x (6 B+7 C)+\frac{C \sin (2 c) \cos (2 d x)}{d}+\frac{C \cos (2 c) \sin (2 d x)}{d}\right)","\frac{a^3 (3 B+C) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{(2 B-C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{2 d}+\frac{1}{2} a^3 x (6 B+7 C)+\frac{5 a^3 C \sin (c+d x)}{2 d}+\frac{a B \tan (c+d x) (a \cos (c+d x)+a)^2}{d}",1,"(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(2*(6*B + 7*C)*x - (4*(3*B + C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (4*(3*B + C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (4*(B + 3*C)*Cos[d*x]*Sin[c])/d + (C*Cos[2*d*x]*Sin[2*c])/d + (4*(B + 3*C)*Cos[c]*Sin[d*x])/d + (C*Cos[2*c]*Sin[2*d*x])/d + (4*B*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (4*B*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/32","B",1
249,1,208,114,1.8416483,"\int (a+a \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a^3 \left(4 (3 B+C) \tan (c+d x)+\frac{B}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{B}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-14 B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+14 B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+4 B c+4 B d x+4 C \sin (c+d x)-12 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+12 c C+12 C d x\right)}{4 d}","\frac{a^3 (7 B+6 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(2 B+C) \tan (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{d}-\frac{5 a^3 B \sin (c+d x)}{2 d}+a^3 x (B+3 C)+\frac{a B \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}",1,"(a^3*(4*B*c + 12*c*C + 4*B*d*x + 12*C*d*x - 14*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 12*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 14*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 12*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + B/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 - B/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + 4*C*Sin[c + d*x] + 4*(3*B + C)*Tan[c + d*x]))/(4*d)","A",1
250,1,786,125,6.375807,"\int (a+a \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{\sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \left(11 B \sin \left(\frac{d x}{2}\right)+9 C \sin \left(\frac{d x}{2}\right)\right)}{24 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \left(11 B \sin \left(\frac{d x}{2}\right)+9 C \sin \left(\frac{d x}{2}\right)\right)}{24 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \left(-8 B \sin \left(\frac{c}{2}\right)+10 B \cos \left(\frac{c}{2}\right)-3 C \sin \left(\frac{c}{2}\right)+3 C \cos \left(\frac{c}{2}\right)\right)}{96 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \left(-8 B \sin \left(\frac{c}{2}\right)-10 B \cos \left(\frac{c}{2}\right)-3 C \sin \left(\frac{c}{2}\right)-3 C \cos \left(\frac{c}{2}\right)\right)}{96 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{(-5 B-7 C) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{16 d}+\frac{(5 B+7 C) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{16 d}+\frac{B \sin \left(\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3}{48 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{B \sin \left(\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3}{48 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{1}{8} C x \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3","\frac{5 a^3 (B+C) \tan (c+d x)}{2 d}+\frac{a^3 (5 B+7 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(5 B+3 C) \tan (c+d x) \sec (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{6 d}+a^3 C x+\frac{a B \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^2}{3 d}",1,"(C*x*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6)/8 + ((-5*B - 7*C)*(a + a*Cos[c + d*x])^3*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^6)/(16*d) + ((5*B + 7*C)*(a + a*Cos[c + d*x])^3*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^6)/(16*d) + (B*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*Sin[(d*x)/2])/(48*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + ((a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(10*B*Cos[c/2] + 3*C*Cos[c/2] - 8*B*Sin[c/2] - 3*C*Sin[c/2]))/(96*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + ((a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(11*B*Sin[(d*x)/2] + 9*C*Sin[(d*x)/2]))/(24*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (B*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*Sin[(d*x)/2])/(48*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + ((a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-10*B*Cos[c/2] - 3*C*Cos[c/2] - 8*B*Sin[c/2] - 3*C*Sin[c/2]))/(96*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + ((a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(11*B*Sin[(d*x)/2] + 9*C*Sin[(d*x)/2]))/(24*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",1
251,1,273,154,1.2664412,"\int (a+a \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","-\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \left(120 (3 B+4 C) \cos ^4(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-24 (9 B+11 C) \sin (c)+69 B \sin (2 c+d x)+264 B \sin (c+2 d x)-24 B \sin (3 c+2 d x)+45 B \sin (2 c+3 d x)+45 B \sin (4 c+3 d x)+72 B \sin (3 c+4 d x)+(69 B+36 C) \sin (d x)+36 C \sin (2 c+d x)+280 C \sin (c+2 d x)-72 C \sin (3 c+2 d x)+36 C \sin (2 c+3 d x)+36 C \sin (4 c+3 d x)+88 C \sin (3 c+4 d x))\right)}{1536 d}","\frac{a^3 (9 B+11 C) \tan (c+d x)}{3 d}+\frac{5 a^3 (3 B+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 (27 B+28 C) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{(3 B+2 C) \tan (c+d x) \sec ^2(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{6 d}+\frac{a B \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}",1,"-1/1536*(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*Sec[c + d*x]^4*(120*(3*B + 4*C)*Cos[c + d*x]^4*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(-24*(9*B + 11*C)*Sin[c] + (69*B + 36*C)*Sin[d*x] + 69*B*Sin[2*c + d*x] + 36*C*Sin[2*c + d*x] + 264*B*Sin[c + 2*d*x] + 280*C*Sin[c + 2*d*x] - 24*B*Sin[3*c + 2*d*x] - 72*C*Sin[3*c + 2*d*x] + 45*B*Sin[2*c + 3*d*x] + 36*C*Sin[2*c + 3*d*x] + 45*B*Sin[4*c + 3*d*x] + 36*C*Sin[4*c + 3*d*x] + 72*B*Sin[3*c + 4*d*x] + 88*C*Sin[3*c + 4*d*x])))/d","A",1
252,1,294,185,1.4496247,"\int (a+a \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","-\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \left(240 (13 B+15 C) \cos ^5(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-240 (3 B+5 C) \sin (2 c+d x)+750 B \sin (c+2 d x)+750 B \sin (3 c+2 d x)+1520 B \sin (2 c+3 d x)+195 B \sin (3 c+4 d x)+195 B \sin (5 c+4 d x)+304 B \sin (4 c+5 d x)+80 (29 B+30 C) \sin (d x)+570 C \sin (c+2 d x)+570 C \sin (3 c+2 d x)+1680 C \sin (2 c+3 d x)-120 C \sin (4 c+3 d x)+225 C \sin (3 c+4 d x)+225 C \sin (5 c+4 d x)+360 C \sin (4 c+5 d x))\right)}{15360 d}","\frac{a^3 (38 B+45 C) \tan (c+d x)}{15 d}+\frac{a^3 (13 B+15 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 (43 B+45 C) \tan (c+d x) \sec ^2(c+d x)}{60 d}+\frac{a^3 (13 B+15 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{(7 B+5 C) \tan (c+d x) \sec ^3(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{20 d}+\frac{a B \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}",1,"-1/15360*(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*Sec[c + d*x]^5*(240*(13*B + 15*C)*Cos[c + d*x]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(80*(29*B + 30*C)*Sin[d*x] - 240*(3*B + 5*C)*Sin[2*c + d*x] + 750*B*Sin[c + 2*d*x] + 570*C*Sin[c + 2*d*x] + 750*B*Sin[3*c + 2*d*x] + 570*C*Sin[3*c + 2*d*x] + 1520*B*Sin[2*c + 3*d*x] + 1680*C*Sin[2*c + 3*d*x] - 120*C*Sin[4*c + 3*d*x] + 195*B*Sin[3*c + 4*d*x] + 225*C*Sin[3*c + 4*d*x] + 195*B*Sin[5*c + 4*d*x] + 225*C*Sin[5*c + 4*d*x] + 304*B*Sin[4*c + 5*d*x] + 360*C*Sin[4*c + 5*d*x])))/d","A",1
253,1,249,122,0.5663682,"\int \frac{\cos ^2(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(36 d x (B-C) \cos \left(c+\frac{d x}{2}\right)-12 B \sin \left(c+\frac{d x}{2}\right)-9 B \sin \left(c+\frac{3 d x}{2}\right)-9 B \sin \left(2 c+\frac{3 d x}{2}\right)+3 B \sin \left(2 c+\frac{5 d x}{2}\right)+3 B \sin \left(3 c+\frac{5 d x}{2}\right)+36 d x (B-C) \cos \left(\frac{d x}{2}\right)-60 B \sin \left(\frac{d x}{2}\right)+21 C \sin \left(c+\frac{d x}{2}\right)+18 C \sin \left(c+\frac{3 d x}{2}\right)+18 C \sin \left(2 c+\frac{3 d x}{2}\right)-2 C \sin \left(2 c+\frac{5 d x}{2}\right)-2 C \sin \left(3 c+\frac{5 d x}{2}\right)+C \sin \left(3 c+\frac{7 d x}{2}\right)+C \sin \left(4 c+\frac{7 d x}{2}\right)+69 C \sin \left(\frac{d x}{2}\right)\right)}{24 a d (\cos (c+d x)+1)}","\frac{(3 B-4 C) \sin ^3(c+d x)}{3 a d}-\frac{(3 B-4 C) \sin (c+d x)}{a d}+\frac{(B-C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}+\frac{3 (B-C) \sin (c+d x) \cos (c+d x)}{2 a d}+\frac{3 x (B-C)}{2 a}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(36*(B - C)*d*x*Cos[(d*x)/2] + 36*(B - C)*d*x*Cos[c + (d*x)/2] - 60*B*Sin[(d*x)/2] + 69*C*Sin[(d*x)/2] - 12*B*Sin[c + (d*x)/2] + 21*C*Sin[c + (d*x)/2] - 9*B*Sin[c + (3*d*x)/2] + 18*C*Sin[c + (3*d*x)/2] - 9*B*Sin[2*c + (3*d*x)/2] + 18*C*Sin[2*c + (3*d*x)/2] + 3*B*Sin[2*c + (5*d*x)/2] - 2*C*Sin[2*c + (5*d*x)/2] + 3*B*Sin[3*c + (5*d*x)/2] - 2*C*Sin[3*c + (5*d*x)/2] + C*Sin[3*c + (7*d*x)/2] + C*Sin[4*c + (7*d*x)/2]))/(24*a*d*(1 + Cos[c + d*x]))","B",1
254,1,197,99,0.4411452,"\int \frac{\cos (c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-4 d x (2 B-3 C) \cos \left(c+\frac{d x}{2}\right)+4 B \sin \left(c+\frac{d x}{2}\right)+4 B \sin \left(c+\frac{3 d x}{2}\right)+4 B \sin \left(2 c+\frac{3 d x}{2}\right)-4 d x (2 B-3 C) \cos \left(\frac{d x}{2}\right)+20 B \sin \left(\frac{d x}{2}\right)-4 C \sin \left(c+\frac{d x}{2}\right)-3 C \sin \left(c+\frac{3 d x}{2}\right)-3 C \sin \left(2 c+\frac{3 d x}{2}\right)+C \sin \left(2 c+\frac{5 d x}{2}\right)+C \sin \left(3 c+\frac{5 d x}{2}\right)-20 C \sin \left(\frac{d x}{2}\right)\right)}{8 a d (\cos (c+d x)+1)}","\frac{2 (B-C) \sin (c+d x)}{a d}+\frac{(B-C) \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(2 B-3 C) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{x (2 B-3 C)}{2 a}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(-4*(2*B - 3*C)*d*x*Cos[(d*x)/2] - 4*(2*B - 3*C)*d*x*Cos[c + (d*x)/2] + 20*B*Sin[(d*x)/2] - 20*C*Sin[(d*x)/2] + 4*B*Sin[c + (d*x)/2] - 4*C*Sin[c + (d*x)/2] + 4*B*Sin[c + (3*d*x)/2] - 3*C*Sin[c + (3*d*x)/2] + 4*B*Sin[2*c + (3*d*x)/2] - 3*C*Sin[2*c + (3*d*x)/2] + C*Sin[2*c + (5*d*x)/2] + C*Sin[3*c + (5*d*x)/2]))/(8*a*d*(1 + Cos[c + d*x]))","A",1
255,1,126,54,0.2437169,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(2 d x (B-C) \cos \left(c+\frac{d x}{2}\right)+2 d x (B-C) \cos \left(\frac{d x}{2}\right)-4 B \sin \left(\frac{d x}{2}\right)+C \sin \left(c+\frac{d x}{2}\right)+C \sin \left(c+\frac{3 d x}{2}\right)+C \sin \left(2 c+\frac{3 d x}{2}\right)+5 C \sin \left(\frac{d x}{2}\right)\right)}{2 a d (\cos (c+d x)+1)}","-\frac{(B-C) \sin (c+d x)}{a d (\cos (c+d x)+1)}+\frac{x (B-C)}{a}+\frac{C \sin (c+d x)}{a d}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(2*(B - C)*d*x*Cos[(d*x)/2] + 2*(B - C)*d*x*Cos[c + (d*x)/2] - 4*B*Sin[(d*x)/2] + 5*C*Sin[(d*x)/2] + C*Sin[c + (d*x)/2] + C*Sin[c + (3*d*x)/2] + C*Sin[2*c + (3*d*x)/2]))/(2*a*d*(1 + Cos[c + d*x]))","B",1
256,1,72,34,0.1240489,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(2 (B-C) \sin \left(\frac{d x}{2}\right)+C d x \cos \left(c+\frac{d x}{2}\right)+C d x \cos \left(\frac{d x}{2}\right)\right)}{a d (\cos (c+d x)+1)}","\frac{(B-C) \sin (c+d x)}{d (a \cos (c+d x)+a)}+\frac{C x}{a}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(C*d*x*Cos[(d*x)/2] + C*d*x*Cos[c + (d*x)/2] + 2*(B - C)*Sin[(d*x)/2]))/(a*d*(1 + Cos[c + d*x]))","B",1
257,1,109,44,0.2505639,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x]),x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left((C-B) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+B \cos \left(\frac{1}{2} (c+d x)\right) \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{a d (\cos (c+d x)+1)}","\frac{B \tanh ^{-1}(\sin (c+d x))}{a d}-\frac{(B-C) \sin (c+d x)}{d (a \cos (c+d x)+a)}",1,"(2*Cos[(c + d*x)/2]*(B*Cos[(c + d*x)/2]*(-Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (-B + C)*Sec[c/2]*Sin[(d*x)/2]))/(a*d*(1 + Cos[c + d*x]))","B",1
258,1,201,69,1.1510776,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x]),x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left((B-C) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+\cos \left(\frac{1}{2} (c+d x)\right) \left((B-C) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+\frac{B \sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}\right)\right)}{a d (\cos (c+d x)+1)}","\frac{(2 B-C) \tan (c+d x)}{a d}-\frac{(B-C) \tanh ^{-1}(\sin (c+d x))}{a d}-\frac{(B-C) \tan (c+d x)}{d (a \cos (c+d x)+a)}",1,"(2*Cos[(c + d*x)/2]*((B - C)*Sec[c/2]*Sin[(d*x)/2] + Cos[(c + d*x)/2]*((B - C)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (B*Sin[d*x])/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))))/(a*d*(1 + Cos[c + d*x]))","B",1
259,1,289,107,3.231407,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x]),x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(4 (C-B) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+\cos \left(\frac{1}{2} (c+d x)\right) \left(-\frac{4 (B-C) \sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+(4 C-6 B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+\frac{B}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{B}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+6 B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-4 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{2 a d (\cos (c+d x)+1)}","-\frac{2 (B-C) \tan (c+d x)}{a d}+\frac{(3 B-2 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{(3 B-2 C) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(B-C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}",1,"(Cos[(c + d*x)/2]*(4*(-B + C)*Sec[c/2]*Sin[(d*x)/2] + Cos[(c + d*x)/2]*((-6*B + 4*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 4*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + B/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 - B/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - (4*(B - C)*Sin[d*x])/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))))/(2*a*d*(1 + Cos[c + d*x]))","B",1
260,1,490,131,4.4835094,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5)/(a + a*Cos[c + d*x]),x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(\sec \left(\frac{c}{2}\right) \sec (c) \sec ^3(c+d x) \left(-24 B \sin \left(c-\frac{d x}{2}\right)-6 B \sin \left(c+\frac{d x}{2}\right)-24 B \sin \left(2 c+\frac{d x}{2}\right)+21 B \sin \left(c+\frac{3 d x}{2}\right)+9 B \sin \left(2 c+\frac{3 d x}{2}\right)-9 B \sin \left(3 c+\frac{3 d x}{2}\right)+7 B \sin \left(c+\frac{5 d x}{2}\right)+B \sin \left(2 c+\frac{5 d x}{2}\right)-3 B \sin \left(3 c+\frac{5 d x}{2}\right)-9 B \sin \left(4 c+\frac{5 d x}{2}\right)+16 B \sin \left(2 c+\frac{7 d x}{2}\right)+10 B \sin \left(3 c+\frac{7 d x}{2}\right)+6 B \sin \left(4 c+\frac{7 d x}{2}\right)+6 (B+C) \sin \left(\frac{d x}{2}\right)+3 (13 B-9 C) \sin \left(\frac{3 d x}{2}\right)+12 C \sin \left(c-\frac{d x}{2}\right)+6 C \sin \left(c+\frac{d x}{2}\right)+24 C \sin \left(2 c+\frac{d x}{2}\right)-9 C \sin \left(c+\frac{3 d x}{2}\right)-9 C \sin \left(2 c+\frac{3 d x}{2}\right)+9 C \sin \left(3 c+\frac{3 d x}{2}\right)-3 C \sin \left(c+\frac{5 d x}{2}\right)+3 C \sin \left(2 c+\frac{5 d x}{2}\right)+3 C \sin \left(3 c+\frac{5 d x}{2}\right)+9 C \sin \left(4 c+\frac{5 d x}{2}\right)-12 C \sin \left(2 c+\frac{7 d x}{2}\right)-6 C \sin \left(3 c+\frac{7 d x}{2}\right)-6 C \sin \left(4 c+\frac{7 d x}{2}\right)\right)+144 (B-C) \cos \left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{48 a d (\cos (c+d x)+1)}","\frac{(4 B-3 C) \tan ^3(c+d x)}{3 a d}+\frac{(4 B-3 C) \tan (c+d x)}{a d}-\frac{3 (B-C) \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{3 (B-C) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(B-C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}",1,"(Cos[(c + d*x)/2]*(144*(B - C)*Cos[(c + d*x)/2]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c/2]*Sec[c]*Sec[c + d*x]^3*(6*(B + C)*Sin[(d*x)/2] + 3*(13*B - 9*C)*Sin[(3*d*x)/2] - 24*B*Sin[c - (d*x)/2] + 12*C*Sin[c - (d*x)/2] - 6*B*Sin[c + (d*x)/2] + 6*C*Sin[c + (d*x)/2] - 24*B*Sin[2*c + (d*x)/2] + 24*C*Sin[2*c + (d*x)/2] + 21*B*Sin[c + (3*d*x)/2] - 9*C*Sin[c + (3*d*x)/2] + 9*B*Sin[2*c + (3*d*x)/2] - 9*C*Sin[2*c + (3*d*x)/2] - 9*B*Sin[3*c + (3*d*x)/2] + 9*C*Sin[3*c + (3*d*x)/2] + 7*B*Sin[c + (5*d*x)/2] - 3*C*Sin[c + (5*d*x)/2] + B*Sin[2*c + (5*d*x)/2] + 3*C*Sin[2*c + (5*d*x)/2] - 3*B*Sin[3*c + (5*d*x)/2] + 3*C*Sin[3*c + (5*d*x)/2] - 9*B*Sin[4*c + (5*d*x)/2] + 9*C*Sin[4*c + (5*d*x)/2] + 16*B*Sin[2*c + (7*d*x)/2] - 12*C*Sin[2*c + (7*d*x)/2] + 10*B*Sin[3*c + (7*d*x)/2] - 6*C*Sin[3*c + (7*d*x)/2] + 6*B*Sin[4*c + (7*d*x)/2] - 6*C*Sin[4*c + (7*d*x)/2])))/(48*a*d*(1 + Cos[c + d*x]))","B",1
261,1,369,170,0.6308478,"\int \frac{\cos ^3(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(36 d x (7 B-10 C) \cos \left(c+\frac{d x}{2}\right)+147 B \sin \left(c+\frac{d x}{2}\right)-239 B \sin \left(c+\frac{3 d x}{2}\right)-63 B \sin \left(2 c+\frac{3 d x}{2}\right)-15 B \sin \left(2 c+\frac{5 d x}{2}\right)-15 B \sin \left(3 c+\frac{5 d x}{2}\right)+3 B \sin \left(3 c+\frac{7 d x}{2}\right)+3 B \sin \left(4 c+\frac{7 d x}{2}\right)+84 B d x \cos \left(c+\frac{3 d x}{2}\right)+84 B d x \cos \left(2 c+\frac{3 d x}{2}\right)+36 d x (7 B-10 C) \cos \left(\frac{d x}{2}\right)-381 B \sin \left(\frac{d x}{2}\right)-156 C \sin \left(c+\frac{d x}{2}\right)+342 C \sin \left(c+\frac{3 d x}{2}\right)+118 C \sin \left(2 c+\frac{3 d x}{2}\right)+30 C \sin \left(2 c+\frac{5 d x}{2}\right)+30 C \sin \left(3 c+\frac{5 d x}{2}\right)-3 C \sin \left(3 c+\frac{7 d x}{2}\right)-3 C \sin \left(4 c+\frac{7 d x}{2}\right)+C \sin \left(4 c+\frac{9 d x}{2}\right)+C \sin \left(5 c+\frac{9 d x}{2}\right)-120 C d x \cos \left(c+\frac{3 d x}{2}\right)-120 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+516 C \sin \left(\frac{d x}{2}\right)\right)}{48 a^2 d (\cos (c+d x)+1)^2}","\frac{4 (2 B-3 C) \sin ^3(c+d x)}{3 a^2 d}-\frac{4 (2 B-3 C) \sin (c+d x)}{a^2 d}+\frac{(7 B-10 C) \sin (c+d x) \cos ^3(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{(7 B-10 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{x (7 B-10 C)}{2 a^2}+\frac{(B-C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(36*(7*B - 10*C)*d*x*Cos[(d*x)/2] + 36*(7*B - 10*C)*d*x*Cos[c + (d*x)/2] + 84*B*d*x*Cos[c + (3*d*x)/2] - 120*C*d*x*Cos[c + (3*d*x)/2] + 84*B*d*x*Cos[2*c + (3*d*x)/2] - 120*C*d*x*Cos[2*c + (3*d*x)/2] - 381*B*Sin[(d*x)/2] + 516*C*Sin[(d*x)/2] + 147*B*Sin[c + (d*x)/2] - 156*C*Sin[c + (d*x)/2] - 239*B*Sin[c + (3*d*x)/2] + 342*C*Sin[c + (3*d*x)/2] - 63*B*Sin[2*c + (3*d*x)/2] + 118*C*Sin[2*c + (3*d*x)/2] - 15*B*Sin[2*c + (5*d*x)/2] + 30*C*Sin[2*c + (5*d*x)/2] - 15*B*Sin[3*c + (5*d*x)/2] + 30*C*Sin[3*c + (5*d*x)/2] + 3*B*Sin[3*c + (7*d*x)/2] - 3*C*Sin[3*c + (7*d*x)/2] + 3*B*Sin[4*c + (7*d*x)/2] - 3*C*Sin[4*c + (7*d*x)/2] + C*Sin[4*c + (9*d*x)/2] + C*Sin[5*c + (9*d*x)/2]))/(48*a^2*d*(1 + Cos[c + d*x])^2)","B",1
262,1,315,147,0.8699061,"\int \frac{\cos ^2(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-36 d x (4 B-7 C) \cos \left(c+\frac{d x}{2}\right)-120 B \sin \left(c+\frac{d x}{2}\right)+164 B \sin \left(c+\frac{3 d x}{2}\right)+36 B \sin \left(2 c+\frac{3 d x}{2}\right)+12 B \sin \left(2 c+\frac{5 d x}{2}\right)+12 B \sin \left(3 c+\frac{5 d x}{2}\right)-48 B d x \cos \left(c+\frac{3 d x}{2}\right)-48 B d x \cos \left(2 c+\frac{3 d x}{2}\right)-36 d x (4 B-7 C) \cos \left(\frac{d x}{2}\right)+264 B \sin \left(\frac{d x}{2}\right)+147 C \sin \left(c+\frac{d x}{2}\right)-239 C \sin \left(c+\frac{3 d x}{2}\right)-63 C \sin \left(2 c+\frac{3 d x}{2}\right)-15 C \sin \left(2 c+\frac{5 d x}{2}\right)-15 C \sin \left(3 c+\frac{5 d x}{2}\right)+3 C \sin \left(3 c+\frac{7 d x}{2}\right)+3 C \sin \left(4 c+\frac{7 d x}{2}\right)+84 C d x \cos \left(c+\frac{3 d x}{2}\right)+84 C d x \cos \left(2 c+\frac{3 d x}{2}\right)-381 C \sin \left(\frac{d x}{2}\right)\right)}{48 a^2 d (\cos (c+d x)+1)^2}","\frac{2 (5 B-8 C) \sin (c+d x)}{3 a^2 d}+\frac{(5 B-8 C) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(4 B-7 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{x (4 B-7 C)}{2 a^2}+\frac{(B-C) \sin (c+d x) \cos ^3(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(-36*(4*B - 7*C)*d*x*Cos[(d*x)/2] - 36*(4*B - 7*C)*d*x*Cos[c + (d*x)/2] - 48*B*d*x*Cos[c + (3*d*x)/2] + 84*C*d*x*Cos[c + (3*d*x)/2] - 48*B*d*x*Cos[2*c + (3*d*x)/2] + 84*C*d*x*Cos[2*c + (3*d*x)/2] + 264*B*Sin[(d*x)/2] - 381*C*Sin[(d*x)/2] - 120*B*Sin[c + (d*x)/2] + 147*C*Sin[c + (d*x)/2] + 164*B*Sin[c + (3*d*x)/2] - 239*C*Sin[c + (3*d*x)/2] + 36*B*Sin[2*c + (3*d*x)/2] - 63*C*Sin[2*c + (3*d*x)/2] + 12*B*Sin[2*c + (5*d*x)/2] - 15*C*Sin[2*c + (5*d*x)/2] + 12*B*Sin[3*c + (5*d*x)/2] - 15*C*Sin[3*c + (5*d*x)/2] + 3*C*Sin[3*c + (7*d*x)/2] + 3*C*Sin[4*c + (7*d*x)/2]))/(48*a^2*d*(1 + Cos[c + d*x])^2)","B",1
263,1,137,99,0.7042521,"\int \frac{\cos (c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(6 \cos ^3\left(\frac{1}{2} (c+d x)\right) (d x (B-2 C)+C \sin (c+d x))+(B-C) \tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)+(B-C) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)-2 (5 B-8 C) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{3 a^2 d (\cos (c+d x)+1)^2}","-\frac{(B-4 C) \sin (c+d x)}{3 a^2 d}-\frac{(B-2 C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{x (B-2 C)}{a^2}+\frac{(B-C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(2*Cos[(c + d*x)/2]*((B - C)*Sec[c/2]*Sin[(d*x)/2] - 2*(5*B - 8*C)*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + 6*Cos[(c + d*x)/2]^3*((B - 2*C)*d*x + C*Sin[c + d*x]) + (B - C)*Cos[(c + d*x)/2]*Tan[c/2]))/(3*a^2*d*(1 + Cos[c + d*x])^2)","A",1
264,1,153,70,0.349733,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \left(-6 B \sin \left(c+\frac{d x}{2}\right)+4 B \sin \left(c+\frac{3 d x}{2}\right)+6 B \sin \left(\frac{d x}{2}\right)+12 C \sin \left(c+\frac{d x}{2}\right)-10 C \sin \left(c+\frac{3 d x}{2}\right)+9 C d x \cos \left(c+\frac{d x}{2}\right)+3 C d x \cos \left(c+\frac{3 d x}{2}\right)+3 C d x \cos \left(2 c+\frac{3 d x}{2}\right)-18 C \sin \left(\frac{d x}{2}\right)+9 C d x \cos \left(\frac{d x}{2}\right)\right)}{24 a^2 d}","\frac{(2 B-5 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{C x}{a^2}-\frac{(B-C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^3*(9*C*d*x*Cos[(d*x)/2] + 9*C*d*x*Cos[c + (d*x)/2] + 3*C*d*x*Cos[c + (3*d*x)/2] + 3*C*d*x*Cos[2*c + (3*d*x)/2] + 6*B*Sin[(d*x)/2] - 18*C*Sin[(d*x)/2] - 6*B*Sin[c + (d*x)/2] + 12*C*Sin[c + (d*x)/2] + 4*B*Sin[c + (3*d*x)/2] - 10*C*Sin[c + (3*d*x)/2]))/(24*a^2*d)","B",1
265,1,76,65,0.180426,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left((B+2 C) \sin \left(c+\frac{3 d x}{2}\right)+3 (B+C) \sin \left(\frac{d x}{2}\right)-3 C \sin \left(c+\frac{d x}{2}\right)\right)}{3 a^2 d (\cos (c+d x)+1)^2}","\frac{(B+2 C) \sin (c+d x)}{3 d \left(a^2 \cos (c+d x)+a^2\right)}+\frac{(B-C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(3*(B + C)*Sin[(d*x)/2] - 3*C*Sin[c + (d*x)/2] + (B + 2*C)*Sin[c + (3*d*x)/2]))/(3*a^2*d*(1 + Cos[c + d*x])^2)","A",1
266,1,170,79,0.5078265,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^2,x]","-\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left((B-C) \tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)+(B-C) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+2 (4 B-C) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)+6 B \cos ^3\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{3 a^2 d (\cos (c+d x)+1)^2}","-\frac{(4 B-C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{B \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(B-C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(-2*Cos[(c + d*x)/2]*(6*B*Cos[(c + d*x)/2]^3*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (B - C)*Sec[c/2]*Sin[(d*x)/2] + 2*(4*B - C)*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + (B - C)*Cos[(c + d*x)/2]*Tan[c/2]))/(3*a^2*d*(1 + Cos[c + d*x])^2)","B",1
267,1,264,107,1.622962,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^2,x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left((B-C) \tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)+(B-C) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+6 \cos ^3\left(\frac{1}{2} (c+d x)\right) \left((2 B-C) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+\frac{B \sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}\right)+2 (7 B-4 C) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{3 a^2 d (\cos (c+d x)+1)^2}","\frac{2 (5 B-2 C) \tan (c+d x)}{3 a^2 d}-\frac{(2 B-C) \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(2 B-C) \tan (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{(B-C) \tan (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(2*Cos[(c + d*x)/2]*((B - C)*Sec[c/2]*Sin[(d*x)/2] + 2*(7*B - 4*C)*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + 6*Cos[(c + d*x)/2]^3*((2*B - C)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (B*Sin[d*x])/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))) + (B - C)*Cos[(c + d*x)/2]*Tan[c/2]))/(3*a^2*d*(1 + Cos[c + d*x])^2)","B",1
268,1,496,152,3.1923203,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^2,x]","-\frac{96 (7 B-4 C) \cos ^4\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+\sec \left(\frac{c}{2}\right) \sec (c) \cos \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(-126 B \sin \left(c-\frac{d x}{2}\right)+42 B \sin \left(c+\frac{d x}{2}\right)-98 B \sin \left(2 c+\frac{d x}{2}\right)-3 B \sin \left(c+\frac{3 d x}{2}\right)+37 B \sin \left(2 c+\frac{3 d x}{2}\right)-63 B \sin \left(3 c+\frac{3 d x}{2}\right)+75 B \sin \left(c+\frac{5 d x}{2}\right)+15 B \sin \left(2 c+\frac{5 d x}{2}\right)+39 B \sin \left(3 c+\frac{5 d x}{2}\right)-21 B \sin \left(4 c+\frac{5 d x}{2}\right)+32 B \sin \left(2 c+\frac{7 d x}{2}\right)+12 B \sin \left(3 c+\frac{7 d x}{2}\right)+20 B \sin \left(4 c+\frac{7 d x}{2}\right)-14 (B-C) \sin \left(\frac{d x}{2}\right)+(97 B-64 C) \sin \left(\frac{3 d x}{2}\right)+84 C \sin \left(c-\frac{d x}{2}\right)-42 C \sin \left(c+\frac{d x}{2}\right)+56 C \sin \left(2 c+\frac{d x}{2}\right)+6 C \sin \left(c+\frac{3 d x}{2}\right)-34 C \sin \left(2 c+\frac{3 d x}{2}\right)+36 C \sin \left(3 c+\frac{3 d x}{2}\right)-48 C \sin \left(c+\frac{5 d x}{2}\right)-6 C \sin \left(2 c+\frac{5 d x}{2}\right)-30 C \sin \left(3 c+\frac{5 d x}{2}\right)+12 C \sin \left(4 c+\frac{5 d x}{2}\right)-20 C \sin \left(2 c+\frac{7 d x}{2}\right)-6 C \sin \left(3 c+\frac{7 d x}{2}\right)-14 C \sin \left(4 c+\frac{7 d x}{2}\right)\right)}{48 a^2 d (\cos (c+d x)+1)^2}","-\frac{2 (8 B-5 C) \tan (c+d x)}{3 a^2 d}+\frac{(7 B-4 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac{(7 B-4 C) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{(8 B-5 C) \tan (c+d x) \sec (c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(B-C) \tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"-1/48*(96*(7*B - 4*C)*Cos[(c + d*x)/2]^4*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Cos[(c + d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]^2*(-14*(B - C)*Sin[(d*x)/2] + (97*B - 64*C)*Sin[(3*d*x)/2] - 126*B*Sin[c - (d*x)/2] + 84*C*Sin[c - (d*x)/2] + 42*B*Sin[c + (d*x)/2] - 42*C*Sin[c + (d*x)/2] - 98*B*Sin[2*c + (d*x)/2] + 56*C*Sin[2*c + (d*x)/2] - 3*B*Sin[c + (3*d*x)/2] + 6*C*Sin[c + (3*d*x)/2] + 37*B*Sin[2*c + (3*d*x)/2] - 34*C*Sin[2*c + (3*d*x)/2] - 63*B*Sin[3*c + (3*d*x)/2] + 36*C*Sin[3*c + (3*d*x)/2] + 75*B*Sin[c + (5*d*x)/2] - 48*C*Sin[c + (5*d*x)/2] + 15*B*Sin[2*c + (5*d*x)/2] - 6*C*Sin[2*c + (5*d*x)/2] + 39*B*Sin[3*c + (5*d*x)/2] - 30*C*Sin[3*c + (5*d*x)/2] - 21*B*Sin[4*c + (5*d*x)/2] + 12*C*Sin[4*c + (5*d*x)/2] + 32*B*Sin[2*c + (7*d*x)/2] - 20*C*Sin[2*c + (7*d*x)/2] + 12*B*Sin[3*c + (7*d*x)/2] - 6*C*Sin[3*c + (7*d*x)/2] + 20*B*Sin[4*c + (7*d*x)/2] - 14*C*Sin[4*c + (7*d*x)/2]))/(a^2*d*(1 + Cos[c + d*x])^2)","B",1
269,1,435,193,0.8488075,"\int \frac{\cos ^3(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-600 d x (6 B-13 C) \cos \left(c+\frac{d x}{2}\right)-4500 B \sin \left(c+\frac{d x}{2}\right)+4860 B \sin \left(c+\frac{3 d x}{2}\right)-900 B \sin \left(2 c+\frac{3 d x}{2}\right)+1452 B \sin \left(2 c+\frac{5 d x}{2}\right)+300 B \sin \left(3 c+\frac{5 d x}{2}\right)+60 B \sin \left(3 c+\frac{7 d x}{2}\right)+60 B \sin \left(4 c+\frac{7 d x}{2}\right)-1800 B d x \cos \left(c+\frac{3 d x}{2}\right)-1800 B d x \cos \left(2 c+\frac{3 d x}{2}\right)-360 B d x \cos \left(2 c+\frac{5 d x}{2}\right)-360 B d x \cos \left(3 c+\frac{5 d x}{2}\right)-600 d x (6 B-13 C) \cos \left(\frac{d x}{2}\right)+7020 B \sin \left(\frac{d x}{2}\right)+7560 C \sin \left(c+\frac{d x}{2}\right)-9230 C \sin \left(c+\frac{3 d x}{2}\right)+930 C \sin \left(2 c+\frac{3 d x}{2}\right)-2782 C \sin \left(2 c+\frac{5 d x}{2}\right)-750 C \sin \left(3 c+\frac{5 d x}{2}\right)-105 C \sin \left(3 c+\frac{7 d x}{2}\right)-105 C \sin \left(4 c+\frac{7 d x}{2}\right)+15 C \sin \left(4 c+\frac{9 d x}{2}\right)+15 C \sin \left(5 c+\frac{9 d x}{2}\right)+3900 C d x \cos \left(c+\frac{3 d x}{2}\right)+3900 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+780 C d x \cos \left(2 c+\frac{5 d x}{2}\right)+780 C d x \cos \left(3 c+\frac{5 d x}{2}\right)-12760 C \sin \left(\frac{d x}{2}\right)\right)}{480 a^3 d (\cos (c+d x)+1)^3}","\frac{8 (9 B-19 C) \sin (c+d x)}{15 a^3 d}+\frac{4 (9 B-19 C) \sin (c+d x) \cos ^2(c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(6 B-13 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{x (6 B-13 C)}{2 a^3}+\frac{(B-C) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(6 B-11 C) \sin (c+d x) \cos ^3(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(-600*(6*B - 13*C)*d*x*Cos[(d*x)/2] - 600*(6*B - 13*C)*d*x*Cos[c + (d*x)/2] - 1800*B*d*x*Cos[c + (3*d*x)/2] + 3900*C*d*x*Cos[c + (3*d*x)/2] - 1800*B*d*x*Cos[2*c + (3*d*x)/2] + 3900*C*d*x*Cos[2*c + (3*d*x)/2] - 360*B*d*x*Cos[2*c + (5*d*x)/2] + 780*C*d*x*Cos[2*c + (5*d*x)/2] - 360*B*d*x*Cos[3*c + (5*d*x)/2] + 780*C*d*x*Cos[3*c + (5*d*x)/2] + 7020*B*Sin[(d*x)/2] - 12760*C*Sin[(d*x)/2] - 4500*B*Sin[c + (d*x)/2] + 7560*C*Sin[c + (d*x)/2] + 4860*B*Sin[c + (3*d*x)/2] - 9230*C*Sin[c + (3*d*x)/2] - 900*B*Sin[2*c + (3*d*x)/2] + 930*C*Sin[2*c + (3*d*x)/2] + 1452*B*Sin[2*c + (5*d*x)/2] - 2782*C*Sin[2*c + (5*d*x)/2] + 300*B*Sin[3*c + (5*d*x)/2] - 750*C*Sin[3*c + (5*d*x)/2] + 60*B*Sin[3*c + (7*d*x)/2] - 105*C*Sin[3*c + (7*d*x)/2] + 60*B*Sin[4*c + (7*d*x)/2] - 105*C*Sin[4*c + (7*d*x)/2] + 15*C*Sin[4*c + (9*d*x)/2] + 15*C*Sin[5*c + (9*d*x)/2]))/(480*a^3*d*(1 + Cos[c + d*x])^3)","B",1
270,1,361,147,0.8690297,"\int \frac{\cos ^2(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(300 d x (B-3 C) \cos \left(c+\frac{d x}{2}\right)+540 B \sin \left(c+\frac{d x}{2}\right)-460 B \sin \left(c+\frac{3 d x}{2}\right)+180 B \sin \left(2 c+\frac{3 d x}{2}\right)-128 B \sin \left(2 c+\frac{5 d x}{2}\right)+150 B d x \cos \left(c+\frac{3 d x}{2}\right)+150 B d x \cos \left(2 c+\frac{3 d x}{2}\right)+30 B d x \cos \left(2 c+\frac{5 d x}{2}\right)+30 B d x \cos \left(3 c+\frac{5 d x}{2}\right)+300 d x (B-3 C) \cos \left(\frac{d x}{2}\right)-740 B \sin \left(\frac{d x}{2}\right)-1125 C \sin \left(c+\frac{d x}{2}\right)+1215 C \sin \left(c+\frac{3 d x}{2}\right)-225 C \sin \left(2 c+\frac{3 d x}{2}\right)+363 C \sin \left(2 c+\frac{5 d x}{2}\right)+75 C \sin \left(3 c+\frac{5 d x}{2}\right)+15 C \sin \left(3 c+\frac{7 d x}{2}\right)+15 C \sin \left(4 c+\frac{7 d x}{2}\right)-450 C d x \cos \left(c+\frac{3 d x}{2}\right)-450 C d x \cos \left(2 c+\frac{3 d x}{2}\right)-90 C d x \cos \left(2 c+\frac{5 d x}{2}\right)-90 C d x \cos \left(3 c+\frac{5 d x}{2}\right)+1755 C \sin \left(\frac{d x}{2}\right)\right)}{120 a^3 d (\cos (c+d x)+1)^3}","-\frac{(7 B-27 C) \sin (c+d x)}{15 a^3 d}-\frac{(B-3 C) \sin (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{x (B-3 C)}{a^3}+\frac{(B-C) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(4 B-9 C) \sin (c+d x) \cos ^2(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(300*(B - 3*C)*d*x*Cos[(d*x)/2] + 300*(B - 3*C)*d*x*Cos[c + (d*x)/2] + 150*B*d*x*Cos[c + (3*d*x)/2] - 450*C*d*x*Cos[c + (3*d*x)/2] + 150*B*d*x*Cos[2*c + (3*d*x)/2] - 450*C*d*x*Cos[2*c + (3*d*x)/2] + 30*B*d*x*Cos[2*c + (5*d*x)/2] - 90*C*d*x*Cos[2*c + (5*d*x)/2] + 30*B*d*x*Cos[3*c + (5*d*x)/2] - 90*C*d*x*Cos[3*c + (5*d*x)/2] - 740*B*Sin[(d*x)/2] + 1755*C*Sin[(d*x)/2] + 540*B*Sin[c + (d*x)/2] - 1125*C*Sin[c + (d*x)/2] - 460*B*Sin[c + (3*d*x)/2] + 1215*C*Sin[c + (3*d*x)/2] + 180*B*Sin[2*c + (3*d*x)/2] - 225*C*Sin[2*c + (3*d*x)/2] - 128*B*Sin[2*c + (5*d*x)/2] + 363*C*Sin[2*c + (5*d*x)/2] + 75*C*Sin[3*c + (5*d*x)/2] + 15*C*Sin[3*c + (7*d*x)/2] + 15*C*Sin[4*c + (7*d*x)/2]))/(120*a^3*d*(1 + Cos[c + d*x])^3)","B",1
271,1,241,116,0.548503,"\int \frac{\cos (c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \left(-60 B \sin \left(c+\frac{d x}{2}\right)+40 B \sin \left(c+\frac{3 d x}{2}\right)-30 B \sin \left(2 c+\frac{3 d x}{2}\right)+14 B \sin \left(2 c+\frac{5 d x}{2}\right)+80 B \sin \left(\frac{d x}{2}\right)+270 C \sin \left(c+\frac{d x}{2}\right)-230 C \sin \left(c+\frac{3 d x}{2}\right)+90 C \sin \left(2 c+\frac{3 d x}{2}\right)-64 C \sin \left(2 c+\frac{5 d x}{2}\right)+150 C d x \cos \left(c+\frac{d x}{2}\right)+75 C d x \cos \left(c+\frac{3 d x}{2}\right)+75 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+15 C d x \cos \left(2 c+\frac{5 d x}{2}\right)+15 C d x \cos \left(3 c+\frac{5 d x}{2}\right)-370 C \sin \left(\frac{d x}{2}\right)+150 C d x \cos \left(\frac{d x}{2}\right)\right)}{480 a^3 d}","\frac{(4 B-29 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{C x}{a^3}+\frac{(B-C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(2 B-7 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^5*(150*C*d*x*Cos[(d*x)/2] + 150*C*d*x*Cos[c + (d*x)/2] + 75*C*d*x*Cos[c + (3*d*x)/2] + 75*C*d*x*Cos[2*c + (3*d*x)/2] + 15*C*d*x*Cos[2*c + (5*d*x)/2] + 15*C*d*x*Cos[3*c + (5*d*x)/2] + 80*B*Sin[(d*x)/2] - 370*C*Sin[(d*x)/2] - 60*B*Sin[c + (d*x)/2] + 270*C*Sin[c + (d*x)/2] + 40*B*Sin[c + (3*d*x)/2] - 230*C*Sin[c + (3*d*x)/2] - 30*B*Sin[2*c + (3*d*x)/2] + 90*C*Sin[2*c + (3*d*x)/2] + 14*B*Sin[2*c + (5*d*x)/2] - 64*C*Sin[2*c + (5*d*x)/2]))/(480*a^3*d)","B",1
272,1,135,102,0.3398981,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-15 (B+2 C) \sin \left(c+\frac{d x}{2}\right)+15 B \sin \left(c+\frac{3 d x}{2}\right)+3 B \sin \left(2 c+\frac{5 d x}{2}\right)+5 (3 B+8 C) \sin \left(\frac{d x}{2}\right)+20 C \sin \left(c+\frac{3 d x}{2}\right)-15 C \sin \left(2 c+\frac{3 d x}{2}\right)+7 C \sin \left(2 c+\frac{5 d x}{2}\right)\right)}{30 a^3 d (\cos (c+d x)+1)^3}","\frac{(3 B+7 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(3 B-8 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(B-C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(5*(3*B + 8*C)*Sin[(d*x)/2] - 15*(B + 2*C)*Sin[c + (d*x)/2] + 15*B*Sin[c + (3*d*x)/2] + 20*C*Sin[c + (3*d*x)/2] - 15*C*Sin[2*c + (3*d*x)/2] + 3*B*Sin[2*c + (5*d*x)/2] + 7*C*Sin[2*c + (5*d*x)/2]))/(30*a^3*d*(1 + Cos[c + d*x])^3)","A",1
273,1,96,102,0.2710989,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left((2 B+3 C) \left(5 \sin \left(c+\frac{3 d x}{2}\right)+\sin \left(2 c+\frac{5 d x}{2}\right)\right)+5 (4 B+3 C) \sin \left(\frac{d x}{2}\right)-15 C \sin \left(c+\frac{d x}{2}\right)\right)}{30 a^3 d (\cos (c+d x)+1)^3}","\frac{(2 B+3 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(2 B+3 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}+\frac{(B-C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(5*(4*B + 3*C)*Sin[(d*x)/2] - 15*C*Sin[c + (d*x)/2] + (2*B + 3*C)*(5*Sin[c + (3*d*x)/2] + Sin[2*c + (5*d*x)/2])))/(30*a^3*d*(1 + Cos[c + d*x])^3)","A",1
274,1,197,117,0.920642,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(75 B \sin \left(c+\frac{d x}{2}\right)-95 B \sin \left(c+\frac{3 d x}{2}\right)+15 B \sin \left(2 c+\frac{3 d x}{2}\right)-22 B \sin \left(2 c+\frac{5 d x}{2}\right)-5 (29 B-4 C) \sin \left(\frac{d x}{2}\right)+10 C \sin \left(c+\frac{3 d x}{2}\right)+2 C \sin \left(2 c+\frac{5 d x}{2}\right)\right)-240 B \cos ^6\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{30 a^3 d (\cos (c+d x)+1)^3}","-\frac{2 (11 B-C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{B \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{(7 B-2 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(B-C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"(-240*B*Cos[(c + d*x)/2]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Cos[(c + d*x)/2]*Sec[c/2]*(-5*(29*B - 4*C)*Sin[(d*x)/2] + 75*B*Sin[c + (d*x)/2] - 95*B*Sin[c + (3*d*x)/2] + 10*C*Sin[c + (3*d*x)/2] + 15*B*Sin[2*c + (3*d*x)/2] - 22*B*Sin[2*c + (5*d*x)/2] + 2*C*Sin[2*c + (5*d*x)/2]))/(30*a^3*d*(1 + Cos[c + d*x])^3)","A",1
275,1,482,145,3.0013198,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^3,x]","\frac{960 (3 B-C) \cos ^6\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+\sec \left(\frac{c}{2}\right) \sec (c) \cos \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(-600 B \sin \left(c-\frac{d x}{2}\right)+375 B \sin \left(c+\frac{d x}{2}\right)-480 B \sin \left(2 c+\frac{d x}{2}\right)-60 B \sin \left(c+\frac{3 d x}{2}\right)+402 B \sin \left(2 c+\frac{3 d x}{2}\right)-225 B \sin \left(3 c+\frac{3 d x}{2}\right)+315 B \sin \left(c+\frac{5 d x}{2}\right)+30 B \sin \left(2 c+\frac{5 d x}{2}\right)+240 B \sin \left(3 c+\frac{5 d x}{2}\right)-45 B \sin \left(4 c+\frac{5 d x}{2}\right)+72 B \sin \left(2 c+\frac{7 d x}{2}\right)+15 B \sin \left(3 c+\frac{7 d x}{2}\right)+57 B \sin \left(4 c+\frac{7 d x}{2}\right)-5 (51 B-32 C) \sin \left(\frac{d x}{2}\right)+(567 B-167 C) \sin \left(\frac{3 d x}{2}\right)+170 C \sin \left(c-\frac{d x}{2}\right)-170 C \sin \left(c+\frac{d x}{2}\right)+160 C \sin \left(2 c+\frac{d x}{2}\right)+75 C \sin \left(c+\frac{3 d x}{2}\right)-167 C \sin \left(2 c+\frac{3 d x}{2}\right)+75 C \sin \left(3 c+\frac{3 d x}{2}\right)-95 C \sin \left(c+\frac{5 d x}{2}\right)+15 C \sin \left(2 c+\frac{5 d x}{2}\right)-95 C \sin \left(3 c+\frac{5 d x}{2}\right)+15 C \sin \left(4 c+\frac{5 d x}{2}\right)-22 C \sin \left(2 c+\frac{7 d x}{2}\right)-22 C \sin \left(4 c+\frac{7 d x}{2}\right)\right)}{120 a^3 d (\cos (c+d x)+1)^3}","\frac{2 (36 B-11 C) \tan (c+d x)}{15 a^3 d}-\frac{(3 B-C) \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{(3 B-C) \tan (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(9 B-4 C) \tan (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(B-C) \tan (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"(960*(3*B - C)*Cos[(c + d*x)/2]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Cos[(c + d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]*(-5*(51*B - 32*C)*Sin[(d*x)/2] + (567*B - 167*C)*Sin[(3*d*x)/2] - 600*B*Sin[c - (d*x)/2] + 170*C*Sin[c - (d*x)/2] + 375*B*Sin[c + (d*x)/2] - 170*C*Sin[c + (d*x)/2] - 480*B*Sin[2*c + (d*x)/2] + 160*C*Sin[2*c + (d*x)/2] - 60*B*Sin[c + (3*d*x)/2] + 75*C*Sin[c + (3*d*x)/2] + 402*B*Sin[2*c + (3*d*x)/2] - 167*C*Sin[2*c + (3*d*x)/2] - 225*B*Sin[3*c + (3*d*x)/2] + 75*C*Sin[3*c + (3*d*x)/2] + 315*B*Sin[c + (5*d*x)/2] - 95*C*Sin[c + (5*d*x)/2] + 30*B*Sin[2*c + (5*d*x)/2] + 15*C*Sin[2*c + (5*d*x)/2] + 240*B*Sin[3*c + (5*d*x)/2] - 95*C*Sin[3*c + (5*d*x)/2] - 45*B*Sin[4*c + (5*d*x)/2] + 15*C*Sin[4*c + (5*d*x)/2] + 72*B*Sin[2*c + (7*d*x)/2] - 22*C*Sin[2*c + (7*d*x)/2] + 15*B*Sin[3*c + (7*d*x)/2] + 57*B*Sin[4*c + (7*d*x)/2] - 22*C*Sin[4*c + (7*d*x)/2]))/(120*a^3*d*(1 + Cos[c + d*x])^3)","B",1
276,1,610,196,4.7746325,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^3,x]","-\frac{1920 (13 B-6 C) \cos ^6\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+\sec \left(\frac{c}{2}\right) \sec (c) \cos \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(-4329 B \sin \left(c-\frac{d x}{2}\right)+1989 B \sin \left(c+\frac{d x}{2}\right)-3575 B \sin \left(2 c+\frac{d x}{2}\right)-475 B \sin \left(c+\frac{3 d x}{2}\right)+2005 B \sin \left(2 c+\frac{3 d x}{2}\right)-2275 B \sin \left(3 c+\frac{3 d x}{2}\right)+2673 B \sin \left(c+\frac{5 d x}{2}\right)+105 B \sin \left(2 c+\frac{5 d x}{2}\right)+1593 B \sin \left(3 c+\frac{5 d x}{2}\right)-975 B \sin \left(4 c+\frac{5 d x}{2}\right)+1325 B \sin \left(2 c+\frac{7 d x}{2}\right)+255 B \sin \left(3 c+\frac{7 d x}{2}\right)+875 B \sin \left(4 c+\frac{7 d x}{2}\right)-195 B \sin \left(5 c+\frac{7 d x}{2}\right)+304 B \sin \left(3 c+\frac{9 d x}{2}\right)+90 B \sin \left(4 c+\frac{9 d x}{2}\right)+214 B \sin \left(5 c+\frac{9 d x}{2}\right)+(870 C-1235 B) \sin \left(\frac{d x}{2}\right)+5 (761 B-366 C) \sin \left(\frac{3 d x}{2}\right)+2094 C \sin \left(c-\frac{d x}{2}\right)-1314 C \sin \left(c+\frac{d x}{2}\right)+1650 C \sin \left(2 c+\frac{d x}{2}\right)+450 C \sin \left(c+\frac{3 d x}{2}\right)-1230 C \sin \left(2 c+\frac{3 d x}{2}\right)+1050 C \sin \left(3 c+\frac{3 d x}{2}\right)-1278 C \sin \left(c+\frac{5 d x}{2}\right)+90 C \sin \left(2 c+\frac{5 d x}{2}\right)-918 C \sin \left(3 c+\frac{5 d x}{2}\right)+450 C \sin \left(4 c+\frac{5 d x}{2}\right)-630 C \sin \left(2 c+\frac{7 d x}{2}\right)-60 C \sin \left(3 c+\frac{7 d x}{2}\right)-480 C \sin \left(4 c+\frac{7 d x}{2}\right)+90 C \sin \left(5 c+\frac{7 d x}{2}\right)-144 C \sin \left(3 c+\frac{9 d x}{2}\right)-30 C \sin \left(4 c+\frac{9 d x}{2}\right)-114 C \sin \left(5 c+\frac{9 d x}{2}\right)\right)}{480 a^3 d (\cos (c+d x)+1)^3}","-\frac{8 (19 B-9 C) \tan (c+d x)}{15 a^3 d}+\frac{(13 B-6 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{(13 B-6 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{4 (19 B-9 C) \tan (c+d x) \sec (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(11 B-6 C) \tan (c+d x) \sec (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(B-C) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"-1/480*(1920*(13*B - 6*C)*Cos[(c + d*x)/2]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Cos[(c + d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]^2*((-1235*B + 870*C)*Sin[(d*x)/2] + 5*(761*B - 366*C)*Sin[(3*d*x)/2] - 4329*B*Sin[c - (d*x)/2] + 2094*C*Sin[c - (d*x)/2] + 1989*B*Sin[c + (d*x)/2] - 1314*C*Sin[c + (d*x)/2] - 3575*B*Sin[2*c + (d*x)/2] + 1650*C*Sin[2*c + (d*x)/2] - 475*B*Sin[c + (3*d*x)/2] + 450*C*Sin[c + (3*d*x)/2] + 2005*B*Sin[2*c + (3*d*x)/2] - 1230*C*Sin[2*c + (3*d*x)/2] - 2275*B*Sin[3*c + (3*d*x)/2] + 1050*C*Sin[3*c + (3*d*x)/2] + 2673*B*Sin[c + (5*d*x)/2] - 1278*C*Sin[c + (5*d*x)/2] + 105*B*Sin[2*c + (5*d*x)/2] + 90*C*Sin[2*c + (5*d*x)/2] + 1593*B*Sin[3*c + (5*d*x)/2] - 918*C*Sin[3*c + (5*d*x)/2] - 975*B*Sin[4*c + (5*d*x)/2] + 450*C*Sin[4*c + (5*d*x)/2] + 1325*B*Sin[2*c + (7*d*x)/2] - 630*C*Sin[2*c + (7*d*x)/2] + 255*B*Sin[3*c + (7*d*x)/2] - 60*C*Sin[3*c + (7*d*x)/2] + 875*B*Sin[4*c + (7*d*x)/2] - 480*C*Sin[4*c + (7*d*x)/2] - 195*B*Sin[5*c + (7*d*x)/2] + 90*C*Sin[5*c + (7*d*x)/2] + 304*B*Sin[3*c + (9*d*x)/2] - 144*C*Sin[3*c + (9*d*x)/2] + 90*B*Sin[4*c + (9*d*x)/2] - 30*C*Sin[4*c + (9*d*x)/2] + 214*B*Sin[5*c + (9*d*x)/2] - 114*C*Sin[5*c + (9*d*x)/2]))/(a^3*d*(1 + Cos[c + d*x])^3)","B",1
277,1,64,101,0.2479351,"\int \sqrt{a+a \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (2 (5 B+4 C) \cos (c+d x)+20 B+3 C \cos (2 (c+d x))+19 C)}{15 d}","\frac{2 (5 B-2 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 a (5 B+7 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(20*B + 19*C + 2*(5*B + 4*C)*Cos[c + d*x] + 3*C*Cos[2*(c + d*x)])*Tan[(c + d*x)/2])/(15*d)","A",1
278,1,81,138,0.3960445,"\int (a+a \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((252 B+253 C) \cos (c+d x)+6 (7 B+13 C) \cos (2 (c+d x))+546 B+15 C \cos (3 (c+d x))+494 C)}{210 d}","\frac{8 a^2 (21 B+19 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (7 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 a (21 B+19 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 a d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(546*B + 494*C + (252*B + 253*C)*Cos[c + d*x] + 6*(7*B + 13*C)*Cos[2*(c + d*x)] + 15*C*Cos[3*(c + d*x)])*Tan[(c + d*x)/2])/(210*d)","A",1
279,1,105,175,0.7083565,"\int (a+a \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((3030 B+3116 C) \cos (c+d x)+8 (90 B+127 C) \cos (2 (c+d x))+90 B \cos (3 (c+d x))+6240 B+260 C \cos (3 (c+d x))+35 C \cos (4 (c+d x))+5653 C)}{1260 d}","\frac{64 a^3 (15 B+13 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (15 B+13 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 (9 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{63 d}+\frac{2 a (15 B+13 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(6240*B + 5653*C + (3030*B + 3116*C)*Cos[c + d*x] + 8*(90*B + 127*C)*Cos[2*(c + d*x)] + 90*B*Cos[3*(c + d*x)] + 260*C*Cos[3*(c + d*x)] + 35*C*Cos[4*(c + d*x)])*Tan[(c + d*x)/2])/(1260*d)","A",1
280,1,78,118,0.1681618,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(-3 (B-C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 B \sin \left(\frac{1}{2} (c+d x)\right)-4 C \sin ^3\left(\frac{1}{2} (c+d x)\right)\right)}{3 d \sqrt{a (\cos (c+d x)+1)}}","\frac{2 (3 B-2 C) \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (B-C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 a d}",1,"(2*Cos[(c + d*x)/2]*(-3*(B - C)*ArcTanh[Sin[(c + d*x)/2]] + 6*B*Sin[(c + d*x)/2] - 4*C*Sin[(c + d*x)/2]^3))/(3*d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
281,1,104,118,0.4402806,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2),x]","\frac{\sin \left(\frac{1}{2} (c+d x)\right) \cos ^3\left(\frac{1}{2} (c+d x)\right) (B-4 C \cos (c+d x)-5 C)-(3 B-7 C) \cos ^5\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d \left(\sin ^2\left(\frac{1}{2} (c+d x)\right)-1\right) (a (\cos (c+d x)+1))^{3/2}}","\frac{(3 B-7 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(B-C) \sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{2 C \sin (c+d x)}{a d \sqrt{a \cos (c+d x)+a}}",1,"(-((3*B - 7*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^5) + Cos[(c + d*x)/2]^3*(B - 5*C - 4*C*Cos[c + d*x])*Sin[(c + d*x)/2])/(d*(a*(1 + Cos[c + d*x]))^(3/2)*(-1 + Sin[(c + d*x)/2]^2))","A",1
282,1,87,126,0.5890384,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) ((5 B-13 C) \cos (c+d x)+B-9 C)+2 (5 B+19 C) \cos ^3\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{16 a d (a (\cos (c+d x)+1))^{3/2}}","\frac{(5 B+19 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(5 B-13 C) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(B-C) \sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(2*(5*B + 19*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 + (B - 9*C + (5*B - 13*C)*Cos[c + d*x])*Tan[(c + d*x)/2])/(16*a*d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
283,1,77,111,0.4917946,"\int \cos ^{\frac{3}{2}}(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} (42 B \cos (c+d x)+15 C \cos (2 (c+d x))+65 C)+126 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+50 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{105 d}","\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{10 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{10 C \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}",1,"(126*B*EllipticE[(c + d*x)/2, 2] + 50*C*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(65*C + 42*B*Cos[c + d*x] + 15*C*Cos[2*(c + d*x)])*Sin[c + d*x])/(105*d)","A",1
284,1,66,87,0.2230756,"\int \sqrt{\cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 \left(\sin (c+d x) \sqrt{\cos (c+d x)} (5 B+3 C \cos (c+d x))+5 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+9 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d}","\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{6 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(2*(9*C*EllipticE[(c + d*x)/2, 2] + 5*B*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(5*B + 3*C*Cos[c + d*x])*Sin[c + d*x]))/(15*d)","A",1
285,1,53,61,0.1104445,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sqrt[Cos[c + d*x]],x]","\frac{2 \left(3 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \left(F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)}\right)\right)}{3 d}","\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(2*(3*B*EllipticE[(c + d*x)/2, 2] + C*(EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*Sin[c + d*x])))/(3*d)","A",1
286,1,35,35,0.0633435,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(3/2),x]","\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}","\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*C*EllipticE[(c + d*x)/2, 2])/d + (2*B*EllipticF[(c + d*x)/2, 2])/d","A",1
287,1,51,57,0.1350483,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(5/2),x]","\frac{2 \left(-B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{B \sin (c+d x)}{\sqrt{\cos (c+d x)}}+C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d}","-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*(-(B*EllipticE[(c + d*x)/2, 2]) + C*EllipticF[(c + d*x)/2, 2] + (B*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/d","A",1
288,1,65,83,0.4022275,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(7/2),x]","\frac{\frac{2 \sin (c+d x) (B+3 C \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)}+2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}","\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 C \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(-6*C*EllipticE[(c + d*x)/2, 2] + 2*B*EllipticF[(c + d*x)/2, 2] + (2*(B + 3*C*Cos[c + d*x])*Sin[c + d*x])/Cos[c + d*x]^(3/2))/(3*d)","A",1
289,1,95,111,0.2952305,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(9/2),x]","\frac{9 B \sin (2 (c+d x))+6 B \tan (c+d x)-18 B \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+10 C \sin (c+d x)+10 C \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{6 B \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 C \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-18*B*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 10*C*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 10*C*Sin[c + d*x] + 9*B*Sin[2*(c + d*x)] + 6*B*Tan[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",1
290,1,102,132,0.2757164,"\int \cos ^4(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{5 ((48 A+45 C) \sin (2 (c+d x))+(6 A+9 C) \sin (4 (c+d x))+72 A c+72 A d x+C \sin (6 (c+d x))+60 c C+60 C d x)+192 B \sin ^5(c+d x)-640 B \sin ^3(c+d x)+960 B \sin (c+d x)}{960 d}","\frac{(6 A+5 C) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{(6 A+5 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x (6 A+5 C)+\frac{B \sin ^5(c+d x)}{5 d}-\frac{2 B \sin ^3(c+d x)}{3 d}+\frac{B \sin (c+d x)}{d}+\frac{C \sin (c+d x) \cos ^5(c+d x)}{6 d}",1,"(960*B*Sin[c + d*x] - 640*B*Sin[c + d*x]^3 + 192*B*Sin[c + d*x]^5 + 5*(72*A*c + 60*c*C + 72*A*d*x + 60*C*d*x + (48*A + 45*C)*Sin[2*(c + d*x)] + (6*A + 9*C)*Sin[4*(c + d*x)] + C*Sin[6*(c + d*x)]))/(960*d)","A",1
291,1,87,113,0.1760274,"\int \cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{60 (6 A+5 C) \sin (c+d x)+40 A \sin (3 (c+d x))+120 B \sin (2 (c+d x))+15 B \sin (4 (c+d x))+180 B c+180 B d x+50 C \sin (3 (c+d x))+6 C \sin (5 (c+d x))}{480 d}","-\frac{(5 A+4 C) \sin ^3(c+d x)}{15 d}+\frac{(5 A+4 C) \sin (c+d x)}{5 d}+\frac{B \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 B \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 B x}{8}+\frac{C \sin (c+d x) \cos ^4(c+d x)}{5 d}",1,"(180*B*c + 180*B*d*x + 60*(6*A + 5*C)*Sin[c + d*x] + 120*B*Sin[2*(c + d*x)] + 40*A*Sin[3*(c + d*x)] + 50*C*Sin[3*(c + d*x)] + 15*B*Sin[4*(c + d*x)] + 6*C*Sin[5*(c + d*x)])/(480*d)","A",1
292,1,70,88,0.155265,"\int \cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{24 (A+C) \sin (2 (c+d x))+48 A c+48 A d x-32 B \sin ^3(c+d x)+96 B \sin (c+d x)+3 C \sin (4 (c+d x))+36 c C+36 C d x}{96 d}","\frac{(4 A+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x (4 A+3 C)-\frac{B \sin ^3(c+d x)}{3 d}+\frac{B \sin (c+d x)}{d}+\frac{C \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"(48*A*c + 36*c*C + 48*A*d*x + 36*C*d*x + 96*B*Sin[c + d*x] - 32*B*Sin[c + d*x]^3 + 24*(A + C)*Sin[2*(c + d*x)] + 3*C*Sin[4*(c + d*x)])/(96*d)","A",1
293,1,53,69,0.0946719,"\int \cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{3 (4 A+3 C) \sin (c+d x)+3 B \sin (2 (c+d x))+6 B c+6 B d x+C \sin (3 (c+d x))}{12 d}","\frac{(3 A+2 C) \sin (c+d x)}{3 d}+\frac{B \sin (c+d x) \cos (c+d x)}{2 d}+\frac{B x}{2}+\frac{C \sin (c+d x) \cos ^2(c+d x)}{3 d}",1,"(6*B*c + 6*B*d*x + 3*(4*A + 3*C)*Sin[c + d*x] + 3*B*Sin[2*(c + d*x)] + C*Sin[3*(c + d*x)])/(12*d)","A",1
294,1,55,41,0.044304,"\int \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[A + B*Cos[c + d*x] + C*Cos[c + d*x]^2,x]","A x+\frac{B \sin (c) \cos (d x)}{d}+\frac{B \cos (c) \sin (d x)}{d}+\frac{C (c+d x)}{2 d}+\frac{C \sin (2 (c+d x))}{4 d}","A x+\frac{B \sin (c+d x)}{d}+\frac{C \sin (c+d x) \cos (c+d x)}{2 d}+\frac{C x}{2}",1,"A*x + (C*(c + d*x))/(2*d) + (B*Cos[d*x]*Sin[c])/d + (B*Cos[c]*Sin[d*x])/d + (C*Sin[2*(c + d*x)])/(4*d)","A",1
295,1,38,27,0.0190885,"\int \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{A \tanh ^{-1}(\sin (c+d x))}{d}+B x+\frac{C \sin (c) \cos (d x)}{d}+\frac{C \cos (c) \sin (d x)}{d}","\frac{A \tanh ^{-1}(\sin (c+d x))}{d}+B x+\frac{C \sin (c+d x)}{d}",1,"B*x + (A*ArcTanh[Sin[c + d*x]])/d + (C*Cos[d*x]*Sin[c])/d + (C*Cos[c]*Sin[d*x])/d","A",1
296,1,27,27,0.0188995,"\int \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{A \tan (c+d x)}{d}+\frac{B \tanh ^{-1}(\sin (c+d x))}{d}+C x","\frac{A \tan (c+d x)}{d}+\frac{B \tanh ^{-1}(\sin (c+d x))}{d}+C x",1,"C*x + (B*ArcTanh[Sin[c + d*x]])/d + (A*Tan[c + d*x])/d","A",1
297,1,59,51,0.0170099,"\int \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{A \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A \tan (c+d x) \sec (c+d x)}{2 d}+\frac{B \tan (c+d x)}{d}+\frac{C \tanh ^{-1}(\sin (c+d x))}{d}","\frac{(A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A \tan (c+d x) \sec (c+d x)}{2 d}+\frac{B \tan (c+d x)}{d}",1,"(A*ArcTanh[Sin[c + d*x]])/(2*d) + (C*ArcTanh[Sin[c + d*x]])/d + (B*Tan[c + d*x])/d + (A*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
298,1,51,78,0.2126165,"\int \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\tan (c+d x) \left(2 A \tan ^2(c+d x)+6 (A+C)+3 B \sec (c+d x)\right)+3 B \tanh ^{-1}(\sin (c+d x))}{6 d}","\frac{(2 A+3 C) \tan (c+d x)}{3 d}+\frac{A \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{B \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{B \tan (c+d x) \sec (c+d x)}{2 d}",1,"(3*B*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(6*(A + C) + 3*B*Sec[c + d*x] + 2*A*Tan[c + d*x]^2))/(6*d)","A",1
299,1,71,97,0.2702762,"\int \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{\tan (c+d x) \left(3 (3 A+4 C) \sec (c+d x)+6 A \sec ^3(c+d x)+8 B \left(\tan ^2(c+d x)+3\right)\right)+3 (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{24 d}","\frac{(3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(3 A+4 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{A \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{B \tan ^3(c+d x)}{3 d}+\frac{B \tan (c+d x)}{d}",1,"(3*(3*A + 4*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(3*(3*A + 4*C)*Sec[c + d*x] + 6*A*Sec[c + d*x]^3 + 8*B*(3 + Tan[c + d*x]^2)))/(24*d)","A",1
300,1,80,122,0.5788356,"\int \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{\tan (c+d x) \left(8 \left(5 (2 A+C) \tan ^2(c+d x)+3 A \tan ^4(c+d x)+15 (A+C)\right)+30 B \sec ^3(c+d x)+45 B \sec (c+d x)\right)+45 B \tanh ^{-1}(\sin (c+d x))}{120 d}","\frac{(4 A+5 C) \tan ^3(c+d x)}{15 d}+\frac{(4 A+5 C) \tan (c+d x)}{5 d}+\frac{A \tan (c+d x) \sec ^4(c+d x)}{5 d}+\frac{3 B \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{B \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 B \tan (c+d x) \sec (c+d x)}{8 d}",1,"(45*B*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(45*B*Sec[c + d*x] + 30*B*Sec[c + d*x]^3 + 8*(15*(A + C) + 5*(2*A + C)*Tan[c + d*x]^2 + 3*A*Tan[c + d*x]^4)))/(120*d)","A",1
301,1,93,143,0.4372035,"\int \cos ^2(c+d x) (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a \left(-160 (A+B+2 C) \sin ^3(c+d x)+480 (A+B+C) \sin (c+d x)+15 (4 (4 A+3 (B+C)) (c+d x)+8 (A+B+C) \sin (2 (c+d x))+(B+C) \sin (4 (c+d x)))+96 C \sin ^5(c+d x)\right)}{480 d}","-\frac{a (5 A+5 B+4 C) \sin ^3(c+d x)}{15 d}+\frac{a (5 A+5 B+4 C) \sin (c+d x)}{5 d}+\frac{a (4 A+3 (B+C)) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 A+3 (B+C))+\frac{a (B+C) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{a C \sin (c+d x) \cos ^4(c+d x)}{5 d}",1,"(a*(480*(A + B + C)*Sin[c + d*x] - 160*(A + B + 2*C)*Sin[c + d*x]^3 + 96*C*Sin[c + d*x]^5 + 15*(4*(4*A + 3*(B + C))*(c + d*x) + 8*(A + B + C)*Sin[2*(c + d*x)] + (B + C)*Sin[4*(c + d*x)])))/(480*d)","A",1
302,1,96,118,0.4033754,"\int \cos (c+d x) (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a (24 (4 A+3 (B+C)) \sin (c+d x)+24 (A+B+C) \sin (2 (c+d x))+48 A d x+8 B \sin (3 (c+d x))+48 B c+48 B d x+8 C \sin (3 (c+d x))+3 C \sin (4 (c+d x))+24 c C+36 C d x)}{96 d}","\frac{a (3 A+2 (B+C)) \sin (c+d x)}{3 d}+\frac{a (4 A+4 B+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 A+4 B+3 C)+\frac{a (B+C) \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac{a C \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"(a*(48*B*c + 24*c*C + 48*A*d*x + 48*B*d*x + 36*C*d*x + 24*(4*A + 3*(B + C))*Sin[c + d*x] + 24*(A + B + C)*Sin[2*(c + d*x)] + 8*B*Sin[3*(c + d*x)] + 8*C*Sin[3*(c + d*x)] + 3*C*Sin[4*(c + d*x)]))/(96*d)","A",1
303,1,65,91,0.2312774,"\int (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a (3 (4 A+4 B+3 C) \sin (c+d x)+12 A d x+3 (B+C) \sin (2 (c+d x))+6 B d x+C \sin (3 (c+d x))+6 C d x)}{12 d}","\frac{a (3 A+3 B+C) \sin (c+d x)}{3 d}+\frac{1}{2} a x (2 A+B+C)+\frac{a (3 B-C) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 a d}",1,"(a*(12*A*d*x + 6*B*d*x + 6*C*d*x + 3*(4*A + 4*B + 3*C)*Sin[c + d*x] + 3*(B + C)*Sin[2*(c + d*x)] + C*Sin[3*(c + d*x)]))/(12*d)","A",1
304,1,59,63,0.13775,"\int (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{a \left(4 A \tanh ^{-1}(\sin (c+d x))+4 A d x+4 (B+C) \sin (c+d x)+4 B d x+C \sin (2 (c+d x))+2 c C+2 C d x\right)}{4 d}","\frac{1}{2} a x (2 A+2 B+C)+\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a (B+C) \sin (c+d x)}{d}+\frac{a C \sin (c+d x) \cos (c+d x)}{2 d}",1,"(a*(2*c*C + 4*A*d*x + 4*B*d*x + 2*C*d*x + 4*A*ArcTanh[Sin[c + d*x]] + 4*(B + C)*Sin[c + d*x] + C*Sin[2*(c + d*x)]))/(4*d)","A",1
305,1,71,46,0.0281253,"\int (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{a A \tan (c+d x)}{d}+\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+a B x+\frac{a C \sin (c) \cos (d x)}{d}+\frac{a C \cos (c) \sin (d x)}{d}+a C x","\frac{a (A+B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A \tan (c+d x)}{d}+a x (B+C)+\frac{a C \sin (c+d x)}{d}",1,"a*B*x + a*C*x + (a*A*ArcTanh[Sin[c + d*x]])/d + (a*B*ArcTanh[Sin[c + d*x]])/d + (a*C*Cos[d*x]*Sin[c])/d + (a*C*Cos[c]*Sin[d*x])/d + (a*A*Tan[c + d*x])/d","A",1
306,1,92,62,0.0340855,"\int (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{a A \tan (c+d x)}{d}+\frac{a A \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a B \tan (c+d x)}{d}+\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d}+a C x","\frac{a (A+2 (B+C)) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (A+B) \tan (c+d x)}{d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}+a C x",1,"a*C*x + (a*A*ArcTanh[Sin[c + d*x]])/(2*d) + (a*B*ArcTanh[Sin[c + d*x]])/d + (a*C*ArcTanh[Sin[c + d*x]])/d + (a*A*Tan[c + d*x])/d + (a*B*Tan[c + d*x])/d + (a*A*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
307,1,60,91,0.378976,"\int (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a \left(3 (A+B+2 C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(3 (A+B) \sec (c+d x)+6 (A+B+C)+2 A \tan ^2(c+d x)\right)\right)}{6 d}","\frac{a (2 A+3 (B+C)) \tan (c+d x)}{3 d}+\frac{a (A+B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (A+B) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"(a*(3*(A + B + 2*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(6*(A + B + C) + 3*(A + B)*Sec[c + d*x] + 2*A*Tan[c + d*x]^2)))/(6*d)","A",1
308,1,84,125,0.6044026,"\int (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{a \left(3 (3 A+4 (B+C)) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(8 \left((A+B) \tan ^2(c+d x)+3 (A+B+C)\right)+3 (3 A+4 (B+C)) \sec (c+d x)+6 A \sec ^3(c+d x)\right)\right)}{24 d}","-\frac{a (-3 (A+B+C)+A+B) \tan (c+d x)}{3 d}+\frac{a (3 A+4 (B+C)) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (3 A+4 (B+C)) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a (A+B) \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"(a*(3*(3*A + 4*(B + C))*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(3*(3*A + 4*(B + C))*Sec[c + d*x] + 6*A*Sec[c + d*x]^3 + 8*(3*(A + B + C) + (A + B)*Tan[c + d*x]^2))))/(24*d)","A",1
309,1,171,213,0.7257905,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^2 (120 (12 A+11 B+10 C) \sin (c+d x)+15 (32 A+32 B+31 C) \sin (2 (c+d x))+160 A \sin (3 (c+d x))+30 A \sin (4 (c+d x))+840 A d x+180 B \sin (3 (c+d x))+60 B \sin (4 (c+d x))+12 B \sin (5 (c+d x))+720 B c+720 B d x+200 C \sin (3 (c+d x))+75 C \sin (4 (c+d x))+24 C \sin (5 (c+d x))+5 C \sin (6 (c+d x))+420 c C+660 C d x)}{960 d}","-\frac{a^2 (10 A+9 B+8 C) \sin ^3(c+d x)}{15 d}+\frac{a^2 (10 A+9 B+8 C) \sin (c+d x)}{5 d}+\frac{a^2 (10 A+12 B+9 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{a^2 (14 A+12 B+11 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^2 x (14 A+12 B+11 C)+\frac{(3 B+C) \sin (c+d x) \cos ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{15 d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}",1,"(a^2*(720*B*c + 420*c*C + 840*A*d*x + 720*B*d*x + 660*C*d*x + 120*(12*A + 11*B + 10*C)*Sin[c + d*x] + 15*(32*A + 32*B + 31*C)*Sin[2*(c + d*x)] + 160*A*Sin[3*(c + d*x)] + 180*B*Sin[3*(c + d*x)] + 200*C*Sin[3*(c + d*x)] + 30*A*Sin[4*(c + d*x)] + 60*B*Sin[4*(c + d*x)] + 75*C*Sin[4*(c + d*x)] + 12*B*Sin[5*(c + d*x)] + 24*C*Sin[5*(c + d*x)] + 5*C*Sin[6*(c + d*x)]))/(960*d)","A",1
310,1,132,181,0.6219363,"\int \cos (c+d x) (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^2 (60 (14 A+12 B+11 C) \sin (c+d x)+240 (A+B+C) \sin (2 (c+d x))+40 A \sin (3 (c+d x))+480 A d x+80 B \sin (3 (c+d x))+15 B \sin (4 (c+d x))+420 B c+420 B d x+90 C \sin (3 (c+d x))+30 C \sin (4 (c+d x))+6 C \sin (5 (c+d x))+240 c C+360 C d x)}{480 d}","\frac{a^2 (8 A+7 B+6 C) \sin (c+d x)}{6 d}+\frac{a^2 (8 A+7 B+6 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} a^2 x (8 A+7 B+6 C)+\frac{(20 A-5 B+6 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{60 d}+\frac{(5 B+2 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 a d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^2}{5 d}",1,"(a^2*(420*B*c + 240*c*C + 480*A*d*x + 420*B*d*x + 360*C*d*x + 60*(14*A + 12*B + 11*C)*Sin[c + d*x] + 240*(A + B + C)*Sin[2*(c + d*x)] + 40*A*Sin[3*(c + d*x)] + 80*B*Sin[3*(c + d*x)] + 90*C*Sin[3*(c + d*x)] + 15*B*Sin[4*(c + d*x)] + 30*C*Sin[4*(c + d*x)] + 6*C*Sin[5*(c + d*x)]))/(480*d)","A",1
311,1,94,138,0.3923878,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^2 (24 (8 A+7 B+6 C) \sin (c+d x)+24 (A+2 (B+C)) \sin (2 (c+d x))+144 A d x+8 B \sin (3 (c+d x))+96 B d x+16 C \sin (3 (c+d x))+3 C \sin (4 (c+d x))+84 C d x)}{96 d}","\frac{a^2 (12 A+8 B+7 C) \sin (c+d x)}{6 d}+\frac{a^2 (12 A+8 B+7 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} a^2 x (12 A+8 B+7 C)+\frac{(4 B-C) \sin (c+d x) (a \cos (c+d x)+a)^2}{12 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d}",1,"(a^2*(144*A*d*x + 96*B*d*x + 84*C*d*x + 24*(8*A + 7*B + 6*C)*Sin[c + d*x] + 24*(A + 2*(B + C))*Sin[2*(c + d*x)] + 8*B*Sin[3*(c + d*x)] + 16*C*Sin[3*(c + d*x)] + 3*C*Sin[4*(c + d*x)]))/(96*d)","A",1
312,1,121,120,0.3209181,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{a^2 \left(3 (4 A+8 B+7 C) \sin (c+d x)-12 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+24 A d x+3 (B+2 C) \sin (2 (c+d x))+18 B d x+C \sin (3 (c+d x))+12 C d x\right)}{12 d}","\frac{a^2 (2 A+3 B+2 C) \sin (c+d x)}{2 d}+\frac{1}{2} a^2 x (4 A+3 B+2 C)+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(3 B+2 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{6 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}",1,"(a^2*(24*A*d*x + 18*B*d*x + 12*C*d*x - 12*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 3*(4*A + 8*B + 7*C)*Sin[c + d*x] + 3*(B + 2*C)*Sin[2*(c + d*x)] + C*Sin[3*(c + d*x)]))/(12*d)","A",1
313,1,174,121,0.6019061,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{a^2 \left(4 A \tan (c+d x)-8 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+8 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+4 A c+4 A d x+4 (B+2 C) \sin (c+d x)-4 B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+8 B c+8 B d x+C \sin (2 (c+d x))+6 c C+6 C d x\right)}{4 d}","-\frac{a^2 (2 A-2 B-3 C) \sin (c+d x)}{2 d}+\frac{a^2 (2 A+B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^2 x (2 A+4 B+3 C)-\frac{(2 A-C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{2 d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^2}{d}",1,"(a^2*(4*A*c + 8*B*c + 6*c*C + 4*A*d*x + 8*B*d*x + 6*C*d*x - 8*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 4*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 8*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 4*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 4*(B + 2*C)*Sin[c + d*x] + C*Sin[2*(c + d*x)] + 4*A*Tan[c + d*x]))/(4*d)","A",1
314,1,259,123,1.4631094,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(-2 (3 A+4 B+2 C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 (3 A+4 B+2 C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{4 (2 A+B) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 (2 A+B) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+\frac{A}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{A}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+4 (B+2 C) (c+d x)+4 C \sin (c+d x)\right)}{16 d}","-\frac{a^2 (3 A+2 B-2 C) \sin (c+d x)}{2 d}+\frac{a^2 (3 A+4 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(A+B) \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{d}+a^2 x (B+2 C)+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}",1,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(4*(B + 2*C)*(c + d*x) - 2*(3*A + 4*B + 2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(3*A + 4*B + 2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + A/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (4*(2*A + B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - A/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*(2*A + B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 4*C*Sin[c + d*x]))/(16*d)","B",1
315,1,315,134,4.9918176,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(-6 (2 A+3 B+4 C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \tan (c+d x) \sec ^2(c+d x) ((5 A+6 B+3 C) \cos (2 (c+d x))+A (-\cos (c+d x))+7 A+6 B+3 C)+\frac{7 A}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{7 A}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+12 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{3 B}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{3 B}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+18 B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+24 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+12 c C+12 C d x\right)}{48 d}","\frac{a^2 (2 A+3 B+2 C) \tan (c+d x)}{2 d}+\frac{a^2 (2 A+3 B+4 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(2 A+3 B) \tan (c+d x) \sec (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{6 d}+a^2 C x+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^2}{3 d}",1,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(12*c*C + 12*C*d*x - 6*(2*A + 3*B + 4*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 18*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 24*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (7*A)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (3*B)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 - (7*A)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - (3*B)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + 2*(7*A + 6*B + 3*C - A*Cos[c + d*x] + (5*A + 6*B + 3*C)*Cos[2*(c + d*x)])*Sec[c + d*x]^2*Tan[c + d*x]))/(48*d)","B",1
316,1,404,160,3.7503945,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(\frac{16 (4 A+5 B+6 C) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{16 (4 A+5 B+6 C) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+\frac{29 A+28 B+12 C}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{-29 A-4 (7 B+3 C)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-6 (7 A+8 B+12 C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 (7 A+8 B+12 C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{8 (2 A+B) \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{8 (2 A+B) \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{3 A}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}-\frac{3 A}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}\right)}{192 d}","\frac{a^2 (4 A+5 B+6 C) \tan (c+d x)}{3 d}+\frac{a^2 (7 A+8 B+12 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (11 A+16 B+12 C) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{(A+2 B) \tan (c+d x) \sec ^2(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{6 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^2}{4 d}",1,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(-6*(7*A + 8*B + 12*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*(7*A + 8*B + 12*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (3*A)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4 + (29*A + 28*B + 12*C)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (8*(2*A + B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (16*(4*A + 5*B + 6*C)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (3*A)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 + (8*(2*A + B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + (-29*A - 4*(7*B + 3*C))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (16*(4*A + 5*B + 6*C)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(192*d)","B",1
317,1,502,196,5.3459112,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(\frac{16 (18 A+20 B+25 C) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{2 (39 A+20 (2 B+C)) \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{16 (18 A+20 B+25 C) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+\frac{129 A+145 B+140 C}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{-129 A-5 (29 B+28 C)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{2 (39 A+20 (2 B+C)) \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}-30 (6 A+7 B+8 C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+30 (6 A+7 B+8 C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{3 (12 A+5 B)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}-\frac{3 (12 A+5 B)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}+\frac{12 A \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5}+\frac{12 A \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}\right)}{960 d}","\frac{a^2 (18 A+20 B+25 C) \tan (c+d x)}{15 d}+\frac{a^2 (6 A+7 B+8 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (18 A+25 B+20 C) \tan (c+d x) \sec ^2(c+d x)}{60 d}+\frac{a^2 (6 A+7 B+8 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{(2 A+5 B) \tan (c+d x) \sec ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{20 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^2}{5 d}",1,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(-30*(6*A + 7*B + 8*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 30*(6*A + 7*B + 8*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (3*(12*A + 5*B))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4 + (129*A + 145*B + 140*C)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (12*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5 + (2*(39*A + 20*(2*B + C))*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (16*(18*A + 20*B + 25*C)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (12*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5 - (3*(12*A + 5*B))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 + (2*(39*A + 20*(2*B + C))*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + (-129*A - 5*(29*B + 28*C))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (16*(18*A + 20*B + 25*C)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(960*d)","B",1
318,1,204,265,1.0636081,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^3 (105 (184 A+168 B+155 C) \sin (c+d x)+105 (64 A+63 B+61 C) \sin (2 (c+d x))+2380 A \sin (3 (c+d x))+630 A \sin (4 (c+d x))+84 A \sin (5 (c+d x))+10920 A d x+2660 B \sin (3 (c+d x))+945 B \sin (4 (c+d x))+252 B \sin (5 (c+d x))+35 B \sin (6 (c+d x))+9660 B c+9660 B d x+2835 C \sin (3 (c+d x))+1155 C \sin (4 (c+d x))+399 C \sin (5 (c+d x))+105 C \sin (6 (c+d x))+15 C \sin (7 (c+d x))+5460 c C+8820 C d x)}{6720 d}","-\frac{a^3 (133 A+119 B+108 C) \sin ^3(c+d x)}{105 d}+\frac{a^3 (133 A+119 B+108 C) \sin (c+d x)}{35 d}+\frac{a^3 (154 A+147 B+129 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{(3 A+4 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d}+\frac{a^3 (26 A+23 B+21 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^3 x (26 A+23 B+21 C)+\frac{(7 B+3 C) \sin (c+d x) \cos ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{42 a d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}",1,"(a^3*(9660*B*c + 5460*c*C + 10920*A*d*x + 9660*B*d*x + 8820*C*d*x + 105*(184*A + 168*B + 155*C)*Sin[c + d*x] + 105*(64*A + 63*B + 61*C)*Sin[2*(c + d*x)] + 2380*A*Sin[3*(c + d*x)] + 2660*B*Sin[3*(c + d*x)] + 2835*C*Sin[3*(c + d*x)] + 630*A*Sin[4*(c + d*x)] + 945*B*Sin[4*(c + d*x)] + 1155*C*Sin[4*(c + d*x)] + 84*A*Sin[5*(c + d*x)] + 252*B*Sin[5*(c + d*x)] + 399*C*Sin[5*(c + d*x)] + 35*B*Sin[6*(c + d*x)] + 105*C*Sin[6*(c + d*x)] + 15*C*Sin[7*(c + d*x)]))/(6720*d)","A",1
319,1,171,207,0.6397045,"\int \cos (c+d x) (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^3 (120 (26 A+23 B+21 C) \sin (c+d x)+15 (64 A+64 B+63 C) \sin (2 (c+d x))+240 A \sin (3 (c+d x))+30 A \sin (4 (c+d x))+1800 A d x+340 B \sin (3 (c+d x))+90 B \sin (4 (c+d x))+12 B \sin (5 (c+d x))+1560 B c+1560 B d x+380 C \sin (3 (c+d x))+135 C \sin (4 (c+d x))+36 C \sin (5 (c+d x))+5 C \sin (6 (c+d x))+900 c C+1380 C d x)}{960 d}","-\frac{a^3 (30 A+26 B+23 C) \sin ^3(c+d x)}{120 d}+\frac{a^3 (30 A+26 B+23 C) \sin (c+d x)}{10 d}+\frac{3 a^3 (30 A+26 B+23 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{1}{16} a^3 x (30 A+26 B+23 C)+\frac{(30 A-6 B+7 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{120 d}+\frac{(2 B+C) \sin (c+d x) (a \cos (c+d x)+a)^4}{10 a d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^3}{6 d}",1,"(a^3*(1560*B*c + 900*c*C + 1800*A*d*x + 1560*B*d*x + 1380*C*d*x + 120*(26*A + 23*B + 21*C)*Sin[c + d*x] + 15*(64*A + 64*B + 63*C)*Sin[2*(c + d*x)] + 240*A*Sin[3*(c + d*x)] + 340*B*Sin[3*(c + d*x)] + 380*C*Sin[3*(c + d*x)] + 30*A*Sin[4*(c + d*x)] + 90*B*Sin[4*(c + d*x)] + 135*C*Sin[4*(c + d*x)] + 12*B*Sin[5*(c + d*x)] + 36*C*Sin[5*(c + d*x)] + 5*C*Sin[6*(c + d*x)]))/(960*d)","A",1
320,1,129,166,0.485792,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^3 (60 (30 A+26 B+23 C) \sin (c+d x)+120 (3 A+4 (B+C)) \sin (2 (c+d x))+40 A \sin (3 (c+d x))+1200 A d x+120 B \sin (3 (c+d x))+15 B \sin (4 (c+d x))+900 B d x+170 C \sin (3 (c+d x))+45 C \sin (4 (c+d x))+6 C \sin (5 (c+d x))+780 C d x)}{480 d}","-\frac{a^3 (20 A+15 B+13 C) \sin ^3(c+d x)}{60 d}+\frac{a^3 (20 A+15 B+13 C) \sin (c+d x)}{5 d}+\frac{3 a^3 (20 A+15 B+13 C) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{1}{8} a^3 x (20 A+15 B+13 C)+\frac{(5 B-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}",1,"(a^3*(1200*A*d*x + 900*B*d*x + 780*C*d*x + 60*(30*A + 26*B + 23*C)*Sin[c + d*x] + 120*(3*A + 4*(B + C))*Sin[2*(c + d*x)] + 40*A*Sin[3*(c + d*x)] + 120*B*Sin[3*(c + d*x)] + 170*C*Sin[3*(c + d*x)] + 15*B*Sin[4*(c + d*x)] + 45*C*Sin[4*(c + d*x)] + 6*C*Sin[5*(c + d*x)]))/(480*d)","A",1
321,1,147,162,0.4857734,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{a^3 \left(24 (12 A+15 B+13 C) \sin (c+d x)+24 (A+3 B+4 C) \sin (2 (c+d x))-96 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+96 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+336 A d x+8 B \sin (3 (c+d x))+240 B d x+24 C \sin (3 (c+d x))+3 C \sin (4 (c+d x))+180 C d x\right)}{96 d}","\frac{5 a^3 (4 A+4 B+3 C) \sin (c+d x)}{8 d}+\frac{(12 A+20 B+15 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{24 d}+\frac{1}{8} a^3 x (28 A+20 B+15 C)+\frac{a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(4 B+3 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{12 a d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}",1,"(a^3*(336*A*d*x + 240*B*d*x + 180*C*d*x - 96*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 96*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 24*(12*A + 15*B + 13*C)*Sin[c + d*x] + 24*(A + 3*B + 4*C)*Sin[2*(c + d*x)] + 8*B*Sin[3*(c + d*x)] + 24*C*Sin[3*(c + d*x)] + 3*C*Sin[4*(c + d*x)]))/(96*d)","A",1
322,1,227,156,0.9778276,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(6 (6 A+7 B+5 C) (c+d x)+3 (4 A+12 B+15 C) \sin (c+d x)-12 (3 A+B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 (3 A+B) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{12 A \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{12 A \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+3 (B+3 C) \sin (2 (c+d x))+C \sin (3 (c+d x))\right)}{96 d}","-\frac{(6 A-3 B-5 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{6 d}+\frac{a^3 (3 A+B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^3 x (6 A+7 B+5 C)+\frac{5 a^3 (B+C) \sin (c+d x)}{2 d}-\frac{(3 A-C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{3 a d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}",1,"(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(6*(6*A + 7*B + 5*C)*(c + d*x) - 12*(3*A + B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*(3*A + B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (12*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (12*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 3*(4*A + 12*B + 15*C)*Sin[c + d*x] + 3*(B + 3*C)*Sin[2*(c + d*x)] + C*Sin[3*(c + d*x)]))/(96*d)","A",1
323,1,256,175,1.4924805,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{a^3 \left(2 (2 A+6 B+7 C) (c+d x)-2 (7 A+6 B+2 C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 (7 A+6 B+2 C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{4 (3 A+B) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 (3 A+B) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+\frac{A}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{A}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+4 (B+3 C) \sin (c+d x)+C \sin (2 (c+d x))\right)}{4 d}","\frac{a^3 (7 A+6 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(4 A+2 B-C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{2 d}+\frac{1}{2} a^3 x (2 A+6 B+7 C)-\frac{5 a^3 (A-C) \sin (c+d x)}{2 d}+\frac{(3 A+2 B) \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{2 a d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^3}{2 d}",1,"(a^3*(2*(2*A + 6*B + 7*C)*(c + d*x) - 2*(7*A + 6*B + 2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(7*A + 6*B + 2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + A/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (4*(3*A + B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - A/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*(3*A + B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 4*(B + 3*C)*Sin[c + d*x] + C*Sin[2*(c + d*x)]))/(4*d)","A",1
324,1,354,169,4.1217941,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(\frac{4 (11 A+9 B+3 C) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 (11 A+9 B+3 C) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}-6 (5 A+7 B+6 C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 (5 A+7 B+6 C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{-10 A-3 B}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{10 A+3 B}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{2 A \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2 A \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+12 (B+3 C) (c+d x)+12 C \sin (c+d x)\right)}{96 d}","\frac{a^3 (5 A+7 B+6 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(5 A+6 B+3 C) \tan (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{3 d}-\frac{5 a^3 (A+B) \sin (c+d x)}{2 d}+a^3 x (B+3 C)+\frac{(A+B) \tan (c+d x) \sec (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{2 a d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}",1,"(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(12*(B + 3*C)*(c + d*x) - 6*(5*A + 7*B + 6*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*(5*A + 7*B + 6*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (10*A + 3*B)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (2*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (4*(11*A + 9*B + 3*C)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (2*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + (-10*A - 3*B)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*(11*A + 9*B + 3*C)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 12*C*Sin[c + d*x]))/(96*d)","B",1
325,1,793,183,6.1899518,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{\sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \left(9 A \sin \left(\frac{1}{2} (c+d x)\right)+11 B \sin \left(\frac{1}{2} (c+d x)\right)+9 C \sin \left(\frac{1}{2} (c+d x)\right)\right)}{24 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{\sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \left(9 A \sin \left(\frac{1}{2} (c+d x)\right)+11 B \sin \left(\frac{1}{2} (c+d x)\right)+9 C \sin \left(\frac{1}{2} (c+d x)\right)\right)}{24 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{(57 A+40 B+12 C) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3}{384 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{(-57 A-40 B-12 C) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3}{384 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{(-15 A-20 B-28 C) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{64 d}+\frac{(15 A+20 B+28 C) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{64 d}+\frac{\sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \left(3 A \sin \left(\frac{1}{2} (c+d x)\right)+B \sin \left(\frac{1}{2} (c+d x)\right)\right)}{48 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{\sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \left(3 A \sin \left(\frac{1}{2} (c+d x)\right)+B \sin \left(\frac{1}{2} (c+d x)\right)\right)}{48 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{A \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3}{128 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}-\frac{A \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3}{128 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}+\frac{C (c+d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3}{8 d}","\frac{5 a^3 (3 A+4 (B+C)) \tan (c+d x)}{8 d}+\frac{a^3 (15 A+20 B+28 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(15 A+20 B+12 C) \tan (c+d x) \sec (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{24 d}+a^3 C x+\frac{(3 A+4 B) \tan (c+d x) \sec ^2(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{12 a d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^3}{4 d}",1,"(C*(c + d*x)*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6)/(8*d) + ((-15*A - 20*B - 28*C)*(a + a*Cos[c + d*x])^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sec[c/2 + (d*x)/2]^6)/(64*d) + ((15*A + 20*B + 28*C)*(a + a*Cos[c + d*x])^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sec[c/2 + (d*x)/2]^6)/(64*d) + (A*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6)/(128*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4) + ((57*A + 40*B + 12*C)*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6)/(384*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) - (A*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6)/(128*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4) + ((-57*A - 40*B - 12*C)*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6)/(384*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + ((a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(3*A*Sin[(c + d*x)/2] + B*Sin[(c + d*x)/2]))/(48*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + ((a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(3*A*Sin[(c + d*x)/2] + B*Sin[(c + d*x)/2]))/(48*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + ((a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(9*A*Sin[(c + d*x)/2] + 11*B*Sin[(c + d*x)/2] + 9*C*Sin[(c + d*x)/2]))/(24*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + ((a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(9*A*Sin[(c + d*x)/2] + 11*B*Sin[(c + d*x)/2] + 9*C*Sin[(c + d*x)/2]))/(24*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","B",1
326,1,931,212,6.2113427,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{(-13 A-15 B-20 C) (\cos (c+d x) a+a)^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{64 d}+\frac{(13 A+15 B+20 C) (\cos (c+d x) a+a)^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{64 d}+\frac{A (\cos (c+d x) a+a)^3 \sin \left(\frac{1}{2} (c+d x)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{160 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5}+\frac{(\cos (c+d x) a+a)^3 \left(79 A \sin \left(\frac{1}{2} (c+d x)\right)+60 B \sin \left(\frac{1}{2} (c+d x)\right)+20 C \sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{960 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{(\cos (c+d x) a+a)^3 \left(79 A \sin \left(\frac{1}{2} (c+d x)\right)+60 B \sin \left(\frac{1}{2} (c+d x)\right)+20 C \sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{960 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{(\cos (c+d x) a+a)^3 \left(38 A \sin \left(\frac{1}{2} (c+d x)\right)+45 B \sin \left(\frac{1}{2} (c+d x)\right)+55 C \sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{120 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{(\cos (c+d x) a+a)^3 \left(38 A \sin \left(\frac{1}{2} (c+d x)\right)+45 B \sin \left(\frac{1}{2} (c+d x)\right)+55 C \sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{120 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{(274 A+285 B+200 C) (\cos (c+d x) a+a)^3 \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{1920 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{(-274 A-285 B-200 C) (\cos (c+d x) a+a)^3 \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{1920 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{(17 A+5 B) (\cos (c+d x) a+a)^3 \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{640 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}+\frac{(-17 A-5 B) (\cos (c+d x) a+a)^3 \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{640 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}+\frac{A (\cos (c+d x) a+a)^3 \sin \left(\frac{1}{2} (c+d x)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{160 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^5}","\frac{a^3 (38 A+45 B+55 C) \tan (c+d x)}{15 d}+\frac{a^3 (13 A+15 B+20 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 (109 A+135 B+140 C) \tan (c+d x) \sec (c+d x)}{120 d}+\frac{(11 A+15 B+10 C) \tan (c+d x) \sec ^2(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{30 d}+\frac{(3 A+5 B) \tan (c+d x) \sec ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{20 a d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}",1,"((-13*A - 15*B - 20*C)*(a + a*Cos[c + d*x])^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sec[c/2 + (d*x)/2]^6)/(64*d) + ((13*A + 15*B + 20*C)*(a + a*Cos[c + d*x])^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sec[c/2 + (d*x)/2]^6)/(64*d) + ((17*A + 5*B)*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6)/(640*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4) + ((274*A + 285*B + 200*C)*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6)/(1920*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (A*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*Sin[(c + d*x)/2])/(160*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5) + (A*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*Sin[(c + d*x)/2])/(160*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5) + ((-17*A - 5*B)*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6)/(640*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4) + ((-274*A - 285*B - 200*C)*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6)/(1920*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + ((a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(79*A*Sin[(c + d*x)/2] + 60*B*Sin[(c + d*x)/2] + 20*C*Sin[(c + d*x)/2]))/(960*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + ((a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(79*A*Sin[(c + d*x)/2] + 60*B*Sin[(c + d*x)/2] + 20*C*Sin[(c + d*x)/2]))/(960*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + ((a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(38*A*Sin[(c + d*x)/2] + 45*B*Sin[(c + d*x)/2] + 55*C*Sin[(c + d*x)/2]))/(120*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + ((a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(38*A*Sin[(c + d*x)/2] + 45*B*Sin[(c + d*x)/2] + 55*C*Sin[(c + d*x)/2]))/(120*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","B",1
327,1,265,244,1.9505931,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","-\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \sec ^6(c+d x) \left(240 (23 A+26 B+30 C) \cos ^6(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-2 \sin (c+d x) (16 (344 A+328 B+315 C) \cos (c+d x)+20 (115 A+114 B+102 C) \cos (2 (c+d x))+1904 A \cos (3 (c+d x))+345 A \cos (4 (c+d x))+272 A \cos (5 (c+d x))+2275 A+2128 B \cos (3 (c+d x))+390 B \cos (4 (c+d x))+304 B \cos (5 (c+d x))+1890 B+2280 C \cos (3 (c+d x))+450 C \cos (4 (c+d x))+360 C \cos (5 (c+d x))+1590 C)\right)}{30720 d}","\frac{a^3 (34 A+38 B+45 C) \tan (c+d x)}{15 d}+\frac{a^3 (23 A+26 B+30 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 (73 A+86 B+90 C) \tan (c+d x) \sec ^2(c+d x)}{120 d}+\frac{a^3 (23 A+26 B+30 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(31 A+42 B+30 C) \tan (c+d x) \sec ^3(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{120 d}+\frac{(A+2 B) \tan (c+d x) \sec ^4(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{10 a d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 d}",1,"-1/30720*(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*Sec[c + d*x]^6*(240*(23*A + 26*B + 30*C)*Cos[c + d*x]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 2*(2275*A + 1890*B + 1590*C + 16*(344*A + 328*B + 315*C)*Cos[c + d*x] + 20*(115*A + 114*B + 102*C)*Cos[2*(c + d*x)] + 1904*A*Cos[3*(c + d*x)] + 2128*B*Cos[3*(c + d*x)] + 2280*C*Cos[3*(c + d*x)] + 345*A*Cos[4*(c + d*x)] + 390*B*Cos[4*(c + d*x)] + 450*C*Cos[4*(c + d*x)] + 272*A*Cos[5*(c + d*x)] + 304*B*Cos[5*(c + d*x)] + 360*C*Cos[5*(c + d*x)])*Sin[c + d*x]))/d","A",1
328,1,237,304,1.4411279,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^4 (1680 (352 A+323 B+300 C) \sin (c+d x)+1680 (127 A+124 B+120 C) \sin (2 (c+d x))+80640 A \sin (3 (c+d x))+25200 A \sin (4 (c+d x))+5376 A \sin (5 (c+d x))+560 A \sin (6 (c+d x))+329280 A d x+87920 B \sin (3 (c+d x))+33600 B \sin (4 (c+d x))+10416 B \sin (5 (c+d x))+2240 B \sin (6 (c+d x))+240 B \sin (7 (c+d x))+295680 B c+295680 B d x+91840 C \sin (3 (c+d x))+39480 C \sin (4 (c+d x))+14784 C \sin (5 (c+d x))+4480 C \sin (6 (c+d x))+960 C \sin (7 (c+d x))+105 C \sin (8 (c+d x))+164640 c C+271320 C d x)}{107520 d}","-\frac{a^4 (252 A+227 B+208 C) \sin ^3(c+d x)}{105 d}+\frac{a^4 (252 A+227 B+208 C) \sin (c+d x)}{35 d}+\frac{a^4 (2408 A+2208 B+2007 C) \sin (c+d x) \cos ^3(c+d x)}{2240 d}+\frac{7 (8 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{120 d}+\frac{a^4 (392 A+352 B+323 C) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{1}{128} a^4 x (392 A+352 B+323 C)+\frac{(56 A+80 B+61 C) \sin (c+d x) \cos ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{336 d}+\frac{a (2 B+C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{14 d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}",1,"(a^4*(295680*B*c + 164640*c*C + 329280*A*d*x + 295680*B*d*x + 271320*C*d*x + 1680*(352*A + 323*B + 300*C)*Sin[c + d*x] + 1680*(127*A + 124*B + 120*C)*Sin[2*(c + d*x)] + 80640*A*Sin[3*(c + d*x)] + 87920*B*Sin[3*(c + d*x)] + 91840*C*Sin[3*(c + d*x)] + 25200*A*Sin[4*(c + d*x)] + 33600*B*Sin[4*(c + d*x)] + 39480*C*Sin[4*(c + d*x)] + 5376*A*Sin[5*(c + d*x)] + 10416*B*Sin[5*(c + d*x)] + 14784*C*Sin[5*(c + d*x)] + 560*A*Sin[6*(c + d*x)] + 2240*B*Sin[6*(c + d*x)] + 4480*C*Sin[6*(c + d*x)] + 240*B*Sin[7*(c + d*x)] + 960*C*Sin[7*(c + d*x)] + 105*C*Sin[8*(c + d*x)]))/(107520*d)","A",1
329,1,204,243,0.9389425,"\int \cos (c+d x) (a+a \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^4 (105 (392 A+352 B+323 C) \sin (c+d x)+105 (128 A+127 B+124 C) \sin (2 (c+d x))+4060 A \sin (3 (c+d x))+840 A \sin (4 (c+d x))+84 A \sin (5 (c+d x))+23520 A d x+5040 B \sin (3 (c+d x))+1575 B \sin (4 (c+d x))+336 B \sin (5 (c+d x))+35 B \sin (6 (c+d x))+20580 B c+20580 B d x+5495 C \sin (3 (c+d x))+2100 C \sin (4 (c+d x))+651 C \sin (5 (c+d x))+140 C \sin (6 (c+d x))+15 C \sin (7 (c+d x))+11760 c C+18480 C d x)}{6720 d}","-\frac{2 a^4 (56 A+49 B+44 C) \sin ^3(c+d x)}{105 d}+\frac{4 a^4 (56 A+49 B+44 C) \sin (c+d x)}{35 d}+\frac{a^4 (56 A+49 B+44 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{27 a^4 (56 A+49 B+44 C) \sin (c+d x) \cos (c+d x)}{560 d}+\frac{1}{16} a^4 x (56 A+49 B+44 C)+\frac{(42 A-7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^4}{210 d}+\frac{(7 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^5}{42 a d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d}",1,"(a^4*(20580*B*c + 11760*c*C + 23520*A*d*x + 20580*B*d*x + 18480*C*d*x + 105*(392*A + 352*B + 323*C)*Sin[c + d*x] + 105*(128*A + 127*B + 124*C)*Sin[2*(c + d*x)] + 4060*A*Sin[3*(c + d*x)] + 5040*B*Sin[3*(c + d*x)] + 5495*C*Sin[3*(c + d*x)] + 840*A*Sin[4*(c + d*x)] + 1575*B*Sin[4*(c + d*x)] + 2100*C*Sin[4*(c + d*x)] + 84*A*Sin[5*(c + d*x)] + 336*B*Sin[5*(c + d*x)] + 651*C*Sin[5*(c + d*x)] + 35*B*Sin[6*(c + d*x)] + 140*C*Sin[6*(c + d*x)] + 15*C*Sin[7*(c + d*x)]))/(6720*d)","A",1
330,1,163,200,0.5414705,"\int (a+a \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^4 (120 (56 A+49 B+44 C) \sin (c+d x)+15 (112 A+128 B+127 C) \sin (2 (c+d x))+320 A \sin (3 (c+d x))+30 A \sin (4 (c+d x))+4200 A d x+580 B \sin (3 (c+d x))+120 B \sin (4 (c+d x))+12 B \sin (5 (c+d x))+3360 B d x+720 C \sin (3 (c+d x))+225 C \sin (4 (c+d x))+48 C \sin (5 (c+d x))+5 C \sin (6 (c+d x))+2940 C d x)}{960 d}","-\frac{2 a^4 (10 A+8 B+7 C) \sin ^3(c+d x)}{15 d}+\frac{4 a^4 (10 A+8 B+7 C) \sin (c+d x)}{5 d}+\frac{a^4 (10 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{27 a^4 (10 A+8 B+7 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{7}{16} a^4 x (10 A+8 B+7 C)+\frac{(6 B-C) \sin (c+d x) (a \cos (c+d x)+a)^4}{30 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^5}{6 a d}",1,"(a^4*(4200*A*d*x + 3360*B*d*x + 2940*C*d*x + 120*(56*A + 49*B + 44*C)*Sin[c + d*x] + 15*(112*A + 128*B + 127*C)*Sin[2*(c + d*x)] + 320*A*Sin[3*(c + d*x)] + 580*B*Sin[3*(c + d*x)] + 720*C*Sin[3*(c + d*x)] + 30*A*Sin[4*(c + d*x)] + 120*B*Sin[4*(c + d*x)] + 225*C*Sin[4*(c + d*x)] + 12*B*Sin[5*(c + d*x)] + 48*C*Sin[5*(c + d*x)] + 5*C*Sin[6*(c + d*x)]))/(960*d)","A",1
331,1,182,195,0.7535889,"\int (a+a \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{a^4 \left(60 (54 A+56 B+49 C) \sin (c+d x)+120 (4 A+7 B+8 C) \sin (2 (c+d x))+40 A \sin (3 (c+d x))-480 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+480 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2880 A d x+160 B \sin (3 (c+d x))+15 B \sin (4 (c+d x))+2100 B d x+290 C \sin (3 (c+d x))+60 C \sin (4 (c+d x))+6 C \sin (5 (c+d x))+1680 C d x\right)}{480 d}","\frac{a^4 (40 A+35 B+28 C) \sin (c+d x)}{8 d}+\frac{(32 A+35 B+28 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{24 d}+\frac{1}{8} a^4 x (48 A+35 B+28 C)+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(20 A+35 B+28 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{60 d}+\frac{a (5 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}",1,"(a^4*(2880*A*d*x + 2100*B*d*x + 1680*C*d*x - 480*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 480*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 60*(54*A + 56*B + 49*C)*Sin[c + d*x] + 120*(4*A + 7*B + 8*C)*Sin[2*(c + d*x)] + 40*A*Sin[3*(c + d*x)] + 160*B*Sin[3*(c + d*x)] + 290*C*Sin[3*(c + d*x)] + 15*B*Sin[4*(c + d*x)] + 60*C*Sin[4*(c + d*x)] + 6*C*Sin[5*(c + d*x)]))/(480*d)","A",1
332,1,246,196,1.906052,"\int (a+a \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \left(12 (52 A+48 B+35 C) (c+d x)+24 (16 A+27 B+28 C) \sin (c+d x)+24 (A+4 B+7 C) \sin (2 (c+d x))-96 (4 A+B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+96 (4 A+B) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{96 A \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{96 A \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+8 (B+4 C) \sin (3 (c+d x))+3 C \sin (4 (c+d x))\right)}{1536 d}","\frac{5 a^4 (4 A+8 B+7 C) \sin (c+d x)}{8 d}-\frac{(12 A-32 B-35 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{24 d}+\frac{a^4 (4 A+B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{8} a^4 x (52 A+48 B+35 C)-\frac{(12 A-4 B-7 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{12 d}-\frac{a (4 A-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^4}{d}",1,"(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*(12*(52*A + 48*B + 35*C)*(c + d*x) - 96*(4*A + B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 96*(4*A + B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (96*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (96*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 24*(16*A + 27*B + 28*C)*Sin[c + d*x] + 24*(A + 4*B + 7*C)*Sin[2*(c + d*x)] + 8*(B + 4*C)*Sin[3*(c + d*x)] + 3*C*Sin[4*(c + d*x)]))/(1536*d)","A",1
333,1,299,206,3.527754,"\int (a+a \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \left(6 (8 A+13 B+12 C) (c+d x)+3 (4 A+16 B+27 C) \sin (c+d x)-6 (13 A+8 B+2 C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 (13 A+8 B+2 C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{12 (4 A+B) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{12 (4 A+B) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+\frac{3 A}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{3 A}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+3 (B+4 C) \sin (2 (c+d x))+C \sin (3 (c+d x))\right)}{192 d}","-\frac{5 a^4 (A-B-2 C) \sin (c+d x)}{2 d}+\frac{a^4 (13 A+8 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(18 A+3 B-8 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{6 d}+\frac{1}{2} a^4 x (8 A+13 B+12 C)-\frac{(15 A+6 B-2 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{6 d}+\frac{a (2 A+B) \tan (c+d x) (a \cos (c+d x)+a)^3}{d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^4}{2 d}",1,"(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*(6*(8*A + 13*B + 12*C)*(c + d*x) - 6*(13*A + 8*B + 2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*(13*A + 8*B + 2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (3*A)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (12*(4*A + B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (3*A)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (12*(4*A + B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 3*(4*A + 16*B + 27*C)*Sin[c + d*x] + 3*(B + 4*C)*Sin[2*(c + d*x)] + C*Sin[3*(c + d*x)]))/(192*d)","A",1
334,1,354,219,5.9862163,"\int (a+a \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a^4 \left(6 (2 A+8 B+13 C) (c+d x)+\frac{4 (20 A+3 (4 B+C)) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 (20 A+3 (4 B+C)) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}-6 (12 A+13 B+8 C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 (12 A+13 B+8 C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{-13 A-3 B}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{13 A+3 B}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{2 A \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2 A \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+12 (B+4 C) \sin (c+d x)+3 C \sin (2 (c+d x))\right)}{12 d}","-\frac{5 a^4 (2 A+B-C) \sin (c+d x)}{2 d}+\frac{a^4 (12 A+13 B+8 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(22 A+18 B+3 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{6 d}+\frac{1}{2} a^4 x (2 A+8 B+13 C)+\frac{(16 A+15 B+6 C) \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{6 d}+\frac{a (4 A+3 B) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^3}{6 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^4}{3 d}",1,"(a^4*(6*(2*A + 8*B + 13*C)*(c + d*x) - 6*(12*A + 13*B + 8*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*(12*A + 13*B + 8*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (13*A + 3*B)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (2*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (4*(20*A + 3*(4*B + C))*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (2*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + (-13*A - 3*B)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*(20*A + 3*(4*B + C))*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 12*(B + 4*C)*Sin[c + d*x] + 3*C*Sin[2*(c + d*x)]))/(12*d)","A",1
335,1,838,217,6.2032375,"\int (a+a \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{(B+4 C) (c+d x) (\cos (c+d x) a+a)^4 \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 d}+\frac{(-35 A-48 B-52 C) (\cos (c+d x) a+a)^4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{128 d}+\frac{(35 A+48 B+52 C) (\cos (c+d x) a+a)^4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{128 d}+\frac{(\cos (c+d x) a+a)^4 \left(4 A \sin \left(\frac{1}{2} (c+d x)\right)+B \sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{96 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{(\cos (c+d x) a+a)^4 \left(4 A \sin \left(\frac{1}{2} (c+d x)\right)+B \sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{96 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{(\cos (c+d x) a+a)^4 \left(5 A \sin \left(\frac{1}{2} (c+d x)\right)+5 B \sin \left(\frac{1}{2} (c+d x)\right)+3 C \sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{12 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{(\cos (c+d x) a+a)^4 \left(5 A \sin \left(\frac{1}{2} (c+d x)\right)+5 B \sin \left(\frac{1}{2} (c+d x)\right)+3 C \sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{12 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{C (\cos (c+d x) a+a)^4 \sin (c+d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 d}+\frac{(97 A+52 B+12 C) (\cos (c+d x) a+a)^4 \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{768 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{(-97 A-52 B-12 C) (\cos (c+d x) a+a)^4 \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{768 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{A (\cos (c+d x) a+a)^4 \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{256 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}-\frac{A (\cos (c+d x) a+a)^4 \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{256 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}","-\frac{5 a^4 (7 A+8 B+4 C) \sin (c+d x)}{8 d}+\frac{a^4 (35 A+48 B+52 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(35 A+44 B+36 C) \tan (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{12 d}+a^4 x (B+4 C)+\frac{(7 A+8 B+4 C) \tan (c+d x) \sec (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{8 d}+\frac{a (A+B) \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}",1,"((B + 4*C)*(c + d*x)*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8)/(16*d) + ((-35*A - 48*B - 52*C)*(a + a*Cos[c + d*x])^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sec[c/2 + (d*x)/2]^8)/(128*d) + ((35*A + 48*B + 52*C)*(a + a*Cos[c + d*x])^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sec[c/2 + (d*x)/2]^8)/(128*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8)/(256*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4) + ((97*A + 52*B + 12*C)*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8)/(768*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) - (A*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8)/(256*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4) + ((-97*A - 52*B - 12*C)*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8)/(768*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + ((a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(4*A*Sin[(c + d*x)/2] + B*Sin[(c + d*x)/2]))/(96*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + ((a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(4*A*Sin[(c + d*x)/2] + B*Sin[(c + d*x)/2]))/(96*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + ((a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(5*A*Sin[(c + d*x)/2] + 5*B*Sin[(c + d*x)/2] + 3*C*Sin[(c + d*x)/2]))/(12*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + ((a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(5*A*Sin[(c + d*x)/2] + 5*B*Sin[(c + d*x)/2] + 3*C*Sin[(c + d*x)/2]))/(12*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (C*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*Sin[c + d*x])/(16*d)","B",1
336,1,971,225,6.2191072,"\int (a+a \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{C (c+d x) (\cos (c+d x) a+a)^4 \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 d}+\frac{(-28 A-35 B-48 C) (\cos (c+d x) a+a)^4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{128 d}+\frac{(28 A+35 B+48 C) (\cos (c+d x) a+a)^4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{128 d}+\frac{A (\cos (c+d x) a+a)^4 \sin \left(\frac{1}{2} (c+d x)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{320 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5}+\frac{(\cos (c+d x) a+a)^4 \left(139 A \sin \left(\frac{1}{2} (c+d x)\right)+80 B \sin \left(\frac{1}{2} (c+d x)\right)+20 C \sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{1920 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{(\cos (c+d x) a+a)^4 \left(139 A \sin \left(\frac{1}{2} (c+d x)\right)+80 B \sin \left(\frac{1}{2} (c+d x)\right)+20 C \sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{1920 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{(\cos (c+d x) a+a)^4 \left(83 A \sin \left(\frac{1}{2} (c+d x)\right)+100 B \sin \left(\frac{1}{2} (c+d x)\right)+100 C \sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{240 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{(\cos (c+d x) a+a)^4 \left(83 A \sin \left(\frac{1}{2} (c+d x)\right)+100 B \sin \left(\frac{1}{2} (c+d x)\right)+100 C \sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{240 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{(559 A+485 B+260 C) (\cos (c+d x) a+a)^4 \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{3840 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{(-559 A-485 B-260 C) (\cos (c+d x) a+a)^4 \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{3840 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{(22 A+5 B) (\cos (c+d x) a+a)^4 \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{1280 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}+\frac{(-22 A-5 B) (\cos (c+d x) a+a)^4 \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{1280 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}+\frac{A (\cos (c+d x) a+a)^4 \sin \left(\frac{1}{2} (c+d x)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{320 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^5}","\frac{a^4 (28 A+35 B+40 C) \tan (c+d x)}{8 d}+\frac{a^4 (28 A+35 B+48 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(28 A+35 B+32 C) \tan (c+d x) \sec (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{24 d}+a^4 C x+\frac{(28 A+35 B+20 C) \tan (c+d x) \sec ^2(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{60 d}+\frac{a (4 A+5 B) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^3}{20 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^4}{5 d}",1,"(C*(c + d*x)*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8)/(16*d) + ((-28*A - 35*B - 48*C)*(a + a*Cos[c + d*x])^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sec[c/2 + (d*x)/2]^8)/(128*d) + ((28*A + 35*B + 48*C)*(a + a*Cos[c + d*x])^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sec[c/2 + (d*x)/2]^8)/(128*d) + ((22*A + 5*B)*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8)/(1280*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4) + ((559*A + 485*B + 260*C)*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8)/(3840*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (A*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*Sin[(c + d*x)/2])/(320*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5) + (A*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*Sin[(c + d*x)/2])/(320*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5) + ((-22*A - 5*B)*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8)/(1280*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4) + ((-559*A - 485*B - 260*C)*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8)/(3840*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + ((a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(139*A*Sin[(c + d*x)/2] + 80*B*Sin[(c + d*x)/2] + 20*C*Sin[(c + d*x)/2]))/(1920*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + ((a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(139*A*Sin[(c + d*x)/2] + 80*B*Sin[(c + d*x)/2] + 20*C*Sin[(c + d*x)/2]))/(1920*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + ((a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(83*A*Sin[(c + d*x)/2] + 100*B*Sin[(c + d*x)/2] + 100*C*Sin[(c + d*x)/2]))/(240*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + ((a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(83*A*Sin[(c + d*x)/2] + 100*B*Sin[(c + d*x)/2] + 100*C*Sin[(c + d*x)/2]))/(240*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","B",1
337,1,265,253,2.0905804,"\int (a+a \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","-\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \sec ^6(c+d x) \left(840 (7 A+8 B+10 C) \cos ^6(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sin (c+d x) (16 (672 A+643 B+620 C) \cos (c+d x)+20 (229 A+216 B+174 C) \cos (2 (c+d x))+4032 A \cos (3 (c+d x))+735 A \cos (4 (c+d x))+576 A \cos (5 (c+d x))+4165 A+4408 B \cos (3 (c+d x))+840 B \cos (4 (c+d x))+664 B \cos (5 (c+d x))+3480 B+4640 C \cos (3 (c+d x))+810 C \cos (4 (c+d x))+800 C \cos (5 (c+d x))+2670 C)\right)}{30720 d}","\frac{a^4 (72 A+83 B+100 C) \tan (c+d x)}{15 d}+\frac{7 a^4 (7 A+8 B+10 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 (417 A+488 B+550 C) \tan (c+d x) \sec (c+d x)}{240 d}+\frac{(43 A+52 B+50 C) \tan (c+d x) \sec ^2(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{60 d}+\frac{(37 A+48 B+30 C) \tan (c+d x) \sec ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{120 d}+\frac{a (2 A+3 B) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{15 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^4}{6 d}",1,"-1/30720*(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*Sec[c + d*x]^6*(840*(7*A + 8*B + 10*C)*Cos[c + d*x]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - (4165*A + 3480*B + 2670*C + 16*(672*A + 643*B + 620*C)*Cos[c + d*x] + 20*(229*A + 216*B + 174*C)*Cos[2*(c + d*x)] + 4032*A*Cos[3*(c + d*x)] + 4408*B*Cos[3*(c + d*x)] + 4640*C*Cos[3*(c + d*x)] + 735*A*Cos[4*(c + d*x)] + 840*B*Cos[4*(c + d*x)] + 810*C*Cos[4*(c + d*x)] + 576*A*Cos[5*(c + d*x)] + 664*B*Cos[5*(c + d*x)] + 800*C*Cos[5*(c + d*x)])*Sin[c + d*x]))/d","A",1
338,1,298,287,3.4670137,"\int (a+a \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^8(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^8,x]","-\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \sec ^7(c+d x) \left(3360 (44 A+49 B+56 C) \cos ^7(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-2 \sin (c+d x) (70 (1444 A+1291 B+1128 C) \cos (c+d x)+8 (12746 A+12936 B+12859 C) \cos (2 (c+d x))+35420 A \cos (3 (c+d x))+29056 A \cos (4 (c+d x))+4620 A \cos (5 (c+d x))+3632 A \cos (6 (c+d x))+80384 A+37205 B \cos (3 (c+d x))+32256 B \cos (4 (c+d x))+5145 B \cos (5 (c+d x))+4032 B \cos (6 (c+d x))+75264 B+36120 C \cos (3 (c+d x))+35504 C \cos (4 (c+d x))+5880 C \cos (5 (c+d x))+4648 C \cos (6 (c+d x))+72016 C)\right)}{860160 d}","\frac{a^4 (454 A+504 B+581 C) \tan (c+d x)}{105 d}+\frac{a^4 (44 A+49 B+56 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 (988 A+1113 B+1232 C) \tan (c+d x) \sec ^2(c+d x)}{840 d}+\frac{a^4 (44 A+49 B+56 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(436 A+511 B+504 C) \tan (c+d x) \sec ^3(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{840 d}+\frac{(16 A+21 B+14 C) \tan (c+d x) \sec ^4(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{70 d}+\frac{a (4 A+7 B) \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{42 d}+\frac{A \tan (c+d x) \sec ^6(c+d x) (a \cos (c+d x)+a)^4}{7 d}",1,"-1/860160*(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*Sec[c + d*x]^7*(3360*(44*A + 49*B + 56*C)*Cos[c + d*x]^7*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 2*(80384*A + 75264*B + 72016*C + 70*(1444*A + 1291*B + 1128*C)*Cos[c + d*x] + 8*(12746*A + 12936*B + 12859*C)*Cos[2*(c + d*x)] + 35420*A*Cos[3*(c + d*x)] + 37205*B*Cos[3*(c + d*x)] + 36120*C*Cos[3*(c + d*x)] + 29056*A*Cos[4*(c + d*x)] + 32256*B*Cos[4*(c + d*x)] + 35504*C*Cos[4*(c + d*x)] + 4620*A*Cos[5*(c + d*x)] + 5145*B*Cos[5*(c + d*x)] + 5880*C*Cos[5*(c + d*x)] + 3632*A*Cos[6*(c + d*x)] + 4032*B*Cos[6*(c + d*x)] + 4648*C*Cos[6*(c + d*x)])*Sin[c + d*x]))/d","A",1
339,1,393,174,0.7763533,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(72 d x (4 A-4 B+5 C) \cos \left(c+\frac{d x}{2}\right)+72 d x (4 A-4 B+5 C) \cos \left(\frac{d x}{2}\right)-96 A \sin \left(c+\frac{d x}{2}\right)-72 A \sin \left(c+\frac{3 d x}{2}\right)-72 A \sin \left(2 c+\frac{3 d x}{2}\right)+24 A \sin \left(2 c+\frac{5 d x}{2}\right)+24 A \sin \left(3 c+\frac{5 d x}{2}\right)-480 A \sin \left(\frac{d x}{2}\right)+168 B \sin \left(c+\frac{d x}{2}\right)+144 B \sin \left(c+\frac{3 d x}{2}\right)+144 B \sin \left(2 c+\frac{3 d x}{2}\right)-16 B \sin \left(2 c+\frac{5 d x}{2}\right)-16 B \sin \left(3 c+\frac{5 d x}{2}\right)+8 B \sin \left(3 c+\frac{7 d x}{2}\right)+8 B \sin \left(4 c+\frac{7 d x}{2}\right)+552 B \sin \left(\frac{d x}{2}\right)-168 C \sin \left(c+\frac{d x}{2}\right)-120 C \sin \left(c+\frac{3 d x}{2}\right)-120 C \sin \left(2 c+\frac{3 d x}{2}\right)+40 C \sin \left(2 c+\frac{5 d x}{2}\right)+40 C \sin \left(3 c+\frac{5 d x}{2}\right)-5 C \sin \left(3 c+\frac{7 d x}{2}\right)-5 C \sin \left(4 c+\frac{7 d x}{2}\right)+3 C \sin \left(4 c+\frac{9 d x}{2}\right)+3 C \sin \left(5 c+\frac{9 d x}{2}\right)-552 C \sin \left(\frac{d x}{2}\right)\right)}{192 a d (\cos (c+d x)+1)}","\frac{(3 A-4 B+4 C) \sin ^3(c+d x)}{3 a d}-\frac{(3 A-4 B+4 C) \sin (c+d x)}{a d}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(4 A-4 B+5 C) \sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac{3 (4 A-4 B+5 C) \sin (c+d x) \cos (c+d x)}{8 a d}+\frac{3 x (4 A-4 B+5 C)}{8 a}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(72*(4*A - 4*B + 5*C)*d*x*Cos[(d*x)/2] + 72*(4*A - 4*B + 5*C)*d*x*Cos[c + (d*x)/2] - 480*A*Sin[(d*x)/2] + 552*B*Sin[(d*x)/2] - 552*C*Sin[(d*x)/2] - 96*A*Sin[c + (d*x)/2] + 168*B*Sin[c + (d*x)/2] - 168*C*Sin[c + (d*x)/2] - 72*A*Sin[c + (3*d*x)/2] + 144*B*Sin[c + (3*d*x)/2] - 120*C*Sin[c + (3*d*x)/2] - 72*A*Sin[2*c + (3*d*x)/2] + 144*B*Sin[2*c + (3*d*x)/2] - 120*C*Sin[2*c + (3*d*x)/2] + 24*A*Sin[2*c + (5*d*x)/2] - 16*B*Sin[2*c + (5*d*x)/2] + 40*C*Sin[2*c + (5*d*x)/2] + 24*A*Sin[3*c + (5*d*x)/2] - 16*B*Sin[3*c + (5*d*x)/2] + 40*C*Sin[3*c + (5*d*x)/2] + 8*B*Sin[3*c + (7*d*x)/2] - 5*C*Sin[3*c + (7*d*x)/2] + 8*B*Sin[4*c + (7*d*x)/2] - 5*C*Sin[4*c + (7*d*x)/2] + 3*C*Sin[4*c + (9*d*x)/2] + 3*C*Sin[5*c + (9*d*x)/2]))/(192*a*d*(1 + Cos[c + d*x]))","B",1
340,1,307,139,0.8221292,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-12 d x (2 A-3 B+3 C) \cos \left(c+\frac{d x}{2}\right)-12 d x (2 A-3 B+3 C) \cos \left(\frac{d x}{2}\right)+12 A \sin \left(c+\frac{d x}{2}\right)+12 A \sin \left(c+\frac{3 d x}{2}\right)+12 A \sin \left(2 c+\frac{3 d x}{2}\right)+60 A \sin \left(\frac{d x}{2}\right)-12 B \sin \left(c+\frac{d x}{2}\right)-9 B \sin \left(c+\frac{3 d x}{2}\right)-9 B \sin \left(2 c+\frac{3 d x}{2}\right)+3 B \sin \left(2 c+\frac{5 d x}{2}\right)+3 B \sin \left(3 c+\frac{5 d x}{2}\right)-60 B \sin \left(\frac{d x}{2}\right)+21 C \sin \left(c+\frac{d x}{2}\right)+18 C \sin \left(c+\frac{3 d x}{2}\right)+18 C \sin \left(2 c+\frac{3 d x}{2}\right)-2 C \sin \left(2 c+\frac{5 d x}{2}\right)-2 C \sin \left(3 c+\frac{5 d x}{2}\right)+C \sin \left(3 c+\frac{7 d x}{2}\right)+C \sin \left(4 c+\frac{7 d x}{2}\right)+69 C \sin \left(\frac{d x}{2}\right)\right)}{24 a d (\cos (c+d x)+1)}","-\frac{(3 A-3 B+4 C) \sin ^3(c+d x)}{3 a d}+\frac{(3 A-3 B+4 C) \sin (c+d x)}{a d}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(2 A-3 B+3 C) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{x (2 A-3 B+3 C)}{2 a}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(-12*(2*A - 3*B + 3*C)*d*x*Cos[(d*x)/2] - 12*(2*A - 3*B + 3*C)*d*x*Cos[c + (d*x)/2] + 60*A*Sin[(d*x)/2] - 60*B*Sin[(d*x)/2] + 69*C*Sin[(d*x)/2] + 12*A*Sin[c + (d*x)/2] - 12*B*Sin[c + (d*x)/2] + 21*C*Sin[c + (d*x)/2] + 12*A*Sin[c + (3*d*x)/2] - 9*B*Sin[c + (3*d*x)/2] + 18*C*Sin[c + (3*d*x)/2] + 12*A*Sin[2*c + (3*d*x)/2] - 9*B*Sin[2*c + (3*d*x)/2] + 18*C*Sin[2*c + (3*d*x)/2] + 3*B*Sin[2*c + (5*d*x)/2] - 2*C*Sin[2*c + (5*d*x)/2] + 3*B*Sin[3*c + (5*d*x)/2] - 2*C*Sin[3*c + (5*d*x)/2] + C*Sin[3*c + (7*d*x)/2] + C*Sin[4*c + (7*d*x)/2]))/(24*a*d*(1 + Cos[c + d*x]))","B",1
341,1,213,110,0.476763,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(4 d x (2 A-2 B+3 C) \cos \left(c+\frac{d x}{2}\right)+4 d x (2 A-2 B+3 C) \cos \left(\frac{d x}{2}\right)-16 A \sin \left(\frac{d x}{2}\right)+4 B \sin \left(c+\frac{d x}{2}\right)+4 B \sin \left(c+\frac{3 d x}{2}\right)+4 B \sin \left(2 c+\frac{3 d x}{2}\right)+20 B \sin \left(\frac{d x}{2}\right)-4 C \sin \left(c+\frac{d x}{2}\right)-3 C \sin \left(c+\frac{3 d x}{2}\right)-3 C \sin \left(2 c+\frac{3 d x}{2}\right)+C \sin \left(2 c+\frac{5 d x}{2}\right)+C \sin \left(3 c+\frac{5 d x}{2}\right)-20 C \sin \left(\frac{d x}{2}\right)\right)}{8 a d (\cos (c+d x)+1)}","-\frac{(A-2 B+2 C) \sin (c+d x)}{a d}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(2 A-2 B+3 C) \sin (c+d x) \cos (c+d x)}{2 a d}+\frac{x (2 A-2 B+3 C)}{2 a}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(4*(2*A - 2*B + 3*C)*d*x*Cos[(d*x)/2] + 4*(2*A - 2*B + 3*C)*d*x*Cos[c + (d*x)/2] - 16*A*Sin[(d*x)/2] + 20*B*Sin[(d*x)/2] - 20*C*Sin[(d*x)/2] + 4*B*Sin[c + (d*x)/2] - 4*C*Sin[c + (d*x)/2] + 4*B*Sin[c + (3*d*x)/2] - 3*C*Sin[c + (3*d*x)/2] + 4*B*Sin[2*c + (3*d*x)/2] - 3*C*Sin[2*c + (3*d*x)/2] + C*Sin[2*c + (5*d*x)/2] + C*Sin[3*c + (5*d*x)/2]))/(8*a*d*(1 + Cos[c + d*x]))","A",1
342,1,136,54,0.2849118,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(4 A \sin \left(\frac{d x}{2}\right)+2 d x (B-C) \cos \left(c+\frac{d x}{2}\right)+2 d x (B-C) \cos \left(\frac{d x}{2}\right)-4 B \sin \left(\frac{d x}{2}\right)+C \sin \left(c+\frac{d x}{2}\right)+C \sin \left(c+\frac{3 d x}{2}\right)+C \sin \left(2 c+\frac{3 d x}{2}\right)+5 C \sin \left(\frac{d x}{2}\right)\right)}{2 a d (\cos (c+d x)+1)}","\frac{(A-B+C) \sin (c+d x)}{a d (\cos (c+d x)+1)}+\frac{x (B-C)}{a}+\frac{C \sin (c+d x)}{a d}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(2*(B - C)*d*x*Cos[(d*x)/2] + 2*(B - C)*d*x*Cos[c + (d*x)/2] + 4*A*Sin[(d*x)/2] - 4*B*Sin[(d*x)/2] + 5*C*Sin[(d*x)/2] + C*Sin[c + (d*x)/2] + C*Sin[c + (3*d*x)/2] + C*Sin[2*c + (3*d*x)/2]))/(2*a*d*(1 + Cos[c + d*x]))","B",1
343,1,163,51,0.5487246,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x]),x]","\frac{4 \cos \left(\frac{1}{2} (c+d x)\right) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right) \left(-A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+C d x\right)-\sec \left(\frac{c}{2}\right) (A-B+C) \sin \left(\frac{d x}{2}\right)\right)}{a d (\cos (c+d x)+1) (2 A+2 B \cos (c+d x)+C \cos (2 (c+d x))+C)}","-\frac{(A-B+C) \sin (c+d x)}{d (a \cos (c+d x)+a)}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a d}+\frac{C x}{a}",1,"(4*Cos[(c + d*x)/2]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(Cos[(c + d*x)/2]*(C*d*x - A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - (A - B + C)*Sec[c/2]*Sin[(d*x)/2]))/(a*d*(1 + Cos[c + d*x])*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*(c + d*x)]))","B",1
344,1,256,71,1.394424,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x]),x]","\frac{4 \cos \left(\frac{1}{2} (c+d x)\right) \cos ^2(c+d x) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(\sec \left(\frac{c}{2}\right) (A-B+C) \sin \left(\frac{d x}{2}\right)+\cos \left(\frac{1}{2} (c+d x)\right) \left((A-B) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+\frac{A \sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}\right)\right)}{a d (\cos (c+d x)+1) (2 A+2 B \cos (c+d x)+C \cos (2 (c+d x))+C)}","\frac{(2 A-B+C) \tan (c+d x)}{a d}-\frac{(A-B+C) \tan (c+d x)}{d (a \cos (c+d x)+a)}-\frac{(A-B) \tanh ^{-1}(\sin (c+d x))}{a d}",1,"(4*Cos[(c + d*x)/2]*Cos[c + d*x]^2*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*((A - B + C)*Sec[c/2]*Sin[(d*x)/2] + Cos[(c + d*x)/2]*((A - B)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (A*Sin[d*x])/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))))/(a*d*(1 + Cos[c + d*x])*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*(c + d*x)]))","B",1
345,1,256,117,1.4559995,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x]),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(-4 (A-B+C) \tan \left(\frac{1}{2} (c+d x)\right)-2 (3 A-2 B+2 C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 (3 A-2 B+2 C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{4 (B-A) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 (B-A) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+\frac{A}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{A}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}\right)}{2 a d (\cos (c+d x)+1)}","-\frac{(2 A-2 B+C) \tan (c+d x)}{a d}+\frac{(3 A-2 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{(3 A-2 B+2 C) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(A-B+C) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}",1,"(Cos[(c + d*x)/2]^2*(-2*(3*A - 2*B + 2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(3*A - 2*B + 2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + A/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (4*(-A + B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - A/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*(-A + B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 4*(A - B + C)*Tan[(c + d*x)/2]))/(2*a*d*(1 + Cos[c + d*x]))","B",1
346,1,351,148,3.7784441,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x]),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(12 (A-B+C) \tan \left(\frac{1}{2} (c+d x)\right)+\frac{4 (5 A-3 B+3 C) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 (5 A-3 B+3 C) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+6 (3 A-3 B+2 C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-6 (3 A-3 B+2 C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{2 A-3 B}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{3 B-2 A}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{2 A \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2 A \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}\right)}{6 a d (\cos (c+d x)+1)}","\frac{(4 A-3 B+3 C) \tan ^3(c+d x)}{3 a d}+\frac{(4 A-3 B+3 C) \tan (c+d x)}{a d}-\frac{(3 A-3 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{(3 A-3 B+2 C) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}",1,"(Cos[(c + d*x)/2]^2*(6*(3*A - 3*B + 2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 6*(3*A - 3*B + 2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (-2*A + 3*B)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (2*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (4*(5*A - 3*B + 3*C)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (2*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + (2*A - 3*B)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*(5*A - 3*B + 3*C)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 12*(A - B + C)*Tan[(c + d*x)/2]))/(6*a*d*(1 + Cos[c + d*x]))","B",1
347,1,481,185,0.9541256,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-36 d x (4 A-7 B+10 C) \cos \left(c+\frac{d x}{2}\right)-36 d x (4 A-7 B+10 C) \cos \left(\frac{d x}{2}\right)-120 A \sin \left(c+\frac{d x}{2}\right)+164 A \sin \left(c+\frac{3 d x}{2}\right)+36 A \sin \left(2 c+\frac{3 d x}{2}\right)+12 A \sin \left(2 c+\frac{5 d x}{2}\right)+12 A \sin \left(3 c+\frac{5 d x}{2}\right)-48 A d x \cos \left(c+\frac{3 d x}{2}\right)-48 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+264 A \sin \left(\frac{d x}{2}\right)+147 B \sin \left(c+\frac{d x}{2}\right)-239 B \sin \left(c+\frac{3 d x}{2}\right)-63 B \sin \left(2 c+\frac{3 d x}{2}\right)-15 B \sin \left(2 c+\frac{5 d x}{2}\right)-15 B \sin \left(3 c+\frac{5 d x}{2}\right)+3 B \sin \left(3 c+\frac{7 d x}{2}\right)+3 B \sin \left(4 c+\frac{7 d x}{2}\right)+84 B d x \cos \left(c+\frac{3 d x}{2}\right)+84 B d x \cos \left(2 c+\frac{3 d x}{2}\right)-381 B \sin \left(\frac{d x}{2}\right)-156 C \sin \left(c+\frac{d x}{2}\right)+342 C \sin \left(c+\frac{3 d x}{2}\right)+118 C \sin \left(2 c+\frac{3 d x}{2}\right)+30 C \sin \left(2 c+\frac{5 d x}{2}\right)+30 C \sin \left(3 c+\frac{5 d x}{2}\right)-3 C \sin \left(3 c+\frac{7 d x}{2}\right)-3 C \sin \left(4 c+\frac{7 d x}{2}\right)+C \sin \left(4 c+\frac{9 d x}{2}\right)+C \sin \left(5 c+\frac{9 d x}{2}\right)-120 C d x \cos \left(c+\frac{3 d x}{2}\right)-120 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+516 C \sin \left(\frac{d x}{2}\right)\right)}{48 a^2 d (\cos (c+d x)+1)^2}","-\frac{(5 A-8 B+12 C) \sin ^3(c+d x)}{3 a^2 d}+\frac{(5 A-8 B+12 C) \sin (c+d x)}{a^2 d}-\frac{(4 A-7 B+10 C) \sin (c+d x) \cos ^3(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(4 A-7 B+10 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{x (4 A-7 B+10 C)}{2 a^2}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(-36*(4*A - 7*B + 10*C)*d*x*Cos[(d*x)/2] - 36*(4*A - 7*B + 10*C)*d*x*Cos[c + (d*x)/2] - 48*A*d*x*Cos[c + (3*d*x)/2] + 84*B*d*x*Cos[c + (3*d*x)/2] - 120*C*d*x*Cos[c + (3*d*x)/2] - 48*A*d*x*Cos[2*c + (3*d*x)/2] + 84*B*d*x*Cos[2*c + (3*d*x)/2] - 120*C*d*x*Cos[2*c + (3*d*x)/2] + 264*A*Sin[(d*x)/2] - 381*B*Sin[(d*x)/2] + 516*C*Sin[(d*x)/2] - 120*A*Sin[c + (d*x)/2] + 147*B*Sin[c + (d*x)/2] - 156*C*Sin[c + (d*x)/2] + 164*A*Sin[c + (3*d*x)/2] - 239*B*Sin[c + (3*d*x)/2] + 342*C*Sin[c + (3*d*x)/2] + 36*A*Sin[2*c + (3*d*x)/2] - 63*B*Sin[2*c + (3*d*x)/2] + 118*C*Sin[2*c + (3*d*x)/2] + 12*A*Sin[2*c + (5*d*x)/2] - 15*B*Sin[2*c + (5*d*x)/2] + 30*C*Sin[2*c + (5*d*x)/2] + 12*A*Sin[3*c + (5*d*x)/2] - 15*B*Sin[3*c + (5*d*x)/2] + 30*C*Sin[3*c + (5*d*x)/2] + 3*B*Sin[3*c + (7*d*x)/2] - 3*C*Sin[3*c + (7*d*x)/2] + 3*B*Sin[4*c + (7*d*x)/2] - 3*C*Sin[4*c + (7*d*x)/2] + C*Sin[4*c + (9*d*x)/2] + C*Sin[5*c + (9*d*x)/2]))/(48*a^2*d*(1 + Cos[c + d*x])^2)","B",1
348,1,385,160,0.8423298,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(36 d x (2 A-4 B+7 C) \cos \left(c+\frac{d x}{2}\right)+36 d x (2 A-4 B+7 C) \cos \left(\frac{d x}{2}\right)+96 A \sin \left(c+\frac{d x}{2}\right)-80 A \sin \left(c+\frac{3 d x}{2}\right)+24 A d x \cos \left(c+\frac{3 d x}{2}\right)+24 A d x \cos \left(2 c+\frac{3 d x}{2}\right)-144 A \sin \left(\frac{d x}{2}\right)-120 B \sin \left(c+\frac{d x}{2}\right)+164 B \sin \left(c+\frac{3 d x}{2}\right)+36 B \sin \left(2 c+\frac{3 d x}{2}\right)+12 B \sin \left(2 c+\frac{5 d x}{2}\right)+12 B \sin \left(3 c+\frac{5 d x}{2}\right)-48 B d x \cos \left(c+\frac{3 d x}{2}\right)-48 B d x \cos \left(2 c+\frac{3 d x}{2}\right)+264 B \sin \left(\frac{d x}{2}\right)+147 C \sin \left(c+\frac{d x}{2}\right)-239 C \sin \left(c+\frac{3 d x}{2}\right)-63 C \sin \left(2 c+\frac{3 d x}{2}\right)-15 C \sin \left(2 c+\frac{5 d x}{2}\right)-15 C \sin \left(3 c+\frac{5 d x}{2}\right)+3 C \sin \left(3 c+\frac{7 d x}{2}\right)+3 C \sin \left(4 c+\frac{7 d x}{2}\right)+84 C d x \cos \left(c+\frac{3 d x}{2}\right)+84 C d x \cos \left(2 c+\frac{3 d x}{2}\right)-381 C \sin \left(\frac{d x}{2}\right)\right)}{48 a^2 d (\cos (c+d x)+1)^2}","-\frac{2 (2 A-5 B+8 C) \sin (c+d x)}{3 a^2 d}-\frac{(2 A-5 B+8 C) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{(2 A-4 B+7 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{x (2 A-4 B+7 C)}{2 a^2}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(36*(2*A - 4*B + 7*C)*d*x*Cos[(d*x)/2] + 36*(2*A - 4*B + 7*C)*d*x*Cos[c + (d*x)/2] + 24*A*d*x*Cos[c + (3*d*x)/2] - 48*B*d*x*Cos[c + (3*d*x)/2] + 84*C*d*x*Cos[c + (3*d*x)/2] + 24*A*d*x*Cos[2*c + (3*d*x)/2] - 48*B*d*x*Cos[2*c + (3*d*x)/2] + 84*C*d*x*Cos[2*c + (3*d*x)/2] - 144*A*Sin[(d*x)/2] + 264*B*Sin[(d*x)/2] - 381*C*Sin[(d*x)/2] + 96*A*Sin[c + (d*x)/2] - 120*B*Sin[c + (d*x)/2] + 147*C*Sin[c + (d*x)/2] - 80*A*Sin[c + (3*d*x)/2] + 164*B*Sin[c + (3*d*x)/2] - 239*C*Sin[c + (3*d*x)/2] + 36*B*Sin[2*c + (3*d*x)/2] - 63*C*Sin[2*c + (3*d*x)/2] + 12*B*Sin[2*c + (5*d*x)/2] - 15*C*Sin[2*c + (5*d*x)/2] + 12*B*Sin[3*c + (5*d*x)/2] - 15*C*Sin[3*c + (5*d*x)/2] + 3*C*Sin[3*c + (7*d*x)/2] + 3*C*Sin[4*c + (7*d*x)/2]))/(48*a^2*d*(1 + Cos[c + d*x])^2)","B",1
349,1,275,103,0.6893494,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-12 A \sin \left(c+\frac{d x}{2}\right)+8 A \sin \left(c+\frac{3 d x}{2}\right)+12 A \sin \left(\frac{d x}{2}\right)+18 d x (B-2 C) \cos \left(c+\frac{d x}{2}\right)+24 B \sin \left(c+\frac{d x}{2}\right)-20 B \sin \left(c+\frac{3 d x}{2}\right)+6 B d x \cos \left(c+\frac{3 d x}{2}\right)+6 B d x \cos \left(2 c+\frac{3 d x}{2}\right)+18 d x (B-2 C) \cos \left(\frac{d x}{2}\right)-36 B \sin \left(\frac{d x}{2}\right)-30 C \sin \left(c+\frac{d x}{2}\right)+41 C \sin \left(c+\frac{3 d x}{2}\right)+9 C \sin \left(2 c+\frac{3 d x}{2}\right)+3 C \sin \left(2 c+\frac{5 d x}{2}\right)+3 C \sin \left(3 c+\frac{5 d x}{2}\right)-12 C d x \cos \left(c+\frac{3 d x}{2}\right)-12 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+66 C \sin \left(\frac{d x}{2}\right)\right)}{12 a^2 d (\cos (c+d x)+1)^2}","\frac{(A-B+4 C) \sin (c+d x)}{3 a^2 d}-\frac{(B-2 C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{x (B-2 C)}{a^2}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(18*(B - 2*C)*d*x*Cos[(d*x)/2] + 18*(B - 2*C)*d*x*Cos[c + (d*x)/2] + 6*B*d*x*Cos[c + (3*d*x)/2] - 12*C*d*x*Cos[c + (3*d*x)/2] + 6*B*d*x*Cos[2*c + (3*d*x)/2] - 12*C*d*x*Cos[2*c + (3*d*x)/2] + 12*A*Sin[(d*x)/2] - 36*B*Sin[(d*x)/2] + 66*C*Sin[(d*x)/2] - 12*A*Sin[c + (d*x)/2] + 24*B*Sin[c + (d*x)/2] - 30*C*Sin[c + (d*x)/2] + 8*A*Sin[c + (3*d*x)/2] - 20*B*Sin[c + (3*d*x)/2] + 41*C*Sin[c + (3*d*x)/2] + 9*C*Sin[2*c + (3*d*x)/2] + 3*C*Sin[2*c + (5*d*x)/2] + 3*C*Sin[3*c + (5*d*x)/2]))/(12*a^2*d*(1 + Cos[c + d*x])^2)","B",1
350,1,175,72,0.4188858,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \left(2 A \sin \left(c+\frac{3 d x}{2}\right)+6 A \sin \left(\frac{d x}{2}\right)-6 B \sin \left(c+\frac{d x}{2}\right)+4 B \sin \left(c+\frac{3 d x}{2}\right)+6 B \sin \left(\frac{d x}{2}\right)+12 C \sin \left(c+\frac{d x}{2}\right)-10 C \sin \left(c+\frac{3 d x}{2}\right)+9 C d x \cos \left(c+\frac{d x}{2}\right)+3 C d x \cos \left(c+\frac{3 d x}{2}\right)+3 C d x \cos \left(2 c+\frac{3 d x}{2}\right)-18 C \sin \left(\frac{d x}{2}\right)+9 C d x \cos \left(\frac{d x}{2}\right)\right)}{24 a^2 d}","\frac{(A+2 B-5 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{C x}{a^2}+\frac{(A-B+C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^3*(9*C*d*x*Cos[(d*x)/2] + 9*C*d*x*Cos[c + (d*x)/2] + 3*C*d*x*Cos[c + (3*d*x)/2] + 3*C*d*x*Cos[2*c + (3*d*x)/2] + 6*A*Sin[(d*x)/2] + 6*B*Sin[(d*x)/2] - 18*C*Sin[(d*x)/2] - 6*B*Sin[c + (d*x)/2] + 12*C*Sin[c + (d*x)/2] + 2*A*Sin[c + (3*d*x)/2] + 4*B*Sin[c + (3*d*x)/2] - 10*C*Sin[c + (3*d*x)/2]))/(24*a^2*d)","B",1
351,1,221,83,0.8668385,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^2,x]","-\frac{4 \cos \left(\frac{1}{2} (c+d x)\right) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \left(\tan \left(\frac{c}{2}\right) (A-B+C) \cos \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{c}{2}\right) (A-B+C) \sin \left(\frac{d x}{2}\right)+2 \sec \left(\frac{c}{2}\right) (4 A-B-2 C) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)+6 A \cos ^3\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{3 a^2 d (\cos (c+d x)+1)^2 (2 A+2 B \cos (c+d x)+C \cos (2 (c+d x))+C)}","-\frac{(4 A-B-2 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(A-B+C) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(-4*Cos[(c + d*x)/2]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(6*A*Cos[(c + d*x)/2]^3*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (A - B + C)*Sec[c/2]*Sin[(d*x)/2] + 2*(4*A - B - 2*C)*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + (A - B + C)*Cos[(c + d*x)/2]*Tan[c/2]))/(3*a^2*d*(1 + Cos[c + d*x])^2*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*(c + d*x)]))","B",1
352,1,321,109,2.0635606,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^2,x]","\frac{4 \cos \left(\frac{1}{2} (c+d x)\right) \cos ^2(c+d x) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(\tan \left(\frac{c}{2}\right) (A-B+C) \cos \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{c}{2}\right) (A-B+C) \sin \left(\frac{d x}{2}\right)+2 \sec \left(\frac{c}{2}\right) (7 A-4 B+C) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)+6 \cos ^3\left(\frac{1}{2} (c+d x)\right) \left((2 A-B) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+\frac{A \sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}\right)\right)}{3 a^2 d (\cos (c+d x)+1)^2 (2 A+2 B \cos (c+d x)+C \cos (2 (c+d x))+C)}","\frac{(10 A-4 B+C) \tan (c+d x)}{3 a^2 d}-\frac{(2 A-B) \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(2 A-B) \tan (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{(A-B+C) \tan (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(4*Cos[(c + d*x)/2]*Cos[c + d*x]^2*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*((A - B + C)*Sec[c/2]*Sin[(d*x)/2] + 2*(7*A - 4*B + C)*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + 6*Cos[(c + d*x)/2]^3*((2*A - B)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (A*Sin[d*x])/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))) + (A - B + C)*Cos[(c + d*x)/2]*Tan[c/2]))/(3*a^2*d*(1 + Cos[c + d*x])^2*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*(c + d*x)]))","B",1
353,1,578,165,6.1796239,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^2,x]","-\frac{2 (7 A-4 B+2 C) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d (a \cos (c+d x)+a)^2}+\frac{2 (7 A-4 B+2 C) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d (a \cos (c+d x)+a)^2}-\frac{2 \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \left(A \sin \left(\frac{1}{2} (c+d x)\right)-B \sin \left(\frac{1}{2} (c+d x)\right)+C \sin \left(\frac{1}{2} (c+d x)\right)\right)}{3 d (a \cos (c+d x)+a)^2}-\frac{4 \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(10 A \sin \left(\frac{1}{2} (c+d x)\right)-7 B \sin \left(\frac{1}{2} (c+d x)\right)+4 C \sin \left(\frac{1}{2} (c+d x)\right)\right)}{3 d (a \cos (c+d x)+a)^2}-\frac{4 \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \left(2 A \sin \left(\frac{1}{2} (c+d x)\right)-B \sin \left(\frac{1}{2} (c+d x)\right)\right)}{d (a \cos (c+d x)+a)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{4 \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \left(2 A \sin \left(\frac{1}{2} (c+d x)\right)-B \sin \left(\frac{1}{2} (c+d x)\right)\right)}{d (a \cos (c+d x)+a)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{A \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a \cos (c+d x)+a)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{A \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a \cos (c+d x)+a)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}","-\frac{2 (8 A-5 B+2 C) \tan (c+d x)}{3 a^2 d}+\frac{(7 A-4 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac{(7 A-4 B+2 C) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{(8 A-5 B+2 C) \tan (c+d x) \sec (c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(A-B+C) \tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(-2*(7*A - 4*B + 2*C)*Cos[c/2 + (d*x)/2]^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(d*(a + a*Cos[c + d*x])^2) + (2*(7*A - 4*B + 2*C)*Cos[c/2 + (d*x)/2]^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(d*(a + a*Cos[c + d*x])^2) + (A*Cos[c/2 + (d*x)/2]^4)/(d*(a + a*Cos[c + d*x])^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) - (A*Cos[c/2 + (d*x)/2]^4)/(d*(a + a*Cos[c + d*x])^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) - (4*Cos[c/2 + (d*x)/2]^4*(2*A*Sin[(c + d*x)/2] - B*Sin[(c + d*x)/2]))/(d*(a + a*Cos[c + d*x])^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - (4*Cos[c/2 + (d*x)/2]^4*(2*A*Sin[(c + d*x)/2] - B*Sin[(c + d*x)/2]))/(d*(a + a*Cos[c + d*x])^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) - (2*Cos[c/2 + (d*x)/2]^4*Sec[(c + d*x)/2]^3*(A*Sin[(c + d*x)/2] - B*Sin[(c + d*x)/2] + C*Sin[(c + d*x)/2]))/(3*d*(a + a*Cos[c + d*x])^2) - (4*Cos[c/2 + (d*x)/2]^4*Sec[(c + d*x)/2]*(10*A*Sin[(c + d*x)/2] - 7*B*Sin[(c + d*x)/2] + 4*C*Sin[(c + d*x)/2]))/(3*d*(a + a*Cos[c + d*x])^2)","B",1
354,1,763,194,6.2189738,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^2,x]","\frac{4 \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \left(11 A \sin \left(\frac{1}{2} (c+d x)\right)-6 B \sin \left(\frac{1}{2} (c+d x)\right)+3 C \sin \left(\frac{1}{2} (c+d x)\right)\right)}{3 d (a \cos (c+d x)+a)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{4 \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \left(11 A \sin \left(\frac{1}{2} (c+d x)\right)-6 B \sin \left(\frac{1}{2} (c+d x)\right)+3 C \sin \left(\frac{1}{2} (c+d x)\right)\right)}{3 d (a \cos (c+d x)+a)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{2 (10 A-7 B+4 C) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d (a \cos (c+d x)+a)^2}-\frac{2 (10 A-7 B+4 C) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d (a \cos (c+d x)+a)^2}+\frac{2 \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \left(A \sin \left(\frac{1}{2} (c+d x)\right)-B \sin \left(\frac{1}{2} (c+d x)\right)+C \sin \left(\frac{1}{2} (c+d x)\right)\right)}{3 d (a \cos (c+d x)+a)^2}+\frac{4 \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(13 A \sin \left(\frac{1}{2} (c+d x)\right)-10 B \sin \left(\frac{1}{2} (c+d x)\right)+7 C \sin \left(\frac{1}{2} (c+d x)\right)\right)}{3 d (a \cos (c+d x)+a)^2}+\frac{(3 B-5 A) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (a \cos (c+d x)+a)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{(5 A-3 B) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (a \cos (c+d x)+a)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{2 A \sin \left(\frac{1}{2} (c+d x)\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (a \cos (c+d x)+a)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2 A \sin \left(\frac{1}{2} (c+d x)\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (a \cos (c+d x)+a)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","\frac{(12 A-8 B+5 C) \tan ^3(c+d x)}{3 a^2 d}+\frac{(12 A-8 B+5 C) \tan (c+d x)}{a^2 d}-\frac{(10 A-7 B+4 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}-\frac{(10 A-7 B+4 C) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{(10 A-7 B+4 C) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(2*(10*A - 7*B + 4*C)*Cos[c/2 + (d*x)/2]^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(d*(a + a*Cos[c + d*x])^2) - (2*(10*A - 7*B + 4*C)*Cos[c/2 + (d*x)/2]^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(d*(a + a*Cos[c + d*x])^2) + ((-5*A + 3*B)*Cos[c/2 + (d*x)/2]^4)/(3*d*(a + a*Cos[c + d*x])^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (2*A*Cos[c/2 + (d*x)/2]^4*Sin[(c + d*x)/2])/(3*d*(a + a*Cos[c + d*x])^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + (2*A*Cos[c/2 + (d*x)/2]^4*Sin[(c + d*x)/2])/(3*d*(a + a*Cos[c + d*x])^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + ((5*A - 3*B)*Cos[c/2 + (d*x)/2]^4)/(3*d*(a + a*Cos[c + d*x])^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (2*Cos[c/2 + (d*x)/2]^4*Sec[(c + d*x)/2]^3*(A*Sin[(c + d*x)/2] - B*Sin[(c + d*x)/2] + C*Sin[(c + d*x)/2]))/(3*d*(a + a*Cos[c + d*x])^2) + (4*Cos[c/2 + (d*x)/2]^4*(11*A*Sin[(c + d*x)/2] - 6*B*Sin[(c + d*x)/2] + 3*C*Sin[(c + d*x)/2]))/(3*d*(a + a*Cos[c + d*x])^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (4*Cos[c/2 + (d*x)/2]^4*(11*A*Sin[(c + d*x)/2] - 6*B*Sin[(c + d*x)/2] + 3*C*Sin[(c + d*x)/2]))/(3*d*(a + a*Cos[c + d*x])^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (4*Cos[c/2 + (d*x)/2]^4*Sec[(c + d*x)/2]*(13*A*Sin[(c + d*x)/2] - 10*B*Sin[(c + d*x)/2] + 7*C*Sin[(c + d*x)/2]))/(3*d*(a + a*Cos[c + d*x])^2)","B",1
355,1,663,237,1.9850365,"\int \frac{\cos ^4(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-600 d x (6 A-13 B+23 C) \cos \left(c+\frac{d x}{2}\right)-600 d x (6 A-13 B+23 C) \cos \left(\frac{d x}{2}\right)-4500 A \sin \left(c+\frac{d x}{2}\right)+4860 A \sin \left(c+\frac{3 d x}{2}\right)-900 A \sin \left(2 c+\frac{3 d x}{2}\right)+1452 A \sin \left(2 c+\frac{5 d x}{2}\right)+300 A \sin \left(3 c+\frac{5 d x}{2}\right)+60 A \sin \left(3 c+\frac{7 d x}{2}\right)+60 A \sin \left(4 c+\frac{7 d x}{2}\right)-1800 A d x \cos \left(c+\frac{3 d x}{2}\right)-1800 A d x \cos \left(2 c+\frac{3 d x}{2}\right)-360 A d x \cos \left(2 c+\frac{5 d x}{2}\right)-360 A d x \cos \left(3 c+\frac{5 d x}{2}\right)+7020 A \sin \left(\frac{d x}{2}\right)+7560 B \sin \left(c+\frac{d x}{2}\right)-9230 B \sin \left(c+\frac{3 d x}{2}\right)+930 B \sin \left(2 c+\frac{3 d x}{2}\right)-2782 B \sin \left(2 c+\frac{5 d x}{2}\right)-750 B \sin \left(3 c+\frac{5 d x}{2}\right)-105 B \sin \left(3 c+\frac{7 d x}{2}\right)-105 B \sin \left(4 c+\frac{7 d x}{2}\right)+15 B \sin \left(4 c+\frac{9 d x}{2}\right)+15 B \sin \left(5 c+\frac{9 d x}{2}\right)+3900 B d x \cos \left(c+\frac{3 d x}{2}\right)+3900 B d x \cos \left(2 c+\frac{3 d x}{2}\right)+780 B d x \cos \left(2 c+\frac{5 d x}{2}\right)+780 B d x \cos \left(3 c+\frac{5 d x}{2}\right)-12760 B \sin \left(\frac{d x}{2}\right)-11110 C \sin \left(c+\frac{d x}{2}\right)+15380 C \sin \left(c+\frac{3 d x}{2}\right)-380 C \sin \left(2 c+\frac{3 d x}{2}\right)+4777 C \sin \left(2 c+\frac{5 d x}{2}\right)+1625 C \sin \left(3 c+\frac{5 d x}{2}\right)+230 C \sin \left(3 c+\frac{7 d x}{2}\right)+230 C \sin \left(4 c+\frac{7 d x}{2}\right)-20 C \sin \left(4 c+\frac{9 d x}{2}\right)-20 C \sin \left(5 c+\frac{9 d x}{2}\right)+5 C \sin \left(5 c+\frac{11 d x}{2}\right)+5 C \sin \left(6 c+\frac{11 d x}{2}\right)-6900 C d x \cos \left(c+\frac{3 d x}{2}\right)-6900 C d x \cos \left(2 c+\frac{3 d x}{2}\right)-1380 C d x \cos \left(2 c+\frac{5 d x}{2}\right)-1380 C d x \cos \left(3 c+\frac{5 d x}{2}\right)+20410 C \sin \left(\frac{d x}{2}\right)\right)}{480 a^3 d (\cos (c+d x)+1)^3}","-\frac{4 (9 A-19 B+34 C) \sin ^3(c+d x)}{15 a^3 d}+\frac{4 (9 A-19 B+34 C) \sin (c+d x)}{5 a^3 d}-\frac{(6 A-13 B+23 C) \sin (c+d x) \cos ^3(c+d x)}{3 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(6 A-13 B+23 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{x (6 A-13 B+23 C)}{2 a^3}-\frac{(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(3 A-8 B+13 C) \sin (c+d x) \cos ^4(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(-600*(6*A - 13*B + 23*C)*d*x*Cos[(d*x)/2] - 600*(6*A - 13*B + 23*C)*d*x*Cos[c + (d*x)/2] - 1800*A*d*x*Cos[c + (3*d*x)/2] + 3900*B*d*x*Cos[c + (3*d*x)/2] - 6900*C*d*x*Cos[c + (3*d*x)/2] - 1800*A*d*x*Cos[2*c + (3*d*x)/2] + 3900*B*d*x*Cos[2*c + (3*d*x)/2] - 6900*C*d*x*Cos[2*c + (3*d*x)/2] - 360*A*d*x*Cos[2*c + (5*d*x)/2] + 780*B*d*x*Cos[2*c + (5*d*x)/2] - 1380*C*d*x*Cos[2*c + (5*d*x)/2] - 360*A*d*x*Cos[3*c + (5*d*x)/2] + 780*B*d*x*Cos[3*c + (5*d*x)/2] - 1380*C*d*x*Cos[3*c + (5*d*x)/2] + 7020*A*Sin[(d*x)/2] - 12760*B*Sin[(d*x)/2] + 20410*C*Sin[(d*x)/2] - 4500*A*Sin[c + (d*x)/2] + 7560*B*Sin[c + (d*x)/2] - 11110*C*Sin[c + (d*x)/2] + 4860*A*Sin[c + (3*d*x)/2] - 9230*B*Sin[c + (3*d*x)/2] + 15380*C*Sin[c + (3*d*x)/2] - 900*A*Sin[2*c + (3*d*x)/2] + 930*B*Sin[2*c + (3*d*x)/2] - 380*C*Sin[2*c + (3*d*x)/2] + 1452*A*Sin[2*c + (5*d*x)/2] - 2782*B*Sin[2*c + (5*d*x)/2] + 4777*C*Sin[2*c + (5*d*x)/2] + 300*A*Sin[3*c + (5*d*x)/2] - 750*B*Sin[3*c + (5*d*x)/2] + 1625*C*Sin[3*c + (5*d*x)/2] + 60*A*Sin[3*c + (7*d*x)/2] - 105*B*Sin[3*c + (7*d*x)/2] + 230*C*Sin[3*c + (7*d*x)/2] + 60*A*Sin[4*c + (7*d*x)/2] - 105*B*Sin[4*c + (7*d*x)/2] + 230*C*Sin[4*c + (7*d*x)/2] + 15*B*Sin[4*c + (9*d*x)/2] - 20*C*Sin[4*c + (9*d*x)/2] + 15*B*Sin[5*c + (9*d*x)/2] - 20*C*Sin[5*c + (9*d*x)/2] + 5*C*Sin[5*c + (11*d*x)/2] + 5*C*Sin[6*c + (11*d*x)/2]))/(480*a^3*d*(1 + Cos[c + d*x])^3)","B",1
356,1,565,207,0.9826222,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(600 d x (2 A-6 B+13 C) \cos \left(c+\frac{d x}{2}\right)+600 d x (2 A-6 B+13 C) \cos \left(\frac{d x}{2}\right)+2160 A \sin \left(c+\frac{d x}{2}\right)-1840 A \sin \left(c+\frac{3 d x}{2}\right)+720 A \sin \left(2 c+\frac{3 d x}{2}\right)-512 A \sin \left(2 c+\frac{5 d x}{2}\right)+600 A d x \cos \left(c+\frac{3 d x}{2}\right)+600 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+120 A d x \cos \left(2 c+\frac{5 d x}{2}\right)+120 A d x \cos \left(3 c+\frac{5 d x}{2}\right)-2960 A \sin \left(\frac{d x}{2}\right)-4500 B \sin \left(c+\frac{d x}{2}\right)+4860 B \sin \left(c+\frac{3 d x}{2}\right)-900 B \sin \left(2 c+\frac{3 d x}{2}\right)+1452 B \sin \left(2 c+\frac{5 d x}{2}\right)+300 B \sin \left(3 c+\frac{5 d x}{2}\right)+60 B \sin \left(3 c+\frac{7 d x}{2}\right)+60 B \sin \left(4 c+\frac{7 d x}{2}\right)-1800 B d x \cos \left(c+\frac{3 d x}{2}\right)-1800 B d x \cos \left(2 c+\frac{3 d x}{2}\right)-360 B d x \cos \left(2 c+\frac{5 d x}{2}\right)-360 B d x \cos \left(3 c+\frac{5 d x}{2}\right)+7020 B \sin \left(\frac{d x}{2}\right)+7560 C \sin \left(c+\frac{d x}{2}\right)-9230 C \sin \left(c+\frac{3 d x}{2}\right)+930 C \sin \left(2 c+\frac{3 d x}{2}\right)-2782 C \sin \left(2 c+\frac{5 d x}{2}\right)-750 C \sin \left(3 c+\frac{5 d x}{2}\right)-105 C \sin \left(3 c+\frac{7 d x}{2}\right)-105 C \sin \left(4 c+\frac{7 d x}{2}\right)+15 C \sin \left(4 c+\frac{9 d x}{2}\right)+15 C \sin \left(5 c+\frac{9 d x}{2}\right)+3900 C d x \cos \left(c+\frac{3 d x}{2}\right)+3900 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+780 C d x \cos \left(2 c+\frac{5 d x}{2}\right)+780 C d x \cos \left(3 c+\frac{5 d x}{2}\right)-12760 C \sin \left(\frac{d x}{2}\right)\right)}{480 a^3 d (\cos (c+d x)+1)^3}","-\frac{2 (11 A-36 B+76 C) \sin (c+d x)}{15 a^3 d}-\frac{(11 A-36 B+76 C) \sin (c+d x) \cos ^2(c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(2 A-6 B+13 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac{x (2 A-6 B+13 C)}{2 a^3}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(A-6 B+11 C) \sin (c+d x) \cos ^3(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(600*(2*A - 6*B + 13*C)*d*x*Cos[(d*x)/2] + 600*(2*A - 6*B + 13*C)*d*x*Cos[c + (d*x)/2] + 600*A*d*x*Cos[c + (3*d*x)/2] - 1800*B*d*x*Cos[c + (3*d*x)/2] + 3900*C*d*x*Cos[c + (3*d*x)/2] + 600*A*d*x*Cos[2*c + (3*d*x)/2] - 1800*B*d*x*Cos[2*c + (3*d*x)/2] + 3900*C*d*x*Cos[2*c + (3*d*x)/2] + 120*A*d*x*Cos[2*c + (5*d*x)/2] - 360*B*d*x*Cos[2*c + (5*d*x)/2] + 780*C*d*x*Cos[2*c + (5*d*x)/2] + 120*A*d*x*Cos[3*c + (5*d*x)/2] - 360*B*d*x*Cos[3*c + (5*d*x)/2] + 780*C*d*x*Cos[3*c + (5*d*x)/2] - 2960*A*Sin[(d*x)/2] + 7020*B*Sin[(d*x)/2] - 12760*C*Sin[(d*x)/2] + 2160*A*Sin[c + (d*x)/2] - 4500*B*Sin[c + (d*x)/2] + 7560*C*Sin[c + (d*x)/2] - 1840*A*Sin[c + (3*d*x)/2] + 4860*B*Sin[c + (3*d*x)/2] - 9230*C*Sin[c + (3*d*x)/2] + 720*A*Sin[2*c + (3*d*x)/2] - 900*B*Sin[2*c + (3*d*x)/2] + 930*C*Sin[2*c + (3*d*x)/2] - 512*A*Sin[2*c + (5*d*x)/2] + 1452*B*Sin[2*c + (5*d*x)/2] - 2782*C*Sin[2*c + (5*d*x)/2] + 300*B*Sin[3*c + (5*d*x)/2] - 750*C*Sin[3*c + (5*d*x)/2] + 60*B*Sin[3*c + (7*d*x)/2] - 105*C*Sin[3*c + (7*d*x)/2] + 60*B*Sin[4*c + (7*d*x)/2] - 105*C*Sin[4*c + (7*d*x)/2] + 15*C*Sin[4*c + (9*d*x)/2] + 15*C*Sin[5*c + (9*d*x)/2]))/(480*a^3*d*(1 + Cos[c + d*x])^3)","B",1
357,1,423,152,1.0314544,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-120 A \sin \left(c+\frac{d x}{2}\right)+80 A \sin \left(c+\frac{3 d x}{2}\right)-60 A \sin \left(2 c+\frac{3 d x}{2}\right)+28 A \sin \left(2 c+\frac{5 d x}{2}\right)+160 A \sin \left(\frac{d x}{2}\right)+300 d x (B-3 C) \cos \left(c+\frac{d x}{2}\right)+540 B \sin \left(c+\frac{d x}{2}\right)-460 B \sin \left(c+\frac{3 d x}{2}\right)+180 B \sin \left(2 c+\frac{3 d x}{2}\right)-128 B \sin \left(2 c+\frac{5 d x}{2}\right)+150 B d x \cos \left(c+\frac{3 d x}{2}\right)+150 B d x \cos \left(2 c+\frac{3 d x}{2}\right)+30 B d x \cos \left(2 c+\frac{5 d x}{2}\right)+30 B d x \cos \left(3 c+\frac{5 d x}{2}\right)+300 d x (B-3 C) \cos \left(\frac{d x}{2}\right)-740 B \sin \left(\frac{d x}{2}\right)-1125 C \sin \left(c+\frac{d x}{2}\right)+1215 C \sin \left(c+\frac{3 d x}{2}\right)-225 C \sin \left(2 c+\frac{3 d x}{2}\right)+363 C \sin \left(2 c+\frac{5 d x}{2}\right)+75 C \sin \left(3 c+\frac{5 d x}{2}\right)+15 C \sin \left(3 c+\frac{7 d x}{2}\right)+15 C \sin \left(4 c+\frac{7 d x}{2}\right)-450 C d x \cos \left(c+\frac{3 d x}{2}\right)-450 C d x \cos \left(2 c+\frac{3 d x}{2}\right)-90 C d x \cos \left(2 c+\frac{5 d x}{2}\right)-90 C d x \cos \left(3 c+\frac{5 d x}{2}\right)+1755 C \sin \left(\frac{d x}{2}\right)\right)}{120 a^3 d (\cos (c+d x)+1)^3}","\frac{(2 A-7 B+27 C) \sin (c+d x)}{15 a^3 d}-\frac{(B-3 C) \sin (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{x (B-3 C)}{a^3}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(A+4 B-9 C) \sin (c+d x) \cos ^2(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(300*(B - 3*C)*d*x*Cos[(d*x)/2] + 300*(B - 3*C)*d*x*Cos[c + (d*x)/2] + 150*B*d*x*Cos[c + (3*d*x)/2] - 450*C*d*x*Cos[c + (3*d*x)/2] + 150*B*d*x*Cos[2*c + (3*d*x)/2] - 450*C*d*x*Cos[2*c + (3*d*x)/2] + 30*B*d*x*Cos[2*c + (5*d*x)/2] - 90*C*d*x*Cos[2*c + (5*d*x)/2] + 30*B*d*x*Cos[3*c + (5*d*x)/2] - 90*C*d*x*Cos[3*c + (5*d*x)/2] + 160*A*Sin[(d*x)/2] - 740*B*Sin[(d*x)/2] + 1755*C*Sin[(d*x)/2] - 120*A*Sin[c + (d*x)/2] + 540*B*Sin[c + (d*x)/2] - 1125*C*Sin[c + (d*x)/2] + 80*A*Sin[c + (3*d*x)/2] - 460*B*Sin[c + (3*d*x)/2] + 1215*C*Sin[c + (3*d*x)/2] - 60*A*Sin[2*c + (3*d*x)/2] + 180*B*Sin[2*c + (3*d*x)/2] - 225*C*Sin[2*c + (3*d*x)/2] + 28*A*Sin[2*c + (5*d*x)/2] - 128*B*Sin[2*c + (5*d*x)/2] + 363*C*Sin[2*c + (5*d*x)/2] + 75*C*Sin[3*c + (5*d*x)/2] + 15*C*Sin[3*c + (7*d*x)/2] + 15*C*Sin[4*c + (7*d*x)/2]))/(120*a^3*d*(1 + Cos[c + d*x])^3)","B",1
358,1,289,123,0.7560526,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \left(-30 A \sin \left(c+\frac{d x}{2}\right)+30 A \sin \left(c+\frac{3 d x}{2}\right)+6 A \sin \left(2 c+\frac{5 d x}{2}\right)+30 A \sin \left(\frac{d x}{2}\right)-60 B \sin \left(c+\frac{d x}{2}\right)+40 B \sin \left(c+\frac{3 d x}{2}\right)-30 B \sin \left(2 c+\frac{3 d x}{2}\right)+14 B \sin \left(2 c+\frac{5 d x}{2}\right)+80 B \sin \left(\frac{d x}{2}\right)+270 C \sin \left(c+\frac{d x}{2}\right)-230 C \sin \left(c+\frac{3 d x}{2}\right)+90 C \sin \left(2 c+\frac{3 d x}{2}\right)-64 C \sin \left(2 c+\frac{5 d x}{2}\right)+150 C d x \cos \left(c+\frac{d x}{2}\right)+75 C d x \cos \left(c+\frac{3 d x}{2}\right)+75 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+15 C d x \cos \left(2 c+\frac{5 d x}{2}\right)+15 C d x \cos \left(3 c+\frac{5 d x}{2}\right)-370 C \sin \left(\frac{d x}{2}\right)+150 C d x \cos \left(\frac{d x}{2}\right)\right)}{480 a^3 d}","\frac{(6 A+4 B-29 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{C x}{a^3}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(3 A+2 B-7 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^5*(150*C*d*x*Cos[(d*x)/2] + 150*C*d*x*Cos[c + (d*x)/2] + 75*C*d*x*Cos[c + (3*d*x)/2] + 75*C*d*x*Cos[2*c + (3*d*x)/2] + 15*C*d*x*Cos[2*c + (5*d*x)/2] + 15*C*d*x*Cos[3*c + (5*d*x)/2] + 30*A*Sin[(d*x)/2] + 80*B*Sin[(d*x)/2] - 370*C*Sin[(d*x)/2] - 30*A*Sin[c + (d*x)/2] - 60*B*Sin[c + (d*x)/2] + 270*C*Sin[c + (d*x)/2] + 30*A*Sin[c + (3*d*x)/2] + 40*B*Sin[c + (3*d*x)/2] - 230*C*Sin[c + (3*d*x)/2] - 30*B*Sin[2*c + (3*d*x)/2] + 90*C*Sin[2*c + (3*d*x)/2] + 6*A*Sin[2*c + (5*d*x)/2] + 14*B*Sin[2*c + (5*d*x)/2] - 64*C*Sin[2*c + (5*d*x)/2]))/(480*a^3*d)","B",1
359,1,164,109,0.4218917,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(5 (4 A+3 B+8 C) \sin \left(\frac{d x}{2}\right)+10 A \sin \left(c+\frac{3 d x}{2}\right)+2 A \sin \left(2 c+\frac{5 d x}{2}\right)-15 (B+2 C) \sin \left(c+\frac{d x}{2}\right)+15 B \sin \left(c+\frac{3 d x}{2}\right)+3 B \sin \left(2 c+\frac{5 d x}{2}\right)+20 C \sin \left(c+\frac{3 d x}{2}\right)-15 C \sin \left(2 c+\frac{3 d x}{2}\right)+7 C \sin \left(2 c+\frac{5 d x}{2}\right)\right)}{30 a^3 d (\cos (c+d x)+1)^3}","\frac{(2 A+3 B+7 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(2 A+3 B-8 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}+\frac{(A-B+C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(5*(4*A + 3*B + 8*C)*Sin[(d*x)/2] - 15*(B + 2*C)*Sin[c + (d*x)/2] + 10*A*Sin[c + (3*d*x)/2] + 15*B*Sin[c + (3*d*x)/2] + 20*C*Sin[c + (3*d*x)/2] - 15*C*Sin[2*c + (3*d*x)/2] + 2*A*Sin[2*c + (5*d*x)/2] + 3*B*Sin[2*c + (5*d*x)/2] + 7*C*Sin[2*c + (5*d*x)/2]))/(30*a^3*d*(1 + Cos[c + d*x])^3)","A",1
360,1,276,124,1.6895165,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^3,x]","-\frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \left(\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(5 (29 A-4 B-3 C) \sin \left(\frac{d x}{2}\right)+15 (C-5 A) \sin \left(c+\frac{d x}{2}\right)+95 A \sin \left(c+\frac{3 d x}{2}\right)-15 A \sin \left(2 c+\frac{3 d x}{2}\right)+22 A \sin \left(2 c+\frac{5 d x}{2}\right)-10 B \sin \left(c+\frac{3 d x}{2}\right)-2 B \sin \left(2 c+\frac{5 d x}{2}\right)-15 C \sin \left(c+\frac{3 d x}{2}\right)-3 C \sin \left(2 c+\frac{5 d x}{2}\right)\right)+240 A \cos ^6\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{15 a^3 d (\cos (c+d x)+1)^3 (2 A+2 B \cos (c+d x)+C \cos (2 (c+d x))+C)}","-\frac{(22 A-2 B-3 C) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{(7 A-2 B-3 C) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"-1/15*((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(240*A*Cos[(c + d*x)/2]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Cos[(c + d*x)/2]*Sec[c/2]*(5*(29*A - 4*B - 3*C)*Sin[(d*x)/2] + 15*(-5*A + C)*Sin[c + (d*x)/2] + 95*A*Sin[c + (3*d*x)/2] - 10*B*Sin[c + (3*d*x)/2] - 15*C*Sin[c + (3*d*x)/2] - 15*A*Sin[2*c + (3*d*x)/2] + 22*A*Sin[2*c + (5*d*x)/2] - 2*B*Sin[2*c + (5*d*x)/2] - 3*C*Sin[2*c + (5*d*x)/2])))/(a^3*d*(1 + Cos[c + d*x])^3*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*(c + d*x)]))","B",1
361,1,839,150,6.3663101,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^3,x]","\frac{\frac{16 (3 A-B) \cos ^2(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x)+1)^3 (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}-\frac{16 (3 A-B) \cos ^2(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x)+1)^3 (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}+\frac{\cos (c+d x) \sec \left(\frac{c}{2}\right) \sec (c) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(-255 A \sin \left(\frac{d x}{2}\right)+160 B \sin \left(\frac{d x}{2}\right)-20 C \sin \left(\frac{d x}{2}\right)+567 A \sin \left(\frac{3 d x}{2}\right)-167 B \sin \left(\frac{3 d x}{2}\right)+22 C \sin \left(\frac{3 d x}{2}\right)-600 A \sin \left(c-\frac{d x}{2}\right)+170 B \sin \left(c-\frac{d x}{2}\right)-10 C \sin \left(c-\frac{d x}{2}\right)+375 A \sin \left(c+\frac{d x}{2}\right)-170 B \sin \left(c+\frac{d x}{2}\right)+10 C \sin \left(c+\frac{d x}{2}\right)-480 A \sin \left(2 c+\frac{d x}{2}\right)+160 B \sin \left(2 c+\frac{d x}{2}\right)-20 C \sin \left(2 c+\frac{d x}{2}\right)-60 A \sin \left(c+\frac{3 d x}{2}\right)+75 B \sin \left(c+\frac{3 d x}{2}\right)+402 A \sin \left(2 c+\frac{3 d x}{2}\right)-167 B \sin \left(2 c+\frac{3 d x}{2}\right)+22 C \sin \left(2 c+\frac{3 d x}{2}\right)-225 A \sin \left(3 c+\frac{3 d x}{2}\right)+75 B \sin \left(3 c+\frac{3 d x}{2}\right)+315 A \sin \left(c+\frac{5 d x}{2}\right)-95 B \sin \left(c+\frac{5 d x}{2}\right)+10 C \sin \left(c+\frac{5 d x}{2}\right)+30 A \sin \left(2 c+\frac{5 d x}{2}\right)+15 B \sin \left(2 c+\frac{5 d x}{2}\right)+240 A \sin \left(3 c+\frac{5 d x}{2}\right)-95 B \sin \left(3 c+\frac{5 d x}{2}\right)+10 C \sin \left(3 c+\frac{5 d x}{2}\right)-45 A \sin \left(4 c+\frac{5 d x}{2}\right)+15 B \sin \left(4 c+\frac{5 d x}{2}\right)+72 A \sin \left(2 c+\frac{7 d x}{2}\right)-22 B \sin \left(2 c+\frac{7 d x}{2}\right)+2 C \sin \left(2 c+\frac{7 d x}{2}\right)+15 A \sin \left(3 c+\frac{7 d x}{2}\right)+57 A \sin \left(4 c+\frac{7 d x}{2}\right)-22 B \sin \left(4 c+\frac{7 d x}{2}\right)+2 C \sin \left(4 c+\frac{7 d x}{2}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{60 d (\cos (c+d x)+1)^3 (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}}{a^3}","\frac{2 (36 A-11 B+C) \tan (c+d x)}{15 a^3 d}-\frac{(3 A-B) \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{(3 A-B) \tan (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(9 A-4 B-C) \tan (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \tan (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"((16*(3*A - B)*Cos[c/2 + (d*x)/2]^6*Cos[c + d*x]^2*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2))/(d*(1 + Cos[c + d*x])^3*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])) - (16*(3*A - B)*Cos[c/2 + (d*x)/2]^6*Cos[c + d*x]^2*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2))/(d*(1 + Cos[c + d*x])^3*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])) + (Cos[c/2 + (d*x)/2]*Cos[c + d*x]*Sec[c/2]*Sec[c]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*(-255*A*Sin[(d*x)/2] + 160*B*Sin[(d*x)/2] - 20*C*Sin[(d*x)/2] + 567*A*Sin[(3*d*x)/2] - 167*B*Sin[(3*d*x)/2] + 22*C*Sin[(3*d*x)/2] - 600*A*Sin[c - (d*x)/2] + 170*B*Sin[c - (d*x)/2] - 10*C*Sin[c - (d*x)/2] + 375*A*Sin[c + (d*x)/2] - 170*B*Sin[c + (d*x)/2] + 10*C*Sin[c + (d*x)/2] - 480*A*Sin[2*c + (d*x)/2] + 160*B*Sin[2*c + (d*x)/2] - 20*C*Sin[2*c + (d*x)/2] - 60*A*Sin[c + (3*d*x)/2] + 75*B*Sin[c + (3*d*x)/2] + 402*A*Sin[2*c + (3*d*x)/2] - 167*B*Sin[2*c + (3*d*x)/2] + 22*C*Sin[2*c + (3*d*x)/2] - 225*A*Sin[3*c + (3*d*x)/2] + 75*B*Sin[3*c + (3*d*x)/2] + 315*A*Sin[c + (5*d*x)/2] - 95*B*Sin[c + (5*d*x)/2] + 10*C*Sin[c + (5*d*x)/2] + 30*A*Sin[2*c + (5*d*x)/2] + 15*B*Sin[2*c + (5*d*x)/2] + 240*A*Sin[3*c + (5*d*x)/2] - 95*B*Sin[3*c + (5*d*x)/2] + 10*C*Sin[3*c + (5*d*x)/2] - 45*A*Sin[4*c + (5*d*x)/2] + 15*B*Sin[4*c + (5*d*x)/2] + 72*A*Sin[2*c + (7*d*x)/2] - 22*B*Sin[2*c + (7*d*x)/2] + 2*C*Sin[2*c + (7*d*x)/2] + 15*A*Sin[3*c + (7*d*x)/2] + 57*A*Sin[4*c + (7*d*x)/2] - 22*B*Sin[4*c + (7*d*x)/2] + 2*C*Sin[4*c + (7*d*x)/2]))/(60*d*(1 + Cos[c + d*x])^3*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])))/a^3","B",1
362,1,206,210,1.5293675,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^3,x]","\frac{2 \cos ^6\left(\frac{1}{2} (c+d x)\right) \left(-4 (107 A-57 B+22 C) \tan \left(\frac{1}{2} (c+d x)\right)-96 (A-B+C) \sin ^6\left(\frac{1}{2} (c+d x)\right) \csc ^5(c+d x)-16 (17 A-12 B+7 C) \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)-30 (13 A-6 B+2 C) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-60 (3 A-B) \tan (c+d x)+30 A \tan (c+d x) \sec (c+d x)\right)}{15 a^3 d (\cos (c+d x)+1)^3}","-\frac{2 (76 A-36 B+11 C) \tan (c+d x)}{15 a^3 d}+\frac{(13 A-6 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{(13 A-6 B+2 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{(76 A-36 B+11 C) \tan (c+d x) \sec (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(11 A-6 B+C) \tan (c+d x) \sec (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"(2*Cos[(c + d*x)/2]^6*(-30*(13*A - 6*B + 2*C)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 16*(17*A - 12*B + 7*C)*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 96*(A - B + C)*Csc[c + d*x]^5*Sin[(c + d*x)/2]^6 - 4*(107*A - 57*B + 22*C)*Tan[(c + d*x)/2] - 60*(3*A - B)*Tan[c + d*x] + 30*A*Sec[c + d*x]*Tan[c + d*x]))/(15*a^3*d*(1 + Cos[c + d*x])^3)","A",1
363,1,270,246,1.0218803,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^3,x]","\frac{960 (23 A-13 B+6 C) \cos ^6\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \cos \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) ((7814 A-4274 B+2124 C) \cos (c+d x)+8 (691 A-381 B+186 C) \cos (2 (c+d x))+3098 A \cos (3 (c+d x))+1287 A \cos (4 (c+d x))+272 A \cos (5 (c+d x))+4321 A-1718 B \cos (3 (c+d x))-717 B \cos (4 (c+d x))-152 B \cos (5 (c+d x))-2331 B+828 C \cos (3 (c+d x))+342 C \cos (4 (c+d x))+72 C \cos (5 (c+d x))+1146 C)}{240 a^3 d (\cos (c+d x)+1)^3}","\frac{4 (34 A-19 B+9 C) \tan ^3(c+d x)}{15 a^3 d}+\frac{4 (34 A-19 B+9 C) \tan (c+d x)}{5 a^3 d}-\frac{(23 A-13 B+6 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{(23 A-13 B+6 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{(23 A-13 B+6 C) \tan (c+d x) \sec ^2(c+d x)}{3 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(13 A-8 B+3 C) \tan (c+d x) \sec ^2(c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"(960*(23*A - 13*B + 6*C)*Cos[(c + d*x)/2]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 2*Cos[(c + d*x)/2]*(4321*A - 2331*B + 1146*C + (7814*A - 4274*B + 2124*C)*Cos[c + d*x] + 8*(691*A - 381*B + 186*C)*Cos[2*(c + d*x)] + 3098*A*Cos[3*(c + d*x)] - 1718*B*Cos[3*(c + d*x)] + 828*C*Cos[3*(c + d*x)] + 1287*A*Cos[4*(c + d*x)] - 717*B*Cos[4*(c + d*x)] + 342*C*Cos[4*(c + d*x)] + 272*A*Cos[5*(c + d*x)] - 152*B*Cos[5*(c + d*x)] + 72*C*Cos[5*(c + d*x)])*Sec[c + d*x]^3*Sin[(c + d*x)/2])/(240*a^3*d*(1 + Cos[c + d*x])^3)","A",1
364,1,299,245,2.8742118,"\int \frac{\cos ^4(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^4,x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(210 \cos ^7\left(\frac{1}{2} (c+d x)\right) (2 d x (2 A-8 B+21 C)+4 (B-4 C) \sin (c+d x)+C \sin (2 (c+d x)))+4 \tan \left(\frac{c}{2}\right) (160 A-286 B+447 C) \cos ^5\left(\frac{1}{2} (c+d x)\right)-6 \tan \left(\frac{c}{2}\right) (25 A-32 B+39 C) \cos ^3\left(\frac{1}{2} (c+d x)\right)+15 \tan \left(\frac{c}{2}\right) (A-B+C) \cos \left(\frac{1}{2} (c+d x)\right)+15 \sec \left(\frac{c}{2}\right) (A-B+C) \sin \left(\frac{d x}{2}\right)-8 \sec \left(\frac{c}{2}\right) (260 A-764 B+1653 C) \sin \left(\frac{d x}{2}\right) \cos ^6\left(\frac{1}{2} (c+d x)\right)+4 \sec \left(\frac{c}{2}\right) (160 A-286 B+447 C) \sin \left(\frac{d x}{2}\right) \cos ^4\left(\frac{1}{2} (c+d x)\right)-6 \sec \left(\frac{c}{2}\right) (25 A-32 B+39 C) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{105 a^4 d (\cos (c+d x)+1)^4}","-\frac{8 (20 A-83 B+216 C) \sin (c+d x)}{105 a^4 d}-\frac{(10 A-52 B+129 C) \sin (c+d x) \cos ^3(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{4 (20 A-83 B+216 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)}+\frac{(2 A-8 B+21 C) \sin (c+d x) \cos (c+d x)}{2 a^4 d}+\frac{x (2 A-8 B+21 C)}{2 a^4}-\frac{(A-B+C) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(B-2 C) \sin (c+d x) \cos ^4(c+d x)}{5 a d (a \cos (c+d x)+a)^3}",1,"(2*Cos[(c + d*x)/2]*(15*(A - B + C)*Sec[c/2]*Sin[(d*x)/2] - 6*(25*A - 32*B + 39*C)*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + 4*(160*A - 286*B + 447*C)*Cos[(c + d*x)/2]^4*Sec[c/2]*Sin[(d*x)/2] - 8*(260*A - 764*B + 1653*C)*Cos[(c + d*x)/2]^6*Sec[c/2]*Sin[(d*x)/2] + 210*Cos[(c + d*x)/2]^7*(2*(2*A - 8*B + 21*C)*d*x + 4*(B - 4*C)*Sin[c + d*x] + C*Sin[2*(c + d*x)]) + 15*(A - B + C)*Cos[(c + d*x)/2]*Tan[c/2] - 6*(25*A - 32*B + 39*C)*Cos[(c + d*x)/2]^3*Tan[c/2] + 4*(160*A - 286*B + 447*C)*Cos[(c + d*x)/2]^5*Tan[c/2]))/(105*a^4*d*(1 + Cos[c + d*x])^4)","A",1
365,1,571,195,1.1414946,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-2520 A \sin \left(c+\frac{d x}{2}\right)+1764 A \sin \left(c+\frac{3 d x}{2}\right)-1260 A \sin \left(2 c+\frac{3 d x}{2}\right)+588 A \sin \left(2 c+\frac{5 d x}{2}\right)-420 A \sin \left(3 c+\frac{5 d x}{2}\right)+144 A \sin \left(3 c+\frac{7 d x}{2}\right)+2520 A \sin \left(\frac{d x}{2}\right)+7350 d x (B-4 C) \cos \left(c+\frac{d x}{2}\right)+16520 B \sin \left(c+\frac{d x}{2}\right)-14280 B \sin \left(c+\frac{3 d x}{2}\right)+7560 B \sin \left(2 c+\frac{3 d x}{2}\right)-5600 B \sin \left(2 c+\frac{5 d x}{2}\right)+1680 B \sin \left(3 c+\frac{5 d x}{2}\right)-1040 B \sin \left(3 c+\frac{7 d x}{2}\right)+4410 B d x \cos \left(c+\frac{3 d x}{2}\right)+4410 B d x \cos \left(2 c+\frac{3 d x}{2}\right)+1470 B d x \cos \left(2 c+\frac{5 d x}{2}\right)+1470 B d x \cos \left(3 c+\frac{5 d x}{2}\right)+210 B d x \cos \left(3 c+\frac{7 d x}{2}\right)+210 B d x \cos \left(4 c+\frac{7 d x}{2}\right)+7350 d x (B-4 C) \cos \left(\frac{d x}{2}\right)-19880 B \sin \left(\frac{d x}{2}\right)-46130 C \sin \left(c+\frac{d x}{2}\right)+46116 C \sin \left(c+\frac{3 d x}{2}\right)-18060 C \sin \left(2 c+\frac{3 d x}{2}\right)+19292 C \sin \left(2 c+\frac{5 d x}{2}\right)-2100 C \sin \left(3 c+\frac{5 d x}{2}\right)+3791 C \sin \left(3 c+\frac{7 d x}{2}\right)+735 C \sin \left(4 c+\frac{7 d x}{2}\right)+105 C \sin \left(4 c+\frac{9 d x}{2}\right)+105 C \sin \left(5 c+\frac{9 d x}{2}\right)-17640 C d x \cos \left(c+\frac{3 d x}{2}\right)-17640 C d x \cos \left(2 c+\frac{3 d x}{2}\right)-5880 C d x \cos \left(2 c+\frac{5 d x}{2}\right)-5880 C d x \cos \left(3 c+\frac{5 d x}{2}\right)-840 C d x \cos \left(3 c+\frac{7 d x}{2}\right)-840 C d x \cos \left(4 c+\frac{7 d x}{2}\right)+60830 C \sin \left(\frac{d x}{2}\right)\right)}{1680 a^4 d (\cos (c+d x)+1)^4}","\frac{(6 A-55 B+244 C) \sin (c+d x)}{105 a^4 d}+\frac{(3 A+25 B-88 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(B-4 C) \sin (c+d x)}{a^4 d (\cos (c+d x)+1)}+\frac{x (B-4 C)}{a^4}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(2 A+5 B-12 C) \sin (c+d x) \cos ^3(c+d x)}{35 a d (a \cos (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(7350*(B - 4*C)*d*x*Cos[(d*x)/2] + 7350*(B - 4*C)*d*x*Cos[c + (d*x)/2] + 4410*B*d*x*Cos[c + (3*d*x)/2] - 17640*C*d*x*Cos[c + (3*d*x)/2] + 4410*B*d*x*Cos[2*c + (3*d*x)/2] - 17640*C*d*x*Cos[2*c + (3*d*x)/2] + 1470*B*d*x*Cos[2*c + (5*d*x)/2] - 5880*C*d*x*Cos[2*c + (5*d*x)/2] + 1470*B*d*x*Cos[3*c + (5*d*x)/2] - 5880*C*d*x*Cos[3*c + (5*d*x)/2] + 210*B*d*x*Cos[3*c + (7*d*x)/2] - 840*C*d*x*Cos[3*c + (7*d*x)/2] + 210*B*d*x*Cos[4*c + (7*d*x)/2] - 840*C*d*x*Cos[4*c + (7*d*x)/2] + 2520*A*Sin[(d*x)/2] - 19880*B*Sin[(d*x)/2] + 60830*C*Sin[(d*x)/2] - 2520*A*Sin[c + (d*x)/2] + 16520*B*Sin[c + (d*x)/2] - 46130*C*Sin[c + (d*x)/2] + 1764*A*Sin[c + (3*d*x)/2] - 14280*B*Sin[c + (3*d*x)/2] + 46116*C*Sin[c + (3*d*x)/2] - 1260*A*Sin[2*c + (3*d*x)/2] + 7560*B*Sin[2*c + (3*d*x)/2] - 18060*C*Sin[2*c + (3*d*x)/2] + 588*A*Sin[2*c + (5*d*x)/2] - 5600*B*Sin[2*c + (5*d*x)/2] + 19292*C*Sin[2*c + (5*d*x)/2] - 420*A*Sin[3*c + (5*d*x)/2] + 1680*B*Sin[3*c + (5*d*x)/2] - 2100*C*Sin[3*c + (5*d*x)/2] + 144*A*Sin[3*c + (7*d*x)/2] - 1040*B*Sin[3*c + (7*d*x)/2] + 3791*C*Sin[3*c + (7*d*x)/2] + 735*C*Sin[4*c + (7*d*x)/2] + 105*C*Sin[4*c + (9*d*x)/2] + 105*C*Sin[5*c + (9*d*x)/2]))/(1680*a^4*d*(1 + Cos[c + d*x])^4)","B",1
366,1,405,164,0.9347864,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^7\left(\frac{1}{2} (c+d x)\right) \left(-350 A \sin \left(c+\frac{d x}{2}\right)+336 A \sin \left(c+\frac{3 d x}{2}\right)-210 A \sin \left(2 c+\frac{3 d x}{2}\right)+182 A \sin \left(2 c+\frac{5 d x}{2}\right)+26 A \sin \left(3 c+\frac{7 d x}{2}\right)+560 A \sin \left(\frac{d x}{2}\right)-1260 B \sin \left(c+\frac{d x}{2}\right)+882 B \sin \left(c+\frac{3 d x}{2}\right)-630 B \sin \left(2 c+\frac{3 d x}{2}\right)+294 B \sin \left(2 c+\frac{5 d x}{2}\right)-210 B \sin \left(3 c+\frac{5 d x}{2}\right)+72 B \sin \left(3 c+\frac{7 d x}{2}\right)+1260 B \sin \left(\frac{d x}{2}\right)+8260 C \sin \left(c+\frac{d x}{2}\right)-7140 C \sin \left(c+\frac{3 d x}{2}\right)+3780 C \sin \left(2 c+\frac{3 d x}{2}\right)-2800 C \sin \left(2 c+\frac{5 d x}{2}\right)+840 C \sin \left(3 c+\frac{5 d x}{2}\right)-520 C \sin \left(3 c+\frac{7 d x}{2}\right)+3675 C d x \cos \left(c+\frac{d x}{2}\right)+2205 C d x \cos \left(c+\frac{3 d x}{2}\right)+2205 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+735 C d x \cos \left(2 c+\frac{5 d x}{2}\right)+735 C d x \cos \left(3 c+\frac{5 d x}{2}\right)+105 C d x \cos \left(3 c+\frac{7 d x}{2}\right)+105 C d x \cos \left(4 c+\frac{7 d x}{2}\right)-9940 C \sin \left(\frac{d x}{2}\right)+3675 C d x \cos \left(\frac{d x}{2}\right)\right)}{13440 a^4 d}","\frac{(16 A+12 B-215 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(8 A+6 B-55 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{C x}{a^4}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(4 A+3 B-10 C) \sin (c+d x) \cos ^2(c+d x)}{35 a d (a \cos (c+d x)+a)^3}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^7*(3675*C*d*x*Cos[(d*x)/2] + 3675*C*d*x*Cos[c + (d*x)/2] + 2205*C*d*x*Cos[c + (3*d*x)/2] + 2205*C*d*x*Cos[2*c + (3*d*x)/2] + 735*C*d*x*Cos[2*c + (5*d*x)/2] + 735*C*d*x*Cos[3*c + (5*d*x)/2] + 105*C*d*x*Cos[3*c + (7*d*x)/2] + 105*C*d*x*Cos[4*c + (7*d*x)/2] + 560*A*Sin[(d*x)/2] + 1260*B*Sin[(d*x)/2] - 9940*C*Sin[(d*x)/2] - 350*A*Sin[c + (d*x)/2] - 1260*B*Sin[c + (d*x)/2] + 8260*C*Sin[c + (d*x)/2] + 336*A*Sin[c + (3*d*x)/2] + 882*B*Sin[c + (3*d*x)/2] - 7140*C*Sin[c + (3*d*x)/2] - 210*A*Sin[2*c + (3*d*x)/2] - 630*B*Sin[2*c + (3*d*x)/2] + 3780*C*Sin[2*c + (3*d*x)/2] + 182*A*Sin[2*c + (5*d*x)/2] + 294*B*Sin[2*c + (5*d*x)/2] - 2800*C*Sin[2*c + (5*d*x)/2] - 210*B*Sin[3*c + (5*d*x)/2] + 840*C*Sin[3*c + (5*d*x)/2] + 26*A*Sin[3*c + (7*d*x)/2] + 72*B*Sin[3*c + (7*d*x)/2] - 520*C*Sin[3*c + (7*d*x)/2]))/(13440*a^4*d)","B",1
367,1,239,148,0.5303031,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-35 (4 A+5 B+18 C) \sin \left(c+\frac{d x}{2}\right)+70 (2 A+4 B+9 C) \sin \left(\frac{d x}{2}\right)+168 A \sin \left(c+\frac{3 d x}{2}\right)+56 A \sin \left(2 c+\frac{5 d x}{2}\right)+8 A \sin \left(3 c+\frac{7 d x}{2}\right)+168 B \sin \left(c+\frac{3 d x}{2}\right)-105 B \sin \left(2 c+\frac{3 d x}{2}\right)+91 B \sin \left(2 c+\frac{5 d x}{2}\right)+13 B \sin \left(3 c+\frac{7 d x}{2}\right)+441 C \sin \left(c+\frac{3 d x}{2}\right)-315 C \sin \left(2 c+\frac{3 d x}{2}\right)+147 C \sin \left(2 c+\frac{5 d x}{2}\right)-105 C \sin \left(3 c+\frac{5 d x}{2}\right)+36 C \sin \left(3 c+\frac{7 d x}{2}\right)\right)}{420 a^4 d (\cos (c+d x)+1)^4}","\frac{(8 A+13 B+36 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}+\frac{(23 A-2 B-54 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{(6 A+B-8 C) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(70*(2*A + 4*B + 9*C)*Sin[(d*x)/2] - 35*(4*A + 5*B + 18*C)*Sin[c + (d*x)/2] + 168*A*Sin[c + (3*d*x)/2] + 168*B*Sin[c + (3*d*x)/2] + 441*C*Sin[c + (3*d*x)/2] - 105*B*Sin[2*c + (3*d*x)/2] - 315*C*Sin[2*c + (3*d*x)/2] + 56*A*Sin[2*c + (5*d*x)/2] + 91*B*Sin[2*c + (5*d*x)/2] + 147*C*Sin[2*c + (5*d*x)/2] - 105*C*Sin[3*c + (5*d*x)/2] + 8*A*Sin[3*c + (7*d*x)/2] + 13*B*Sin[3*c + (7*d*x)/2] + 36*C*Sin[3*c + (7*d*x)/2]))/(420*a^4*d*(1 + Cos[c + d*x])^4)","A",1
368,1,208,148,0.4813095,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(70 (3 A+2 B+4 C) \sin \left(\frac{d x}{2}\right)+126 A \sin \left(c+\frac{3 d x}{2}\right)+42 A \sin \left(2 c+\frac{5 d x}{2}\right)+6 A \sin \left(3 c+\frac{7 d x}{2}\right)-35 (4 B+5 C) \sin \left(c+\frac{d x}{2}\right)+168 B \sin \left(c+\frac{3 d x}{2}\right)+56 B \sin \left(2 c+\frac{5 d x}{2}\right)+8 B \sin \left(3 c+\frac{7 d x}{2}\right)+168 C \sin \left(c+\frac{3 d x}{2}\right)-105 C \sin \left(2 c+\frac{3 d x}{2}\right)+91 C \sin \left(2 c+\frac{5 d x}{2}\right)+13 C \sin \left(3 c+\frac{7 d x}{2}\right)\right)}{420 a^4 d (\cos (c+d x)+1)^4}","\frac{(6 A+8 B+13 C) \sin (c+d x)}{105 d \left(a^4 \cos (c+d x)+a^4\right)}+\frac{(6 A+8 B+13 C) \sin (c+d x)}{105 d \left(a^2 \cos (c+d x)+a^2\right)^2}+\frac{(3 A+4 B-11 C) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac{(A-B+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(70*(3*A + 2*B + 4*C)*Sin[(d*x)/2] - 35*(4*B + 5*C)*Sin[c + (d*x)/2] + 126*A*Sin[c + (3*d*x)/2] + 168*B*Sin[c + (3*d*x)/2] + 168*C*Sin[c + (3*d*x)/2] - 105*C*Sin[2*c + (3*d*x)/2] + 42*A*Sin[2*c + (5*d*x)/2] + 56*B*Sin[2*c + (5*d*x)/2] + 91*C*Sin[2*c + (5*d*x)/2] + 6*A*Sin[3*c + (7*d*x)/2] + 8*B*Sin[3*c + (7*d*x)/2] + 13*C*Sin[3*c + (7*d*x)/2]))/(420*a^4*d*(1 + Cos[c + d*x])^4)","A",1
369,1,334,157,2.6182262,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+a \cos (c+d x))^4} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^4,x]","-\frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \left(\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(70 (49 A-3 B-2 C) \sin \left(\frac{d x}{2}\right)-70 (31 A-2 C) \sin \left(c+\frac{d x}{2}\right)+2625 A \sin \left(c+\frac{3 d x}{2}\right)-735 A \sin \left(2 c+\frac{3 d x}{2}\right)+1015 A \sin \left(2 c+\frac{5 d x}{2}\right)-105 A \sin \left(3 c+\frac{5 d x}{2}\right)+160 A \sin \left(3 c+\frac{7 d x}{2}\right)-126 B \sin \left(c+\frac{3 d x}{2}\right)-42 B \sin \left(2 c+\frac{5 d x}{2}\right)-6 B \sin \left(3 c+\frac{7 d x}{2}\right)-168 C \sin \left(c+\frac{3 d x}{2}\right)-56 C \sin \left(2 c+\frac{5 d x}{2}\right)-8 C \sin \left(3 c+\frac{7 d x}{2}\right)\right)+6720 A \cos ^8\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{210 a^4 d (\cos (c+d x)+1)^4 (2 A+2 B \cos (c+d x)+C \cos (2 (c+d x))+C)}","-\frac{2 (80 A-3 B-4 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(55 A-6 B-8 C) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{(10 A-3 B-4 C) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A-B+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"-1/210*((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*(6720*A*Cos[(c + d*x)/2]^8*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Cos[(c + d*x)/2]*Sec[c/2]*(70*(49*A - 3*B - 2*C)*Sin[(d*x)/2] - 70*(31*A - 2*C)*Sin[c + (d*x)/2] + 2625*A*Sin[c + (3*d*x)/2] - 126*B*Sin[c + (3*d*x)/2] - 168*C*Sin[c + (3*d*x)/2] - 735*A*Sin[2*c + (3*d*x)/2] + 1015*A*Sin[2*c + (5*d*x)/2] - 42*B*Sin[2*c + (5*d*x)/2] - 56*C*Sin[2*c + (5*d*x)/2] - 105*A*Sin[3*c + (5*d*x)/2] + 160*A*Sin[3*c + (7*d*x)/2] - 6*B*Sin[3*c + (7*d*x)/2] - 8*C*Sin[3*c + (7*d*x)/2])))/(a^4*d*(1 + Cos[c + d*x])^4*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*(c + d*x)]))","B",1
370,1,1190,185,6.3995542,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^4,x]","\frac{\frac{32 (4 A-B) \cos ^2(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x)+1)^4 (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}-\frac{32 (4 A-B) \cos ^2(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x)+1)^4 (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}+\frac{32 A \cos (c+d x) \sec (c) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \sin (d x) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x)+1)^4 (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}+\frac{32 \cos ^2(c+d x) \sec \left(\frac{c}{2}\right) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(559 A \sin \left(\frac{d x}{2}\right)-160 B \sin \left(\frac{d x}{2}\right)+6 C \sin \left(\frac{d x}{2}\right)\right) \cos ^7\left(\frac{c}{2}+\frac{d x}{2}\right)}{105 d (\cos (c+d x)+1)^4 (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}+\frac{16 \cos ^2(c+d x) \sec \left(\frac{c}{2}\right) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(139 A \sin \left(\frac{c}{2}\right)-55 B \sin \left(\frac{c}{2}\right)+6 C \sin \left(\frac{c}{2}\right)\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{105 d (\cos (c+d x)+1)^4 (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}+\frac{16 \cos ^2(c+d x) \sec \left(\frac{c}{2}\right) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(139 A \sin \left(\frac{d x}{2}\right)-55 B \sin \left(\frac{d x}{2}\right)+6 C \sin \left(\frac{d x}{2}\right)\right) \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{105 d (\cos (c+d x)+1)^4 (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}+\frac{8 \cos ^2(c+d x) \sec \left(\frac{c}{2}\right) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(17 A \sin \left(\frac{c}{2}\right)-10 B \sin \left(\frac{c}{2}\right)+3 C \sin \left(\frac{c}{2}\right)\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{35 d (\cos (c+d x)+1)^4 (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}+\frac{8 \cos ^2(c+d x) \sec \left(\frac{c}{2}\right) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(17 A \sin \left(\frac{d x}{2}\right)-10 B \sin \left(\frac{d x}{2}\right)+3 C \sin \left(\frac{d x}{2}\right)\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{35 d (\cos (c+d x)+1)^4 (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}+\frac{4 \cos ^2(c+d x) \sec \left(\frac{c}{2}\right) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(A \sin \left(\frac{c}{2}\right)-B \sin \left(\frac{c}{2}\right)+C \sin \left(\frac{c}{2}\right)\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d (\cos (c+d x)+1)^4 (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}+\frac{4 \cos ^2(c+d x) \sec \left(\frac{c}{2}\right) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d (\cos (c+d x)+1)^4 (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}}{a^4}","\frac{2 (332 A-80 B+3 C) \tan (c+d x)}{105 a^4 d}-\frac{(88 A-25 B-3 C) \tan (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(4 A-B) \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{(4 A-B) \tan (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac{(12 A-5 B-2 C) \tan (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A-B+C) \tan (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"((32*(4*A - B)*Cos[c/2 + (d*x)/2]^8*Cos[c + d*x]^2*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2))/(d*(1 + Cos[c + d*x])^4*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])) - (32*(4*A - B)*Cos[c/2 + (d*x)/2]^8*Cos[c + d*x]^2*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2))/(d*(1 + Cos[c + d*x])^4*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])) + (4*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]^2*Sec[c/2]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*(A*Sin[c/2] - B*Sin[c/2] + C*Sin[c/2]))/(7*d*(1 + Cos[c + d*x])^4*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])) + (8*Cos[c/2 + (d*x)/2]^4*Cos[c + d*x]^2*Sec[c/2]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*(17*A*Sin[c/2] - 10*B*Sin[c/2] + 3*C*Sin[c/2]))/(35*d*(1 + Cos[c + d*x])^4*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])) + (16*Cos[c/2 + (d*x)/2]^6*Cos[c + d*x]^2*Sec[c/2]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*(139*A*Sin[c/2] - 55*B*Sin[c/2] + 6*C*Sin[c/2]))/(105*d*(1 + Cos[c + d*x])^4*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])) + (4*Cos[c/2 + (d*x)/2]*Cos[c + d*x]^2*Sec[c/2]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(7*d*(1 + Cos[c + d*x])^4*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])) + (8*Cos[c/2 + (d*x)/2]^3*Cos[c + d*x]^2*Sec[c/2]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*(17*A*Sin[(d*x)/2] - 10*B*Sin[(d*x)/2] + 3*C*Sin[(d*x)/2]))/(35*d*(1 + Cos[c + d*x])^4*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])) + (32*Cos[c/2 + (d*x)/2]^7*Cos[c + d*x]^2*Sec[c/2]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*(559*A*Sin[(d*x)/2] - 160*B*Sin[(d*x)/2] + 6*C*Sin[(d*x)/2]))/(105*d*(1 + Cos[c + d*x])^4*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])) + (16*Cos[c/2 + (d*x)/2]^5*Cos[c + d*x]^2*Sec[c/2]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*(139*A*Sin[(d*x)/2] - 55*B*Sin[(d*x)/2] + 6*C*Sin[(d*x)/2]))/(105*d*(1 + Cos[c + d*x])^4*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])) + (32*A*Cos[c/2 + (d*x)/2]^8*Cos[c + d*x]*Sec[c]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*Sin[d*x])/(d*(1 + Cos[c + d*x])^4*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])))/a^4","B",1
371,1,271,248,1.5261985,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^4,x]","-\frac{13440 (21 A-8 B+2 C) \cos ^8\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \cos \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) (8 (12813 A-4994 B+1130 C) \cos (c+d x)+60 (1177 A-456 B+106 C) \cos (2 (c+d x))+35928 A \cos (3 (c+d x))+11619 A \cos (4 (c+d x))+1728 A \cos (5 (c+d x))+58161 A-13864 B \cos (3 (c+d x))-4472 B \cos (4 (c+d x))-664 B \cos (5 (c+d x))-22888 B+3280 C \cos (3 (c+d x))+1070 C \cos (4 (c+d x))+160 C \cos (5 (c+d x))+5290 C)}{1680 a^4 d (\cos (c+d x)+1)^4}","-\frac{8 (216 A-83 B+20 C) \tan (c+d x)}{105 a^4 d}+\frac{(21 A-8 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{(21 A-8 B+2 C) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{4 (216 A-83 B+20 C) \tan (c+d x) \sec (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(129 A-52 B+10 C) \tan (c+d x) \sec (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(A-B+C) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{(2 A-B) \tan (c+d x) \sec (c+d x)}{5 a d (a \cos (c+d x)+a)^3}",1,"-1/1680*(13440*(21*A - 8*B + 2*C)*Cos[(c + d*x)/2]^8*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 2*Cos[(c + d*x)/2]*(58161*A - 22888*B + 5290*C + 8*(12813*A - 4994*B + 1130*C)*Cos[c + d*x] + 60*(1177*A - 456*B + 106*C)*Cos[2*(c + d*x)] + 35928*A*Cos[3*(c + d*x)] - 13864*B*Cos[3*(c + d*x)] + 3280*C*Cos[3*(c + d*x)] + 11619*A*Cos[4*(c + d*x)] - 4472*B*Cos[4*(c + d*x)] + 1070*C*Cos[4*(c + d*x)] + 1728*A*Cos[5*(c + d*x)] - 664*B*Cos[5*(c + d*x)] + 160*C*Cos[5*(c + d*x)])*Sec[c + d*x]^2*Sin[(c + d*x)/2])/(a^4*d*(1 + Cos[c + d*x])^4)","A",1
372,1,304,287,1.7062239,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^4,x]","\frac{26880 (44 A-21 B+8 C) \cos ^8\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \cos \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) (14 (28252 A-13353 B+5224 C) \cos (c+d x)+56 (5218 A-2472 B+961 C) \cos (2 (c+d x))+173316 A \cos (3 (c+d x))+79264 A \cos (4 (c+d x))+24436 A \cos (5 (c+d x))+3632 A \cos (6 (c+d x))+217696 A-82239 B \cos (3 (c+d x))-37656 B \cos (4 (c+d x))-11619 B \cos (5 (c+d x))-1728 B \cos (6 (c+d x))-102504 B+31832 C \cos (3 (c+d x))+14528 C \cos (4 (c+d x))+4472 C \cos (5 (c+d x))+664 C \cos (6 (c+d x))+39952 C)}{3360 a^4 d (\cos (c+d x)+1)^4}","\frac{4 (454 A-216 B+83 C) \tan ^3(c+d x)}{105 a^4 d}+\frac{4 (454 A-216 B+83 C) \tan (c+d x)}{35 a^4 d}-\frac{(44 A-21 B+8 C) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{(44 A-21 B+8 C) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{(44 A-21 B+8 C) \tan (c+d x) \sec ^2(c+d x)}{3 a^4 d (\cos (c+d x)+1)}-\frac{(178 A-87 B+31 C) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(16 A-9 B+2 C) \tan (c+d x) \sec ^2(c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"(26880*(44*A - 21*B + 8*C)*Cos[(c + d*x)/2]^8*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 2*Cos[(c + d*x)/2]*(217696*A - 102504*B + 39952*C + 14*(28252*A - 13353*B + 5224*C)*Cos[c + d*x] + 56*(5218*A - 2472*B + 961*C)*Cos[2*(c + d*x)] + 173316*A*Cos[3*(c + d*x)] - 82239*B*Cos[3*(c + d*x)] + 31832*C*Cos[3*(c + d*x)] + 79264*A*Cos[4*(c + d*x)] - 37656*B*Cos[4*(c + d*x)] + 14528*C*Cos[4*(c + d*x)] + 24436*A*Cos[5*(c + d*x)] - 11619*B*Cos[5*(c + d*x)] + 4472*C*Cos[5*(c + d*x)] + 3632*A*Cos[6*(c + d*x)] - 1728*B*Cos[6*(c + d*x)] + 664*C*Cos[6*(c + d*x)])*Sec[c + d*x]^3*Sin[(c + d*x)/2])/(3360*a^4*d*(1 + Cos[c + d*x])^4)","A",1
373,1,145,239,1.2987749,"\int \cos ^3(c+d x) \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (2 (9306 A+8272 B+9095 C) \cos (c+d x)+8 (594 A+913 B+830 C) \cos (2 (c+d x))+1980 A \cos (3 (c+d x))+30096 A+1760 B \cos (3 (c+d x))+770 B \cos (4 (c+d x))+29062 B+3175 C \cos (3 (c+d x))+700 C \cos (4 (c+d x))+315 C \cos (5 (c+d x))+26420 C)}{27720 d}","\frac{2 a (99 A+88 B+80 C) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{4 (99 A+88 B+80 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 a d}-\frac{8 (99 A+88 B+80 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3465 d}+\frac{4 a (99 A+88 B+80 C) \sin (c+d x)}{495 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (11 B+C) \sin (c+d x) \cos ^4(c+d x)}{99 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \cos ^4(c+d x) \sqrt{a \cos (c+d x)+a}}{11 d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(30096*A + 29062*B + 26420*C + 2*(9306*A + 8272*B + 9095*C)*Cos[c + d*x] + 8*(594*A + 913*B + 830*C)*Cos[2*(c + d*x)] + 1980*A*Cos[3*(c + d*x)] + 1760*B*Cos[3*(c + d*x)] + 3175*C*Cos[3*(c + d*x)] + 770*B*Cos[4*(c + d*x)] + 700*C*Cos[4*(c + d*x)] + 315*C*Cos[5*(c + d*x)])*Tan[(c + d*x)/2])/(27720*d)","A",1
374,1,114,193,0.7163873,"\int \cos ^2(c+d x) \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((672 A+94 (9 B+8 C)) \cos (c+d x)+4 (63 A+54 B+83 C) \cos (2 (c+d x))+1596 A+90 B \cos (3 (c+d x))+1368 B+80 C \cos (3 (c+d x))+35 C \cos (4 (c+d x))+1321 C)}{1260 d}","\frac{2 (21 A+18 B+16 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 a d}-\frac{4 (21 A+18 B+16 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 a (21 A+18 B+16 C) \sin (c+d x)}{45 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (9 B+C) \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{9 d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(1596*A + 1368*B + 1321*C + (672*A + 94*(9*B + 8*C))*Cos[c + d*x] + 4*(63*A + 54*B + 83*C)*Cos[2*(c + d*x)] + 90*B*Cos[3*(c + d*x)] + 80*C*Cos[3*(c + d*x)] + 35*C*Cos[4*(c + d*x)])*Tan[(c + d*x)/2])/(1260*d)","A",1
375,1,86,147,0.3997079,"\int \cos (c+d x) \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((140 A+112 B+141 C) \cos (c+d x)+280 A+6 (7 B+6 C) \cos (2 (c+d x))+266 B+15 C \cos (3 (c+d x))+228 C)}{210 d}","\frac{2 (35 A-14 B+18 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 a (35 A+49 B+27 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (7 B+C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 a d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{7 d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(280*A + 266*B + 228*C + (140*A + 112*B + 141*C)*Cos[c + d*x] + 6*(7*B + 6*C)*Cos[2*(c + d*x)] + 15*C*Cos[3*(c + d*x)])*Tan[(c + d*x)/2])/(210*d)","A",1
376,1,67,104,0.1767104,"\int \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (30 A+2 (5 B+4 C) \cos (c+d x)+20 B+3 C \cos (2 (c+d x))+19 C)}{15 d}","\frac{2 a (15 A+5 B+7 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (5 B-2 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(30*A + 20*B + 19*C + 2*(5*B + 4*C)*Cos[c + d*x] + 3*C*Cos[2*(c + d*x)])*Tan[(c + d*x)/2])/(15*d)","A",1
377,1,84,100,0.1911247,"\int \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} A \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) (3 B+C \cos (c+d x)+2 C)\right)}{3 d}","\frac{2 \sqrt{a} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a (3 B+C) \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(3*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*(3*B + 2*C + C*Cos[c + d*x])*Sin[(c + d*x)/2]))/(3*d)","A",1
378,1,95,98,0.2881728,"\int \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt{a (\cos (c+d x)+1)} \left(\sqrt{2} (A+2 B) \cos (c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) (A+2 C \cos (c+d x))\right)}{2 d}","\frac{\sqrt{a} (A+2 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a (A-2 C) \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]*(Sqrt[2]*(A + 2*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x] + 2*(A + 2*C*Cos[c + d*x])*Sin[(c + d*x)/2]))/(2*d)","A",1
379,1,111,117,0.5535768,"\int \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(\sqrt{2} (3 A+4 B+8 C) \cos ^2(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) ((3 A+4 B) \cos (c+d x)+2 A)\right)}{8 d}","\frac{\sqrt{a} (3 A+4 B+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a (A+4 B) \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^2*(Sqrt[2]*(3*A + 4*B + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^2 + 2*(2*A + (3*A + 4*B)*Cos[c + d*x])*Sin[(c + d*x)/2]))/(8*d)","A",1
380,1,138,163,1.1643111,"\int \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (3 (5 A+6 B+8 C) \cos (2 (c+d x))+4 (5 A+6 B) \cos (c+d x)+31 A+18 B+24 C)+3 \sqrt{2} (5 A+6 B+8 C) \cos ^3(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{48 d}","\frac{a (5 A+6 B+8 C) \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (5 A+6 B+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a (A+6 B) \tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^3*(3*Sqrt[2]*(5*A + 6*B + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^3 + (31*A + 18*B + 24*C + 4*(5*A + 6*B)*Cos[c + d*x] + 3*(5*A + 6*B + 8*C)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
381,1,168,209,1.5258663,"\int \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{\sqrt{a (\cos (c+d x)+1)} \left(\frac{1}{2} \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) ((539 A+616 B+432 C) \cos (c+d x)+4 (35 A+40 B+48 C) \cos (2 (c+d x))+105 A \cos (3 (c+d x))+332 A+120 B \cos (3 (c+d x))+160 B+144 C \cos (3 (c+d x))+192 C)+3 \sqrt{2} (35 A+40 B+48 C) \sec \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{384 d}","\frac{a (35 A+40 B+48 C) \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (35 A+40 B+48 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a (35 A+40 B+48 C) \tan (c+d x) \sec (c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a (A+8 B) \tan (c+d x) \sec ^2(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(3*Sqrt[2]*(35*A + 40*B + 48*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Sec[(c + d*x)/2] + ((332*A + 160*B + 192*C + (539*A + 616*B + 432*C)*Cos[c + d*x] + 4*(35*A + 40*B + 48*C)*Cos[2*(c + d*x)] + 105*A*Cos[3*(c + d*x)] + 120*B*Cos[3*(c + d*x)] + 144*C*Cos[3*(c + d*x)])*Sec[c + d*x]^4*Tan[(c + d*x)/2])/2))/(384*d)","A",1
382,1,145,243,1.3062007,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((33396 A+35156 B+34734 C) \cos (c+d x)+8 (1287 A+1507 B+1743 C) \cos (2 (c+d x))+1980 A \cos (3 (c+d x))+65208 A+3740 B \cos (3 (c+d x))+770 B \cos (4 (c+d x))+59158 B+4935 C \cos (3 (c+d x))+1470 C \cos (4 (c+d x))+315 C \cos (5 (c+d x))+55482 C)}{27720 d}","\frac{2 a^2 (99 A+110 B+84 C) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (429 A+374 B+336 C) \sin (c+d x)}{495 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (429 A+374 B+336 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 d}-\frac{4 a (429 A+374 B+336 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3465 d}+\frac{2 a (11 B+3 C) \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{99 d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(65208*A + 59158*B + 55482*C + (33396*A + 35156*B + 34734*C)*Cos[c + d*x] + 8*(1287*A + 1507*B + 1743*C)*Cos[2*(c + d*x)] + 1980*A*Cos[3*(c + d*x)] + 3740*B*Cos[3*(c + d*x)] + 4935*C*Cos[3*(c + d*x)] + 770*B*Cos[4*(c + d*x)] + 1470*C*Cos[4*(c + d*x)] + 315*C*Cos[5*(c + d*x)])*Tan[(c + d*x)/2])/(27720*d)","A",1
383,1,113,187,0.7654568,"\int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (2 (756 A+759 B+799 C) \cos (c+d x)+4 (63 A+117 B+137 C) \cos (2 (c+d x))+3276 A+90 B \cos (3 (c+d x))+2964 B+170 C \cos (3 (c+d x))+35 C \cos (4 (c+d x))+2689 C)}{1260 d}","\frac{8 a^2 (63 A+57 B+47 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (63 A-18 B+22 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{315 d}+\frac{2 a (63 A+57 B+47 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 (3 B+C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{21 a d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(3276*A + 2964*B + 2689*C + 2*(756*A + 759*B + 799*C)*Cos[c + d*x] + 4*(63*A + 117*B + 137*C)*Cos[2*(c + d*x)] + 90*B*Cos[3*(c + d*x)] + 170*C*Cos[3*(c + d*x)] + 35*C*Cos[4*(c + d*x)])*Tan[(c + d*x)/2])/(1260*d)","A",1
384,1,87,144,0.3475781,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((140 A+252 B+253 C) \cos (c+d x)+700 A+6 (7 B+13 C) \cos (2 (c+d x))+546 B+15 C \cos (3 (c+d x))+494 C)}{210 d}","\frac{8 a^2 (35 A+21 B+19 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (35 A+21 B+19 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 (7 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 a d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(700*A + 546*B + 494*C + (140*A + 252*B + 253*C)*Cos[c + d*x] + 6*(7*B + 13*C)*Cos[2*(c + d*x)] + 15*C*Cos[3*(c + d*x)])*Tan[(c + d*x)/2])/(210*d)","A",1
385,1,105,142,0.4270172,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (30 A+2 (5 B+9 C) \cos (c+d x)+50 B+3 C \cos (2 (c+d x))+39 C)+15 \sqrt{2} A \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{15 d}","\frac{2 a^{3/2} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a^2 (15 A+20 B+12 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (5 B+3 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(15*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + (30*A + 50*B + 39*C + 2*(5*B + 9*C)*Cos[c + d*x] + 3*C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(15*d)","A",1
386,1,118,144,0.5126884,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) (3 A+2 (3 B+5 C) \cos (c+d x)+C \cos (2 (c+d x))+C)+3 \sqrt{2} (3 A+2 B) \cos (c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{6 d}","\frac{a^{3/2} (3 A+2 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a^2 (3 A-6 B-8 C) \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{a (3 A-2 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^{3/2}}{d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]*(3*Sqrt[2]*(3*A + 2*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x] + 2*(3*A + C + 2*(3*B + 5*C)*Cos[c + d*x] + C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(6*d)","A",1
387,1,127,159,0.7727702,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) ((7 A+4 B) \cos (c+d x)+2 (A+2 C \cos (2 (c+d x))+2 C))+\sqrt{2} (7 A+12 B+8 C) \cos ^2(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{8 d}","\frac{a^{3/2} (7 A+12 B+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^2 (5 A+4 B-8 C) \sin (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{a (3 A+4 B) \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^{3/2}}{2 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^2*(Sqrt[2]*(7*A + 12*B + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^2 + 2*((7*A + 4*B)*Cos[c + d*x] + 2*(A + 2*C + 2*C*Cos[2*(c + d*x)]))*Sin[(c + d*x)/2]))/(8*d)","A",1
388,1,139,165,1.276029,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (3 (11 A+14 B+8 C) \cos (2 (c+d x))+4 (11 A+6 B) \cos (c+d x)+49 A+42 B+24 C)+3 \sqrt{2} (11 A+14 B+24 C) \cos ^3(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{48 d}","\frac{a^{3/2} (11 A+14 B+24 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 (19 A+30 B+24 C) \tan (c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a (A+2 B) \tan (c+d x) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^3*(3*Sqrt[2]*(11*A + 14*B + 24*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^3 + (49*A + 42*B + 24*C + 4*(11*A + 6*B)*Cos[c + d*x] + 3*(11*A + 14*B + 8*C)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
389,1,174,215,2.0902652,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) ((1155 A+1048 B+1008 C) \cos (c+d x)+4 (75 A+88 B+48 C) \cos (2 (c+d x))+225 A \cos (3 (c+d x))+492 A+264 B \cos (3 (c+d x))+352 B+336 C \cos (3 (c+d x))+192 C)+6 \sqrt{2} (75 A+88 B+112 C) \cos ^4(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{768 d}","\frac{a^{3/2} (75 A+88 B+112 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (75 A+88 B+112 C) \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (39 A+56 B+48 C) \tan (c+d x) \sec (c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a (3 A+8 B) \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{24 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^4*(6*Sqrt[2]*(75*A + 88*B + 112*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^4 + (492*A + 352*B + 192*C + (1155*A + 1048*B + 1008*C)*Cos[c + d*x] + 4*(75*A + 88*B + 48*C)*Cos[2*(c + d*x)] + 225*A*Cos[3*(c + d*x)] + 264*B*Cos[3*(c + d*x)] + 336*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(768*d)","A",1
390,1,208,263,3.2116531,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (12 (1273 A+1070 B+880 C) \cos (c+d x)+4 (3059 A+3450 B+3280 C) \cos (2 (c+d x))+2660 A \cos (3 (c+d x))+1995 A \cos (4 (c+d x))+13313 A+3000 B \cos (3 (c+d x))+2250 B \cos (4 (c+d x))+11550 B+3520 C \cos (3 (c+d x))+2640 C \cos (4 (c+d x))+10480 C)+60 \sqrt{2} (133 A+150 B+176 C) \cos ^5(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{15360 d}","\frac{a^{3/2} (133 A+150 B+176 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^2 (133 A+150 B+176 C) \tan (c+d x)}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (67 A+90 B+80 C) \tan (c+d x) \sec ^2(c+d x)}{240 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (133 A+150 B+176 C) \tan (c+d x) \sec (c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a (3 A+10 B) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{40 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^5*(60*Sqrt[2]*(133*A + 150*B + 176*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^5 + (13313*A + 11550*B + 10480*C + 12*(1273*A + 1070*B + 880*C)*Cos[c + d*x] + 4*(3059*A + 3450*B + 3280*C)*Cos[2*(c + d*x)] + 2660*A*Cos[3*(c + d*x)] + 3000*B*Cos[3*(c + d*x)] + 3520*C*Cos[3*(c + d*x)] + 1995*A*Cos[4*(c + d*x)] + 2250*B*Cos[4*(c + d*x)] + 2640*C*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]))/(15360*d)","A",1
391,1,180,294,1.7348912,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (4 (445588 A+454285 B+453146 C) \cos (c+d x)+(581152 A+676000 B+746519 C) \cos (2 (c+d x))+148720 A \cos (3 (c+d x))+20020 A \cos (4 (c+d x))+3233516 A+225550 B \cos (3 (c+d x))+58240 B \cos (4 (c+d x))+8190 B \cos (5 (c+d x))+2980640 B+287060 C \cos (3 (c+d x))+94010 C \cos (4 (c+d x))+23940 C \cos (5 (c+d x))+3465 C \cos (6 (c+d x))+2798182 C)}{720720 d}","\frac{2 a^3 (2717 A+2522 B+2224 C) \sin (c+d x) \cos ^3(c+d x)}{9009 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (10439 A+9230 B+8368 C) \sin (c+d x)}{6435 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (143 A+182 B+136 C) \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{1287 d}-\frac{4 a^2 (10439 A+9230 B+8368 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{45045 d}+\frac{2 a (10439 A+9230 B+8368 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{15015 d}+\frac{2 a (13 B+5 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(3233516*A + 2980640*B + 2798182*C + 4*(445588*A + 454285*B + 453146*C)*Cos[c + d*x] + (581152*A + 676000*B + 746519*C)*Cos[2*(c + d*x)] + 148720*A*Cos[3*(c + d*x)] + 225550*B*Cos[3*(c + d*x)] + 287060*C*Cos[3*(c + d*x)] + 20020*A*Cos[4*(c + d*x)] + 58240*B*Cos[4*(c + d*x)] + 94010*C*Cos[4*(c + d*x)] + 8190*B*Cos[5*(c + d*x)] + 23940*C*Cos[5*(c + d*x)] + 3465*C*Cos[6*(c + d*x)])*Tan[(c + d*x)/2])/(720720*d)","A",1
392,1,147,229,1.2624918,"\int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((66660 A+68552 B+69890 C) \cos (c+d x)+16 (990 A+1397 B+1625 C) \cos (2 (c+d x))+1980 A \cos (3 (c+d x))+137280 A+5720 B \cos (3 (c+d x))+770 B \cos (4 (c+d x))+124366 B+8675 C \cos (3 (c+d x))+2240 C \cos (4 (c+d x))+315 C \cos (5 (c+d x))+114640 C)}{27720 d}","\frac{64 a^3 (165 A+143 B+125 C) \sin (c+d x)}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (165 A+143 B+125 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3465 d}+\frac{2 (99 A-22 B+26 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{693 d}+\frac{2 a (165 A+143 B+125 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 d}+\frac{2 (11 B+5 C) \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{99 a d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(137280*A + 124366*B + 114640*C + (66660*A + 68552*B + 69890*C)*Cos[c + d*x] + 16*(990*A + 1397*B + 1625*C)*Cos[2*(c + d*x)] + 1980*A*Cos[3*(c + d*x)] + 5720*B*Cos[3*(c + d*x)] + 8675*C*Cos[3*(c + d*x)] + 770*B*Cos[4*(c + d*x)] + 2240*C*Cos[4*(c + d*x)] + 315*C*Cos[5*(c + d*x)])*Tan[(c + d*x)/2])/(27720*d)","A",1
393,1,114,184,0.682425,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((2352 A+3030 B+3116 C) \cos (c+d x)+4 (63 A+180 B+254 C) \cos (2 (c+d x))+7476 A+90 B \cos (3 (c+d x))+6240 B+260 C \cos (3 (c+d x))+35 C \cos (4 (c+d x))+5653 C)}{1260 d}","\frac{64 a^3 (21 A+15 B+13 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (21 A+15 B+13 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 a (21 A+15 B+13 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}+\frac{2 (9 B-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{63 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(7476*A + 6240*B + 5653*C + (2352*A + 3030*B + 3116*C)*Cos[c + d*x] + 4*(63*A + 180*B + 254*C)*Cos[2*(c + d*x)] + 90*B*Cos[3*(c + d*x)] + 260*C*Cos[3*(c + d*x)] + 35*C*Cos[4*(c + d*x)])*Tan[(c + d*x)/2])/(1260*d)","A",1
394,1,127,182,0.7440873,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) ((140 A+392 B+505 C) \cos (c+d x)+1120 A+6 (7 B+20 C) \cos (2 (c+d x))+1246 B+15 C \cos (3 (c+d x))+1040 C)+420 \sqrt{2} A \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{420 d}","\frac{2 a^{5/2} A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a^3 (245 A+224 B+160 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (35 A+56 B+40 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 a (7 B+5 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(420*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*(1120*A + 1246*B + 1040*C + (140*A + 392*B + 505*C)*Cos[c + d*x] + 6*(7*B + 20*C)*Cos[2*(c + d*x)] + 15*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(420*d)","A",1
395,1,145,184,0.7920531,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) ((60 A+160 B+181 C) \cos (c+d x)+30 A+2 (5 B+14 C) \cos (2 (c+d x))+10 B+3 C \cos (3 (c+d x))+28 C)+30 \sqrt{2} (5 A+2 B) \cos (c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{60 d}","\frac{a^{5/2} (5 A+2 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{a^3 (15 A+70 B+64 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (15 A-10 B-16 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}-\frac{a (5 A-2 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}+\frac{A \tan (c+d x) (a \cos (c+d x)+a)^{5/2}}{d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]*(30*Sqrt[2]*(5*A + 2*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x] + 2*(30*A + 10*B + 28*C + (60*A + 160*B + 181*C)*Cos[c + d*x] + 2*(5*B + 14*C)*Cos[2*(c + d*x)] + 3*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(60*d)","A",1
396,1,153,199,1.1008702,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(4 \sin \left(\frac{1}{2} (c+d x)\right) (3 (11 A+4 B+2 C) \cos (c+d x)+6 A+4 (3 B+8 C) \cos (2 (c+d x))+12 B+2 C \cos (3 (c+d x))+32 C)+6 \sqrt{2} (19 A+20 B+8 C) \cos ^2(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{48 d}","\frac{a^{5/2} (19 A+20 B+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^3 (27 A-12 B-56 C) \sin (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (21 A+12 B-8 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{12 d}+\frac{a (5 A+4 B) \tan (c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d}+\frac{A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^{5/2}}{2 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^2*(6*Sqrt[2]*(19*A + 20*B + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^2 + 4*(6*A + 12*B + 32*C + 3*(11*A + 4*B + 2*C)*Cos[c + d*x] + 4*(3*B + 8*C)*Cos[2*(c + d*x)] + 2*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
397,1,156,207,1.5362894,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (4 (17 A+6 (B+3 C)) \cos (c+d x)+3 (25 A+22 B+8 C) \cos (2 (c+d x))+91 A+66 B+24 C \cos (3 (c+d x))+24 C)+3 \sqrt{2} (25 A+38 B+40 C) \cos ^3(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{48 d}","\frac{a^{5/2} (25 A+38 B+40 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}-\frac{a^3 (49 A+54 B-24 C) \sin (c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (31 A+42 B+24 C) \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{24 d}+\frac{a (5 A+6 B) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^{3/2}}{12 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^{5/2}}{3 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^3*(3*Sqrt[2]*(25*A + 38*B + 40*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^3 + (91*A + 66*B + 24*C + 4*(17*A + 6*(B + 3*C))*Cos[c + d*x] + 3*(25*A + 22*B + 8*C)*Cos[2*(c + d*x)] + 24*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
398,1,176,215,1.9857149,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) ((2203 A+2056 B+1584 C) \cos (c+d x)+4 (163 A+136 B+48 C) \cos (2 (c+d x))+489 A \cos (3 (c+d x))+844 A+600 B \cos (3 (c+d x))+544 B+528 C \cos (3 (c+d x))+192 C)+6 \sqrt{2} (163 A+200 B+304 C) \cos ^4(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{768 d}","\frac{a^{5/2} (163 A+200 B+304 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^3 (299 A+392 B+432 C) \tan (c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (17 A+24 B+16 C) \tan (c+d x) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{32 d}+\frac{a (5 A+8 B) \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^{3/2}}{24 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^{5/2}}{4 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^4*(6*Sqrt[2]*(163*A + 200*B + 304*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^4 + (844*A + 544*B + 192*C + (2203*A + 2056*B + 1584*C)*Cos[c + d*x] + 4*(163*A + 136*B + 48*C)*Cos[2*(c + d*x)] + 489*A*Cos[3*(c + d*x)] + 600*B*Cos[3*(c + d*x)] + 528*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(768*d)","A",1
399,1,210,261,2.8646216,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (12 (2343 A+1950 B+1360 C) \cos (c+d x)+4 (6509 A+6730 B+6640 C) \cos (2 (c+d x))+5660 A \cos (3 (c+d x))+4245 A \cos (4 (c+d x))+24863 A+6520 B \cos (3 (c+d x))+4890 B \cos (4 (c+d x))+22030 B+5440 C \cos (3 (c+d x))+6000 C \cos (4 (c+d x))+20560 C)+60 \sqrt{2} (283 A+326 B+400 C) \cos ^5(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{15360 d}","\frac{a^{5/2} (283 A+326 B+400 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^3 (283 A+326 B+400 C) \tan (c+d x)}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (787 A+950 B+1040 C) \tan (c+d x) \sec (c+d x)}{960 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (79 A+110 B+80 C) \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{240 d}+\frac{a (A+2 B) \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{8 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^5*(60*Sqrt[2]*(283*A + 326*B + 400*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^5 + (24863*A + 22030*B + 20560*C + 12*(2343*A + 1950*B + 1360*C)*Cos[c + d*x] + 4*(6509*A + 6730*B + 6640*C)*Cos[2*(c + d*x)] + 5660*A*Cos[3*(c + d*x)] + 6520*B*Cos[3*(c + d*x)] + 5440*C*Cos[3*(c + d*x)] + 4245*A*Cos[4*(c + d*x)] + 4890*B*Cos[4*(c + d*x)] + 6000*C*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]))/(15360*d)","A",1
400,1,242,311,4.0090112,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^6(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) ((321370 A+303048 B+283920 C) \cos (c+d x)+16 (8555 A+8444 B+7480 C) \cos (2 (c+d x))+108605 A \cos (3 (c+d x))+20300 A \cos (4 (c+d x))+15225 A \cos (5 (c+d x))+137060 A+121124 B \cos (3 (c+d x))+22640 B \cos (4 (c+d x))+16980 B \cos (5 (c+d x))+112464 B+127240 C \cos (3 (c+d x))+26080 C \cos (4 (c+d x))+19560 C \cos (5 (c+d x))+93600 C)+120 \sqrt{2} (1015 A+1132 B+1304 C) \cos ^6(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{122880 d}","\frac{a^{5/2} (1015 A+1132 B+1304 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{512 d}+\frac{a^3 (1015 A+1132 B+1304 C) \tan (c+d x)}{512 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (545 A+628 B+680 C) \tan (c+d x) \sec ^2(c+d x)}{960 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (1015 A+1132 B+1304 C) \tan (c+d x) \sec (c+d x)}{768 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (115 A+156 B+120 C) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{480 d}+\frac{a (5 A+12 B) \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{3/2}}{60 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^6*(120*Sqrt[2]*(1015*A + 1132*B + 1304*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^6 + (137060*A + 112464*B + 93600*C + (321370*A + 303048*B + 283920*C)*Cos[c + d*x] + 16*(8555*A + 8444*B + 7480*C)*Cos[2*(c + d*x)] + 108605*A*Cos[3*(c + d*x)] + 121124*B*Cos[3*(c + d*x)] + 127240*C*Cos[3*(c + d*x)] + 20300*A*Cos[4*(c + d*x)] + 22640*B*Cos[4*(c + d*x)] + 26080*C*Cos[4*(c + d*x)] + 15225*A*Cos[5*(c + d*x)] + 16980*B*Cos[5*(c + d*x)] + 19560*C*Cos[5*(c + d*x)])*Sin[(c + d*x)/2]))/(122880*d)","A",1
401,1,144,254,0.7859244,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(2 \sin \left(\frac{1}{2} (c+d x)\right) (-2 (84 A-507 B+131 C) \cos (c+d x)+4 (63 A-9 B+92 C) \cos (2 (c+d x))+2436 A+90 B \cos (3 (c+d x))-1068 B-10 C \cos (3 (c+d x))+35 C \cos (4 (c+d x))+2389 C)-2520 (A-B+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{1260 d \sqrt{a (\cos (c+d x)+1)}}","\frac{2 (21 A-3 B+19 C) \sin (c+d x) \cos ^2(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (21 A-93 B+29 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 a d}+\frac{4 (147 A-111 B+143 C) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 (9 B-C) \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*(-2520*(A - B + C)*ArcTanh[Sin[(c + d*x)/2]] + 2*(2436*A - 1068*B + 2389*C - 2*(84*A - 507*B + 131*C)*Cos[c + d*x] + 4*(63*A - 9*B + 92*C)*Cos[2*(c + d*x)] + 90*B*Cos[3*(c + d*x)] - 10*C*Cos[3*(c + d*x)] + 35*C*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]))/(1260*d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
402,1,118,208,0.6425666,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(420 (A-B+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) ((140 A-28 B+169 C) \cos (c+d x)-140 A+6 (7 B-C) \cos (2 (c+d x))+406 B+15 C \cos (3 (c+d x))-178 C)\right)}{210 d \sqrt{a (\cos (c+d x)+1)}}","\frac{2 (35 A-7 B+31 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 a d}-\frac{4 (35 A-49 B+37 C) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 (7 B-C) \sin (c+d x) \cos ^2(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*(420*(A - B + C)*ArcTanh[Sin[(c + d*x)/2]] + 2*(-140*A + 406*B - 178*C + (140*A - 28*B + 169*C)*Cos[c + d*x] + 6*(7*B - C)*Cos[2*(c + d*x)] + 15*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(210*d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
403,1,98,164,0.3726631,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right) (30 A+2 (5 B-C) \cos (c+d x)-10 B+3 C \cos (2 (c+d x))+29 C)-15 (A-B+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{15 d \sqrt{a (\cos (c+d x)+1)}}","\frac{2 (15 A-10 B+14 C) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 (5 B-C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 a d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}",1,"(2*Cos[(c + d*x)/2]*(-15*(A - B + C)*ArcTanh[Sin[(c + d*x)/2]] + (30*A - 10*B + 29*C + 2*(5*B - C)*Cos[c + d*x] + 3*C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(15*d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
404,1,79,118,0.1445111,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(3 (A-B+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 B \sin \left(\frac{1}{2} (c+d x)\right)-4 C \sin ^3\left(\frac{1}{2} (c+d x)\right)\right)}{3 d \sqrt{a (\cos (c+d x)+1)}}","\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 (3 B-2 C) \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 a d}",1,"(2*Cos[(c + d*x)/2]*(3*(A - B + C)*ArcTanh[Sin[(c + d*x)/2]] + 6*B*Sin[(c + d*x)/2] - 4*C*Sin[(c + d*x)/2]^3))/(3*d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
405,1,86,118,0.3372635,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(-\left((A-B+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)+\sqrt{2} A \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 C \sin \left(\frac{1}{2} (c+d x)\right)\right)}{d \sqrt{a (\cos (c+d x)+1)}}","-\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 C \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}",1,"(2*Cos[(c + d*x)/2]*(-((A - B + C)*ArcTanh[Sin[(c + d*x)/2]]) + Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*C*Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
406,1,96,120,0.5146864,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(2 (A-B+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\sqrt{2} (A-2 B) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 A \sin \left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)}{d \sqrt{a (\cos (c+d x)+1)}}","\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{(A-2 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{A \tan (c+d x)}{d \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*(2*(A - B + C)*ArcTanh[Sin[(c + d*x)/2]] - Sqrt[2]*(A - 2*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*A*Sec[c + d*x]*Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
407,1,118,169,0.5018778,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(-8 (A-B+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\sqrt{2} (7 A-4 B+8 C) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (2 A \sec (c+d x)-A+4 B)\right)}{4 d \sqrt{a (\cos (c+d x)+1)}}","\frac{(7 A-4 B+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{(A-4 B) \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec (c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*(-8*(A - B + C)*ArcTanh[Sin[(c + d*x)/2]] + Sqrt[2]*(7*A - 4*B + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sec[c + d*x]*(-A + 4*B + 2*A*Sec[c + d*x])*Sin[(c + d*x)/2]))/(4*d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
408,1,147,213,1.3903762,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(48 (A-B+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-3 \sqrt{2} (9 A-14 B+8 C) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+\sin \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) (3 (7 A-2 B+8 C) \cos (2 (c+d x))-4 (A-6 B) \cos (c+d x)+37 A-6 B+24 C)\right)}{24 d \sqrt{a (\cos (c+d x)+1)}}","\frac{(7 A-2 B+8 C) \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}-\frac{(9 A-14 B+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{(A-6 B) \tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*(48*(A - B + C)*ArcTanh[Sin[(c + d*x)/2]] - 3*Sqrt[2]*(9*A - 14*B + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + (37*A - 6*B + 24*C - 4*(A - 6*B)*Cos[c + d*x] + 3*(7*A - 2*B + 8*C)*Cos[2*(c + d*x)])*Sec[c + d*x]^3*Sin[(c + d*x)/2]))/(24*d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
409,1,200,259,2.8246355,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5)/Sqrt[a + a*Cos[c + d*x]],x]","-\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \left(\sin \left(\frac{1}{2} (c+d x)\right) ((221 A-760 B+144 C) \cos (c+d x)-4 (43 A-8 B+48 C) \cos (2 (c+d x))+63 A \cos (3 (c+d x))-364 A-168 B \cos (3 (c+d x))+32 B+48 C \cos (3 (c+d x))-192 C)+768 (A-B+C) \cos ^4(c+d x) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-6 \sqrt{2} (107 A-72 B+112 C) \cos ^4(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{384 d \sqrt{a (\cos (c+d x)+1)}}","-\frac{(21 A-56 B+16 C) \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{(107 A-72 B+112 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 \sqrt{a} d}-\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(43 A-8 B+48 C) \tan (c+d x) \sec (c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}-\frac{(A-8 B) \tan (c+d x) \sec ^2(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{A \tan (c+d x) \sec ^3(c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}",1,"-1/384*(Cos[(c + d*x)/2]*Sec[c + d*x]^4*(768*(A - B + C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[c + d*x]^4 - 6*Sqrt[2]*(107*A - 72*B + 112*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^4 + (-364*A + 32*B - 192*C + (221*A - 760*B + 144*C)*Cos[c + d*x] - 4*(43*A - 8*B + 48*C)*Cos[2*(c + d*x)] + 63*A*Cos[3*(c + d*x)] - 168*B*Cos[3*(c + d*x)] + 48*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
410,1,180,277,1.5410924,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{\frac{1}{2} \sin \left(\frac{1}{2} (c+d x)\right) \cos ^3\left(\frac{1}{2} (c+d x)\right) (6 (140 A-273 B+277 C) \cos (c+d x)-4 (35 A-21 B+64 C) \cos (2 (c+d x))+1190 A-42 B \cos (3 (c+d x))-1974 B+18 C \cos (3 (c+d x))-15 C \cos (4 (c+d x))+2161 C)-105 (11 A-15 B+19 C) \cos ^5\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{105 d \left(\sin ^2\left(\frac{1}{2} (c+d x)\right)-1\right) (a (\cos (c+d x)+1))^{3/2}}","\frac{(11 A-15 B+19 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(245 A-273 B+397 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{210 a^2 d}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(7 A-7 B+11 C) \sin (c+d x) \cos ^3(c+d x)}{14 a d \sqrt{a \cos (c+d x)+a}}-\frac{(35 A-63 B+67 C) \sin (c+d x) \cos ^2(c+d x)}{70 a d \sqrt{a \cos (c+d x)+a}}-\frac{(455 A-651 B+799 C) \sin (c+d x)}{105 a d \sqrt{a \cos (c+d x)+a}}",1,"(-105*(11*A - 15*B + 19*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^5 + (Cos[(c + d*x)/2]^3*(1190*A - 1974*B + 2161*C + 6*(140*A - 273*B + 277*C)*Cos[c + d*x] - 4*(35*A - 21*B + 64*C)*Cos[2*(c + d*x)] - 42*B*Cos[3*(c + d*x)] + 18*C*Cos[3*(c + d*x)] - 15*C*Cos[4*(c + d*x)])*Sin[(c + d*x)/2])/2)/(105*d*(a*(1 + Cos[c + d*x]))^(3/2)*(-1 + Sin[(c + d*x)/2]^2))","A",1
411,1,153,229,0.7415729,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{15 (7 A-11 B+15 C) \cos ^5\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\sin \left(\frac{1}{2} (c+d x)\right) \cos ^3\left(\frac{1}{2} (c+d x)\right) (3 (20 A-20 B+39 C) \cos (c+d x)+75 A+2 (5 B-3 C) \cos (2 (c+d x))-85 B+3 C \cos (3 (c+d x))+141 C)}{15 d \left(\sin ^2\left(\frac{1}{2} (c+d x)\right)-1\right) (a (\cos (c+d x)+1))^{3/2}}","-\frac{(7 A-11 B+15 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(15 A-35 B+39 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{30 a^2 d}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(5 A-5 B+9 C) \sin (c+d x) \cos ^2(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}+\frac{(45 A-65 B+93 C) \sin (c+d x)}{15 a d \sqrt{a \cos (c+d x)+a}}",1,"(15*(7*A - 11*B + 15*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^5 - Cos[(c + d*x)/2]^3*(75*A - 85*B + 141*C + 3*(20*A - 20*B + 39*C)*Cos[c + d*x] + 2*(5*B - 3*C)*Cos[2*(c + d*x)] + 3*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2])/(15*d*(a*(1 + Cos[c + d*x]))^(3/2)*(-1 + Sin[(c + d*x)/2]^2))","A",1
412,1,131,181,0.4848509,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{-\sin \left(\frac{1}{2} (c+d x)\right) \cos ^3\left(\frac{1}{2} (c+d x)\right) (-3 A+12 (B-C) \cos (c+d x)+15 B+2 C \cos (2 (c+d x))-17 C)-3 (3 A-7 B+11 C) \cos ^5\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{3 d \left(\sin ^2\left(\frac{1}{2} (c+d x)\right)-1\right) (a (\cos (c+d x)+1))^{3/2}}","\frac{(3 A-7 B+11 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(3 A-3 B+7 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{6 a^2 d}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(3 A-9 B+13 C) \sin (c+d x)}{3 a d \sqrt{a \cos (c+d x)+a}}",1,"(-3*(3*A - 7*B + 11*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^5 - Cos[(c + d*x)/2]^3*(-3*A + 15*B - 17*C + 12*(B - C)*Cos[c + d*x] + 2*C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2])/(3*d*(a*(1 + Cos[c + d*x]))^(3/2)*(-1 + Sin[(c + d*x)/2]^2))","A",1
413,1,83,120,0.5124578,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) (A-B+4 C \cos (c+d x)+5 C)+(A+3 B-7 C) \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a d \sqrt{a (\cos (c+d x)+1)}}","\frac{(A+3 B-7 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(A-B+C) \sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{2 C \sin (c+d x)}{a d \sqrt{a \cos (c+d x)+a}}",1,"((A + 3*B - 7*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2] + (A - B + 5*C + 4*C*Cos[c + d*x])*Tan[(c + d*x)/2])/(2*a*d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
414,1,135,131,0.9612084,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(A-B+C) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^3\left(\frac{1}{2} (c+d x)\right)+(5 A-B-3 C) \cos ^5\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-4 \sqrt{2} A \cos ^5\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)}{d \left(\sin ^2\left(\frac{1}{2} (c+d x)\right)-1\right) (a (\cos (c+d x)+1))^{3/2}}","-\frac{(5 A-B-3 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A-B+C) \sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"((5*A - B - 3*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^5 - 4*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^5 + (A - B + C)*Cos[(c + d*x)/2]^3*Sin[(c + d*x)/2])/(d*(a*(1 + Cos[c + d*x]))^(3/2)*(-1 + Sin[(c + d*x)/2]^2))","A",1
415,1,196,173,1.6244814,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^(3/2),x]","\frac{\cos ^3\left(\frac{1}{2} (c+d x)\right) \cos ^2(c+d x) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(2 (9 A-5 B+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\frac{4 \sqrt{2} (3 A-2 B) \cos ^2\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right) (2 A \sec (c+d x)+3 A-B+C)}{\sin ^2\left(\frac{1}{2} (c+d x)\right)-1}\right)}{d (a (\cos (c+d x)+1))^{3/2} (2 A+2 B \cos (c+d x)+C \cos (2 (c+d x))+C)}","\frac{(9 A-5 B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(3 A-2 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}+\frac{(3 A-B+C) \tan (c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \tan (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]^3*Cos[c + d*x]^2*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*(2*(9*A - 5*B + C)*ArcTanh[Sin[(c + d*x)/2]] + (4*Sqrt[2]*(3*A - 2*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^2 - 2*(3*A - B + C + 2*A*Sec[c + d*x])*Sin[(c + d*x)/2])/(-1 + Sin[(c + d*x)/2]^2)))/(d*(a*(1 + Cos[c + d*x]))^(3/2)*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*(c + d*x)]))","A",0
416,1,186,232,1.575924,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^(3/2),x]","\frac{\cos ^3\left(\frac{1}{2} (c+d x)\right) \left(\frac{\frac{1}{2} \sin \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) ((7 A-6 B+2 C) \cos (2 (c+d x))+(6 A-8 B) \cos (c+d x)+3 A-6 B+2 C)-\sqrt{2} (19 A-12 B+8 C) \cos ^2\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)}{\sin ^2\left(\frac{1}{2} (c+d x)\right)-1}-2 (13 A-9 B+5 C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{2 d (a (\cos (c+d x)+1))^{3/2}}","\frac{(19 A-12 B+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{(13 A-9 B+5 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(7 A-6 B+2 C) \tan (c+d x)}{4 a d \sqrt{a \cos (c+d x)+a}}+\frac{(2 A-B+C) \tan (c+d x) \sec (c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \tan (c+d x) \sec (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]^3*(-2*(13*A - 9*B + 5*C)*ArcTanh[Sin[(c + d*x)/2]] + (-(Sqrt[2]*(19*A - 12*B + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^2) + ((3*A - 6*B + 2*C + (6*A - 8*B)*Cos[c + d*x] + (7*A - 6*B + 2*C)*Cos[2*(c + d*x)])*Sec[c + d*x]^2*Sin[(c + d*x)/2])/2)/(-1 + Sin[(c + d*x)/2]^2)))/(2*d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
417,1,223,284,2.6972772,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^(3/2),x]","\frac{\cos ^3\left(\frac{1}{2} (c+d x)\right) \left(12 (17 A-13 B+9 C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\frac{3 \sqrt{2} (47 A-38 B+24 C) \cos ^2\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)-\frac{1}{4} \sin \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) (3 (55 A-26 B+36 C) \cos (c+d x)+(74 A-36 B+48 C) \cos (2 (c+d x))+63 A \cos (3 (c+d x))+106 A-42 B \cos (3 (c+d x))-36 B+36 C \cos (3 (c+d x))+48 C)}{\sin ^2\left(\frac{1}{2} (c+d x)\right)-1}\right)}{12 d (a (\cos (c+d x)+1))^{3/2}}","-\frac{(47 A-38 B+24 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 a^{3/2} d}+\frac{(17 A-13 B+9 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(21 A-14 B+12 C) \tan (c+d x)}{8 a d \sqrt{a \cos (c+d x)+a}}+\frac{(5 A-3 B+3 C) \tan (c+d x) \sec ^2(c+d x)}{6 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(13 A-12 B+6 C) \tan (c+d x) \sec (c+d x)}{12 a d \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]^3*(12*(17*A - 13*B + 9*C)*ArcTanh[Sin[(c + d*x)/2]] + (3*Sqrt[2]*(47*A - 38*B + 24*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^2 - ((106*A - 36*B + 48*C + 3*(55*A - 26*B + 36*C)*Cos[c + d*x] + (74*A - 36*B + 48*C)*Cos[2*(c + d*x)] + 63*A*Cos[3*(c + d*x)] - 42*B*Cos[3*(c + d*x)] + 36*C*Cos[3*(c + d*x)])*Sec[c + d*x]^3*Sin[(c + d*x)/2])/4)/(-1 + Sin[(c + d*x)/2]^2)))/(12*d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
418,1,152,277,1.6167917,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) (5 (255 A-479 B+887 C) \cos (c+d x)+16 (15 A-25 B+52 C) \cos (2 (c+d x))+975 A+40 B \cos (3 (c+d x))-1895 B-40 C \cos (3 (c+d x))+12 C \cos (4 (c+d x))+3491 C)-30 (75 A-163 B+283 C) \cos ^3\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{240 a d (a (\cos (c+d x)+1))^{3/2}}","-\frac{(75 A-163 B+283 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(195 A-475 B+787 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{240 a^3 d}+\frac{(45 A-85 B+157 C) \sin (c+d x) \cos ^2(c+d x)}{80 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(465 A-985 B+1729 C) \sin (c+d x)}{120 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{(5 A-13 B+21 C) \sin (c+d x) \cos ^3(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"(-30*(75*A - 163*B + 283*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 + (975*A - 1895*B + 3491*C + 5*(255*A - 479*B + 887*C)*Cos[c + d*x] + 16*(15*A - 25*B + 52*C)*Cos[2*(c + d*x)] + 40*B*Cos[3*(c + d*x)] - 40*C*Cos[3*(c + d*x)] + 12*C*Cos[4*(c + d*x)])*Tan[(c + d*x)/2])/(240*a*d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
419,1,126,227,1.1957428,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) ((-39 A+255 B-479 C) \cos (c+d x)-27 A+16 (3 B-5 C) \cos (2 (c+d x))+195 B+8 C \cos (3 (c+d x))-379 C)+6 (19 A-75 B+163 C) \cos ^3\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{48 a d (a (\cos (c+d x)+1))^{3/2}}","\frac{(19 A-75 B+163 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(15 A-39 B+95 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{48 a^3 d}-\frac{(21 A-93 B+197 C) \sin (c+d x)}{24 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{(A-9 B+17 C) \sin (c+d x) \cos ^2(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"(6*(19*A - 75*B + 163*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 + (-27*A + 195*B - 379*C + (-39*A + 255*B - 479*C)*Cos[c + d*x] + 16*(3*B - 5*C)*Cos[2*(c + d*x)] + 8*C*Cos[3*(c + d*x)])*Tan[(c + d*x)/2])/(48*a*d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
420,1,107,179,0.8198251,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) ((5 A-13 B+85 C) \cos (c+d x)+A-9 B+16 C \cos (2 (c+d x))+65 C)+2 (5 A+19 B-75 C) \cos ^3\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{16 a d (a (\cos (c+d x)+1))^{3/2}}","\frac{(5 A+19 B-75 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(A-B+9 C) \sin (c+d x)}{4 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{(3 A+5 B-13 C) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"(2*(5*A + 19*B - 75*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 + (A - 9*B + 65*C + (5*A - 13*B + 85*C)*Cos[c + d*x] + 16*C*Cos[2*(c + d*x)])*Tan[(c + d*x)/2])/(16*a*d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
421,1,96,133,0.5800077,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) ((3 A+5 B-13 C) \cos (c+d x)+7 A+B-9 C)+2 (3 A+5 B+19 C) \cos ^3\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{16 a d (a (\cos (c+d x)+1))^{3/2}}","\frac{(3 A+5 B+19 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(3 A+5 B-13 C) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B+C) \sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(2*(3*A + 5*B + 19*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 + (7*A + B - 9*C + (3*A + 5*B - 13*C)*Cos[c + d*x])*Tan[(c + d*x)/2])/(16*a*d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
422,1,200,171,1.8052827,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + a*Cos[c + d*x])^(5/2),x]","-\frac{\cos ^5\left(\frac{1}{2} (c+d x)\right) \cos (c+d x) (A \sec (c+d x)+B+C \cos (c+d x)) \left(2 (43 A-3 B-5 C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\frac{\sin \left(\frac{1}{2} (c+d x)\right) ((11 A-3 B-5 C) \cos (c+d x)+15 A-7 B-C)-64 \sqrt{2} A \cos ^4\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)}{\left(\sin ^2\left(\frac{1}{2} (c+d x)\right)-1\right)^2}\right)}{4 d (a (\cos (c+d x)+1))^{5/2} (2 A+2 B \cos (c+d x)+C \cos (2 (c+d x))+C)}","-\frac{(43 A-3 B-5 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 A \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(11 A-3 B-5 C) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"-1/4*(Cos[(c + d*x)/2]^5*Cos[c + d*x]*(B + C*Cos[c + d*x] + A*Sec[c + d*x])*(2*(43*A - 3*B - 5*C)*ArcTanh[Sin[(c + d*x)/2]] + (-64*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^4 + (15*A - 7*B - C + (11*A - 3*B - 5*C)*Cos[c + d*x])*Sin[(c + d*x)/2])/(-1 + Sin[(c + d*x)/2]^2)^2))/(d*(a*(1 + Cos[c + d*x]))^(5/2)*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*(c + d*x)]))","A",1
423,1,189,217,3.7684741,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(\sin \left(\frac{1}{2} (c+d x)\right) (2 (55 A-15 B+7 C) \cos (c+d x)+(35 A-11 B+3 C) \cos (2 (c+d x))+67 A-11 B+3 C)+4 (115 A-43 B+3 C) \cos (c+d x) \cos ^4\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-64 \sqrt{2} (5 A-2 B) \cos (c+d x) \cos ^4\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{32 a d (a (\cos (c+d x)+1))^{3/2}}","\frac{(115 A-43 B+3 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(5 A-2 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}+\frac{(35 A-11 B+3 C) \tan (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(15 A-7 B-C) \tan (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \tan (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(Sec[(c + d*x)/2]*Sec[c + d*x]*(4*(115*A - 43*B + 3*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^4*Cos[c + d*x] - 64*Sqrt[2]*(5*A - 2*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^4*Cos[c + d*x] + (67*A - 11*B + 3*C + 2*(55*A - 15*B + 7*C)*Cos[c + d*x] + (35*A - 11*B + 3*C)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(32*a*d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
424,1,248,280,5.4333296,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\sec ^6\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(-\frac{1}{8} \sin \left(\frac{1}{2} (c+d x)\right) \cos ^5\left(\frac{1}{2} (c+d x)\right) ((269 A-169 B+33 C) \cos (c+d x)+10 (19 A-11 B+3 C) \cos (2 (c+d x))+63 A \cos (3 (c+d x))+158 A-35 B \cos (3 (c+d x))-110 B+11 C \cos (3 (c+d x))+30 C)-\left((219 A-115 B+43 C) \cos ^2(c+d x) \cos ^9\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)+4 \sqrt{2} (39 A-20 B+8 C) \cos ^2(c+d x) \cos ^9\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{8 a d (a (\cos (c+d x)+1))^{3/2}}","\frac{(39 A-20 B+8 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{5/2} d}-\frac{(219 A-115 B+43 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(63 A-35 B+11 C) \tan (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(31 A-15 B+7 C) \tan (c+d x) \sec (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(19 A-11 B+3 C) \tan (c+d x) \sec (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \tan (c+d x) \sec (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(Sec[(c + d*x)/2]^6*Sec[c + d*x]^2*(-((219*A - 115*B + 43*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^9*Cos[c + d*x]^2) + 4*Sqrt[2]*(39*A - 20*B + 8*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^9*Cos[c + d*x]^2 - (Cos[(c + d*x)/2]^5*(158*A - 110*B + 30*C + (269*A - 169*B + 33*C)*Cos[c + d*x] + 10*(19*A - 11*B + 3*C)*Cos[2*(c + d*x)] + 63*A*Cos[3*(c + d*x)] - 35*B*Cos[3*(c + d*x)] + 11*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2])/8))/(8*a*d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
425,1,86,123,0.5591669,"\int \cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} (70 A+42 B \cos (c+d x)+15 C \cos (2 (c+d x))+65 C)+10 (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+126 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{105 d}","\frac{2 (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 (7 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}",1,"(126*B*EllipticE[(c + d*x)/2, 2] + 10*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(70*A + 65*C + 42*B*Cos[c + d*x] + 15*C*Cos[2*(c + d*x)])*Sin[c + d*x])/(105*d)","A",1
426,1,72,93,0.2574384,"\int \sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 \left(3 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} (5 B+3 C \cos (c+d x))+5 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d}","\frac{2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(2*(3*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2] + 5*B*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(5*B + 3*C*Cos[c + d*x])*Sin[c + d*x]))/(15*d)","A",1
427,1,57,65,0.1321379,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sqrt[Cos[c + d*x]],x]","\frac{2 \left((3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (c+d x) \sqrt{\cos (c+d x)}\right)}{3 d}","\frac{2 (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(2*(3*B*EllipticE[(c + d*x)/2, 2] + (3*A + C)*EllipticF[(c + d*x)/2, 2] + C*Sqrt[Cos[c + d*x]]*Sin[c + d*x]))/(3*d)","A",1
428,1,54,61,0.1904643,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(3/2),x]","\frac{2 \left((C-A) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{A \sin (c+d x)}{\sqrt{\cos (c+d x)}}+B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d}","-\frac{2 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*((-A + C)*EllipticE[(c + d*x)/2, 2] + B*EllipticF[(c + d*x)/2, 2] + (A*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/d","A",1
429,1,69,87,0.5399062,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(5/2),x]","\frac{\frac{2 \sin (c+d x) (A+3 B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)}+2 (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}","\frac{2 (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(-6*B*EllipticE[(c + d*x)/2, 2] + 2*(A + 3*C)*EllipticF[(c + d*x)/2, 2] + (2*(A + 3*B*Cos[c + d*x])*Sin[c + d*x])/Cos[c + d*x]^(3/2))/(3*d)","A",1
430,1,112,123,0.4669552,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Cos[c + d*x]^(7/2),x]","\frac{-6 (3 A+5 C) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+9 A \sin (2 (c+d x))+6 A \tan (c+d x)+10 B \sin (c+d x)+10 B \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+15 C \sin (2 (c+d x))}{15 d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{2 (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (3 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-6*(3*A + 5*C)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 10*B*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 10*B*Sin[c + d*x] + 9*A*Sin[2*(c + d*x)] + 15*C*Sin[2*(c + d*x)] + 6*A*Tan[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",1
431,1,1344,211,6.4591961,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(-\frac{(9 A+7 B+7 C) \cot (c)}{15 d}+\frac{(506 A+506 B+435 C) \cos (d x) \sin (c)}{1848 d}+\frac{(18 A+19 B+19 C) \cos (2 d x) \sin (2 c)}{180 d}+\frac{(44 A+44 B+57 C) \cos (3 d x) \sin (3 c)}{1232 d}+\frac{(B+C) \cos (4 d x) \sin (4 c)}{72 d}+\frac{C \cos (5 d x) \sin (5 c)}{176 d}+\frac{(506 A+506 B+435 C) \cos (c) \sin (d x)}{1848 d}+\frac{(18 A+19 B+19 C) \cos (2 c) \sin (2 d x)}{180 d}+\frac{(44 A+44 B+57 C) \cos (3 c) \sin (3 d x)}{1232 d}+\frac{(B+C) \cos (4 c) \sin (4 d x)}{72 d}+\frac{C \cos (5 c) \sin (5 d x)}{176 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{3 A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{7 B (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{30 d}-\frac{7 C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{30 d}-\frac{5 A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d \sqrt{\cot ^2(c)+1}}-\frac{5 B (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d \sqrt{\cot ^2(c)+1}}-\frac{15 C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{77 d \sqrt{\cot ^2(c)+1}}\right)","\frac{10 a (11 A+11 B+9 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 a (9 A+7 (B+C)) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a (11 A+11 B+9 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{2 a (9 A+7 (B+C)) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 a (11 A+11 B+9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 a (B+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{11 d}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-1/15*((9*A + 7*B + 7*C)*Cot[c])/d + ((506*A + 506*B + 435*C)*Cos[d*x]*Sin[c])/(1848*d) + ((18*A + 19*B + 19*C)*Cos[2*d*x]*Sin[2*c])/(180*d) + ((44*A + 44*B + 57*C)*Cos[3*d*x]*Sin[3*c])/(1232*d) + ((B + C)*Cos[4*d*x]*Sin[4*c])/(72*d) + (C*Cos[5*d*x]*Sin[5*c])/(176*d) + ((506*A + 506*B + 435*C)*Cos[c]*Sin[d*x])/(1848*d) + ((18*A + 19*B + 19*C)*Cos[2*c]*Sin[2*d*x])/(180*d) + ((44*A + 44*B + 57*C)*Cos[3*c]*Sin[3*d*x])/(1232*d) + ((B + C)*Cos[4*c]*Sin[4*d*x])/(72*d) + (C*Cos[5*c]*Sin[5*d*x])/(176*d)) - (5*A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (5*B*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (15*C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(77*d*Sqrt[1 + Cot[c]^2]) - (3*A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) - (7*B*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(30*d) - (7*C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(30*d))","C",0
432,1,1292,177,6.3657601,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(-\frac{(9 A+9 B+7 C) \cot (c)}{15 d}+\frac{(28 A+23 B+23 C) \cos (d x) \sin (c)}{84 d}+\frac{(18 A+18 B+19 C) \cos (2 d x) \sin (2 c)}{180 d}+\frac{(B+C) \cos (3 d x) \sin (3 c)}{28 d}+\frac{C \cos (4 d x) \sin (4 c)}{72 d}+\frac{(28 A+23 B+23 C) \cos (c) \sin (d x)}{84 d}+\frac{(18 A+18 B+19 C) \cos (2 c) \sin (2 d x)}{180 d}+\frac{(B+C) \cos (3 c) \sin (3 d x)}{28 d}+\frac{C \cos (4 c) \sin (4 d x)}{72 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{3 A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{3 B (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{7 C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{30 d}-\frac{A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{5 B (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d \sqrt{\cot ^2(c)+1}}-\frac{5 C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (7 A+5 (B+C)) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (9 A+9 B+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a (9 A+9 B+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 a (7 A+5 (B+C)) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a (B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-1/15*((9*A + 9*B + 7*C)*Cot[c])/d + ((28*A + 23*B + 23*C)*Cos[d*x]*Sin[c])/(84*d) + ((18*A + 18*B + 19*C)*Cos[2*d*x]*Sin[2*c])/(180*d) + ((B + C)*Cos[3*d*x]*Sin[3*c])/(28*d) + (C*Cos[4*d*x]*Sin[4*c])/(72*d) + ((28*A + 23*B + 23*C)*Cos[c]*Sin[d*x])/(84*d) + ((18*A + 18*B + 19*C)*Cos[2*c]*Sin[2*d*x])/(180*d) + ((B + C)*Cos[3*c]*Sin[3*d*x])/(28*d) + (C*Cos[4*c]*Sin[4*d*x])/(72*d)) - (A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (5*B*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (5*C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (3*A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) - (3*B*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) - (7*C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(30*d))","C",0
433,1,1240,144,6.3391994,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(-\frac{(5 A+3 B+3 C) \cot (c)}{5 d}+\frac{(28 A+28 B+23 C) \cos (d x) \sin (c)}{84 d}+\frac{(B+C) \cos (2 d x) \sin (2 c)}{10 d}+\frac{C \cos (3 d x) \sin (3 c)}{28 d}+\frac{(28 A+28 B+23 C) \cos (c) \sin (d x)}{84 d}+\frac{(B+C) \cos (2 c) \sin (2 d x)}{10 d}+\frac{C \cos (3 c) \sin (3 d x)}{28 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{3 B (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{3 C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{B (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{5 C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (7 A+7 B+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (5 A+3 (B+C)) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (7 A+7 B+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a (B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-1/5*((5*A + 3*B + 3*C)*Cot[c])/d + ((28*A + 28*B + 23*C)*Cos[d*x]*Sin[c])/(84*d) + ((B + C)*Cos[2*d*x]*Sin[2*c])/(10*d) + (C*Cos[3*d*x]*Sin[3*c])/(28*d) + ((28*A + 28*B + 23*C)*Cos[c]*Sin[d*x])/(84*d) + ((B + C)*Cos[2*c]*Sin[2*d*x])/(10*d) + (C*Cos[3*c]*Sin[3*d*x])/(28*d)) - (A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (B*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (5*C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) - (3*B*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) - (3*C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d))","C",0
434,1,1186,107,6.3752105,"\int \frac{(a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(-\frac{(5 A+5 B+3 C) \cot (c)}{5 d}+\frac{(B+C) \cos (d x) \sin (c)}{3 d}+\frac{C \cos (2 d x) \sin (2 c)}{10 d}+\frac{(B+C) \cos (c) \sin (d x)}{3 d}+\frac{C \cos (2 c) \sin (2 d x)}{10 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{B (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{3 C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{\cot ^2(c)+1}}-\frac{B (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (3 A+B+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (5 A+5 B+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-1/5*((5*A + 5*B + 3*C)*Cot[c])/d + ((B + C)*Cos[d*x]*Sin[c])/(3*d) + (C*Cos[2*d*x]*Sin[2*c])/(10*d) + ((B + C)*Cos[c]*Sin[d*x])/(3*d) + (C*Cos[2*c]*Sin[2*d*x])/(10*d)) - (A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) - (B*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) - (B*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) - (3*C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d))","C",0
435,1,1173,101,6.4644055,"\int \frac{(a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(-\frac{(-2 A+B+C+B \cos (2 c)+C \cos (2 c)) \csc (c) \sec (c)}{2 d}+\frac{A \sec (c+d x) \sin (d x) \sec (c)}{d}+\frac{C \cos (d x) \sin (c)}{3 d}+\frac{C \cos (c) \sin (d x)}{3 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{B (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{\cot ^2(c)+1}}-\frac{B (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (3 A+3 B+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A-B-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 a C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-1/2*((-2*A + B + C + B*Cos[2*c] + C*Cos[2*c])*Csc[c]*Sec[c])/d + (C*Cos[d*x]*Sin[c])/(3*d) + (C*Cos[c]*Sin[d*x])/(3*d) + (A*Sec[c]*Sec[c + d*x]*Sin[d*x])/d) - (A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) - (B*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) - (C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) + (A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) - (B*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) - (C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d))","C",0
436,1,1180,100,6.4799446,"\int \frac{(a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(\frac{A \sec (c) \sin (d x) \sec ^2(c+d x)}{3 d}+\frac{\sec (c) (A \sin (c)+3 A \sin (d x)+3 B \sin (d x)) \sec (c+d x)}{3 d}-\frac{(-2 A-2 B+C+C \cos (2 c)) \csc (c) \sec (c)}{2 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{B (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{B (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (A+3 (B+C)) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A+B-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a (A+B) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-1/2*((-2*A - 2*B + C + C*Cos[2*c])*Csc[c]*Sec[c])/d + (A*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (Sec[c]*Sec[c + d*x]*(A*Sin[c] + 3*A*Sin[d*x] + 3*B*Sin[d*x]))/(3*d)) - (A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (B*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) - (C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) + (A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) + (B*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) - (C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d))","C",0
437,1,1228,139,6.5910614,"\int \frac{(a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(\frac{A \sec (c) \sin (d x) \sec ^3(c+d x)}{5 d}+\frac{\sec (c) (3 A \sin (c)+5 A \sin (d x)+5 B \sin (d x)) \sec ^2(c+d x)}{15 d}+\frac{\sec (c) (5 A \sin (c)+5 B \sin (c)+9 A \sin (d x)+15 B \sin (d x)+15 C \sin (d x)) \sec (c+d x)}{15 d}+\frac{(3 A+5 B+5 C) \csc (c) \sec (c)}{5 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{3 A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}+\frac{B (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{B (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (A+B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (3 A+5 (B+C)) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (3 A+5 (B+C)) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a (A+B) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(((3*A + 5*B + 5*C)*Csc[c]*Sec[c])/(5*d) + (A*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(5*d) + (Sec[c]*Sec[c + d*x]^2*(3*A*Sin[c] + 5*A*Sin[d*x] + 5*B*Sin[d*x]))/(15*d) + (Sec[c]*Sec[c + d*x]*(5*A*Sin[c] + 5*B*Sin[c] + 9*A*Sin[d*x] + 15*B*Sin[d*x] + 15*C*Sin[d*x]))/(15*d)) - (A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (B*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) + (3*A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) + (B*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) + (C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d))","C",0
438,1,1284,177,6.652692,"\int \frac{(a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(\frac{A \sec (c) \sin (d x) \sec ^4(c+d x)}{7 d}+\frac{\sec (c) (5 A \sin (c)+7 A \sin (d x)+7 B \sin (d x)) \sec ^3(c+d x)}{35 d}+\frac{\sec (c) (21 A \sin (c)+21 B \sin (c)+25 A \sin (d x)+35 B \sin (d x)+35 C \sin (d x)) \sec ^2(c+d x)}{105 d}+\frac{\sec (c) (25 A \sin (c)+35 B \sin (c)+35 C \sin (c)+63 A \sin (d x)+63 B \sin (d x)+105 C \sin (d x)) \sec (c+d x)}{105 d}+\frac{(3 A+3 B+5 C) \csc (c) \sec (c)}{5 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{3 A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}+\frac{3 B (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}+\frac{C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{5 A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d \sqrt{\cot ^2(c)+1}}-\frac{B (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (5 A+7 (B+C)) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a (3 A+3 B+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (5 A+7 (B+C)) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a (3 A+3 B+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a (A+B) \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(((3*A + 3*B + 5*C)*Csc[c]*Sec[c])/(5*d) + (A*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(7*d) + (Sec[c]*Sec[c + d*x]^3*(5*A*Sin[c] + 7*A*Sin[d*x] + 7*B*Sin[d*x]))/(35*d) + (Sec[c]*Sec[c + d*x]^2*(21*A*Sin[c] + 21*B*Sin[c] + 25*A*Sin[d*x] + 35*B*Sin[d*x] + 35*C*Sin[d*x]))/(105*d) + (Sec[c]*Sec[c + d*x]*(25*A*Sin[c] + 35*B*Sin[c] + 35*C*Sin[c] + 63*A*Sin[d*x] + 63*B*Sin[d*x] + 105*C*Sin[d*x]))/(105*d)) - (5*A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (B*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) + (3*A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) + (3*B*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) + (C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d))","C",0
439,1,1374,251,6.4320142,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^2 \left(-\frac{(9 A+8 B+7 C) \cot (c)}{15 d}+\frac{(1122 A+1012 B+941 C) \cos (d x) \sin (c)}{3696 d}+\frac{(36 A+37 B+38 C) \cos (2 d x) \sin (2 c)}{360 d}+\frac{(44 A+88 B+101 C) \cos (3 d x) \sin (3 c)}{2464 d}+\frac{(B+2 C) \cos (4 d x) \sin (4 c)}{144 d}+\frac{C \cos (5 d x) \sin (5 c)}{352 d}+\frac{(1122 A+1012 B+941 C) \cos (c) \sin (d x)}{3696 d}+\frac{(36 A+37 B+38 C) \cos (2 c) \sin (2 d x)}{360 d}+\frac{(44 A+88 B+101 C) \cos (3 c) \sin (3 d x)}{2464 d}+\frac{(B+2 C) \cos (4 c) \sin (4 d x)}{144 d}+\frac{C \cos (5 c) \sin (5 d x)}{352 d}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{3 A (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{4 B (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{7 C (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{30 d}-\frac{2 A (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d \sqrt{\cot ^2(c)+1}}-\frac{5 B (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d \sqrt{\cot ^2(c)+1}}-\frac{50 C (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{231 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^2 (66 A+55 B+50 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^2 (9 A+8 B+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 (99 A+121 B+89 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{693 d}+\frac{4 a^2 (9 A+8 B+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{4 a^2 (66 A+55 B+50 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 (11 B+4 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{99 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(-1/15*((9*A + 8*B + 7*C)*Cot[c])/d + ((1122*A + 1012*B + 941*C)*Cos[d*x]*Sin[c])/(3696*d) + ((36*A + 37*B + 38*C)*Cos[2*d*x]*Sin[2*c])/(360*d) + ((44*A + 88*B + 101*C)*Cos[3*d*x]*Sin[3*c])/(2464*d) + ((B + 2*C)*Cos[4*d*x]*Sin[4*c])/(144*d) + (C*Cos[5*d*x]*Sin[5*c])/(352*d) + ((1122*A + 1012*B + 941*C)*Cos[c]*Sin[d*x])/(3696*d) + ((36*A + 37*B + 38*C)*Cos[2*c]*Sin[2*d*x])/(360*d) + ((44*A + 88*B + 101*C)*Cos[3*c]*Sin[3*d*x])/(2464*d) + ((B + 2*C)*Cos[4*c]*Sin[4*d*x])/(144*d) + (C*Cos[5*c]*Sin[5*d*x])/(352*d)) - (2*A*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*Sqrt[1 + Cot[c]^2]) - (5*B*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (50*C*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(231*d*Sqrt[1 + Cot[c]^2]) - (3*A*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) - (4*B*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*d) - (7*C*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(30*d)","C",0
440,1,1322,215,6.3788425,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^2 \left(-\frac{(12 A+9 B+8 C) \cot (c)}{15 d}+\frac{(56 A+51 B+46 C) \cos (d x) \sin (c)}{168 d}+\frac{(18 A+36 B+37 C) \cos (2 d x) \sin (2 c)}{360 d}+\frac{(B+2 C) \cos (3 d x) \sin (3 c)}{56 d}+\frac{C \cos (4 d x) \sin (4 c)}{144 d}+\frac{(56 A+51 B+46 C) \cos (c) \sin (d x)}{168 d}+\frac{(18 A+36 B+37 C) \cos (2 c) \sin (2 d x)}{360 d}+\frac{(B+2 C) \cos (3 c) \sin (3 d x)}{56 d}+\frac{C \cos (4 c) \sin (4 d x)}{144 d}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{2 A (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{3 B (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{4 C (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{A (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{2 B (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d \sqrt{\cot ^2(c)+1}}-\frac{5 C (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^2 (7 A+6 B+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (12 A+9 B+8 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 (21 A+27 B+19 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^2 (7 A+6 B+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 (9 B+4 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(-1/15*((12*A + 9*B + 8*C)*Cot[c])/d + ((56*A + 51*B + 46*C)*Cos[d*x]*Sin[c])/(168*d) + ((18*A + 36*B + 37*C)*Cos[2*d*x]*Sin[2*c])/(360*d) + ((B + 2*C)*Cos[3*d*x]*Sin[3*c])/(56*d) + (C*Cos[4*d*x]*Sin[4*c])/(144*d) + ((56*A + 51*B + 46*C)*Cos[c]*Sin[d*x])/(168*d) + ((18*A + 36*B + 37*C)*Cos[2*c]*Sin[2*d*x])/(360*d) + ((B + 2*C)*Cos[3*c]*Sin[3*d*x])/(56*d) + (C*Cos[4*c]*Sin[4*d*x])/(144*d)) - (A*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (2*B*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*Sqrt[1 + Cot[c]^2]) - (5*C*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (2*A*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d) - (3*B*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) - (4*C*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*d)","C",0
441,1,1270,179,6.4676867,"\int \frac{(a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^2 \left(-\frac{(5 A+4 B+3 C) \cot (c)}{5 d}+\frac{(28 A+56 B+51 C) \cos (d x) \sin (c)}{168 d}+\frac{(B+2 C) \cos (2 d x) \sin (2 c)}{20 d}+\frac{C \cos (3 d x) \sin (3 c)}{56 d}+\frac{(28 A+56 B+51 C) \cos (c) \sin (d x)}{168 d}+\frac{(B+2 C) \cos (2 c) \sin (2 d x)}{20 d}+\frac{C \cos (3 c) \sin (3 d x)}{56 d}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{A (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{2 B (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{3 C (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{2 A (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{B (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{2 C (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^2 (14 A+7 B+6 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (5 A+4 B+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (35 A+49 B+33 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{2 (7 B+4 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)}{35 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(-1/5*((5*A + 4*B + 3*C)*Cot[c])/d + ((28*A + 56*B + 51*C)*Cos[d*x]*Sin[c])/(168*d) + ((B + 2*C)*Cos[2*d*x]*Sin[2*c])/(20*d) + (C*Cos[3*d*x]*Sin[3*c])/(56*d) + ((28*A + 56*B + 51*C)*Cos[c]*Sin[d*x])/(168*d) + ((B + 2*C)*Cos[2*c]*Sin[2*d*x])/(20*d) + (C*Cos[3*c]*Sin[3*d*x])/(56*d)) - (2*A*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (B*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (2*C*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*Sqrt[1 + Cot[c]^2]) - (A*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) - (2*B*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d) - (3*C*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d)","C",0
442,1,1039,172,6.5778105,"\int \frac{(a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^2 \left(-\frac{(5 \cos (2 c) A-5 A+10 B+8 C+10 B \cos (2 c)+8 C \cos (2 c)) \csc (c) \sec (c)}{20 d}+\frac{A \sec (c+d x) \sin (d x) \sec (c)}{2 d}+\frac{(B+2 C) \cos (d x) \sin (c)}{6 d}+\frac{C \cos (2 d x) \sin (2 c)}{20 d}+\frac{(B+2 C) \cos (c) \sin (d x)}{6 d}+\frac{C \cos (2 c) \sin (2 d x)}{20 d}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{B (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{2 C (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{A (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{\cot ^2(c)+1}}-\frac{2 B (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^2 (3 A+2 B+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a^2 (15 A-5 B-7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}-\frac{2 (5 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)}{5 d}+\frac{4 a^2 (5 B+4 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{d \sqrt{\cos (c+d x)}}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(-1/20*((-5*A + 10*B + 8*C + 5*A*Cos[2*c] + 10*B*Cos[2*c] + 8*C*Cos[2*c])*Csc[c]*Sec[c])/d + ((B + 2*C)*Cos[d*x]*Sin[c])/(6*d) + (C*Cos[2*d*x]*Sin[2*c])/(20*d) + ((B + 2*C)*Cos[c]*Sin[d*x])/(6*d) + (A*Sec[c]*Sec[c + d*x]*Sin[d*x])/(2*d) + (C*Cos[2*c]*Sin[2*d*x])/(20*d)) - (A*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) - (2*B*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (C*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (B*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) - (2*C*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d)","C",0
443,1,1025,172,6.6332444,"\int \frac{(a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^2 \left(\frac{A \sec (c) \sin (d x) \sec ^2(c+d x)}{6 d}+\frac{\sec (c) (A \sin (c)+6 A \sin (d x)+3 B \sin (d x)) \sec (c+d x)}{6 d}-\frac{(-4 A-B+2 C+B \cos (2 c)+2 C \cos (2 c)) \csc (c) \sec (c)}{4 d}+\frac{C \cos (d x) \sin (c)}{6 d}+\frac{C \cos (c) \sin (d x)}{6 d}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{A (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{C (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{2 A (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{B (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{\cot ^2(c)+1}}-\frac{2 C (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^2 (2 A+3 B+2 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a^2 (5 A+3 B-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 (4 A+3 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{3 d \sqrt{\cos (c+d x)}}-\frac{4 a^2 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(-1/4*((-4*A - B + 2*C + B*Cos[2*c] + 2*C*Cos[2*c])*Csc[c]*Sec[c])/d + (C*Cos[d*x]*Sin[c])/(6*d) + (C*Cos[c]*Sin[d*x])/(6*d) + (A*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(6*d) + (Sec[c]*Sec[c + d*x]*(A*Sin[c] + 6*A*Sin[d*x] + 3*B*Sin[d*x]))/(6*d)) - (2*A*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (B*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) - (2*C*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) + (A*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) - (C*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d)","C",0
444,1,1041,174,6.73796,"\int \frac{(a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^2 \left(\frac{A \sec (c) \sin (d x) \sec ^3(c+d x)}{10 d}+\frac{\sec (c) (3 A \sin (c)+10 A \sin (d x)+5 B \sin (d x)) \sec ^2(c+d x)}{30 d}+\frac{\sec (c) (10 A \sin (c)+5 B \sin (c)+24 A \sin (d x)+30 B \sin (d x)+15 C \sin (d x)) \sec (c+d x)}{30 d}-\frac{(-16 A-20 B-5 C+5 C \cos (2 c)) \csc (c) \sec (c)}{20 d}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{2 A (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{B (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{A (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{2 B (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{\cot ^2(c)+1}}","\frac{4 a^2 (A+2 B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^2 (17 A+25 B+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}-\frac{4 a^2 (4 A+5 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (4 A+5 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(-1/20*((-16*A - 20*B - 5*C + 5*C*Cos[2*c])*Csc[c]*Sec[c])/d + (A*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(10*d) + (Sec[c]*Sec[c + d*x]^2*(3*A*Sin[c] + 10*A*Sin[d*x] + 5*B*Sin[d*x]))/(30*d) + (Sec[c]*Sec[c + d*x]*(10*A*Sin[c] + 5*B*Sin[c] + 24*A*Sin[d*x] + 30*B*Sin[d*x] + 15*C*Sin[d*x]))/(30*d)) - (A*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (2*B*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (C*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) + (2*A*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d) + (B*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d)","C",0
445,1,1310,215,6.8621351,"\int \frac{(a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^2 \left(\frac{A \sec (c) \sin (d x) \sec ^4(c+d x)}{14 d}+\frac{\sec (c) (5 A \sin (c)+14 A \sin (d x)+7 B \sin (d x)) \sec ^3(c+d x)}{70 d}+\frac{\sec (c) (42 A \sin (c)+21 B \sin (c)+60 A \sin (d x)+70 B \sin (d x)+35 C \sin (d x)) \sec ^2(c+d x)}{210 d}+\frac{\sec (c) (60 A \sin (c)+70 B \sin (c)+35 C \sin (c)+126 A \sin (d x)+168 B \sin (d x)+210 C \sin (d x)) \sec (c+d x)}{210 d}+\frac{(3 A+4 B+5 C) \csc (c) \sec (c)}{5 d}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{3 A (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}+\frac{2 B (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{C (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{2 A (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d \sqrt{\cot ^2(c)+1}}-\frac{B (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{2 C (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^2 (6 A+7 B+14 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (3 A+4 B+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (33 A+49 B+35 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (3 A+4 B+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 (4 A+7 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(((3*A + 4*B + 5*C)*Csc[c]*Sec[c])/(5*d) + (A*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(14*d) + (Sec[c]*Sec[c + d*x]^3*(5*A*Sin[c] + 14*A*Sin[d*x] + 7*B*Sin[d*x]))/(70*d) + (Sec[c]*Sec[c + d*x]^2*(42*A*Sin[c] + 21*B*Sin[c] + 60*A*Sin[d*x] + 70*B*Sin[d*x] + 35*C*Sin[d*x]))/(210*d) + (Sec[c]*Sec[c + d*x]*(60*A*Sin[c] + 70*B*Sin[c] + 35*C*Sin[c] + 126*A*Sin[d*x] + 168*B*Sin[d*x] + 210*C*Sin[d*x]))/(210*d)) - (2*A*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*Sqrt[1 + Cot[c]^2]) - (B*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (2*C*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) + (3*A*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) + (2*B*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d) + (C*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d)","C",0
446,1,1364,251,6.9941936,"\int \frac{(a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^2 \left(\frac{A \sec (c) \sin (d x) \sec ^5(c+d x)}{18 d}+\frac{\sec (c) (7 A \sin (c)+18 A \sin (d x)+9 B \sin (d x)) \sec ^4(c+d x)}{126 d}+\frac{\sec (c) (90 A \sin (c)+45 B \sin (c)+112 A \sin (d x)+126 B \sin (d x)+63 C \sin (d x)) \sec ^3(c+d x)}{630 d}+\frac{\sec (c) (112 A \sin (c)+126 B \sin (c)+63 C \sin (c)+150 A \sin (d x)+180 B \sin (d x)+210 C \sin (d x)) \sec ^2(c+d x)}{630 d}+\frac{\sec (c) (25 A \sin (c)+30 B \sin (c)+35 C \sin (c)+56 A \sin (d x)+63 B \sin (d x)+84 C \sin (d x)) \sec (c+d x)}{105 d}+\frac{(8 A+9 B+12 C) \csc (c) \sec (c)}{15 d}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{4 A (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{3 B (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}+\frac{2 C (\cos (c+d x) a+a)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{5 A (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d \sqrt{\cot ^2(c)+1}}-\frac{2 B (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x) a+a)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^2 (5 A+6 B+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (8 A+9 B+12 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^2 (5 A+6 B+7 C) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (19 A+27 B+21 C) \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2 (8 A+9 B+12 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (4 A+9 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(((8*A + 9*B + 12*C)*Csc[c]*Sec[c])/(15*d) + (A*Sec[c]*Sec[c + d*x]^5*Sin[d*x])/(18*d) + (Sec[c]*Sec[c + d*x]^4*(7*A*Sin[c] + 18*A*Sin[d*x] + 9*B*Sin[d*x]))/(126*d) + (Sec[c]*Sec[c + d*x]^3*(90*A*Sin[c] + 45*B*Sin[c] + 112*A*Sin[d*x] + 126*B*Sin[d*x] + 63*C*Sin[d*x]))/(630*d) + (Sec[c]*Sec[c + d*x]*(25*A*Sin[c] + 30*B*Sin[c] + 35*C*Sin[c] + 56*A*Sin[d*x] + 63*B*Sin[d*x] + 84*C*Sin[d*x]))/(105*d) + (Sec[c]*Sec[c + d*x]^2*(112*A*Sin[c] + 126*B*Sin[c] + 63*C*Sin[c] + 150*A*Sin[d*x] + 180*B*Sin[d*x] + 210*C*Sin[d*x]))/(630*d)) - (5*A*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (2*B*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*Sqrt[1 + Cot[c]^2]) - (C*(a + a*Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) + (4*A*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*d) + (3*B*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) + (2*C*(a + a*Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d)","C",0
447,1,1426,303,6.4806974,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^3 \left(-\frac{(221 A+195 B+175 C) \cot (c)}{390 d}+\frac{(2134 A+1953 B+1811 C) \cos (d x) \sin (c)}{7392 d}+\frac{(7592 A+7800 B+7825 C) \cos (2 d x) \sin (2 c)}{74880 d}+\frac{(132 A+189 B+215 C) \cos (3 d x) \sin (3 c)}{4928 d}+\frac{(13 A+39 B+59 C) \cos (4 d x) \sin (4 c)}{3744 d}+\frac{(B+3 C) \cos (5 d x) \sin (5 c)}{704 d}+\frac{C \cos (6 d x) \sin (6 c)}{1664 d}+\frac{(2134 A+1953 B+1811 C) \cos (c) \sin (d x)}{7392 d}+\frac{(7592 A+7800 B+7825 C) \cos (2 c) \sin (2 d x)}{74880 d}+\frac{(132 A+189 B+215 C) \cos (3 c) \sin (3 d x)}{4928 d}+\frac{(13 A+39 B+59 C) \cos (4 c) \sin (4 d x)}{3744 d}+\frac{(B+3 C) \cos (5 c) \sin (5 d x)}{704 d}+\frac{C \cos (6 c) \sin (6 d x)}{1664 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{17 A (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{60 d}-\frac{B (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}-\frac{35 C (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{156 d}-\frac{11 A (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{42 d \sqrt{\cot ^2(c)+1}}-\frac{5 B (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{22 d \sqrt{\cot ^2(c)+1}}-\frac{95 C (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{462 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (121 A+105 B+95 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (221 A+195 B+175 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{195 d}+\frac{20 a^3 (286 A+273 B+236 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{9009 d}+\frac{4 a^3 (221 A+195 B+175 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{585 d}+\frac{2 (143 A+195 B+145 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{1287 d}+\frac{4 a^3 (121 A+105 B+95 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 (13 B+6 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{143 a d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-1/390*((221*A + 195*B + 175*C)*Cot[c])/d + ((2134*A + 1953*B + 1811*C)*Cos[d*x]*Sin[c])/(7392*d) + ((7592*A + 7800*B + 7825*C)*Cos[2*d*x]*Sin[2*c])/(74880*d) + ((132*A + 189*B + 215*C)*Cos[3*d*x]*Sin[3*c])/(4928*d) + ((13*A + 39*B + 59*C)*Cos[4*d*x]*Sin[4*c])/(3744*d) + ((B + 3*C)*Cos[5*d*x]*Sin[5*c])/(704*d) + (C*Cos[6*d*x]*Sin[6*c])/(1664*d) + ((2134*A + 1953*B + 1811*C)*Cos[c]*Sin[d*x])/(7392*d) + ((7592*A + 7800*B + 7825*C)*Cos[2*c]*Sin[2*d*x])/(74880*d) + ((132*A + 189*B + 215*C)*Cos[3*c]*Sin[3*d*x])/(4928*d) + ((13*A + 39*B + 59*C)*Cos[4*c]*Sin[4*d*x])/(3744*d) + ((B + 3*C)*Cos[5*c]*Sin[5*d*x])/(704*d) + (C*Cos[6*c]*Sin[6*d*x])/(1664*d)) - (11*A*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (5*B*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(22*d*Sqrt[1 + Cot[c]^2]) - (95*C*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(462*d*Sqrt[1 + Cot[c]^2]) - (17*A*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(60*d) - (B*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(4*d) - (35*C*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(156*d)","C",0
448,1,1374,267,6.4326768,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^3 \left(-\frac{(21 A+17 B+15 C) \cot (c)}{30 d}+\frac{(2354 A+2134 B+1953 C) \cos (d x) \sin (c)}{7392 d}+\frac{(54 A+73 B+75 C) \cos (2 d x) \sin (2 c)}{720 d}+\frac{(44 A+132 B+189 C) \cos (3 d x) \sin (3 c)}{4928 d}+\frac{(B+3 C) \cos (4 d x) \sin (4 c)}{288 d}+\frac{C \cos (5 d x) \sin (5 c)}{704 d}+\frac{(2354 A+2134 B+1953 C) \cos (c) \sin (d x)}{7392 d}+\frac{(54 A+73 B+75 C) \cos (2 c) \sin (2 d x)}{720 d}+\frac{(44 A+132 B+189 C) \cos (3 c) \sin (3 d x)}{4928 d}+\frac{(B+3 C) \cos (4 c) \sin (4 d x)}{288 d}+\frac{C \cos (5 c) \sin (5 d x)}{704 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{7 A (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 d}-\frac{17 B (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{60 d}-\frac{C (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}-\frac{13 A (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{42 d \sqrt{\cot ^2(c)+1}}-\frac{11 B (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{42 d \sqrt{\cot ^2(c)+1}}-\frac{5 C (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{22 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (143 A+121 B+105 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (21 A+17 B+15 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^3 (264 A+253 B+210 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{1155 d}+\frac{2 (99 A+143 B+105 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{693 d}+\frac{4 a^3 (143 A+121 B+105 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 (11 B+6 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{99 a d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-1/30*((21*A + 17*B + 15*C)*Cot[c])/d + ((2354*A + 2134*B + 1953*C)*Cos[d*x]*Sin[c])/(7392*d) + ((54*A + 73*B + 75*C)*Cos[2*d*x]*Sin[2*c])/(720*d) + ((44*A + 132*B + 189*C)*Cos[3*d*x]*Sin[3*c])/(4928*d) + ((B + 3*C)*Cos[4*d*x]*Sin[4*c])/(288*d) + (C*Cos[5*d*x]*Sin[5*c])/(704*d) + ((2354*A + 2134*B + 1953*C)*Cos[c]*Sin[d*x])/(7392*d) + ((54*A + 73*B + 75*C)*Cos[2*c]*Sin[2*d*x])/(720*d) + ((44*A + 132*B + 189*C)*Cos[3*c]*Sin[3*d*x])/(4928*d) + ((B + 3*C)*Cos[4*c]*Sin[4*d*x])/(288*d) + (C*Cos[5*c]*Sin[5*d*x])/(704*d)) - (13*A*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (11*B*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (5*C*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(22*d*Sqrt[1 + Cot[c]^2]) - (7*A*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d) - (17*B*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(60*d) - (C*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(4*d)","C",0
449,1,1322,231,6.533483,"\int \frac{(a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^3 \left(-\frac{(27 A+21 B+17 C) \cot (c)}{30 d}+\frac{(84 A+107 B+97 C) \cos (d x) \sin (c)}{336 d}+\frac{(18 A+54 B+73 C) \cos (2 d x) \sin (2 c)}{720 d}+\frac{(B+3 C) \cos (3 d x) \sin (3 c)}{112 d}+\frac{C \cos (4 d x) \sin (4 c)}{288 d}+\frac{(84 A+107 B+97 C) \cos (c) \sin (d x)}{336 d}+\frac{(18 A+54 B+73 C) \cos (2 c) \sin (2 d x)}{720 d}+\frac{(B+3 C) \cos (3 c) \sin (3 d x)}{112 d}+\frac{C \cos (4 c) \sin (4 d x)}{288 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{9 A (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 d}-\frac{7 B (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 d}-\frac{17 C (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{60 d}-\frac{A (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d \sqrt{\cot ^2(c)+1}}-\frac{13 B (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{42 d \sqrt{\cot ^2(c)+1}}-\frac{11 C (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{42 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (21 A+13 B+11 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (27 A+21 B+17 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^3 (42 A+41 B+32 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{2 (63 A+99 B+73 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{315 d}+\frac{2 (3 B+2 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-1/30*((27*A + 21*B + 17*C)*Cot[c])/d + ((84*A + 107*B + 97*C)*Cos[d*x]*Sin[c])/(336*d) + ((18*A + 54*B + 73*C)*Cos[2*d*x]*Sin[2*c])/(720*d) + ((B + 3*C)*Cos[3*d*x]*Sin[3*c])/(112*d) + (C*Cos[4*d*x]*Sin[4*c])/(288*d) + ((84*A + 107*B + 97*C)*Cos[c]*Sin[d*x])/(336*d) + ((18*A + 54*B + 73*C)*Cos[2*c]*Sin[2*d*x])/(720*d) + ((B + 3*C)*Cos[3*c]*Sin[3*d*x])/(112*d) + (C*Cos[4*c]*Sin[4*d*x])/(288*d)) - (A*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(2*d*Sqrt[1 + Cot[c]^2]) - (13*B*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (11*C*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (9*A*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d) - (7*B*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d) - (17*C*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(60*d)","C",0
450,1,1313,229,6.7054925,"\int \frac{(a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^3 \left(-\frac{(15 \cos (2 c) A+5 A+18 B+14 C+18 B \cos (2 c)+14 C \cos (2 c)) \csc (c) \sec (c)}{40 d}+\frac{A \sec (c+d x) \sin (d x) \sec (c)}{4 d}+\frac{(28 A+84 B+107 C) \cos (d x) \sin (c)}{336 d}+\frac{(B+3 C) \cos (2 d x) \sin (2 c)}{40 d}+\frac{C \cos (3 d x) \sin (3 c)}{112 d}+\frac{(28 A+84 B+107 C) \cos (c) \sin (d x)}{336 d}+\frac{(B+3 C) \cos (2 c) \sin (2 d x)}{40 d}+\frac{C \cos (3 c) \sin (3 d x)}{112 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{A (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}-\frac{9 B (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 d}-\frac{7 C (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 d}-\frac{5 A (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{6 d \sqrt{\cot ^2(c)+1}}-\frac{B (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d \sqrt{\cot ^2(c)+1}}-\frac{13 C (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{42 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (35 A+21 B+13 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (5 A+9 B+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{4 a^3 (35 A-42 B-41 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}-\frac{2 (35 A-7 B-11 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{35 d}-\frac{2 (7 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)^2}{7 a d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt{\cos (c+d x)}}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-1/40*((5*A + 18*B + 14*C + 15*A*Cos[2*c] + 18*B*Cos[2*c] + 14*C*Cos[2*c])*Csc[c]*Sec[c])/d + ((28*A + 84*B + 107*C)*Cos[d*x]*Sin[c])/(336*d) + ((B + 3*C)*Cos[2*d*x]*Sin[2*c])/(40*d) + (C*Cos[3*d*x]*Sin[3*c])/(112*d) + ((28*A + 84*B + 107*C)*Cos[c]*Sin[d*x])/(336*d) + (A*Sec[c]*Sec[c + d*x]*Sin[d*x])/(4*d) + ((B + 3*C)*Cos[2*c]*Sin[2*d*x])/(40*d) + (C*Cos[3*c]*Sin[3*d*x])/(112*d)) - (5*A*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(6*d*Sqrt[1 + Cot[c]^2]) - (B*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(2*d*Sqrt[1 + Cot[c]^2]) - (13*C*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (A*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(4*d) - (9*B*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d) - (7*C*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d)","C",0
451,1,1297,227,6.8063967,"\int \frac{(a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^3 \left(\frac{A \sec (c) \sin (d x) \sec ^2(c+d x)}{12 d}+\frac{\sec (c) (A \sin (c)+9 A \sin (d x)+3 B \sin (d x)) \sec (c+d x)}{12 d}-\frac{(5 \cos (2 c) A-25 A+5 B+18 C+15 B \cos (2 c)+18 C \cos (2 c)) \csc (c) \sec (c)}{40 d}+\frac{(B+3 C) \cos (d x) \sin (c)}{12 d}+\frac{C \cos (2 d x) \sin (2 c)}{40 d}+\frac{(B+3 C) \cos (c) \sin (d x)}{12 d}+\frac{C \cos (2 c) \sin (2 d x)}{40 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{A (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}-\frac{B (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}-\frac{9 C (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 d}-\frac{5 A (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{6 d \sqrt{\cot ^2(c)+1}}-\frac{5 B (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{6 d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (5 A+5 B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (5 A-5 B-9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{4 a^3 (20 A+5 B-6 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}-\frac{2 (35 A+15 B-3 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{15 d}+\frac{2 (2 A+B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{a d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-1/40*((-25*A + 5*B + 18*C + 5*A*Cos[2*c] + 15*B*Cos[2*c] + 18*C*Cos[2*c])*Csc[c]*Sec[c])/d + ((B + 3*C)*Cos[d*x]*Sin[c])/(12*d) + (C*Cos[2*d*x]*Sin[2*c])/(40*d) + ((B + 3*C)*Cos[c]*Sin[d*x])/(12*d) + (A*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(12*d) + (Sec[c]*Sec[c + d*x]*(A*Sin[c] + 9*A*Sin[d*x] + 3*B*Sin[d*x]))/(12*d) + (C*Cos[2*c]*Sin[2*d*x])/(40*d)) - (5*A*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(6*d*Sqrt[1 + Cot[c]^2]) - (5*B*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(6*d*Sqrt[1 + Cot[c]^2]) - (C*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(2*d*Sqrt[1 + Cot[c]^2]) + (A*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(4*d) - (B*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(4*d) - (9*C*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d)","C",0
452,1,1298,230,6.8664074,"\int \frac{(a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^3 \left(\frac{A \sec (c) \sin (d x) \sec ^3(c+d x)}{20 d}+\frac{\sec (c) (3 A \sin (c)+15 A \sin (d x)+5 B \sin (d x)) \sec ^2(c+d x)}{60 d}+\frac{\sec (c) (15 A \sin (c)+5 B \sin (c)+54 A \sin (d x)+45 B \sin (d x)+15 C \sin (d x)) \sec (c+d x)}{60 d}-\frac{(-36 A-25 B+5 C+5 B \cos (2 c)+15 C \cos (2 c)) \csc (c) \sec (c)}{40 d}+\frac{C \cos (d x) \sin (c)}{12 d}+\frac{C \cos (c) \sin (d x)}{12 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{9 A (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 d}+\frac{B (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}-\frac{C (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}-\frac{A (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d \sqrt{\cot ^2(c)+1}}-\frac{5 B (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{6 d \sqrt{\cot ^2(c)+1}}-\frac{5 C (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{6 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (3 A+5 (B+C)) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (9 A+5 B-5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{4 a^3 (21 A+20 B+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 (33 A+35 B+15 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (6 A+5 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{15 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-1/40*((-36*A - 25*B + 5*C + 5*B*Cos[2*c] + 15*C*Cos[2*c])*Csc[c]*Sec[c])/d + (C*Cos[d*x]*Sin[c])/(12*d) + (C*Cos[c]*Sin[d*x])/(12*d) + (A*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(20*d) + (Sec[c]*Sec[c + d*x]^2*(3*A*Sin[c] + 15*A*Sin[d*x] + 5*B*Sin[d*x]))/(60*d) + (Sec[c]*Sec[c + d*x]*(15*A*Sin[c] + 5*B*Sin[c] + 54*A*Sin[d*x] + 45*B*Sin[d*x] + 15*C*Sin[d*x]))/(60*d)) - (A*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(2*d*Sqrt[1 + Cot[c]^2]) - (5*B*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(6*d*Sqrt[1 + Cot[c]^2]) - (5*C*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(6*d*Sqrt[1 + Cot[c]^2]) + (9*A*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d) + (B*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(4*d) - (C*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(4*d)","C",0
453,1,1317,231,6.97763,"\int \frac{(a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^3 \left(\frac{A \sec (c) \sin (d x) \sec ^4(c+d x)}{28 d}+\frac{\sec (c) (5 A \sin (c)+21 A \sin (d x)+7 B \sin (d x)) \sec ^3(c+d x)}{140 d}+\frac{\sec (c) (63 A \sin (c)+21 B \sin (c)+130 A \sin (d x)+105 B \sin (d x)+35 C \sin (d x)) \sec ^2(c+d x)}{420 d}+\frac{\sec (c) (130 A \sin (c)+105 B \sin (c)+35 C \sin (c)+294 A \sin (d x)+378 B \sin (d x)+315 C \sin (d x)) \sec (c+d x)}{420 d}-\frac{(-28 A-36 B-25 C+5 C \cos (2 c)) \csc (c) \sec (c)}{40 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{7 A (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 d}+\frac{9 B (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 d}+\frac{C (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}-\frac{13 A (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{42 d \sqrt{\cot ^2(c)+1}}-\frac{B (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d \sqrt{\cot ^2(c)+1}}-\frac{5 C (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{6 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (13 A+21 B+35 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (7 A+9 B+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (7 A+9 B+5 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (106 A+147 B+140 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)}}+\frac{2 (6 A+7 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{35 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-1/40*((-28*A - 36*B - 25*C + 5*C*Cos[2*c])*Csc[c]*Sec[c])/d + (A*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(28*d) + (Sec[c]*Sec[c + d*x]^3*(5*A*Sin[c] + 21*A*Sin[d*x] + 7*B*Sin[d*x]))/(140*d) + (Sec[c]*Sec[c + d*x]^2*(63*A*Sin[c] + 21*B*Sin[c] + 130*A*Sin[d*x] + 105*B*Sin[d*x] + 35*C*Sin[d*x]))/(420*d) + (Sec[c]*Sec[c + d*x]*(130*A*Sin[c] + 105*B*Sin[c] + 35*C*Sin[c] + 294*A*Sin[d*x] + 378*B*Sin[d*x] + 315*C*Sin[d*x]))/(420*d)) - (13*A*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (B*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(2*d*Sqrt[1 + Cot[c]^2]) - (5*C*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(6*d*Sqrt[1 + Cot[c]^2]) + (7*A*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d) + (9*B*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d) + (C*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(4*d)","C",0
454,1,1364,267,7.0284889,"\int \frac{(a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^3 \left(\frac{A \sec (c) \sin (d x) \sec ^5(c+d x)}{36 d}+\frac{\sec (c) (7 A \sin (c)+27 A \sin (d x)+9 B \sin (d x)) \sec ^4(c+d x)}{252 d}+\frac{\sec (c) (135 A \sin (c)+45 B \sin (c)+238 A \sin (d x)+189 B \sin (d x)+63 C \sin (d x)) \sec ^3(c+d x)}{1260 d}+\frac{\sec (c) (238 A \sin (c)+189 B \sin (c)+63 C \sin (c)+330 A \sin (d x)+390 B \sin (d x)+315 C \sin (d x)) \sec ^2(c+d x)}{1260 d}+\frac{\sec (c) (110 A \sin (c)+130 B \sin (c)+105 C \sin (c)+238 A \sin (d x)+294 B \sin (d x)+378 C \sin (d x)) \sec (c+d x)}{420 d}+\frac{(17 A+21 B+27 C) \csc (c) \sec (c)}{30 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{17 A (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{60 d}+\frac{7 B (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 d}+\frac{9 C (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 d}-\frac{11 A (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{42 d \sqrt{\cot ^2(c)+1}}-\frac{13 B (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{42 d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (11 A+13 B+21 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (17 A+21 B+27 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^3 (32 A+41 B+42 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (73 A+99 B+63 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (17 A+21 B+27 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (2 A+3 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(((17*A + 21*B + 27*C)*Csc[c]*Sec[c])/(30*d) + (A*Sec[c]*Sec[c + d*x]^5*Sin[d*x])/(36*d) + (Sec[c]*Sec[c + d*x]^4*(7*A*Sin[c] + 27*A*Sin[d*x] + 9*B*Sin[d*x]))/(252*d) + (Sec[c]*Sec[c + d*x]^3*(135*A*Sin[c] + 45*B*Sin[c] + 238*A*Sin[d*x] + 189*B*Sin[d*x] + 63*C*Sin[d*x]))/(1260*d) + (Sec[c]*Sec[c + d*x]^2*(238*A*Sin[c] + 189*B*Sin[c] + 63*C*Sin[c] + 330*A*Sin[d*x] + 390*B*Sin[d*x] + 315*C*Sin[d*x]))/(1260*d) + (Sec[c]*Sec[c + d*x]*(110*A*Sin[c] + 130*B*Sin[c] + 105*C*Sin[c] + 238*A*Sin[d*x] + 294*B*Sin[d*x] + 378*C*Sin[d*x]))/(420*d)) - (11*A*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (13*B*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (C*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(2*d*Sqrt[1 + Cot[c]^2]) + (17*A*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(60*d) + (7*B*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d) + (9*C*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d)","C",0
455,1,1418,303,7.1979695,"\int \frac{(a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^3 \left(\frac{A \sec (c) \sin (d x) \sec ^6(c+d x)}{44 d}+\frac{\sec (c) (9 A \sin (c)+33 A \sin (d x)+11 B \sin (d x)) \sec ^5(c+d x)}{396 d}+\frac{\sec (c) (231 A \sin (c)+77 B \sin (c)+378 A \sin (d x)+297 B \sin (d x)+99 C \sin (d x)) \sec ^4(c+d x)}{2772 d}+\frac{\sec (c) (1890 A \sin (c)+1485 B \sin (c)+495 C \sin (c)+2310 A \sin (d x)+2618 B \sin (d x)+2079 C \sin (d x)) \sec ^3(c+d x)}{13860 d}+\frac{\sec (c) (2310 A \sin (c)+2618 B \sin (c)+2079 C \sin (c)+3150 A \sin (d x)+3630 B \sin (d x)+4290 C \sin (d x)) \sec ^2(c+d x)}{13860 d}+\frac{\sec (c) (525 A \sin (c)+605 B \sin (c)+715 C \sin (c)+1155 A \sin (d x)+1309 B \sin (d x)+1617 C \sin (d x)) \sec (c+d x)}{2310 d}+\frac{(15 A+17 B+21 C) \csc (c) \sec (c)}{30 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{A (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}+\frac{17 B (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{60 d}+\frac{7 C (\cos (c+d x) a+a)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 d}-\frac{5 A (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{22 d \sqrt{\cot ^2(c)+1}}-\frac{11 B (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{42 d \sqrt{\cot ^2(c)+1}}-\frac{13 C (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{42 d \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (105 A+121 B+143 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}-\frac{4 a^3 (15 A+17 B+21 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^3 (105 A+121 B+143 C) \sin (c+d x)}{231 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (210 A+253 B+264 C) \sin (c+d x)}{1155 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (105 A+143 B+99 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{693 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (15 A+17 B+21 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (6 A+11 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{99 a d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac{11}{2}}(c+d x)}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(((15*A + 17*B + 21*C)*Csc[c]*Sec[c])/(30*d) + (A*Sec[c]*Sec[c + d*x]^6*Sin[d*x])/(44*d) + (Sec[c]*Sec[c + d*x]^5*(9*A*Sin[c] + 33*A*Sin[d*x] + 11*B*Sin[d*x]))/(396*d) + (Sec[c]*Sec[c + d*x]^4*(231*A*Sin[c] + 77*B*Sin[c] + 378*A*Sin[d*x] + 297*B*Sin[d*x] + 99*C*Sin[d*x]))/(2772*d) + (Sec[c]*Sec[c + d*x]*(525*A*Sin[c] + 605*B*Sin[c] + 715*C*Sin[c] + 1155*A*Sin[d*x] + 1309*B*Sin[d*x] + 1617*C*Sin[d*x]))/(2310*d) + (Sec[c]*Sec[c + d*x]^3*(1890*A*Sin[c] + 1485*B*Sin[c] + 495*C*Sin[c] + 2310*A*Sin[d*x] + 2618*B*Sin[d*x] + 2079*C*Sin[d*x]))/(13860*d) + (Sec[c]*Sec[c + d*x]^2*(2310*A*Sin[c] + 2618*B*Sin[c] + 2079*C*Sin[c] + 3150*A*Sin[d*x] + 3630*B*Sin[d*x] + 4290*C*Sin[d*x]))/(13860*d)) - (5*A*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(22*d*Sqrt[1 + Cot[c]^2]) - (11*B*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (13*C*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) + (A*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(4*d) + (17*B*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(60*d) + (7*C*(a + a*Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d)","C",0
456,1,1752,210,6.9622662,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","-\frac{3 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}+\frac{21 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 (\cos (c+d x) a+a)}-\frac{21 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 (\cos (c+d x) a+a)}+\frac{\sqrt{\cos (c+d x)} \left(\frac{2 (10 \cos (c) A+5 A-5 B+5 C-16 B \cos (c)+16 C \cos (c)) \csc (c)}{5 d}+\frac{(28 A-28 B+51 C) \cos (d x) \sin (c)}{21 d}+\frac{2 (B-C) \cos (2 d x) \sin (2 c)}{5 d}+\frac{C \cos (3 d x) \sin (3 c)}{7 d}+\frac{2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}+\frac{(28 A-28 B+51 C) \cos (c) \sin (d x)}{21 d}+\frac{2 (B-C) \cos (2 c) \sin (2 d x)}{5 d}+\frac{C \cos (3 c) \sin (3 d x)}{7 d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (c+d x) a+a}-\frac{5 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}+\frac{5 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}-\frac{15 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}","\frac{5 (7 A-7 B+9 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a d}-\frac{3 (5 A-7 B+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(7 A-7 B+9 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 a d}-\frac{(5 A-7 B+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}+\frac{5 (7 A-7 B+9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 a d}",1,"(((-3*I)/4)*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) + (((21*I)/20)*B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) - (((21*I)/20)*C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*((2*(5*A - 5*B + 5*C + 10*A*Cos[c] - 16*B*Cos[c] + 16*C*Cos[c])*Csc[c])/(5*d) + ((28*A - 28*B + 51*C)*Cos[d*x]*Sin[c])/(21*d) + (2*(B - C)*Cos[2*d*x]*Sin[2*c])/(5*d) + (C*Cos[3*d*x]*Sin[3*c])/(7*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + ((28*A - 28*B + 51*C)*Cos[c]*Sin[d*x])/(21*d) + (2*(B - C)*Cos[2*c]*Sin[2*d*x])/(5*d) + (C*Cos[3*c]*Sin[3*d*x])/(7*d)))/(a + a*Cos[c + d*x]) - (5*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) + (5*B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) - (15*C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2])","C",0
457,1,1697,174,6.8219087,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","\frac{3 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}-\frac{3 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}+\frac{21 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 (\cos (c+d x) a+a)}+\frac{\sqrt{\cos (c+d x)} \left(-\frac{2 (10 \cos (c) A+5 A-5 B+5 C-10 B \cos (c)+16 C \cos (c)) \csc (c)}{5 d}+\frac{4 (B-C) \cos (d x) \sin (c)}{3 d}+\frac{2 C \cos (2 d x) \sin (2 c)}{5 d}-\frac{2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}+\frac{4 (B-C) \cos (c) \sin (d x)}{3 d}+\frac{2 C \cos (2 c) \sin (2 d x)}{5 d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (c+d x) a+a}+\frac{A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}-\frac{5 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}+\frac{5 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}","-\frac{(3 A-5 B+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (5 A-5 B+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(5 A-5 B+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}-\frac{(3 A-5 B+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"(((3*I)/4)*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) - (((3*I)/4)*B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) + (((21*I)/20)*C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*((-2*(5*A - 5*B + 5*C + 10*A*Cos[c] - 10*B*Cos[c] + 16*C*Cos[c])*Csc[c])/(5*d) + (4*(B - C)*Cos[d*x]*Sin[c])/(3*d) + (2*C*Cos[2*d*x]*Sin[2*c])/(5*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (4*(B - C)*Cos[c]*Sin[d*x])/(3*d) + (2*C*Cos[2*c]*Sin[2*d*x])/(5*d)))/(a + a*Cos[c + d*x]) + (A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) - (5*B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) + (5*C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2])","C",0
458,1,1644,134,6.6643916,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+a \cos (c+d x)} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x]),x]","-\frac{i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}+\frac{3 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}-\frac{3 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}+\frac{\sqrt{\cos (c+d x)} \left(-\frac{2 (-A+B-C+2 B \cos (c)-2 C \cos (c)) \csc (c)}{d}+\frac{4 C \cos (d x) \sin (c)}{3 d}+\frac{2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}+\frac{4 C \cos (c) \sin (d x)}{3 d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (c+d x) a+a}-\frac{A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}+\frac{B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}-\frac{5 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}","\frac{(3 A-3 B+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{(A-3 B+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(3 A-3 B+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"((-1/4*I)*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) + (((3*I)/4)*B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) - (((3*I)/4)*C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*((-2*(-A + B - C + 2*B*Cos[c] - 2*C*Cos[c])*Csc[c])/d + (4*C*Cos[d*x]*Sin[c])/(3*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (4*C*Cos[c]*Sin[d*x])/(3*d)))/(a + a*Cos[c + d*x]) - (A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) + (B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) - (5*C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2])","C",1
459,1,1607,90,6.6497836,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])),x]","\frac{i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}-\frac{i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}+\frac{3 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}+\frac{\sqrt{\cos (c+d x)} \left(-\frac{2 (A-B+C+2 C \cos (c)) \csc (c)}{d}-\frac{2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (c+d x) a+a}-\frac{A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}-\frac{B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}+\frac{C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}","\frac{(A+B-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A-B+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \cos (c+d x)+a)}",1,"((I/4)*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) - ((I/4)*B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) + (((3*I)/4)*C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*((-2*(A - B + C + 2*C*Cos[c])*Csc[c])/d - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d))/(a + a*Cos[c + d*x]) - (A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) - (B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) + (C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2])","C",1
460,1,1642,125,6.8325854,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])),x]","-\frac{3 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}+\frac{i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}-\frac{i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}+\frac{\sqrt{\cos (c+d x)} \left(\frac{(\cos (c) A+2 A-B \cos (c)+C \cos (c)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c)}{d}+\frac{4 A \sec (c+d x) \sin (d x) \sec (c)}{d}+\frac{2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (c+d x) a+a}+\frac{A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}-\frac{B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}-\frac{C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}","-\frac{(A-B-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(3 A-B+C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(3 A-B+C) \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)}",1,"(((-3*I)/4)*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) + ((I/4)*B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) - ((I/4)*C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*(((2*A + A*Cos[c] - B*Cos[c] + C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/d + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (4*A*Sec[c]*Sec[c + d*x]*Sin[d*x])/d))/(a + a*Cos[c + d*x]) + (A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) - (B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) - (C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2])","C",1
461,1,1686,165,7.2400386,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])),x]","\frac{3 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}-\frac{3 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}+\frac{i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}+\frac{\sqrt{\cos (c+d x)} \left(\frac{4 A \sec (c) \sin (d x) \sec ^2(c+d x)}{3 d}+\frac{4 \sec (c) (A \sin (c)-3 A \sin (d x)+3 B \sin (d x)) \sec (c+d x)}{3 d}-\frac{(\cos (c) A+2 A-2 B-B \cos (c)+C \cos (c)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c)}{d}-\frac{2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (c+d x) a+a}-\frac{5 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}+\frac{B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}-\frac{C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}","\frac{(5 A-3 B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{(3 A-3 B+C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-B+C) \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{(5 A-3 B+3 C) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(3 A-3 B+C) \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}",1,"(((3*I)/4)*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) - (((3*I)/4)*B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) + ((I/4)*C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*(-(((2*A - 2*B + A*Cos[c] - B*Cos[c] + C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (4*A*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (4*Sec[c]*Sec[c + d*x]*(A*Sin[c] - 3*A*Sin[d*x] + 3*B*Sin[d*x]))/(3*d)))/(a + a*Cos[c + d*x]) - (5*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) + (B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) - (C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2])","C",0
462,1,1745,210,7.6399774,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \cos (c+d x))} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Cos[c + d*x])),x]","-\frac{21 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 (\cos (c+d x) a+a)}+\frac{3 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}-\frac{3 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (c+d x) a+a)}+\frac{\sqrt{\cos (c+d x)} \left(\frac{4 A \sec (c) \sin (d x) \sec ^3(c+d x)}{5 d}+\frac{4 \sec (c) (3 A \sin (c)-5 A \sin (d x)+5 B \sin (d x)) \sec ^2(c+d x)}{15 d}-\frac{4 \sec (c) (5 A \sin (c)-5 B \sin (c)-24 A \sin (d x)+15 B \sin (d x)-15 C \sin (d x)) \sec (c+d x)}{15 d}+\frac{(5 \cos (c) A+16 A-10 B+10 C-5 B \cos (c)+5 C \cos (c)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c)}{5 d}+\frac{2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (c+d x) a+a}+\frac{5 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}-\frac{5 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}+\frac{C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a) \sqrt{\cot ^2(c)+1}}","-\frac{(5 A-5 B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 (7 A-5 B+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{(A-B+C) \sin (c+d x)}{d \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)}-\frac{(5 A-5 B+3 C) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{(7 A-5 B+5 C) \sin (c+d x)}{5 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{3 (7 A-5 B+5 C) \sin (c+d x)}{5 a d \sqrt{\cos (c+d x)}}",1,"(((-21*I)/20)*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) + (((3*I)/4)*B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) - (((3*I)/4)*C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x]) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*(((16*A - 10*B + 10*C + 5*A*Cos[c] - 5*B*Cos[c] + 5*C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/(5*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (4*A*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(5*d) + (4*Sec[c]*Sec[c + d*x]^2*(3*A*Sin[c] - 5*A*Sin[d*x] + 5*B*Sin[d*x]))/(15*d) - (4*Sec[c]*Sec[c + d*x]*(5*A*Sin[c] - 5*B*Sin[c] - 24*A*Sin[d*x] + 15*B*Sin[d*x] - 15*C*Sin[d*x]))/(15*d)))/(a + a*Cos[c + d*x]) + (5*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) - (5*B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) + (C*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])*Sqrt[1 + Cot[c]^2])","C",0
463,1,1789,214,7.1324975,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","\frac{2 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^2}-\frac{7 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (c+d x) a+a)^2}+\frac{28 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 (\cos (c+d x) a+a)^2}+\frac{\sqrt{\cos (c+d x)} \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{2 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(2 A \sin \left(\frac{d x}{2}\right)-3 B \sin \left(\frac{d x}{2}\right)+4 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{4 (10 \cos (c) A+10 A-15 B+20 C-20 B \cos (c)+36 C \cos (c)) \csc (c)}{5 d}+\frac{8 (B-2 C) \cos (d x) \sin (c)}{3 d}+\frac{4 C \cos (2 d x) \sin (2 c)}{5 d}+\frac{8 (B-2 C) \cos (c) \sin (d x)}{3 d}+\frac{4 C \cos (2 c) \sin (2 d x)}{5 d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^2}+\frac{10 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}-\frac{20 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}+\frac{10 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}","-\frac{5 (A-2 B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(20 A-35 B+56 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{(A-2 B+3 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{(20 A-35 B+56 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a^2 d}-\frac{5 (A-2 B+3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((2*I)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^2 - (((7*I)/2)*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^2 + (((28*I)/5)*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^2 + (10*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) - (20*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) + (10*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*((-4*(10*A - 15*B + 20*C + 10*A*Cos[c] - 20*B*Cos[c] + 36*C*Cos[c])*Csc[c])/(5*d) + (8*(B - 2*C)*Cos[d*x]*Sin[c])/(3*d) + (4*C*Cos[2*d*x]*Sin[2*c])/(5*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(2*A*Sin[(d*x)/2] - 3*B*Sin[(d*x)/2] + 4*C*Sin[(d*x)/2]))/d + (8*(B - 2*C)*Cos[c]*Sin[d*x])/(3*d) + (4*C*Cos[2*c]*Sin[2*d*x])/(5*d) + (2*(A - B + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])^2","C",0
464,1,1741,180,6.9533007,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","-\frac{i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (c+d x) a+a)^2}+\frac{2 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^2}-\frac{7 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (c+d x) a+a)^2}+\frac{\sqrt{\cos (c+d x)} \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-2 B \sin \left(\frac{d x}{2}\right)+3 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{4 (-A+2 B-3 C+2 B \cos (c)-4 C \cos (c)) \csc (c)}{d}+\frac{8 C \cos (d x) \sin (c)}{3 d}+\frac{8 C \cos (c) \sin (d x)}{3 d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^2}-\frac{4 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}+\frac{10 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}-\frac{20 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}","\frac{(2 A-5 B+10 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(A-4 B+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-4 B+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{(2 A-5 B+10 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((-1/2*I)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^2 + ((2*I)*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^2 - (((7*I)/2)*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^2 - (4*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) + (10*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) - (20*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*((-4*(-A + 2*B - 3*C + 2*B*Cos[c] - 4*C*Cos[c])*Csc[c])/d + (8*C*Cos[d*x]*Sin[c])/(3*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - 2*B*Sin[(d*x)/2] + 3*C*Sin[(d*x)/2]))/d + (8*C*Cos[c]*Sin[d*x])/(3*d) - (2*(A - B + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])^2","C",0
465,1,1347,139,6.7787679,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^2} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^2,x]","-\frac{i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (c+d x) a+a)^2}+\frac{2 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^2}+\frac{\sqrt{\cos (c+d x)} \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{2 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-2 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{4 (-B+2 C+2 C \cos (c)) \csc (c)}{d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^2}-\frac{2 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}-\frac{4 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}+\frac{10 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}","\frac{(A+2 B-5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(A+2 B-5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d (\cos (c+d x)+1)}-\frac{(B-4 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((-1/2*I)*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^2 + ((2*I)*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^2 - (2*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) - (4*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) + (10*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*((-4*(-B + 2*C + 2*C*Cos[c])*Csc[c])/d + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(B*Sin[(d*x)/2] - 2*C*Sin[(d*x)/2]))/d + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) + (2*(A - B + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])^2","C",1
466,1,1342,133,6.7806514,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2),x]","\frac{i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (c+d x) a+a)^2}-\frac{i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (c+d x) a+a)^2}+\frac{\sqrt{\cos (c+d x)} \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{4 (A-C) \csc (c)}{d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^2}-\frac{4 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}-\frac{2 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}-\frac{4 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}","\frac{(2 A+B+2 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d (\cos (c+d x)+1)}-\frac{(A-B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \cos (c+d x)+a)^2}",1,"((I/2)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^2 - ((I/2)*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^2 - (4*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) - (2*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) - (4*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*((-4*(A - C)*Csc[c])/d - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - C*Sin[(d*x)/2]))/d - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) - (2*(A - B + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])^2","C",1
467,1,1380,175,6.941769,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2),x]","-\frac{2 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^2}+\frac{i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (c+d x) a+a)^2}+\frac{\sqrt{\cos (c+d x)} \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{2 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(2 A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{2 (2 \cos (c) A+2 A-B \cos (c)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c)}{d}+\frac{8 A \sec (c) \sec (c+d x) \sin (d x)}{d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^2}+\frac{10 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}-\frac{4 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}-\frac{2 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}","-\frac{(5 A-2 B-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(5 A-2 B-C) \sin (c+d x)}{3 a^2 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1)}-\frac{(4 A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{(4 A-B) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}",1,"((-2*I)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^2 + ((I/2)*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^2 + (10*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) - (4*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) - (2*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*((2*(2*A + 2*A*Cos[c] - B*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/d + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(2*A*Sin[(d*x)/2] - B*Sin[(d*x)/2]))/d + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) + (8*A*Sec[c]*Sec[c + d*x]*Sin[d*x])/d + (2*(A - B + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])^2","C",0
468,1,1782,211,7.5861032,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^2} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2),x]","\frac{7 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (c+d x) a+a)^2}-\frac{2 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^2}+\frac{i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (c+d x) a+a)^2}+\frac{\sqrt{\cos (c+d x)} \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(3 A \sin \left(\frac{d x}{2}\right)-2 B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{2 (3 \cos (c) A+4 A-2 B-2 B \cos (c)+C \cos (c)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c)}{d}+\frac{8 A \sec (c) \sec ^2(c+d x) \sin (d x)}{3 d}+\frac{8 \sec (c) \sec (c+d x) (A \sin (c)-6 A \sin (d x)+3 B \sin (d x))}{3 d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^2}-\frac{20 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}+\frac{10 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}-\frac{4 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}","\frac{(10 A-5 B+2 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(7 A-4 B+C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(7 A-4 B+C) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x) (\cos (c+d x)+1)}+\frac{(10 A-5 B+2 C) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(7 A-4 B+C) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"(((7*I)/2)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^2 - ((2*I)*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^2 + ((I/2)*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^2 - (20*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) + (10*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) - (4*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^2*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*((-2*(4*A - 2*B + 3*A*Cos[c] - 2*B*Cos[c] + C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/d - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(3*A*Sin[(d*x)/2] - 2*B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) + (8*A*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (8*Sec[c]*Sec[c + d*x]*(A*Sin[c] - 6*A*Sin[d*x] + 3*B*Sin[d*x]))/(3*d) - (2*(A - B + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])^2","C",0
469,1,1888,273,7.4986208,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","\frac{49 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}-\frac{119 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}+\frac{231 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}+\frac{\sqrt{\cos (c+d x)} \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{2 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(14 A \sin \left(\frac{d x}{2}\right)-19 B \sin \left(\frac{d x}{2}\right)+24 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 (14 A-19 B+24 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(29 A \sin \left(\frac{d x}{2}\right)-59 B \sin \left(\frac{d x}{2}\right)+99 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (20 \cos (c) A+29 A-59 B+99 C-60 B \cos (c)+132 C \cos (c)) \csc (c)}{5 d}+\frac{16 (B-3 C) \cos (d x) \sin (c)}{3 d}+\frac{8 C \cos (2 d x) \sin (2 c)}{5 d}+\frac{16 (B-3 C) \cos (c) \sin (d x)}{3 d}+\frac{8 C \cos (2 c) \sin (2 d x)}{5 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^3}+\frac{26 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}-\frac{22 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}+\frac{42 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}","-\frac{(13 A-33 B+63 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{7 (7 A-17 B+33 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(13 A-33 B+63 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{7 (7 A-17 B+33 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 a^3 d}-\frac{(13 A-33 B+63 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 a^3 d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(2 A-7 B+12 C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(((49*I)/10)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 - (((119*I)/10)*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 + (((231*I)/10)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 + (26*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) - (22*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) + (42*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*((-4*(29*A - 59*B + 99*C + 20*A*Cos[c] - 60*B*Cos[c] + 132*C*Cos[c])*Csc[c])/(5*d) + (16*(B - 3*C)*Cos[d*x]*Sin[c])/(3*d) + (8*C*Cos[2*d*x]*Sin[2*c])/(5*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(14*A*Sin[(d*x)/2] - 19*B*Sin[(d*x)/2] + 24*C*Sin[(d*x)/2]))/(15*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(29*A*Sin[(d*x)/2] - 59*B*Sin[(d*x)/2] + 99*C*Sin[(d*x)/2]))/(5*d) + (16*(B - 3*C)*Cos[c]*Sin[d*x])/(3*d) + (8*C*Cos[2*c]*Sin[2*d*x])/(5*d) + (4*(14*A - 19*B + 24*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (2*(A - B + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",0
470,1,1841,232,7.2208754,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","-\frac{9 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}+\frac{49 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}-\frac{119 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}+\frac{\sqrt{\cos (c+d x)} \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{2 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(9 A \sin \left(\frac{d x}{2}\right)-14 B \sin \left(\frac{d x}{2}\right)+19 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 (9 A-14 B+19 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(9 A \sin \left(\frac{d x}{2}\right)-29 B \sin \left(\frac{d x}{2}\right)+59 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (-9 A+29 B-59 C+20 B \cos (c)-60 C \cos (c)) \csc (c)}{5 d}+\frac{16 C \cos (d x) \sin (c)}{3 d}+\frac{16 C \cos (c) \sin (d x)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^3}-\frac{2 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}+\frac{26 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}-\frac{22 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}","\frac{(3 A-13 B+33 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(9 A-49 B+119 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(9 A-49 B+119 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(3 A-13 B+33 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 a^3 d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(B-2 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 a d (a \cos (c+d x)+a)^2}",1,"(((-9*I)/10)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 + (((49*I)/10)*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 - (((119*I)/10)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 - (2*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) + (26*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) - (22*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*((-4*(-9*A + 29*B - 59*C + 20*B*Cos[c] - 60*C*Cos[c])*Csc[c])/(5*d) + (16*C*Cos[d*x]*Sin[c])/(3*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(9*A*Sin[(d*x)/2] - 14*B*Sin[(d*x)/2] + 19*C*Sin[(d*x)/2]))/(15*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(9*A*Sin[(d*x)/2] - 29*B*Sin[(d*x)/2] + 59*C*Sin[(d*x)/2]))/(5*d) + (16*C*Cos[c]*Sin[d*x])/(3*d) - (4*(9*A - 14*B + 19*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (2*(A - B + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",0
471,1,1809,195,7.1561874,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","-\frac{i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}-\frac{9 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}+\frac{49 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}+\frac{\sqrt{\cos (c+d x)} \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{2 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(4 A \sin \left(\frac{d x}{2}\right)-9 B \sin \left(\frac{d x}{2}\right)+14 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 (4 A-9 B+14 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+9 B \sin \left(\frac{d x}{2}\right)-29 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (-A-9 B+29 C+20 C \cos (c)) \csc (c)}{5 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^3}-\frac{2 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}-\frac{2 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}+\frac{26 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}","\frac{(A+3 B-13 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(A+9 B-49 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(A+3 B-13 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(2 A+3 B-8 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"((-1/10*I)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 - (((9*I)/10)*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 + (((49*I)/10)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 - (2*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) - (2*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) + (26*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*((-4*(-A - 9*B + 29*C + 20*C*Cos[c])*Csc[c])/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + 9*B*Sin[(d*x)/2] - 29*C*Sin[(d*x)/2]))/(5*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(4*A*Sin[(d*x)/2] - 9*B*Sin[(d*x)/2] + 14*C*Sin[(d*x)/2]))/(15*d) + (4*(4*A - 9*B + 14*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (2*(A - B + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",0
472,1,1799,191,6.9777279,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^3,x]","\frac{i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}-\frac{i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}-\frac{9 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}+\frac{\sqrt{\cos (c+d x)} \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{2 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+4 B \sin \left(\frac{d x}{2}\right)-9 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 (A+4 B-9 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)-9 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (A-B-9 C) \csc (c)}{5 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^3}-\frac{2 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}-\frac{2 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}-\frac{2 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}","\frac{(A+B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(A-B-9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B-9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(4 A+B-6 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}",1,"((I/10)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 - ((I/10)*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 - (((9*I)/10)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 - (2*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) - (2*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) - (2*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*((-4*(A - B - 9*C)*Csc[c])/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] - 9*C*Sin[(d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + 4*B*Sin[(d*x)/2] - 9*C*Sin[(d*x)/2]))/(15*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (4*(A + 4*B - 9*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (2*(A - B + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",0
473,1,1802,193,6.9429736,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3),x]","\frac{9 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}+\frac{i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}-\frac{i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}+\frac{\sqrt{\cos (c+d x)} \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{2 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(6 A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)-4 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 (6 A-B-4 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(9 A \sin \left(\frac{d x}{2}\right)+B \sin \left(\frac{d x}{2}\right)-C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (9 A+B-C) \csc (c)}{5 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^3}-\frac{2 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}-\frac{2 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}-\frac{2 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}","\frac{(3 A+B+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(9 A+B-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(9 A+B-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(6 A-B-4 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}",1,"(((9*I)/10)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 + ((I/10)*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 - ((I/10)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 - (2*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) - (2*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) - (2*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*((-4*(9*A + B - C)*Csc[c])/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(6*A*Sin[(d*x)/2] - B*Sin[(d*x)/2] - 4*C*Sin[(d*x)/2]))/(15*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(9*A*Sin[(d*x)/2] + B*Sin[(d*x)/2] - C*Sin[(d*x)/2]))/(5*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) - (4*(6*A - B - 4*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (2*(A - B + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",0
474,1,1841,237,7.2897396,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^3} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3),x]","-\frac{49 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}+\frac{9 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}+\frac{i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}+\frac{\sqrt{\cos (c+d x)} \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{2 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(11 A \sin \left(\frac{d x}{2}\right)-6 B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 (11 A-6 B+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(29 A \sin \left(\frac{d x}{2}\right)-9 B \sin \left(\frac{d x}{2}\right)-C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{2 (29 \cos (c) A+20 A-9 B \cos (c)-C \cos (c)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c)}{5 d}+\frac{16 A \sec (c) \sec (c+d x) \sin (d x)}{d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^3}+\frac{26 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}-\frac{2 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}-\frac{2 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}","-\frac{(13 A-3 B-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(49 A-9 B-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(49 A-9 B-C) \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{(13 A-3 B-C) \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(8 A-3 B-2 C) \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}",1,"(((-49*I)/10)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 + (((9*I)/10)*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 + ((I/10)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 + (26*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) - (2*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) - (2*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*((2*(20*A + 29*A*Cos[c] - 9*B*Cos[c] - C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(29*A*Sin[(d*x)/2] - 9*B*Sin[(d*x)/2] - C*Sin[(d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(11*A*Sin[(d*x)/2] - 6*B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(15*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (16*A*Sec[c]*Sec[c + d*x]*Sin[d*x])/d + (4*(11*A - 6*B + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (2*(A - B + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",0
475,1,1883,270,8.0271562,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^3} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^3),x]","\frac{119 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}-\frac{49 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}+\frac{9 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (c+d x) a+a)^3}+\frac{\sqrt{\cos (c+d x)} \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{2 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(16 A \sin \left(\frac{d x}{2}\right)-11 B \sin \left(\frac{d x}{2}\right)+6 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 (16 A-11 B+6 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(59 A \sin \left(\frac{d x}{2}\right)-29 B \sin \left(\frac{d x}{2}\right)+9 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{2 (59 \cos (c) A+60 A-20 B-29 B \cos (c)+9 C \cos (c)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c)}{5 d}+\frac{16 A \sec (c) \sec ^2(c+d x) \sin (d x)}{3 d}+\frac{16 \sec (c) \sec (c+d x) (A \sin (c)-9 A \sin (d x)+3 B \sin (d x))}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^3}-\frac{22 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}+\frac{26 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}-\frac{2 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^3 \sqrt{\cot ^2(c)+1}}","\frac{(33 A-13 B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(119 A-49 B+9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(119 A-49 B+9 C) \sin (c+d x)}{30 d \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(33 A-13 B+3 C) \sin (c+d x)}{6 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(119 A-49 B+9 C) \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}-\frac{(2 A-B) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"(((119*I)/10)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 - (((49*I)/10)*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 + (((9*I)/10)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Cos[c + d*x])^3 - (22*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) + (26*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) - (2*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(a + a*Cos[c + d*x])^3*Sqrt[1 + Cot[c]^2]) + (Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*((-2*(60*A - 20*B + 59*A*Cos[c] - 29*B*Cos[c] + 9*C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/(5*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(16*A*Sin[(d*x)/2] - 11*B*Sin[(d*x)/2] + 6*C*Sin[(d*x)/2]))/(15*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(59*A*Sin[(d*x)/2] - 29*B*Sin[(d*x)/2] + 9*C*Sin[(d*x)/2]))/(5*d) + (16*A*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (16*Sec[c]*Sec[c + d*x]*(A*Sin[c] - 9*A*Sin[d*x] + 3*B*Sin[d*x]))/(3*d) - (4*(16*A - 11*B + 6*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (2*(A - B + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",0
476,1,144,227,0.9124464,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} (48 A+40 B+35 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} (2 (48 A+40 B+53 C) \cos (c+d x)+144 A+4 (8 B+7 C) \cos (2 (c+d x))+152 B+12 C \cos (3 (c+d x))+133 C)\right)}{384 d}","\frac{a (48 A+40 B+35 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (48 A+40 B+35 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a (48 A+40 B+35 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a (8 B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(3*Sqrt[2]*(48*A + 40*B + 35*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(144*A + 152*B + 133*C + 2*(48*A + 40*B + 53*C)*Cos[c + d*x] + 4*(8*B + 7*C)*Cos[2*(c + d*x)] + 12*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(384*d)","A",1
477,1,124,179,0.50756,"\int \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} (8 A+6 B+5 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} (24 A+2 (6 B+5 C) \cos (c+d x)+18 B+4 C \cos (2 (c+d x))+19 C)\right)}{48 d}","\frac{\sqrt{a} (8 A+6 B+5 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a (8 A+6 B+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{a (6 B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(3*Sqrt[2]*(8*A + 6*B + 5*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(24*A + 18*B + 19*C + 2*(6*B + 5*C)*Cos[c + d*x] + 4*C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
478,1,103,131,0.3637637,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(\sqrt{2} (8 A+4 B+3 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} (4 B+2 C \cos (c+d x)+3 C)\right)}{8 d}","\frac{\sqrt{a} (8 A+4 B+3 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a (4 B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(Sqrt[2]*(8*A + 4*B + 3*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(4*B + 3*C + 2*C*Cos[c + d*x])*Sin[(c + d*x)/2]))/(8*d)","A",1
479,1,104,121,0.3106391,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) (2 A+C \cos (c+d x))+\sqrt{2} (2 B+C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}\right)}{2 d \sqrt{\cos (c+d x)}}","-\frac{a (2 A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a} (2 B+C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(Sqrt[2]*(2*B + C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*(2*A + C*Cos[c + d*x])*Sin[(c + d*x)/2]))/(2*d*Sqrt[Cos[c + d*x]])","A",1
480,1,105,120,0.4014695,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) ((2 A+3 B) \cos (c+d x)+A)+3 \sqrt{2} C \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{3}{2}}(c+d x)\right)}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 a (A+3 B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sqrt{a} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(3*Sqrt[2]*C*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(3/2) + 2*(A + (2*A + 3*B)*Cos[c + d*x])*Sin[(c + d*x)/2]))/(3*d*Cos[c + d*x]^(3/2))","A",1
481,1,85,130,0.3723683,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((8 A+10 B+15 C) \cos (2 (c+d x))+2 (4 A+5 B) \cos (c+d x)+14 A+10 B+15 C)}{15 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 a (8 A+10 B+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+5 B) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(14*A + 10*B + 15*C + 2*(4*A + 5*B)*Cos[c + d*x] + (8*A + 10*B + 15*C)*Cos[2*(c + d*x)])*Tan[(c + d*x)/2])/(15*d*Cos[c + d*x]^(5/2))","A",1
482,1,121,178,0.5990131,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (3 (36 A+42 B+35 C) \cos (c+d x)+(24 A+28 B+35 C) \cos (2 (c+d x))+24 A \cos (3 (c+d x))+54 A+28 B \cos (3 (c+d x))+28 B+35 C \cos (3 (c+d x))+35 C)}{105 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 a (24 A+28 B+35 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a (24 A+28 B+35 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+7 B) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(54*A + 28*B + 35*C + 3*(36*A + 42*B + 35*C)*Cos[c + d*x] + (24*A + 28*B + 35*C)*Cos[2*(c + d*x)] + 24*A*Cos[3*(c + d*x)] + 28*B*Cos[3*(c + d*x)] + 35*C*Cos[3*(c + d*x)])*Tan[(c + d*x)/2])/(105*d*Cos[c + d*x]^(7/2))","A",1
483,1,155,226,0.8755794,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (2 (88 A+99 B+63 C) \cos (c+d x)+11 (16 A+18 B+21 C) \cos (2 (c+d x))+32 A \cos (3 (c+d x))+32 A \cos (4 (c+d x))+214 A+36 B \cos (3 (c+d x))+36 B \cos (4 (c+d x))+162 B+42 C \cos (3 (c+d x))+42 C \cos (4 (c+d x))+189 C)}{315 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{8 a (16 A+18 B+21 C) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a (16 A+18 B+21 C) \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a (16 A+18 B+21 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+9 B) \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(214*A + 162*B + 189*C + 2*(88*A + 99*B + 63*C)*Cos[c + d*x] + 11*(16*A + 18*B + 21*C)*Cos[2*(c + d*x)] + 32*A*Cos[3*(c + d*x)] + 36*B*Cos[3*(c + d*x)] + 42*C*Cos[3*(c + d*x)] + 32*A*Cos[4*(c + d*x)] + 36*B*Cos[4*(c + d*x)] + 42*C*Cos[4*(c + d*x)])*Tan[(c + d*x)/2])/(315*d*Cos[c + d*x]^(9/2))","A",1
484,1,170,283,1.6140058,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(15 \sqrt{2} (176 A+150 B+133 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} (2 (880 A+930 B+1007 C) \cos (c+d x)+4 (80 A+150 B+181 C) \cos (2 (c+d x))+2960 A+120 B \cos (3 (c+d x))+2850 B+228 C \cos (3 (c+d x))+48 C \cos (4 (c+d x))+2671 C)\right)}{3840 d}","\frac{a^{3/2} (176 A+150 B+133 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^2 (80 A+90 B+67 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{240 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a (10 B+3 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{40 d}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(15*Sqrt[2]*(176*A + 150*B + 133*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(2960*A + 2850*B + 2671*C + 2*(880*A + 930*B + 1007*C)*Cos[c + d*x] + 4*(80*A + 150*B + 181*C)*Cos[2*(c + d*x)] + 120*B*Cos[3*(c + d*x)] + 228*C*Cos[3*(c + d*x)] + 48*C*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]))/(3840*d)","A",1
485,1,145,233,0.9608985,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} (112 A+88 B+75 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} (2 (48 A+88 B+93 C) \cos (c+d x)+336 A+4 (8 B+15 C) \cos (2 (c+d x))+296 B+12 C \cos (3 (c+d x))+285 C)\right)}{384 d}","\frac{a^{3/2} (112 A+88 B+75 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (48 A+56 B+39 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (112 A+88 B+75 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a (8 B+3 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{24 d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(3*Sqrt[2]*(112*A + 88*B + 75*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(336*A + 296*B + 285*C + 2*(48*A + 88*B + 93*C)*Cos[c + d*x] + 4*(8*B + 15*C)*Cos[2*(c + d*x)] + 12*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(384*d)","A",1
486,1,125,181,0.7481393,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} (24 A+14 B+11 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} (24 A+2 (6 B+11 C) \cos (c+d x)+42 B+4 C \cos (2 (c+d x))+37 C)\right)}{48 d}","\frac{a^{3/2} (24 A+14 B+11 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 (24 A+30 B+19 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a (2 B+C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}{3 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(3*Sqrt[2]*(24*A + 14*B + 11*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(24*A + 42*B + 37*C + 2*(6*B + 11*C)*Cos[c + d*x] + 4*C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
487,1,127,181,0.609851,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(\sqrt{2} (8 A+12 B+7 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+2 \sin \left(\frac{1}{2} (c+d x)\right) (8 A+(4 B+7 C) \cos (c+d x)+C \cos (2 (c+d x))+C)\right)}{8 d \sqrt{\cos (c+d x)}}","\frac{a^{3/2} (8 A+12 B+7 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^2 (8 A-4 B-5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}-\frac{a (4 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{d \sqrt{\cos (c+d x)}}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(Sqrt[2]*(8*A + 12*B + 7*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*(8*A + C + (4*B + 7*C)*Cos[c + d*x] + C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(8*d*Sqrt[Cos[c + d*x]])","A",1
488,1,128,171,0.7501943,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (4 (5 A+3 B) \cos (c+d x)+4 A+3 C \cos (2 (c+d x))+3 C)+3 \sqrt{2} (2 B+3 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{3}{2}}(c+d x)\right)}{6 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{a^{3/2} (2 B+3 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a^2 (8 A+6 B-3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(3*Sqrt[2]*(2*B + 3*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(3/2) + (4*A + 3*C + 4*(5*A + 3*B)*Cos[c + d*x] + 3*C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(6*d*Cos[c + d*x]^(3/2))","A",1
489,1,134,172,0.8627448,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) ((18 A+25 B+15 C) \cos (2 (c+d x))+2 (9 A+5 B) \cos (c+d x)+24 A+25 B+15 C)+30 \sqrt{2} C \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{5}{2}}(c+d x)\right)}{30 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 a^{3/2} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a^2 (12 A+20 B+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (3 A+5 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(30*Sqrt[2]*C*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(5/2) + 2*(24*A + 25*B + 15*C + 2*(9*A + 5*B)*Cos[c + d*x] + (18*A + 25*B + 15*C)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(30*d*Cos[c + d*x]^(5/2))","A",1
490,1,122,184,0.7193168,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((468 A+462 B+525 C) \cos (c+d x)+2 (52 A+63 B+35 C) \cos (2 (c+d x))+104 A \cos (3 (c+d x))+164 A+126 B \cos (3 (c+d x))+126 B+175 C \cos (3 (c+d x))+70 C)}{210 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 a^2 (4 A+6 B+5 C) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (104 A+126 B+175 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (3 A+7 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(164*A + 126*B + 70*C + (468*A + 462*B + 525*C)*Cos[c + d*x] + 2*(52*A + 63*B + 35*C)*Cos[2*(c + d*x)] + 104*A*Cos[3*(c + d*x)] + 126*B*Cos[3*(c + d*x)] + 175*C*Cos[3*(c + d*x)])*Tan[(c + d*x)/2])/(210*d*Cos[c + d*x]^(7/2))","A",1
491,1,157,232,0.9793767,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((748 A+81 (8 B+7 C)) \cos (c+d x)+(748 A+858 B+882 C) \cos (2 (c+d x))+136 A \cos (3 (c+d x))+136 A \cos (4 (c+d x))+752 A+156 B \cos (3 (c+d x))+156 B \cos (4 (c+d x))+702 B+189 C \cos (3 (c+d x))+189 C \cos (4 (c+d x))+693 C)}{630 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{2 a^2 (136 A+156 B+189 C) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (52 A+72 B+63 C) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (136 A+156 B+189 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+3 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(752*A + 702*B + 693*C + (748*A + 81*(8*B + 7*C))*Cos[c + d*x] + (748*A + 858*B + 882*C)*Cos[2*(c + d*x)] + 136*A*Cos[3*(c + d*x)] + 156*B*Cos[3*(c + d*x)] + 189*C*Cos[3*(c + d*x)] + 136*A*Cos[4*(c + d*x)] + 156*B*Cos[4*(c + d*x)] + 189*C*Cos[4*(c + d*x)])*Tan[(c + d*x)/2])/(630*d*Cos[c + d*x]^(9/2))","A",1
492,1,187,284,1.0610819,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((12684 A+12386 B+12441 C) \cos (c+d x)+(4368 A+4862 B+4422 C) \cos (2 (c+d x))+4368 A \cos (3 (c+d x))+672 A \cos (4 (c+d x))+672 A \cos (5 (c+d x))+4956 A+4862 B \cos (3 (c+d x))+748 B \cos (4 (c+d x))+748 B \cos (5 (c+d x))+4114 B+5577 C \cos (3 (c+d x))+858 C \cos (4 (c+d x))+858 C \cos (5 (c+d x))+3564 C)}{6930 d \cos ^{\frac{11}{2}}(c+d x)}","\frac{8 a^2 (336 A+374 B+429 C) \sin (c+d x)}{3465 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (336 A+374 B+429 C) \sin (c+d x)}{1155 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (84 A+110 B+99 C) \sin (c+d x)}{693 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (336 A+374 B+429 C) \sin (c+d x)}{3465 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (3 A+11 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac{11}{2}}(c+d x)}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(4956*A + 4114*B + 3564*C + (12684*A + 12386*B + 12441*C)*Cos[c + d*x] + (4368*A + 4862*B + 4422*C)*Cos[2*(c + d*x)] + 4368*A*Cos[3*(c + d*x)] + 4862*B*Cos[3*(c + d*x)] + 5577*C*Cos[3*(c + d*x)] + 672*A*Cos[4*(c + d*x)] + 748*B*Cos[4*(c + d*x)] + 858*C*Cos[4*(c + d*x)] + 672*A*Cos[5*(c + d*x)] + 748*B*Cos[5*(c + d*x)] + 858*C*Cos[5*(c + d*x)])*Tan[(c + d*x)/2])/(6930*d*Cos[c + d*x]^(11/2))","A",1
493,1,205,333,2.2044293,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(15 \sqrt{2} (1304 A+1132 B+1015 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} (2 (7240 A+7748 B+8085 C) \cos (c+d x)+4 (920 A+1324 B+1575 C) \cos (2 (c+d x))+480 A \cos (3 (c+d x))+23240 A+1392 B \cos (3 (c+d x))+192 B \cos (4 (c+d x))+22084 B+2140 C \cos (3 (c+d x))+560 C \cos (4 (c+d x))+80 C \cos (5 (c+d x))+20965 C)\right)}{15360 d}","\frac{a^{5/2} (1304 A+1132 B+1015 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{512 d}+\frac{a^3 (680 A+628 B+545 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{960 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (1304 A+1132 B+1015 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{768 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (1304 A+1132 B+1015 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{512 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (120 A+156 B+115 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{480 d}+\frac{a (12 B+5 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{60 d}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(15*Sqrt[2]*(1304*A + 1132*B + 1015*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(23240*A + 22084*B + 20965*C + 2*(7240*A + 7748*B + 8085*C)*Cos[c + d*x] + 4*(920*A + 1324*B + 1575*C)*Cos[2*(c + d*x)] + 480*A*Cos[3*(c + d*x)] + 1392*B*Cos[3*(c + d*x)] + 2140*C*Cos[3*(c + d*x)] + 192*B*Cos[4*(c + d*x)] + 560*C*Cos[4*(c + d*x)] + 80*C*Cos[5*(c + d*x)])*Sin[(c + d*x)/2]))/(15360*d)","A",1
494,1,171,281,1.7144455,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(15 \sqrt{2} (400 A+326 B+283 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} ((2720 A+3620 B+3874 C) \cos (c+d x)+4 (80 A+230 B+331 C) \cos (2 (c+d x))+6320 A+120 B \cos (3 (c+d x))+5810 B+348 C \cos (3 (c+d x))+48 C \cos (4 (c+d x))+5521 C)\right)}{3840 d}","\frac{a^{5/2} (400 A+326 B+283 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^3 (1040 A+950 B+787 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{960 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (400 A+326 B+283 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (80 A+110 B+79 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{240 d}+\frac{a (2 B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{8 d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(15*Sqrt[2]*(400*A + 326*B + 283*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(6320*A + 5810*B + 5521*C + (2720*A + 3620*B + 3874*C)*Cos[c + d*x] + 4*(80*A + 230*B + 331*C)*Cos[2*(c + d*x)] + 120*B*Cos[3*(c + d*x)] + 348*C*Cos[3*(c + d*x)] + 48*C*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]))/(3840*d)","A",1
495,1,146,233,1.075205,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} (304 A+200 B+163 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} ((96 A+272 B+362 C) \cos (c+d x)+528 A+4 (8 B+23 C) \cos (2 (c+d x))+632 B+12 C \cos (3 (c+d x))+581 C)\right)}{384 d}","\frac{a^{5/2} (304 A+200 B+163 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^3 (432 A+392 B+299 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (16 A+24 B+17 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{32 d}+\frac{a (8 B+5 C) \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}{24 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}{4 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(3*Sqrt[2]*(304*A + 200*B + 163*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(528*A + 632*B + 581*C + (96*A + 272*B + 362*C)*Cos[c + d*x] + 4*(8*B + 23*C)*Cos[2*(c + d*x)] + 12*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(384*d)","A",1
496,1,156,231,1.0737368,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} (40 A+38 B+25 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+2 \sin \left(\frac{1}{2} (c+d x)\right) (3 (8 A+22 B+27 C) \cos (c+d x)+48 A+(6 B+17 C) \cos (2 (c+d x))+6 B+2 C \cos (3 (c+d x))+17 C)\right)}{48 d \sqrt{\cos (c+d x)}}","\frac{a^{5/2} (40 A+38 B+25 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}-\frac{a^3 (24 A-54 B-49 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{24 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A-2 B-3 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{4 d}-\frac{a (6 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}{3 d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{d \sqrt{\cos (c+d x)}}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(3*Sqrt[2]*(40*A + 38*B + 25*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*(48*A + 6*B + 17*C + 3*(8*A + 22*B + 27*C)*Cos[c + d*x] + (6*B + 17*C)*Cos[2*(c + d*x)] + 2*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d*Sqrt[Cos[c + d*x]])","A",1
497,1,156,233,1.0819727,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(6 \sqrt{2} (8 A+20 B+19 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{3}{2}}(c+d x)+2 \sin \left(\frac{1}{2} (c+d x)\right) ((128 A+48 B+9 C) \cos (c+d x)+16 A+3 (4 B+11 C) \cos (2 (c+d x))+12 B+3 C \cos (3 (c+d x))+33 C)\right)}{48 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{a^{5/2} (8 A+20 B+19 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^3 (56 A+12 B-27 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{12 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A+4 B-C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}+\frac{2 a (5 A+3 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(6*Sqrt[2]*(8*A + 20*B + 19*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(3/2) + 2*(16*A + 12*B + 33*C + (128*A + 48*B + 9*C)*Cos[c + d*x] + 3*(4*B + 11*C)*Cos[2*(c + d*x)] + 3*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d*Cos[c + d*x]^(3/2))","A",1
498,1,156,223,1.334491,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) ((112 A+40 B+45 C) \cos (c+d x)+4 (43 A+40 B+15 C) \cos (2 (c+d x))+196 A+160 B+15 C \cos (3 (c+d x))+60 C)+60 \sqrt{2} (2 B+5 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{5}{2}}(c+d x)\right)}{120 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{a^{5/2} (2 B+5 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a^3 (64 A+70 B+15 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (8 A+10 B+5 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a (A+B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(60*Sqrt[2]*(2*B + 5*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(5/2) + 2*(196*A + 160*B + 60*C + (112*A + 40*B + 45*C)*Cos[c + d*x] + 4*(43*A + 40*B + 15*C)*Cos[2*(c + d*x)] + 15*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(120*d*Cos[c + d*x]^(5/2))","A",1
499,1,172,222,1.5514834,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) ((930 A+987 B+840 C) \cos (c+d x)+2 (115 A+98 B+35 C) \cos (2 (c+d x))+230 A \cos (3 (c+d x))+290 A+301 B \cos (3 (c+d x))+196 B+280 C \cos (3 (c+d x))+70 C)+420 \sqrt{2} C \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{7}{2}}(c+d x)\right)}{420 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 a^{5/2} C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a^3 (160 A+224 B+245 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (40 A+56 B+35 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a (5 A+7 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(420*Sqrt[2]*C*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(7/2) + 2*(290*A + 196*B + 70*C + (930*A + 987*B + 840*C)*Cos[c + d*x] + 2*(115*A + 98*B + 35*C)*Cos[2*(c + d*x)] + 230*A*Cos[3*(c + d*x)] + 301*B*Cos[3*(c + d*x)] + 280*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(420*d*Cos[c + d*x]^(7/2))","A",1
500,1,158,234,1.2234008,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (2 (1396 A+1215 B+882 C) \cos (c+d x)+4 (803 A+870 B+966 C) \cos (2 (c+d x))+584 A \cos (3 (c+d x))+584 A \cos (4 (c+d x))+2908 A+690 B \cos (3 (c+d x))+690 B \cos (4 (c+d x))+2790 B+588 C \cos (3 (c+d x))+903 C \cos (4 (c+d x))+2961 C)}{1260 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{2 a^3 (8 A+10 B+11 C) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (584 A+690 B+903 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (64 A+90 B+63 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a (5 A+9 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(2908*A + 2790*B + 2961*C + 2*(1396*A + 1215*B + 882*C)*Cos[c + d*x] + 4*(803*A + 870*B + 966*C)*Cos[2*(c + d*x)] + 584*A*Cos[3*(c + d*x)] + 690*B*Cos[3*(c + d*x)] + 588*C*Cos[3*(c + d*x)] + 584*A*Cos[4*(c + d*x)] + 690*B*Cos[4*(c + d*x)] + 903*C*Cos[4*(c + d*x)])*Tan[(c + d*x)/2])/(1260*d*Cos[c + d*x]^(9/2))","A",1
501,1,190,284,0.9375672,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((50140 A+49654 B+49830 C) \cos (c+d x)+4 (4615 A+4642 B+4290 C) \cos (2 (c+d x))+18460 A \cos (3 (c+d x))+2840 A \cos (4 (c+d x))+2840 A \cos (5 (c+d x))+18140 A+20878 B \cos (3 (c+d x))+3212 B \cos (4 (c+d x))+3212 B \cos (5 (c+d x))+15356 B+22935 C \cos (3 (c+d x))+3795 C \cos (4 (c+d x))+3795 C \cos (5 (c+d x))+13365 C)}{13860 d \cos ^{\frac{11}{2}}(c+d x)}","\frac{2 a^3 (2840 A+3212 B+3795 C) \sin (c+d x)}{3465 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (1160 A+1364 B+1485 C) \sin (c+d x)}{3465 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a^3 (2840 A+3212 B+3795 C) \sin (c+d x)}{3465 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (32 A+44 B+33 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a (5 A+11 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d \cos ^{\frac{11}{2}}(c+d x)}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(18140*A + 15356*B + 13365*C + (50140*A + 49654*B + 49830*C)*Cos[c + d*x] + 4*(4615*A + 4642*B + 4290*C)*Cos[2*(c + d*x)] + 18460*A*Cos[3*(c + d*x)] + 20878*B*Cos[3*(c + d*x)] + 22935*C*Cos[3*(c + d*x)] + 2840*A*Cos[4*(c + d*x)] + 3212*B*Cos[4*(c + d*x)] + 3795*C*Cos[4*(c + d*x)] + 2840*A*Cos[5*(c + d*x)] + 3212*B*Cos[5*(c + d*x)] + 3795*C*Cos[5*(c + d*x)])*Tan[(c + d*x)/2])/(13860*d*Cos[c + d*x]^(11/2))","A",1
502,1,224,334,1.2180569,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{15}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(15/2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (70 (5552 A+5083 B+4576 C) \cos (c+d x)+14 (30334 A+31850 B+32747 C) \cos (2 (c+d x))+125520 A \cos (3 (c+d x))+125520 A \cos (4 (c+d x))+16736 A \cos (5 (c+d x))+16736 A \cos (6 (c+d x))+343612 A+138450 B \cos (3 (c+d x))+138450 B \cos (4 (c+d x))+18460 B \cos (5 (c+d x))+18460 B \cos (6 (c+d x))+325910 B+141570 C \cos (3 (c+d x))+156585 C \cos (4 (c+d x))+20878 C \cos (5 (c+d x))+20878 C \cos (6 (c+d x))+322751 C)}{180180 d \cos ^{\frac{13}{2}}(c+d x)}","\frac{8 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{45045 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{15015 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (2224 A+2522 B+2717 C) \sin (c+d x)}{9009 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{45045 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (136 A+182 B+143 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{1287 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 a (5 A+13 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d \cos ^{\frac{13}{2}}(c+d x)}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(343612*A + 325910*B + 322751*C + 70*(5552*A + 5083*B + 4576*C)*Cos[c + d*x] + 14*(30334*A + 31850*B + 32747*C)*Cos[2*(c + d*x)] + 125520*A*Cos[3*(c + d*x)] + 138450*B*Cos[3*(c + d*x)] + 141570*C*Cos[3*(c + d*x)] + 125520*A*Cos[4*(c + d*x)] + 138450*B*Cos[4*(c + d*x)] + 156585*C*Cos[4*(c + d*x)] + 16736*A*Cos[5*(c + d*x)] + 18460*B*Cos[5*(c + d*x)] + 20878*C*Cos[5*(c + d*x)] + 16736*A*Cos[6*(c + d*x)] + 18460*B*Cos[6*(c + d*x)] + 20878*C*Cos[6*(c + d*x)])*Tan[(c + d*x)/2])/(180180*d*Cos[c + d*x]^(13/2))","A",1
503,1,449,241,2.9759899,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(4 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} (24 A+2 (6 B-C) \cos (c+d x)-6 B+4 C \cos (2 (c+d x))+25 C)+\frac{3 \sqrt{2} e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \left(-16 i \sqrt{2} (A-B+C) \log \left(1+e^{i (c+d x)}\right)+i (8 A-14 B+9 C) \sinh ^{-1}\left(e^{i (c+d x)}\right)-8 i A \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)+16 i \sqrt{2} A \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)-8 A d x+14 i B \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)-16 i \sqrt{2} B \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)+14 B d x-9 i C \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)+16 i \sqrt{2} C \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)-9 C d x\right)}{\sqrt{1+e^{2 i (c+d x)}}}\right)}{48 d \sqrt{a (\cos (c+d x)+1)}}","-\frac{(8 A-14 B+9 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{(8 A-2 B+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(6 B-C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*((3*Sqrt[2]*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(-8*A*d*x + 14*B*d*x - 9*C*d*x + I*(8*A - 14*B + 9*C)*ArcSinh[E^(I*(c + d*x))] - (16*I)*Sqrt[2]*(A - B + C)*Log[1 + E^(I*(c + d*x))] - (8*I)*A*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] + (14*I)*B*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] - (9*I)*C*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] + (16*I)*Sqrt[2]*A*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]] - (16*I)*Sqrt[2]*B*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]] + (16*I)*Sqrt[2]*C*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]]))/Sqrt[1 + E^((2*I)*(c + d*x))] + 4*Sqrt[Cos[c + d*x]]*(24*A - 6*B + 25*C + 2*(6*B - C)*Cos[c + d*x] + 4*C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d*Sqrt[a*(1 + Cos[c + d*x])])","C",1
504,1,431,195,2.2152346,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(\frac{4 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} (4 B+2 C \cos (c+d x)-C)}{d}+\frac{\sqrt{2} e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \left(8 i \sqrt{2} (A-B+C) \log \left(1+e^{i (c+d x)}\right)-i (8 A-4 B+7 C) \sinh ^{-1}\left(e^{i (c+d x)}\right)+8 i A \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)-8 i \sqrt{2} A \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)+8 A d x-4 i B \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)+8 i \sqrt{2} B \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)-4 B d x+7 i C \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)-8 i \sqrt{2} C \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)+7 C d x\right)}{d \sqrt{1+e^{2 i (c+d x)}}}\right)}{8 \sqrt{a (\cos (c+d x)+1)}}","\frac{(8 A-4 B+7 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(4 B-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*((Sqrt[2]*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(8*A*d*x - 4*B*d*x + 7*C*d*x - I*(8*A - 4*B + 7*C)*ArcSinh[E^(I*(c + d*x))] + (8*I)*Sqrt[2]*(A - B + C)*Log[1 + E^(I*(c + d*x))] + (8*I)*A*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] - (4*I)*B*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] + (7*I)*C*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] - (8*I)*Sqrt[2]*A*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]] + (8*I)*Sqrt[2]*B*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]] - (8*I)*Sqrt[2]*C*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]]))/(d*Sqrt[1 + E^((2*I)*(c + d*x))]) + (4*Sqrt[Cos[c + d*x]]*(4*B - C + 2*C*Cos[c + d*x])*Sin[(c + d*x)/2])/d))/(8*Sqrt[a*(1 + Cos[c + d*x])])","C",1
505,1,112,141,0.2875895,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]),x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(2 (A-B+C) \tan ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)+\sqrt{2} (2 B-C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 C \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)}\right)}{d \sqrt{a (\cos (c+d x)+1)}}","\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(2 B-C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*(Sqrt[2]*(2*B - C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*(A - B + C)*ArcTan[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]] + 2*C*Sqrt[Cos[c + d*x]]*Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
506,1,266,138,5.2161965,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]),x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(-\frac{1}{2} (A-B+C) \csc ^3\left(\frac{1}{2} (c+d x)\right) \left(\sin ^4\left(\frac{1}{2} (c+d x)\right) \sin ^2(c+d x) \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};-\sec (c+d x) \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)-5 \cos ^2(c+d x) (\cos (c+d x)+2) \left(-\cos (c+d x)+\cos (c+d x) \sqrt{2-2 \sec (c+d x)} \tanh ^{-1}\left(\sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right) (-\sec (c+d x))}\right)+1\right)\right)+10 B \sin \left(\frac{1}{2} (c+d x)\right) \cos ^2(c+d x)+5 C \cos ^2(c+d x) \left(\sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)}}","-\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(2*Cos[(c + d*x)/2]*(5*C*Cos[c + d*x]^2*(Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] - 2*Sin[(c + d*x)/2]) + 10*B*Cos[c + d*x]^2*Sin[(c + d*x)/2] - ((A - B + C)*Csc[(c + d*x)/2]^3*(-5*Cos[c + d*x]^2*(2 + Cos[c + d*x])*(1 - Cos[c + d*x] + ArcTanh[Sqrt[-(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]]*Cos[c + d*x]*Sqrt[2 - 2*Sec[c + d*x]]) + Hypergeometric2F1[2, 5/2, 7/2, -(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^4*Sin[c + d*x]^2))/2))/(5*d*Cos[c + d*x]^(5/2)*Sqrt[a*(1 + Cos[c + d*x])])","C",0
507,1,701,143,6.7784605,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]),x]","\frac{2 (A-B+C) \cot \left(\frac{c}{2}+\frac{d x}{2}\right) \csc ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \left(-12 \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right) \cos ^4\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(2,2,\frac{7}{2};1,\frac{9}{2};-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}\right)-12 \left(3 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-7 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+4\right) \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right) \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}\right)+7 \sqrt{-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}} \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3 \left(8 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-20 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+15\right) \left(\left(3-7 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sqrt{-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}-3 \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right) \tanh ^{-1}\left(\sqrt{-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}\right)\right)\right)}{63 d \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{7/2} \sqrt{a (\cos (c+d x)+1)}}+\frac{8 B \sin \left(\frac{c}{2}+\frac{d x}{2}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)} \sqrt{a (\cos (c+d x)+1)}}+\frac{4 B \sin \left(\frac{c}{2}+\frac{d x}{2}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{3/2} \sqrt{a (\cos (c+d x)+1)}}-\frac{8 C \sin ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{3/2} \sqrt{a (\cos (c+d x)+1)}}","\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-3 B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"(4*B*Cos[c/2 + (d*x)/2]*Sin[c/2 + (d*x)/2])/(3*d*Sqrt[a*(1 + Cos[c + d*x])]*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) - (8*C*Cos[c/2 + (d*x)/2]*Sin[c/2 + (d*x)/2]^3)/(3*d*Sqrt[a*(1 + Cos[c + d*x])]*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) + (8*B*Cos[c/2 + (d*x)/2]*Sin[c/2 + (d*x)/2])/(3*d*Sqrt[a*(1 + Cos[c + d*x])]*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) + (2*(A - B + C)*Cot[c/2 + (d*x)/2]*Csc[c/2 + (d*x)/2]^4*(-12*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 7/2}, {1, 9/2}, -(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]*Sin[c/2 + (d*x)/2]^8 - 12*Hypergeometric2F1[2, 7/2, 9/2, -(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]*Sin[c/2 + (d*x)/2]^8*(4 - 7*Sin[c/2 + (d*x)/2]^2 + 3*Sin[c/2 + (d*x)/2]^4) + 7*Sqrt[-(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]*(1 - 2*Sin[c/2 + (d*x)/2]^2)^3*(15 - 20*Sin[c/2 + (d*x)/2]^2 + 8*Sin[c/2 + (d*x)/2]^4)*((3 - 7*Sin[c/2 + (d*x)/2]^2)*Sqrt[-(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))] - 3*ArcTanh[Sqrt[-(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]]*(1 - 2*Sin[c/2 + (d*x)/2]^2))))/(63*d*Sqrt[a*(1 + Cos[c + d*x])]*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2))","C",0
508,1,1950,191,7.7439575,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]),x]","-\frac{2 (A-B+C) \cot \left(\frac{c}{2}+\frac{d x}{2}\right) \left(440 \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right) \sin ^{16}\left(\frac{c}{2}+\frac{d x}{2}\right)+69120 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^{16}\left(\frac{c}{2}+\frac{d x}{2}\right)-42048 \sin ^{16}\left(\frac{c}{2}+\frac{d x}{2}\right)-1500 \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right) \sin ^{14}\left(\frac{c}{2}+\frac{d x}{2}\right)-414720 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^{14}\left(\frac{c}{2}+\frac{d x}{2}\right)+226656 \sin ^{14}\left(\frac{c}{2}+\frac{d x}{2}\right)+1770 \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right) \sin ^{12}\left(\frac{c}{2}+\frac{d x}{2}\right)+1080000 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^{12}\left(\frac{c}{2}+\frac{d x}{2}\right)-518760 \sin ^{12}\left(\frac{c}{2}+\frac{d x}{2}\right)-710 \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right) \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)-40 \cos ^6\left(\frac{1}{2} (c+d x)\right) \, _4F_3\left(2,2,2,\frac{9}{2};1,1,\frac{11}{2};\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right) \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)+60 \cos ^4\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(2,2,\frac{9}{2};1,\frac{11}{2};\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right) \left(4 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-5\right) \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)-1598400 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)+655812 \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)+1458000 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)-486630 \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)-833760 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)+210105 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)+291060 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-48825 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-56700 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+4725 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+4725 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \csc ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{675 d \sqrt{a (\cos (c+d x)+1)} \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{7/2} \left(2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1\right)}+\frac{16 B \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}+\frac{\sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{3/2}}\right)}{15 d \sqrt{a (\cos (c+d x)+1)}}+\frac{C \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{3 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{5/2}}+4 \left(\frac{2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}+\frac{\sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{3/2}}\right)\right)}{15 d \sqrt{a (\cos (c+d x)+1)}}+\frac{4 B \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d \sqrt{a (\cos (c+d x)+1)} \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{5/2}}-\frac{C \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{a (\cos (c+d x)+1)} \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{5/2}}","\frac{2 (13 A-5 B+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-5 B) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"(4*B*Cos[c/2 + (d*x)/2]*Sin[c/2 + (d*x)/2])/(5*d*Sqrt[a*(1 + Cos[c + d*x])]*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2)) - (C*Cos[c/2 + (d*x)/2]*Sin[c/2 + (d*x)/2])/(d*Sqrt[a*(1 + Cos[c + d*x])]*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2)) + (16*B*Cos[c/2 + (d*x)/2]*(Sin[c/2 + (d*x)/2]/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2) + (2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]))/(15*d*Sqrt[a*(1 + Cos[c + d*x])]) - (2*(A - B + C)*Cot[c/2 + (d*x)/2]*Csc[c/2 + (d*x)/2]^6*(4725*Sin[c/2 + (d*x)/2]^2 - 48825*Sin[c/2 + (d*x)/2]^4 + 210105*Sin[c/2 + (d*x)/2]^6 - 486630*Sin[c/2 + (d*x)/2]^8 + 655812*Sin[c/2 + (d*x)/2]^10 - 710*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^10 - 40*Cos[(c + d*x)/2]^6*HypergeometricPFQ[{2, 2, 2, 9/2}, {1, 1, 11/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^10 - 518760*Sin[c/2 + (d*x)/2]^12 + 1770*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12 + 226656*Sin[c/2 + (d*x)/2]^14 - 1500*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14 - 42048*Sin[c/2 + (d*x)/2]^16 + 440*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^16 + 4725*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 56700*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^2*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 291060*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^4*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 833760*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^6*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 1458000*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^8*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 1598400*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^10*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 1080000*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^12*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 414720*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^14*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 69120*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^16*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 60*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 9/2}, {1, 11/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^10*(-5 + 4*Sin[c/2 + (d*x)/2]^2)))/(675*d*Sqrt[a*(1 + Cos[c + d*x])]*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2)*(-1 + 2*Sin[c/2 + (d*x)/2]^2)) + (C*Cos[c/2 + (d*x)/2]*((3*Sin[c/2 + (d*x)/2])/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2) + 4*(Sin[c/2 + (d*x)/2]/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2) + (2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])))/(15*d*Sqrt[a*(1 + Cos[c + d*x])])","C",0
509,1,2716,237,9.8423457,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{9}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*Sqrt[a + a*Cos[c + d*x]]),x]","\text{Result too large to show}","\frac{2 (31 A-7 B+35 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{2 (43 A-91 B+35 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-7 B) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"(4*B*Cos[c/2 + (d*x)/2]*Sin[c/2 + (d*x)/2])/(7*d*Sqrt[a*(1 + Cos[c + d*x])]*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2)) - (2*C*Cos[c/2 + (d*x)/2]*Sin[c/2 + (d*x)/2])/(3*d*Sqrt[a*(1 + Cos[c + d*x])]*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2)) + (2*(A - B + C)*Cot[c/2 + (d*x)/2]*Csc[c/2 + (d*x)/2]^8*(363825*Sin[c/2 + (d*x)/2]^2 - 4729725*Sin[c/2 + (d*x)/2]^4 + 26785605*Sin[c/2 + (d*x)/2]^6 - 86790165*Sin[c/2 + (d*x)/2]^8 + 177677808*Sin[c/2 + (d*x)/2]^10 - 239283044*Sin[c/2 + (d*x)/2]^12 + 52080*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12 + 560*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12 + 213120160*Sin[c/2 + (d*x)/2]^14 - 168280*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14 - 2240*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14 - 121497024*Sin[c/2 + (d*x)/2]^16 + 212520*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^16 + 3360*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^16 + 40125184*Sin[c/2 + (d*x)/2]^18 - 124320*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^18 - 2240*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^18 - 5840384*Sin[c/2 + (d*x)/2]^20 + 28000*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^20 + 560*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^20 + 363825*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 5336100*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^2*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 34636140*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^4*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 131060160*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^6*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 320535600*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^8*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 530671680*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^10*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 604296000*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^12*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 468948480*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^14*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 237726720*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^16*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 70963200*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^18*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 9461760*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^20*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 1120*Cos[(c + d*x)/2]^6*HypergeometricPFQ[{2, 2, 2, 11/2}, {1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12*(-6 + 5*Sin[c/2 + (d*x)/2]^2) + 280*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 11/2}, {1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12*(103 - 164*Sin[c/2 + (d*x)/2]^2 + 70*Sin[c/2 + (d*x)/2]^4)))/(40425*d*Sqrt[a*(1 + Cos[c + d*x])]*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(9/2)*(-1 + 2*Sin[c/2 + (d*x)/2]^2)) + (8*B*Cos[c/2 + (d*x)/2]*((3*Sin[c/2 + (d*x)/2])/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2) + 4*(Sin[c/2 + (d*x)/2]/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2) + (2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])))/(35*d*Sqrt[a*(1 + Cos[c + d*x])]) + (2*C*Cos[c/2 + (d*x)/2]*((5*Sin[c/2 + (d*x)/2])/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2) + 2*((3*Sin[c/2 + (d*x)/2])/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2) + 4*(Sin[c/2 + (d*x)/2]/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2) + (2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]))))/(105*d*Sqrt[a*(1 + Cos[c + d*x])])","C",0
510,1,540,213,3.0071939,"\int \frac{\sqrt{\cos (c+d x)} \left(a A+(A b+a B) \cos (c+d x)+b B \cos ^2(c+d x)\right)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(a*A + (A*b + a*B)*Cos[c + d*x] + b*B*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(\frac{4 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} (4 a B+4 A b+2 b B \cos (c+d x)-b B)}{d}+\frac{\sqrt{2} e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \left(8 i \sqrt{2} (a-b) (A-B) \log \left(1+e^{i (c+d x)}\right)-i (8 a A-4 a B-4 A b+7 b B) \sinh ^{-1}\left(e^{i (c+d x)}\right)+8 i a A \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)-8 i \sqrt{2} a A \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)+8 a A d x-4 i a B \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)+8 i \sqrt{2} a B \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)-4 a B d x-4 i A b \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)+8 i \sqrt{2} A b \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)-4 A b d x+7 i b B \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)-8 i \sqrt{2} b B \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)+7 b B d x\right)}{d \sqrt{1+e^{2 i (c+d x)}}}\right)}{8 \sqrt{a (\cos (c+d x)+1)}}","\frac{(8 a A-4 a B-4 A b+7 b B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 \sqrt{a} d}+\frac{(4 a B+4 A b-b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (a-b) (A-B) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{b B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*((Sqrt[2]*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(8*a*A*d*x - 4*A*b*d*x - 4*a*B*d*x + 7*b*B*d*x - I*(8*a*A - 4*A*b - 4*a*B + 7*b*B)*ArcSinh[E^(I*(c + d*x))] + (8*I)*Sqrt[2]*(a - b)*(A - B)*Log[1 + E^(I*(c + d*x))] + (8*I)*a*A*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] - (4*I)*A*b*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] - (4*I)*a*B*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] + (7*I)*b*B*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] - (8*I)*Sqrt[2]*a*A*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]] + (8*I)*Sqrt[2]*A*b*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]] + (8*I)*Sqrt[2]*a*B*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]] - (8*I)*Sqrt[2]*b*B*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]]))/(d*Sqrt[1 + E^((2*I)*(c + d*x))]) + (4*Sqrt[Cos[c + d*x]]*(4*A*b + 4*a*B - b*B + 2*b*B*Cos[c + d*x])*Sin[(c + d*x)/2])/d))/(8*Sqrt[a*(1 + Cos[c + d*x])])","C",1
511,1,462,260,3.2637546,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{\cos ^3\left(\frac{1}{2} (c+d x)\right) \left(2 \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) (-2 A+(4 B-3 C) \cos (c+d x)+6 B+C \cos (2 (c+d x))-6 C)+\frac{\sqrt{2} e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \left(2 i \sqrt{2} (5 A-9 B+13 C) \log \left(1+e^{i (c+d x)}\right)-i (8 A-12 B+19 C) \sinh ^{-1}\left(e^{i (c+d x)}\right)+8 i A \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)-10 i \sqrt{2} A \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)+8 A d x-12 i B \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)+18 i \sqrt{2} B \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)-12 B d x+19 i C \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)-26 i \sqrt{2} C \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)+19 C d x\right)}{\sqrt{1+e^{2 i (c+d x)}}}\right)}{4 d (a (\cos (c+d x)+1))^{3/2}}","\frac{(8 A-12 B+19 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{(5 A-9 B+13 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B+2 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(2 A-6 B+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 a d \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]^3*((Sqrt[2]*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(8*A*d*x - 12*B*d*x + 19*C*d*x - I*(8*A - 12*B + 19*C)*ArcSinh[E^(I*(c + d*x))] + (2*I)*Sqrt[2]*(5*A - 9*B + 13*C)*Log[1 + E^(I*(c + d*x))] + (8*I)*A*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] - (12*I)*B*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] + (19*I)*C*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] - (10*I)*Sqrt[2]*A*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]] + (18*I)*Sqrt[2]*B*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]] - (26*I)*Sqrt[2]*C*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]]))/Sqrt[1 + E^((2*I)*(c + d*x))] + 2*Sqrt[Cos[c + d*x]]*(-2*A + 6*B - 6*C + (4*B - 3*C)*Cos[c + d*x] + C*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]*Tan[(c + d*x)/2]))/(4*d*(a*(1 + Cos[c + d*x]))^(3/2))","C",1
512,1,413,202,2.4943583,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{\cos ^3\left(\frac{1}{2} (c+d x)\right) \left(\frac{2 \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) (A-B+2 C \cos (c+d x)+3 C)}{d}+\frac{\sqrt{2} e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \left(-i \sqrt{2} (A-5 B+9 C) \log \left(1+e^{i (c+d x)}\right)+i \sqrt{2} A \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)-2 i (2 B-3 C) \sinh ^{-1}\left(e^{i (c+d x)}\right)+4 i B \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)-5 i \sqrt{2} B \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)+4 B d x-6 i C \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)+9 i \sqrt{2} C \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)-6 C d x\right)}{d \sqrt{1+e^{2 i (c+d x)}}}\right)}{2 (a (\cos (c+d x)+1))^{3/2}}","\frac{(A-5 B+9 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(2 B-3 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B+3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a d \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]^3*((Sqrt[2]*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(4*B*d*x - 6*C*d*x - (2*I)*(2*B - 3*C)*ArcSinh[E^(I*(c + d*x))] - I*Sqrt[2]*(A - 5*B + 9*C)*Log[1 + E^(I*(c + d*x))] + (4*I)*B*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] - (6*I)*C*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] + I*Sqrt[2]*A*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]] - (5*I)*Sqrt[2]*B*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]] + (9*I)*Sqrt[2]*C*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]]))/(d*Sqrt[1 + E^((2*I)*(c + d*x))]) + (2*Sqrt[Cos[c + d*x]]*(A - B + 3*C + 2*C*Cos[c + d*x])*Sec[(c + d*x)/2]*Tan[(c + d*x)/2])/d))/(2*(a*(1 + Cos[c + d*x]))^(3/2))","C",1
513,1,366,149,3.12332,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)),x]","\frac{\cos ^3\left(\frac{1}{2} (c+d x)\right) \left(-\frac{2 (A-B+C) \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right)}{d}+\frac{\sqrt{2} e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \left(-i \sqrt{2} (3 A+B-5 C) \log \left(1+e^{i (c+d x)}\right)+3 i \sqrt{2} A \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)+i \sqrt{2} B \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)+4 i C \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)-5 i \sqrt{2} C \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)-4 i C \sinh ^{-1}\left(e^{i (c+d x)}\right)+4 C d x\right)}{d \sqrt{1+e^{2 i (c+d x)}}}\right)}{2 (a (\cos (c+d x)+1))^{3/2}}","\frac{(3 A+B-5 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A-B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]^3*((Sqrt[2]*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(4*C*d*x - (4*I)*C*ArcSinh[E^(I*(c + d*x))] - I*Sqrt[2]*(3*A + B - 5*C)*Log[1 + E^(I*(c + d*x))] + (4*I)*C*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] + (3*I)*Sqrt[2]*A*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]] + I*Sqrt[2]*B*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]] - (5*I)*Sqrt[2]*C*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]]))/(d*Sqrt[1 + E^((2*I)*(c + d*x))]) - (2*(A - B + C)*Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]*Tan[(c + d*x)/2])/d))/(2*(a*(1 + Cos[c + d*x]))^(3/2))","C",1
514,1,455,161,4.778634,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)),x]","\frac{\cos ^3\left(\frac{1}{2} (c+d x)\right) \left(\frac{(A+3 B-7 C) \csc ^3\left(\frac{1}{2} (c+d x)\right) \left(5 (4 \cos (c+d x)+\cos (2 (c+d x))+1) \left(-\cos (c+d x)+\cos (c+d x) \sqrt{2-2 \sec (c+d x)} \tanh ^{-1}\left(\sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right) (-\sec (c+d x))}\right)+1\right)-2 \sin ^4\left(\frac{1}{2} (c+d x)\right) \sin (c+d x) \tan (c+d x) \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};-\sec (c+d x) \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}{2 \cos ^{\frac{3}{2}}(c+d x)}+\frac{5 (A-B+C) \left(2 \sin \left(\frac{1}{2} (c+d x)\right)-1\right)}{\sqrt{\cos (c+d x)} \left(\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)\right)^2}-\frac{5 (A-B+C) \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+1\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)-1\right) \sqrt{\cos (c+d x)}}-\frac{20 (A-B+C) \sqrt{\cos (c+d x)}}{\sin \left(\frac{1}{2} (c+d x)\right)-1}-\frac{20 (A-B+C) \sqrt{\cos (c+d x)}}{\sin \left(\frac{1}{2} (c+d x)\right)+1}+30 (A-B+C) \tan ^{-1}\left(\frac{1-2 \sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)-30 (A-B+C) \tan ^{-1}\left(\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)+1}{\sqrt{\cos (c+d x)}}\right)+\frac{80 C \sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)}{10 d (a (\cos (c+d x)+1))^{3/2}}","-\frac{(7 A-3 B-C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A-B+C) \sin (c+d x)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x)}{2 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]^3*(30*(A - B + C)*ArcTan[(1 - 2*Sin[(c + d*x)/2])/Sqrt[Cos[c + d*x]]] - 30*(A - B + C)*ArcTan[(1 + 2*Sin[(c + d*x)/2])/Sqrt[Cos[c + d*x]]] - (20*(A - B + C)*Sqrt[Cos[c + d*x]])/(-1 + Sin[(c + d*x)/2]) + (80*C*Sin[(c + d*x)/2])/Sqrt[Cos[c + d*x]] - (20*(A - B + C)*Sqrt[Cos[c + d*x]])/(1 + Sin[(c + d*x)/2]) + (5*(A - B + C)*(-1 + 2*Sin[(c + d*x)/2]))/(Sqrt[Cos[c + d*x]]*(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^2) - (5*(A - B + C)*(1 + 2*Sin[(c + d*x)/2]))/(Sqrt[Cos[c + d*x]]*(-1 + Sin[(c + d*x)/2])) + ((A + 3*B - 7*C)*Csc[(c + d*x)/2]^3*(5*(1 + 4*Cos[c + d*x] + Cos[2*(c + d*x)])*(1 - Cos[c + d*x] + ArcTanh[Sqrt[-(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]]*Cos[c + d*x]*Sqrt[2 - 2*Sec[c + d*x]]) - 2*Hypergeometric2F1[2, 5/2, 7/2, -(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^4*Sin[c + d*x]*Tan[c + d*x]))/(2*Cos[c + d*x]^(3/2))))/(10*d*(a*(1 + Cos[c + d*x]))^(3/2))","C",0
515,1,1207,213,6.8575612,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)),x]","-\frac{(A-B+C) \left(5 \tan ^{-1}\left(\frac{1-2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}\right)+\frac{3 \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}{1-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)}+\frac{\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1}{\left(1-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{3/2}}+\frac{(A-B+C) \left(5 \tan ^{-1}\left(\frac{2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1}{\sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}\right)+\frac{3 \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}{\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1}+\frac{1-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1\right) \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{3/2}}+\frac{16 C \sin \left(\frac{c}{2}+\frac{d x}{2}\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (a (\cos (c+d x)+1))^{3/2} \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}+\frac{8 C \sin \left(\frac{c}{2}+\frac{d x}{2}\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (a (\cos (c+d x)+1))^{3/2} \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{3/2}}+\frac{(A-B+C) \left(2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{6 d (a (\cos (c+d x)+1))^{3/2} \left(1-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{3/2}}-\frac{(A-B+C) \left(1-2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{6 d (a (\cos (c+d x)+1))^{3/2} \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1\right) \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{3/2}}+\frac{(A+3 B-7 C) \cot ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \csc ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \left(-12 \cos ^4\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(2,2,\frac{7}{2};1,\frac{9}{2};-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}\right) \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)-12 \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}\right) \left(3 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-7 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+4\right) \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)+7 \sqrt{-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}} \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3 \left(8 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-20 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+15\right) \left(\left(3-7 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sqrt{-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}-3 \tanh ^{-1}\left(\sqrt{-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}\right) \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)\right)\right)}{63 d (a (\cos (c+d x)+1))^{3/2} \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{7/2}}","\frac{(11 A-7 B+3 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(7 A-3 B+3 C) \sin (c+d x)}{6 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(19 A-15 B+3 C) \sin (c+d x)}{6 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(8*C*Cos[c/2 + (d*x)/2]^3*Sin[c/2 + (d*x)/2])/(3*d*(a*(1 + Cos[c + d*x]))^(3/2)*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) - ((A - B + C)*Cos[c/2 + (d*x)/2]^3*(1 - 2*Sin[c/2 + (d*x)/2]))/(6*d*(a*(1 + Cos[c + d*x]))^(3/2)*(1 + Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) + ((A - B + C)*Cos[c/2 + (d*x)/2]^3*(1 + 2*Sin[c/2 + (d*x)/2]))/(6*d*(a*(1 + Cos[c + d*x]))^(3/2)*(1 - Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) + (16*C*Cos[c/2 + (d*x)/2]^3*Sin[c/2 + (d*x)/2])/(3*d*(a*(1 + Cos[c + d*x]))^(3/2)*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) - ((A - B + C)*Cos[c/2 + (d*x)/2]^3*(5*ArcTan[(1 - 2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]] + (1 + Sin[c/2 + (d*x)/2])/((1 - Sin[c/2 + (d*x)/2])*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) + (3*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])/(1 - Sin[c/2 + (d*x)/2])))/(d*(a*(1 + Cos[c + d*x]))^(3/2)) + ((A - B + C)*Cos[c/2 + (d*x)/2]^3*(5*ArcTan[(1 + 2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]] + (1 - Sin[c/2 + (d*x)/2])/((1 + Sin[c/2 + (d*x)/2])*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) + (3*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])/(1 + Sin[c/2 + (d*x)/2])))/(d*(a*(1 + Cos[c + d*x]))^(3/2)) + ((A + 3*B - 7*C)*Cot[c/2 + (d*x)/2]^3*Csc[c/2 + (d*x)/2]^2*(-12*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 7/2}, {1, 9/2}, -(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]*Sin[c/2 + (d*x)/2]^8 - 12*Hypergeometric2F1[2, 7/2, 9/2, -(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]*Sin[c/2 + (d*x)/2]^8*(4 - 7*Sin[c/2 + (d*x)/2]^2 + 3*Sin[c/2 + (d*x)/2]^4) + 7*Sqrt[-(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]*(1 - 2*Sin[c/2 + (d*x)/2]^2)^3*(15 - 20*Sin[c/2 + (d*x)/2]^2 + 8*Sin[c/2 + (d*x)/2]^4)*((3 - 7*Sin[c/2 + (d*x)/2]^2)*Sqrt[-(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))] - 3*ArcTanh[Sqrt[-(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]]*(1 - 2*Sin[c/2 + (d*x)/2]^2))))/(63*d*(a*(1 + Cos[c + d*x]))^(3/2)*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2))","C",0
516,1,2437,263,7.857978,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Cos[c + d*x])^(3/2)),x]","\text{Result too large to show}","-\frac{(15 A-11 B+7 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(39 A-35 B+15 C) \sin (c+d x)}{30 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{(9 A-5 B+5 C) \sin (c+d x)}{10 a d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}+\frac{(147 A-95 B+75 C) \sin (c+d x)}{30 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(8*C*Cos[c/2 + (d*x)/2]^3*Sin[c/2 + (d*x)/2])/(5*d*(a*(1 + Cos[c + d*x]))^(3/2)*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2)) - ((A - B + C)*Cos[c/2 + (d*x)/2]^3*(1 - 2*Sin[c/2 + (d*x)/2]))/(10*d*(a*(1 + Cos[c + d*x]))^(3/2)*(1 + Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2)) + ((A - B + C)*Cos[c/2 + (d*x)/2]^3*(1 + 2*Sin[c/2 + (d*x)/2]))/(10*d*(a*(1 + Cos[c + d*x]))^(3/2)*(1 - Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2)) + (32*C*Cos[c/2 + (d*x)/2]^3*(Sin[c/2 + (d*x)/2]/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2) + (2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]))/(15*d*(a*(1 + Cos[c + d*x]))^(3/2)) + ((A - B + C)*Cos[c/2 + (d*x)/2]^3*(105*ArcTan[(1 - 2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]] - (4 + 3*Sin[c/2 + (d*x)/2])/((1 - Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) + (19 + 29*Sin[c/2 + (d*x)/2])/((1 - Sin[c/2 + (d*x)/2])*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) + (67*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])/(1 - Sin[c/2 + (d*x)/2])))/(15*d*(a*(1 + Cos[c + d*x]))^(3/2)) - ((A - B + C)*Cos[c/2 + (d*x)/2]^3*(105*ArcTan[(1 + 2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]] - (4 - 3*Sin[c/2 + (d*x)/2])/((1 + Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) + (19 - 29*Sin[c/2 + (d*x)/2])/((1 + Sin[c/2 + (d*x)/2])*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) + (67*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])/(1 + Sin[c/2 + (d*x)/2])))/(15*d*(a*(1 + Cos[c + d*x]))^(3/2)) + ((-A - 3*B + 7*C)*Cot[c/2 + (d*x)/2]^3*Csc[c/2 + (d*x)/2]^4*(4725*Sin[c/2 + (d*x)/2]^2 - 48825*Sin[c/2 + (d*x)/2]^4 + 210105*Sin[c/2 + (d*x)/2]^6 - 486630*Sin[c/2 + (d*x)/2]^8 + 655812*Sin[c/2 + (d*x)/2]^10 - 710*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^10 - 40*Cos[(c + d*x)/2]^6*HypergeometricPFQ[{2, 2, 2, 9/2}, {1, 1, 11/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^10 - 518760*Sin[c/2 + (d*x)/2]^12 + 1770*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12 + 226656*Sin[c/2 + (d*x)/2]^14 - 1500*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14 - 42048*Sin[c/2 + (d*x)/2]^16 + 440*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^16 + 4725*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 56700*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^2*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 291060*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^4*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 833760*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^6*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 1458000*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^8*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 1598400*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^10*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 1080000*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^12*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 414720*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^14*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 69120*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^16*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 60*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 9/2}, {1, 11/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^10*(-5 + 4*Sin[c/2 + (d*x)/2]^2)))/(675*d*(a*(1 + Cos[c + d*x]))^(3/2)*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2)*(-1 + 2*Sin[c/2 + (d*x)/2]^2))","C",0
517,1,434,254,4.1039642,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\cos ^5\left(\frac{1}{2} (c+d x)\right) \left(\sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) ((7 A-15 B+55 C) \cos (c+d x)+3 A-11 B+8 C \cos (2 (c+d x))+43 C)+\frac{\sqrt{2} e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \left(-i \sqrt{2} (3 A-43 B+115 C) \log \left(1+e^{i (c+d x)}\right)+3 i \sqrt{2} A \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)-16 i (2 B-5 C) \sinh ^{-1}\left(e^{i (c+d x)}\right)+32 i B \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)-43 i \sqrt{2} B \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)+32 B d x-80 i C \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)+115 i \sqrt{2} C \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)-80 C d x\right)}{\sqrt{1+e^{2 i (c+d x)}}}\right)}{8 d (a (\cos (c+d x)+1))^{5/2}}","\frac{(3 A-43 B+115 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(2 B-5 C) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}+\frac{(3 A-11 B+35 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(A+7 B-15 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]^5*((Sqrt[2]*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(32*B*d*x - 80*C*d*x - (16*I)*(2*B - 5*C)*ArcSinh[E^(I*(c + d*x))] - I*Sqrt[2]*(3*A - 43*B + 115*C)*Log[1 + E^(I*(c + d*x))] + (32*I)*B*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] - (80*I)*C*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] + (3*I)*Sqrt[2]*A*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]] - (43*I)*Sqrt[2]*B*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]] + (115*I)*Sqrt[2]*C*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]]))/Sqrt[1 + E^((2*I)*(c + d*x))] + Sqrt[Cos[c + d*x]]*(3*A - 11*B + 43*C + (7*A - 15*B + 55*C)*Cos[c + d*x] + 8*C*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^3*Tan[(c + d*x)/2]))/(8*d*(a*(1 + Cos[c + d*x]))^(5/2))","C",1
518,1,385,201,3.1744297,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\cos ^5\left(\frac{1}{2} (c+d x)\right) \left(\sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) ((A+7 B-15 C) \cos (c+d x)+5 A+3 B-11 C)+\frac{\sqrt{2} e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \left(-i \sqrt{2} (5 A+3 B-43 C) \log \left(1+e^{i (c+d x)}\right)+5 i \sqrt{2} A \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)+3 i \sqrt{2} B \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)+32 i C \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)-43 i \sqrt{2} C \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)-32 i C \sinh ^{-1}\left(e^{i (c+d x)}\right)+32 C d x\right)}{\sqrt{1+e^{2 i (c+d x)}}}\right)}{8 d (a (\cos (c+d x)+1))^{5/2}}","\frac{(5 A+3 B-43 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 C \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(5 A+3 B-11 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]^5*((Sqrt[2]*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(32*C*d*x - (32*I)*C*ArcSinh[E^(I*(c + d*x))] - I*Sqrt[2]*(5*A + 3*B - 43*C)*Log[1 + E^(I*(c + d*x))] + (32*I)*C*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] + (5*I)*Sqrt[2]*A*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]] + (3*I)*Sqrt[2]*B*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]] - (43*I)*Sqrt[2]*C*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]]))/Sqrt[1 + E^((2*I)*(c + d*x))] + Sqrt[Cos[c + d*x]]*(5*A + 3*B - 11*C + (A + 7*B - 15*C)*Cos[c + d*x])*Sec[(c + d*x)/2]^3*Tan[(c + d*x)/2]))/(8*d*(a*(1 + Cos[c + d*x]))^(5/2))","C",1
519,1,209,163,1.8851067,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)),x]","\frac{\cos ^5\left(\frac{1}{2} (c+d x)\right) \left(-\frac{1}{2} \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) ((9 A-B-7 C) \cos (c+d x)+13 A-5 B-3 C)+\frac{i (19 A+5 B+3 C) e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)}{\sqrt{1+e^{2 i (c+d x)}}}\right)}{4 d (a (\cos (c+d x)+1))^{5/2}}","\frac{(19 A+5 B+3 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(9 A-B-7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(Cos[(c + d*x)/2]^5*((I*(19*A + 5*B + 3*C)*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/Sqrt[1 + E^((2*I)*(c + d*x))] - (Sqrt[Cos[c + d*x]]*(13*A - 5*B - 3*C + (9*A - B - 7*C)*Cos[c + d*x])*Sec[(c + d*x)/2]^3*Tan[(c + d*x)/2])/2))/(4*d*(a*(1 + Cos[c + d*x]))^(5/2))","C",1
520,1,225,211,2.441521,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)),x]","\frac{\cos ^5\left(\frac{1}{2} (c+d x)\right) \left(\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) (2 (85 A-13 B+5 C) \cos (c+d x)+(49 A-9 B+C) \cos (2 (c+d x))+113 A-9 B+C)}{4 \sqrt{\cos (c+d x)}}-\frac{i (75 A-19 B-5 C) e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)}{\sqrt{1+e^{2 i (c+d x)}}}\right)}{4 d (a (\cos (c+d x)+1))^{5/2}}","-\frac{(75 A-19 B-5 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(49 A-9 B+C) \sin (c+d x)}{16 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(13 A-5 B-3 C) \sin (c+d x)}{16 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x)}{4 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}",1,"(Cos[(c + d*x)/2]^5*(((-I)*(75*A - 19*B - 5*C)*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/Sqrt[1 + E^((2*I)*(c + d*x))] + ((113*A - 9*B + C + 2*(85*A - 13*B + 5*C)*Cos[c + d*x] + (49*A - 9*B + C)*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^3*Tan[(c + d*x)/2])/(4*Sqrt[Cos[c + d*x]])))/(4*d*(a*(1 + Cos[c + d*x]))^(5/2))","C",1
521,1,262,261,4.1772276,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(5/2)),x]","\frac{\cos ^5\left(\frac{1}{2} (c+d x)\right) \left(-\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) ((1537 A-825 B+81 C) \cos (c+d x)+2 (503 A-255 B+39 C) \cos (2 (c+d x))+299 A \cos (3 (c+d x))+878 A-147 B \cos (3 (c+d x))-510 B+27 C \cos (3 (c+d x))+78 C)}{8 \cos ^{\frac{3}{2}}(c+d x)}+\frac{3 i (163 A-75 B+19 C) e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)}{\sqrt{1+e^{2 i (c+d x)}}}\right)}{12 d (a (\cos (c+d x)+1))^{5/2}}","\frac{(163 A-75 B+19 C) \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(95 A-39 B+15 C) \sin (c+d x)}{48 a^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(299 A-147 B+27 C) \sin (c+d x)}{48 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(17 A-9 B+C) \sin (c+d x)}{16 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}",1,"(Cos[(c + d*x)/2]^5*(((3*I)*(163*A - 75*B + 19*C)*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/Sqrt[1 + E^((2*I)*(c + d*x))] - ((878*A - 510*B + 78*C + (1537*A - 825*B + 81*C)*Cos[c + d*x] + 2*(503*A - 255*B + 39*C)*Cos[2*(c + d*x)] + 299*A*Cos[3*(c + d*x)] - 147*B*Cos[3*(c + d*x)] + 27*C*Cos[3*(c + d*x)])*Sec[(c + d*x)/2]^3*Tan[(c + d*x)/2])/(8*Cos[c + d*x]^(3/2))))/(12*d*(a*(1 + Cos[c + d*x]))^(5/2))","C",1
522,1,89,131,0.3144292,"\int \cos ^2(c+d x) (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","\frac{15 a (4 (4 A+3 C) (c+d x)+8 (A+C) \sin (2 (c+d x))+C \sin (4 (c+d x)))-160 b (A+2 C) \sin ^3(c+d x)+480 b (A+C) \sin (c+d x)+96 b C \sin ^5(c+d x)}{480 d}","\frac{a (4 A+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 A+3 C)+\frac{a C \sin (c+d x) \cos ^3(c+d x)}{4 d}-\frac{b (5 A+4 C) \sin ^3(c+d x)}{15 d}+\frac{b (5 A+4 C) \sin (c+d x)}{5 d}+\frac{b C \sin (c+d x) \cos ^4(c+d x)}{5 d}",1,"(480*b*(A + C)*Sin[c + d*x] - 160*b*(A + 2*C)*Sin[c + d*x]^3 + 96*b*C*Sin[c + d*x]^5 + 15*a*(4*(4*A + 3*C)*(c + d*x) + 8*(A + C)*Sin[2*(c + d*x)] + C*Sin[4*(c + d*x)]))/(480*d)","A",1
523,1,84,108,0.2174346,"\int \cos (c+d x) (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","\frac{24 a (4 A+3 C) \sin (c+d x)+8 a C \sin (3 (c+d x))+24 b (A+C) \sin (2 (c+d x))+48 A b c+48 A b d x+3 b C \sin (4 (c+d x))+36 b c C+36 b C d x}{96 d}","\frac{a (3 A+2 C) \sin (c+d x)}{3 d}+\frac{a C \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac{b (4 A+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} b x (4 A+3 C)+\frac{b C \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"(48*A*b*c + 36*b*c*C + 48*A*b*d*x + 36*b*C*d*x + 24*a*(4*A + 3*C)*Sin[c + d*x] + 24*b*(A + C)*Sin[2*(c + d*x)] + 8*a*C*Sin[3*(c + d*x)] + 3*b*C*Sin[4*(c + d*x)])/(96*d)","A",1
524,1,64,96,0.1203572,"\int (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","\frac{12 a A d x+3 a C \sin (2 (c+d x))+6 a c C+6 a C d x+3 b (4 A+3 C) \sin (c+d x)+b C \sin (3 (c+d x))}{12 d}","-\frac{\left(a^2 C-b^2 (3 A+2 C)\right) \sin (c+d x)}{3 b d}+\frac{1}{2} a x (2 A+C)+\frac{C \sin (c+d x) (a+b \cos (c+d x))^2}{3 b d}-\frac{a C \sin (c+d x) \cos (c+d x)}{6 d}",1,"(6*a*c*C + 12*a*A*d*x + 6*a*C*d*x + 3*b*(4*A + 3*C)*Sin[c + d*x] + 3*a*C*Sin[2*(c + d*x)] + b*C*Sin[3*(c + d*x)])/(12*d)","A",1
525,1,73,58,0.1327687,"\int (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a C \sin (c) \cos (d x)}{d}+\frac{a C \cos (c) \sin (d x)}{d}+A b x+\frac{b C (c+d x)}{2 d}+\frac{b C \sin (2 (c+d x))}{4 d}","\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a C \sin (c+d x)}{d}+\frac{1}{2} b x (2 A+C)+\frac{b C \sin (c+d x) \cos (c+d x)}{2 d}",1,"A*b*x + (b*C*(c + d*x))/(2*d) + (a*A*ArcTanh[Sin[c + d*x]])/d + (a*C*Cos[d*x]*Sin[c])/d + (a*C*Cos[c]*Sin[d*x])/d + (b*C*Sin[2*(c + d*x)])/(4*d)","A",1
526,1,54,42,0.0199619,"\int (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{a A \tan (c+d x)}{d}+a C x+\frac{A b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b C \sin (c) \cos (d x)}{d}+\frac{b C \cos (c) \sin (d x)}{d}","\frac{a A \tan (c+d x)}{d}+a C x+\frac{A b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b C \sin (c+d x)}{d}",1,"a*C*x + (A*b*ArcTanh[Sin[c + d*x]])/d + (b*C*Cos[d*x]*Sin[c])/d + (b*C*Cos[c]*Sin[d*x])/d + (a*A*Tan[c + d*x])/d","A",1
527,1,67,58,0.0201089,"\int (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{a A \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A b \tan (c+d x)}{d}+b C x","\frac{a (A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}+\frac{A b \tan (c+d x)}{d}+b C x",1,"b*C*x + (a*A*ArcTanh[Sin[c + d*x]])/(2*d) + (a*C*ArcTanh[Sin[c + d*x]])/d + (A*b*Tan[c + d*x])/d + (a*A*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
528,1,59,86,0.3406852,"\int (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\tan (c+d x) \left(2 a A \tan ^2(c+d x)+6 a (A+C)+3 A b \sec (c+d x)\right)+3 b (A+2 C) \tanh ^{-1}(\sin (c+d x))}{6 d}","\frac{a (2 A+3 C) \tan (c+d x)}{3 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{b (A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A b \tan (c+d x) \sec (c+d x)}{2 d}",1,"(3*b*(A + 2*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(6*a*(A + C) + 3*A*b*Sec[c + d*x] + 2*a*A*Tan[c + d*x]^2))/(6*d)","A",1
529,1,80,117,0.4621634,"\int (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{\tan (c+d x) \left(3 a (3 A+4 C) \sec (c+d x)+6 a A \sec ^3(c+d x)+8 b \left(A \tan ^2(c+d x)+3 (A+C)\right)\right)+3 a (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{24 d}","\frac{a (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (3 A+4 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{b (2 A+3 C) \tan (c+d x)}{3 d}+\frac{A b \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"(3*a*(3*A + 4*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(3*a*(3*A + 4*C)*Sec[c + d*x] + 6*a*A*Sec[c + d*x]^3 + 8*b*(3*(A + C) + A*Tan[c + d*x]^2)))/(24*d)","A",1
530,1,96,140,0.8010268,"\int (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{\tan (c+d x) \left(8 a \left(5 (2 A+C) \tan ^2(c+d x)+3 A \tan ^4(c+d x)+15 (A+C)\right)+15 b (3 A+4 C) \sec (c+d x)+30 A b \sec ^3(c+d x)\right)+15 b (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{120 d}","\frac{a (4 A+5 C) \tan ^3(c+d x)}{15 d}+\frac{a (4 A+5 C) \tan (c+d x)}{5 d}+\frac{a A \tan (c+d x) \sec ^4(c+d x)}{5 d}+\frac{b (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b (3 A+4 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{A b \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"(15*b*(3*A + 4*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*b*(3*A + 4*C)*Sec[c + d*x] + 30*A*b*Sec[c + d*x]^3 + 8*a*(15*(A + C) + 5*(2*A + C)*Tan[c + d*x]^2 + 3*A*Tan[c + d*x]^4)))/(120*d)","A",1
531,1,160,214,0.719597,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2),x]","\frac{60 (c+d x) \left(a^2 (8 A+6 C)+b^2 (6 A+5 C)\right)+15 \left(16 a^2 (A+C)+b^2 (16 A+15 C)\right) \sin (2 (c+d x))+15 \left(2 a^2 C+2 A b^2+3 b^2 C\right) \sin (4 (c+d x))+240 a b (6 A+5 C) \sin (c+d x)+40 a b (4 A+5 C) \sin (3 (c+d x))+24 a b C \sin (5 (c+d x))+5 b^2 C \sin (6 (c+d x))}{960 d}","\frac{\left(2 a^2 C+b^2 (6 A+5 C)\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{\left(a^2 (8 A+6 C)+b^2 (6 A+5 C)\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left(a^2 (8 A+6 C)+b^2 (6 A+5 C)\right)-\frac{2 a b (5 A+4 C) \sin ^3(c+d x)}{15 d}+\frac{2 a b (5 A+4 C) \sin (c+d x)}{5 d}+\frac{a b C \sin (c+d x) \cos ^4(c+d x)}{15 d}+\frac{C \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^2}{6 d}",1,"(60*(b^2*(6*A + 5*C) + a^2*(8*A + 6*C))*(c + d*x) + 240*a*b*(6*A + 5*C)*Sin[c + d*x] + 15*(16*a^2*(A + C) + b^2*(16*A + 15*C))*Sin[2*(c + d*x)] + 40*a*b*(4*A + 5*C)*Sin[3*(c + d*x)] + 15*(2*A*b^2 + 2*a^2*C + 3*b^2*C)*Sin[4*(c + d*x)] + 24*a*b*C*Sin[5*(c + d*x)] + 5*b^2*C*Sin[6*(c + d*x)])/(960*d)","A",1
532,1,126,178,0.4339948,"\int \cos (c+d x) (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2),x]","\frac{30 \left(a^2 (8 A+6 C)+b^2 (6 A+5 C)\right) \sin (c+d x)+5 \left(4 a^2 C+4 A b^2+5 b^2 C\right) \sin (3 (c+d x))+60 a b (4 A+3 C) (c+d x)+120 a b (A+C) \sin (2 (c+d x))+15 a b C \sin (4 (c+d x))+3 b^2 C \sin (5 (c+d x))}{240 d}","\frac{\left(5 a^2 (3 A+2 C)+2 b^2 (5 A+4 C)\right) \sin (c+d x)}{15 d}+\frac{\left(2 a^2 C+b^2 (5 A+4 C)\right) \sin (c+d x) \cos ^2(c+d x)}{15 d}+\frac{a b (4 A+3 C) \sin (c+d x) \cos (c+d x)}{4 d}+\frac{1}{4} a b x (4 A+3 C)+\frac{a b C \sin (c+d x) \cos ^3(c+d x)}{10 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{5 d}",1,"(60*a*b*(4*A + 3*C)*(c + d*x) + 30*(b^2*(6*A + 5*C) + a^2*(8*A + 6*C))*Sin[c + d*x] + 120*a*b*(A + C)*Sin[2*(c + d*x)] + 5*(4*A*b^2 + 4*a^2*C + 5*b^2*C)*Sin[3*(c + d*x)] + 15*a*b*C*Sin[4*(c + d*x)] + 3*b^2*C*Sin[5*(c + d*x)])/(240*d)","A",1
533,1,106,161,0.384021,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2),x]","\frac{12 (c+d x) \left(4 a^2 (2 A+C)+b^2 (4 A+3 C)\right)+24 \left(C \left(a^2+b^2\right)+A b^2\right) \sin (2 (c+d x))+48 a b (4 A+3 C) \sin (c+d x)+16 a b C \sin (3 (c+d x))+3 b^2 C \sin (4 (c+d x))}{96 d}","\frac{a \left(a^2 (-C)+12 A b^2+8 b^2 C\right) \sin (c+d x)}{6 b d}-\frac{\left(2 a^2 C-3 b^2 (4 A+3 C)\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(4 a^2 (2 A+C)+b^2 (4 A+3 C)\right)+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d}-\frac{a C \sin (c+d x) (a+b \cos (c+d x))^2}{12 b d}",1,"(12*(4*a^2*(2*A + C) + b^2*(4*A + 3*C))*(c + d*x) + 48*a*b*(4*A + 3*C)*Sin[c + d*x] + 24*(A*b^2 + (a^2 + b^2)*C)*Sin[2*(c + d*x)] + 16*a*b*C*Sin[3*(c + d*x)] + 3*b^2*C*Sin[4*(c + d*x)])/(96*d)","A",1
534,1,145,103,0.253304,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{3 \left(4 a^2 C+4 A b^2+3 b^2 C\right) \sin (c+d x)-12 a^2 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 a^2 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+24 a A b c+24 a A b d x+6 a b C \sin (2 (c+d x))+12 a b c C+12 a b C d x+b^2 C \sin (3 (c+d x))}{12 d}","\frac{\left(2 C \left(a^2+b^2\right)+3 A b^2\right) \sin (c+d x)}{3 d}+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+a b x (2 A+C)+\frac{a b C \sin (c+d x) \cos (c+d x)}{3 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}",1,"(24*a*A*b*c + 12*a*b*c*C + 24*a*A*b*d*x + 12*a*b*C*d*x - 12*a^2*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*a^2*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 3*(4*A*b^2 + 4*a^2*C + 3*b^2*C)*Sin[c + d*x] + 6*a*b*C*Sin[2*(c + d*x)] + b^2*C*Sin[3*(c + d*x)])/(12*d)","A",1
535,1,132,109,0.7420127,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{2 (c+d x) \left(C \left(2 a^2+b^2\right)+2 A b^2\right)+\tan (c+d x) \left(4 a^2 A+b^2 C \cos (2 (c+d x))+b^2 C\right)-8 a A b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+8 a A b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+8 a b C \sin (c+d x)}{4 d}","\frac{1}{2} x \left(C \left(2 a^2+b^2\right)+2 A b^2\right)-\frac{2 a b (A-C) \sin (c+d x)}{d}+\frac{2 a A b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^2}{d}-\frac{b^2 (2 A-C) \sin (c+d x) \cos (c+d x)}{2 d}",1,"(2*(2*A*b^2 + (2*a^2 + b^2)*C)*(c + d*x) - 8*a*A*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 8*a*A*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 8*a*b*C*Sin[c + d*x] + (4*a^2*A + b^2*C + b^2*C*Cos[2*(c + d*x)])*Tan[c + d*x])/(4*d)","A",1
536,1,249,103,1.2881651,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{-2 \left(a^2 (A+2 C)+2 A b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \left(a^2 (A+2 C)+2 A b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{a^2 A}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{a^2 A}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{8 a A b \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{8 a A b \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+8 a b C (c+d x)+4 b^2 C \sin (c+d x)}{4 d}","\frac{\left(a^2 (A+2 C)+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a A b \tan (c+d x)}{d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}+2 a b C x-\frac{b^2 (A-2 C) \sin (c+d x)}{2 d}",1,"(8*a*b*C*(c + d*x) - 2*(2*A*b^2 + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(2*A*b^2 + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a^2*A)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (8*a*A*b*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (a^2*A)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (8*a*A*b*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 4*b^2*C*Sin[c + d*x])/(4*d)","B",1
537,1,76,112,0.45867,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{3 \tan (c+d x) \left(a^2 (A+C)+a A b \sec (c+d x)+A b^2\right)+a^2 A \tan ^3(c+d x)+3 a b (A+2 C) \tanh ^{-1}(\sin (c+d x))+3 b^2 C d x}{3 d}","\frac{\left(a^2 (2 A+3 C)+2 A b^2\right) \tan (c+d x)}{3 d}+\frac{a b (A+2 C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A b \tan (c+d x) \sec (c+d x)}{3 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}+b^2 C x",1,"(3*b^2*C*d*x + 3*a*b*(A + 2*C)*ArcTanh[Sin[c + d*x]] + 3*(A*b^2 + a^2*(A + C) + a*A*b*Sec[c + d*x])*Tan[c + d*x] + a^2*A*Tan[c + d*x]^3)/(3*d)","A",1
538,1,107,154,0.6359796,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{3 \left(a^2 (3 A+4 C)+4 b^2 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(3 \left(a^2 (3 A+4 C)+4 A b^2\right) \sec (c+d x)+6 a^2 A \sec ^3(c+d x)+16 a b \left(A \tan ^2(c+d x)+3 (A+C)\right)\right)}{24 d}","\frac{\left(a^2 (3 A+4 C)+4 b^2 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(a^2 (3 A+4 C)+2 A b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{2 a b (2 A+3 C) \tan (c+d x)}{3 d}+\frac{a A b \tan (c+d x) \sec ^2(c+d x)}{6 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{4 d}",1,"(3*(4*b^2*(A + 2*C) + a^2*(3*A + 4*C))*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(3*(4*A*b^2 + a^2*(3*A + 4*C))*Sec[c + d*x] + 6*a^2*A*Sec[c + d*x]^3 + 16*a*b*(3*(A + C) + A*Tan[c + d*x]^2)))/(24*d)","A",1
539,1,115,187,1.1028353,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{\tan (c+d x) \left(20 \left(a^2 (2 A+C)+A b^2\right) \tan ^2(c+d x)+60 \left(a^2+b^2\right) (A+C)+12 a^2 A \tan ^4(c+d x)+15 a b (3 A+4 C) \sec (c+d x)+30 a A b \sec ^3(c+d x)\right)+15 a b (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{60 d}","\frac{\left(2 a^2 (4 A+5 C)+5 b^2 (2 A+3 C)\right) \tan (c+d x)}{15 d}+\frac{\left(a^2 (4 A+5 C)+2 A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{15 d}+\frac{a b (3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a b (3 A+4 C) \tan (c+d x) \sec (c+d x)}{4 d}+\frac{a A b \tan (c+d x) \sec ^3(c+d x)}{10 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}",1,"(15*a*b*(3*A + 4*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(60*(a^2 + b^2)*(A + C) + 15*a*b*(3*A + 4*C)*Sec[c + d*x] + 30*a*A*b*Sec[c + d*x]^3 + 20*(A*b^2 + a^2*(2*A + C))*Tan[c + d*x]^2 + 12*a^2*A*Tan[c + d*x]^4))/(60*d)","A",1
540,1,252,264,0.7050363,"\int \cos (c+d x) (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2),x]","\frac{80 a^3 C \sin (3 (c+d x))+120 a \left(a^2 (8 A+6 C)+3 b^2 (6 A+5 C)\right) \sin (c+d x)+15 b \left(48 a^2 (A+C)+b^2 (16 A+15 C)\right) \sin (2 (c+d x))+1440 a^2 A b c+1440 a^2 A b d x+90 a^2 b C \sin (4 (c+d x))+1080 a^2 b c C+1080 a^2 b C d x+240 a A b^2 \sin (3 (c+d x))+300 a b^2 C \sin (3 (c+d x))+36 a b^2 C \sin (5 (c+d x))+30 A b^3 \sin (4 (c+d x))+360 A b^3 c+360 A b^3 d x+45 b^3 C \sin (4 (c+d x))+5 b^3 C \sin (6 (c+d x))+300 b^3 c C+300 b^3 C d x}{960 d}","\frac{a \left(5 a^2 (3 A+2 C)+6 b^2 (5 A+4 C)\right) \sin (c+d x)}{15 d}+\frac{b \left(6 a^2 C+5 b^2 (6 A+5 C)\right) \sin (c+d x) \cos ^3(c+d x)}{120 d}+\frac{a \left(C \left(a^2+12 b^2\right)+15 A b^2\right) \sin (c+d x) \cos ^2(c+d x)}{15 d}+\frac{b \left(6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} b x \left(6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right)+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}+\frac{a C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{10 d}",1,"(1440*a^2*A*b*c + 360*A*b^3*c + 1080*a^2*b*c*C + 300*b^3*c*C + 1440*a^2*A*b*d*x + 360*A*b^3*d*x + 1080*a^2*b*C*d*x + 300*b^3*C*d*x + 120*a*(3*b^2*(6*A + 5*C) + a^2*(8*A + 6*C))*Sin[c + d*x] + 15*b*(48*a^2*(A + C) + b^2*(16*A + 15*C))*Sin[2*(c + d*x)] + 240*a*A*b^2*Sin[3*(c + d*x)] + 80*a^3*C*Sin[3*(c + d*x)] + 300*a*b^2*C*Sin[3*(c + d*x)] + 30*A*b^3*Sin[4*(c + d*x)] + 90*a^2*b*C*Sin[4*(c + d*x)] + 45*b^3*C*Sin[4*(c + d*x)] + 36*a*b^2*C*Sin[5*(c + d*x)] + 5*b^3*C*Sin[6*(c + d*x)])/(960*d)","A",1
541,1,160,225,0.7061257,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2),x]","\frac{60 a (c+d x) \left(4 a^2 (2 A+C)+3 b^2 (4 A+3 C)\right)+60 b \left(6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right) \sin (c+d x)+10 b \left(12 a^2 C+4 A b^2+5 b^2 C\right) \sin (3 (c+d x))+120 a \left(C \left(a^2+3 b^2\right)+3 A b^2\right) \sin (2 (c+d x))+45 a b^2 C \sin (4 (c+d x))+6 b^3 C \sin (5 (c+d x))}{480 d}","-\frac{\left(3 a^2 C-4 b^2 (5 A+4 C)\right) \sin (c+d x) (a+b \cos (c+d x))^2}{60 b d}+\frac{a \left(-6 a^2 C+100 A b^2+71 b^2 C\right) \sin (c+d x) \cos (c+d x)}{120 d}+\frac{1}{8} a x \left(4 a^2 (2 A+C)+3 b^2 (4 A+3 C)\right)-\frac{\left(3 a^4 C-4 a^2 b^2 (20 A+13 C)-4 b^4 (5 A+4 C)\right) \sin (c+d x)}{30 b d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^4}{5 b d}-\frac{a C \sin (c+d x) (a+b \cos (c+d x))^3}{20 b d}",1,"(60*a*(4*a^2*(2*A + C) + 3*b^2*(4*A + 3*C))*(c + d*x) + 60*b*(6*a^2*(4*A + 3*C) + b^2*(6*A + 5*C))*Sin[c + d*x] + 120*a*(3*A*b^2 + (a^2 + 3*b^2)*C)*Sin[2*(c + d*x)] + 10*b*(4*A*b^2 + 12*a^2*C + 5*b^2*C)*Sin[3*(c + d*x)] + 45*a*b^2*C*Sin[4*(c + d*x)] + 6*b^3*C*Sin[5*(c + d*x)])/(480*d)","A",1
542,1,180,167,0.5951832,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{-32 a^3 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+32 a^3 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+4 b (c+d x) \left(12 a^2 (2 A+C)+b^2 (4 A+3 C)\right)+8 a \left(4 a^2 C+12 A b^2+9 b^2 C\right) \sin (c+d x)+8 b \left(C \left(3 a^2+b^2\right)+A b^2\right) \sin (2 (c+d x))+8 a b^2 C \sin (3 (c+d x))+b^3 C \sin (4 (c+d x))}{32 d}","\frac{a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a \left(C \left(a^2+4 b^2\right)+6 A b^2\right) \sin (c+d x)}{2 d}+\frac{b \left(2 a^2 C+b^2 (4 A+3 C)\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} b x \left(12 a^2 (2 A+C)+b^2 (4 A+3 C)\right)+\frac{a C \sin (c+d x) (a+b \cos (c+d x))^2}{4 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}",1,"(4*b*(12*a^2*(2*A + C) + b^2*(4*A + 3*C))*(c + d*x) - 32*a^3*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 32*a^3*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 8*a*(12*A*b^2 + 4*a^2*C + 9*b^2*C)*Sin[c + d*x] + 8*b*(A*b^2 + (3*a^2 + b^2)*C)*Sin[2*(c + d*x)] + 8*a*b^2*C*Sin[3*(c + d*x)] + b^3*C*Sin[4*(c + d*x)])/(32*d)","A",1
543,1,185,167,0.8912139,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{12 a^3 A \tan (c+d x)+12 a^3 c C+12 a^3 C d x+3 b \left(3 C \left(4 a^2+b^2\right)+4 A b^2\right) \sin (c+d x)-36 a^2 A b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+36 a^2 A b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+36 a A b^2 c+36 a A b^2 d x+9 a b^2 C \sin (2 (c+d x))+18 a b^2 c C+18 a b^2 C d x+b^3 C \sin (3 (c+d x))}{12 d}","-\frac{b \left(a^2 (6 A-8 C)-b^2 (3 A+2 C)\right) \sin (c+d x)}{3 d}+\frac{1}{2} a x \left(2 a^2 C+6 A b^2+3 b^2 C\right)+\frac{3 a^2 A b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a b^2 (6 A-5 C) \sin (c+d x) \cos (c+d x)}{6 d}-\frac{b (3 A-C) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^3}{d}",1,"(36*a*A*b^2*c + 12*a^3*c*C + 18*a*b^2*c*C + 36*a*A*b^2*d*x + 12*a^3*C*d*x + 18*a*b^2*C*d*x - 36*a^2*A*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 36*a^2*A*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 3*b*(4*A*b^2 + 3*(4*a^2 + b^2)*C)*Sin[c + d*x] + 9*a*b^2*C*Sin[2*(c + d*x)] + b^3*C*Sin[3*(c + d*x)] + 12*a^3*A*Tan[c + d*x])/(12*d)","A",1
544,1,285,168,1.5125138,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{\frac{a^3 A}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{a^3 A}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+2 b (c+d x) \left(C \left(6 a^2+b^2\right)+2 A b^2\right)-2 a \left(a^2 (A+2 C)+6 A b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 a \left(a^2 (A+2 C)+6 A b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{12 a^2 A b \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{12 a^2 A b \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+12 a b^2 C \sin (c+d x)+b^3 C \sin (2 (c+d x))}{4 d}","\frac{a \left(a^2 (A+2 C)+6 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{1}{2} b x \left(C \left(6 a^2+b^2\right)+2 A b^2\right)-\frac{3 a b^2 (3 A-2 C) \sin (c+d x)}{2 d}+\frac{3 A b \tan (c+d x) (a+b \cos (c+d x))^2}{2 d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^3}{2 d}-\frac{b^3 (4 A-C) \sin (c+d x) \cos (c+d x)}{2 d}",1,"(2*b*(2*A*b^2 + (6*a^2 + b^2)*C)*(c + d*x) - 2*a*(6*A*b^2 + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*a*(6*A*b^2 + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a^3*A)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (12*a^2*A*b*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (a^3*A)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (12*a^2*A*b*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 12*a*b^2*C*Sin[c + d*x] + b^3*C*Sin[2*(c + d*x)])/(4*d)","A",1
545,1,377,163,4.2220436,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\frac{2 a^3 A \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2 a^3 A \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{4 a \left(a^2 (2 A+3 C)+9 A b^2\right) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 a \left(a^2 (2 A+3 C)+9 A b^2\right) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}-6 b \left(3 a^2 (A+2 C)+2 A b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 b \left(3 a^2 (A+2 C)+2 A b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{a^2 A (a+9 b)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{a^2 A (a+9 b)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+36 a b^2 C (c+d x)+12 b^3 C \sin (c+d x)}{12 d}","\frac{a \left(a^2 (2 A+3 C)+3 A b^2\right) \tan (c+d x)}{3 d}+\frac{b \left(3 a^2 (A+2 C)+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}+\frac{A b \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}+3 a b^2 C x-\frac{b^3 (5 A-6 C) \sin (c+d x)}{6 d}",1,"(36*a*b^2*C*(c + d*x) - 6*b*(2*A*b^2 + 3*a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*b*(2*A*b^2 + 3*a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a^2*A*(a + 9*b))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (2*a^3*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (4*a*(9*A*b^2 + a^2*(2*A + 3*C))*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (2*a^3*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - (a^2*A*(a + 9*b))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*a*(9*A*b^2 + a^2*(2*A + 3*C))*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 12*b^3*C*Sin[c + d*x])/(12*d)","B",1
546,1,127,182,0.919186,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{a \left(a^2 (3 A+4 C)+12 b^2 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))+8 a^2 A b \tan ^3(c+d x)+\tan (c+d x) \left(2 a^3 A \sec ^3(c+d x)+a \left(a^2 (3 A+4 C)+12 A b^2\right) \sec (c+d x)+8 b \left(3 a^2 (A+C)+A b^2\right)\right)+8 b^3 C d x}{8 d}","\frac{b \left(a^2 (4 A+6 C)+A b^2\right) \tan (c+d x)}{2 d}+\frac{a \left(a^2 (3 A+4 C)+12 b^2 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \left(a^2 (3 A+4 C)+2 A b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac{A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{4 d}+b^3 C x",1,"(8*b^3*C*d*x + a*(12*b^2*(A + 2*C) + a^2*(3*A + 4*C))*ArcTanh[Sin[c + d*x]] + (8*b*(A*b^2 + 3*a^2*(A + C)) + a*(12*A*b^2 + a^2*(3*A + 4*C))*Sec[c + d*x] + 2*a^3*A*Sec[c + d*x]^3)*Tan[c + d*x] + 8*a^2*A*b*Tan[c + d*x]^3)/(8*d)","A",1
547,1,150,227,2.2599584,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{15 b \left(3 a^2 (3 A+4 C)+4 b^2 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(8 a \left(5 \left(a^2 (2 A+C)+3 A b^2\right) \tan ^2(c+d x)+15 \left(a^2+3 b^2\right) (A+C)+3 a^2 A \tan ^4(c+d x)\right)+15 b \left(3 a^2 (3 A+4 C)+4 A b^2\right) \sec (c+d x)+90 a^2 A b \sec ^3(c+d x)\right)}{120 d}","\frac{a \left(2 a^2 (4 A+5 C)+15 b^2 (2 A+3 C)\right) \tan (c+d x)}{15 d}+\frac{b \left(3 a^2 (3 A+4 C)+4 b^2 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \left(2 a^2 (4 A+5 C)+3 A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{30 d}+\frac{3 b \left(5 a^2 (3 A+4 C)+2 A b^2\right) \tan (c+d x) \sec (c+d x)}{40 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{5 d}+\frac{3 A b \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{20 d}",1,"(15*b*(4*b^2*(A + 2*C) + 3*a^2*(3*A + 4*C))*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*b*(4*A*b^2 + 3*a^2*(3*A + 4*C))*Sec[c + d*x] + 90*a^2*A*b*Sec[c + d*x]^3 + 8*a*(15*(a^2 + 3*b^2)*(A + C) + 5*(3*A*b^2 + a^2*(2*A + C))*Tan[c + d*x]^2 + 3*a^2*A*Tan[c + d*x]^4)))/(120*d)","A",1
548,1,184,273,1.713206,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","\frac{15 a \left(a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(40 a^3 A \sec ^5(c+d x)+16 b \left(5 \left(3 a^2 (2 A+C)+A b^2\right) \tan ^2(c+d x)+15 \left(3 a^2+b^2\right) (A+C)+9 a^2 A \tan ^4(c+d x)\right)+10 a \left(a^2 (5 A+6 C)+18 A b^2\right) \sec ^3(c+d x)+15 a \left(a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right) \sec (c+d x)\right)}{240 d}","\frac{b \left(6 a^2 (4 A+5 C)+5 b^2 (2 A+3 C)\right) \tan (c+d x)}{15 d}+\frac{a \left(a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a \left(5 a^2 (5 A+6 C)+6 A b^2\right) \tan (c+d x) \sec ^3(c+d x)}{120 d}+\frac{b \left(3 a^2 (4 A+5 C)+A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{15 d}+\frac{a \left(a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}+\frac{A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{10 d}",1,"(15*a*(6*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*a*(6*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*Sec[c + d*x] + 10*a*(18*A*b^2 + a^2*(5*A + 6*C))*Sec[c + d*x]^3 + 40*a^3*A*Sec[c + d*x]^5 + 16*b*(15*(3*a^2 + b^2)*(A + C) + 5*(A*b^2 + 3*a^2*(2*A + C))*Tan[c + d*x]^2 + 9*a^2*A*Tan[c + d*x]^4)))/(240*d)","A",1
549,1,351,345,0.8465162,"\int \cos (c+d x) (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2),x]","\frac{560 a^4 C \sin (3 (c+d x))+13440 a^3 A b c+13440 a^3 A b d x+840 a^3 b C \sin (4 (c+d x))+10080 a^3 b c C+10080 a^3 b C d x+420 a b \left(16 a^2 (A+C)+b^2 (16 A+15 C)\right) \sin (2 (c+d x))+3360 a^2 A b^2 \sin (3 (c+d x))+4200 a^2 b^2 C \sin (3 (c+d x))+504 a^2 b^2 C \sin (5 (c+d x))+105 \left(16 a^4 (4 A+3 C)+48 a^2 b^2 (6 A+5 C)+5 b^4 (8 A+7 C)\right) \sin (c+d x)+840 a A b^3 \sin (4 (c+d x))+10080 a A b^3 c+10080 a A b^3 d x+1260 a b^3 C \sin (4 (c+d x))+140 a b^3 C \sin (6 (c+d x))+8400 a b^3 c C+8400 a b^3 C d x+700 A b^4 \sin (3 (c+d x))+84 A b^4 \sin (5 (c+d x))+735 b^4 C \sin (3 (c+d x))+147 b^4 C \sin (5 (c+d x))+15 b^4 C \sin (7 (c+d x))}{6720 d}","\frac{a b \left(6 a^2 C+126 A b^2+103 b^2 C\right) \sin (c+d x) \cos ^3(c+d x)}{210 d}+\frac{\left(2 a^2 C+b^2 (7 A+6 C)\right) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{35 d}+\frac{a b \left(a^2 (8 A+6 C)+b^2 (6 A+5 C)\right) \sin (c+d x) \cos (c+d x)}{4 d}+\frac{1}{4} a b x \left(a^2 (8 A+6 C)+b^2 (6 A+5 C)\right)+\frac{\left(35 a^4 (3 A+2 C)+84 a^2 b^2 (5 A+4 C)+8 b^4 (7 A+6 C)\right) \sin (c+d x)}{105 d}+\frac{\left(4 a^4 C+3 a^2 b^2 (63 A+50 C)+4 b^4 (7 A+6 C)\right) \sin (c+d x) \cos ^2(c+d x)}{105 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d}+\frac{2 a C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{21 d}",1,"(13440*a^3*A*b*c + 10080*a*A*b^3*c + 10080*a^3*b*c*C + 8400*a*b^3*c*C + 13440*a^3*A*b*d*x + 10080*a*A*b^3*d*x + 10080*a^3*b*C*d*x + 8400*a*b^3*C*d*x + 105*(16*a^4*(4*A + 3*C) + 48*a^2*b^2*(6*A + 5*C) + 5*b^4*(8*A + 7*C))*Sin[c + d*x] + 420*a*b*(16*a^2*(A + C) + b^2*(16*A + 15*C))*Sin[2*(c + d*x)] + 3360*a^2*A*b^2*Sin[3*(c + d*x)] + 700*A*b^4*Sin[3*(c + d*x)] + 560*a^4*C*Sin[3*(c + d*x)] + 4200*a^2*b^2*C*Sin[3*(c + d*x)] + 735*b^4*C*Sin[3*(c + d*x)] + 840*a*A*b^3*Sin[4*(c + d*x)] + 840*a^3*b*C*Sin[4*(c + d*x)] + 1260*a*b^3*C*Sin[4*(c + d*x)] + 84*A*b^4*Sin[5*(c + d*x)] + 504*a^2*b^2*C*Sin[5*(c + d*x)] + 147*b^4*C*Sin[5*(c + d*x)] + 140*a*b^3*C*Sin[6*(c + d*x)] + 15*b^4*C*Sin[7*(c + d*x)])/(6720*d)","A",1
550,1,301,301,0.8546241,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2),x]","\frac{960 a^4 A c+960 a^4 A d x+480 a^4 c C+480 a^4 C d x+320 a^3 b C \sin (3 (c+d x))+480 a b \left(a^2 (8 A+6 C)+b^2 (6 A+5 C)\right) \sin (c+d x)+2880 a^2 A b^2 c+2880 a^2 A b^2 d x+180 a^2 b^2 C \sin (4 (c+d x))+2160 a^2 b^2 c C+2160 a^2 b^2 C d x+15 \left(16 a^4 C+96 a^2 b^2 (A+C)+b^4 (16 A+15 C)\right) \sin (2 (c+d x))+320 a A b^3 \sin (3 (c+d x))+400 a b^3 C \sin (3 (c+d x))+48 a b^3 C \sin (5 (c+d x))+30 A b^4 \sin (4 (c+d x))+360 A b^4 c+360 A b^4 d x+45 b^4 C \sin (4 (c+d x))+5 b^4 C \sin (6 (c+d x))+300 b^4 c C+300 b^4 C d x}{960 d}","-\frac{\left(4 a^2 C-5 b^2 (6 A+5 C)\right) \sin (c+d x) (a+b \cos (c+d x))^3}{120 b d}+\frac{a \left(-4 a^2 C+70 A b^2+53 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^2}{120 b d}-\frac{a \left(4 a^4 C-a^2 b^2 (190 A+121 C)-32 b^4 (5 A+4 C)\right) \sin (c+d x)}{60 b d}-\frac{\left(8 a^4 C-2 a^2 b^2 (130 A+89 C)-15 b^4 (6 A+5 C)\right) \sin (c+d x) \cos (c+d x)}{240 d}+\frac{1}{16} x \left(8 a^4 (2 A+C)+12 a^2 b^2 (4 A+3 C)+b^4 (6 A+5 C)\right)+\frac{C \sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}-\frac{a C \sin (c+d x) (a+b \cos (c+d x))^4}{30 b d}",1,"(960*a^4*A*c + 2880*a^2*A*b^2*c + 360*A*b^4*c + 480*a^4*c*C + 2160*a^2*b^2*c*C + 300*b^4*c*C + 960*a^4*A*d*x + 2880*a^2*A*b^2*d*x + 360*A*b^4*d*x + 480*a^4*C*d*x + 2160*a^2*b^2*C*d*x + 300*b^4*C*d*x + 480*a*b*(b^2*(6*A + 5*C) + a^2*(8*A + 6*C))*Sin[c + d*x] + 15*(16*a^4*C + 96*a^2*b^2*(A + C) + b^4*(16*A + 15*C))*Sin[2*(c + d*x)] + 320*a*A*b^3*Sin[3*(c + d*x)] + 320*a^3*b*C*Sin[3*(c + d*x)] + 400*a*b^3*C*Sin[3*(c + d*x)] + 30*A*b^4*Sin[4*(c + d*x)] + 180*a^2*b^2*C*Sin[4*(c + d*x)] + 45*b^4*C*Sin[4*(c + d*x)] + 48*a*b^3*C*Sin[5*(c + d*x)] + 5*b^4*C*Sin[6*(c + d*x)])/(960*d)","A",1
551,1,226,227,1.0471237,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{-240 a^4 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+240 a^4 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+120 a b (c+d x) \left(4 a^2 (2 A+C)+b^2 (4 A+3 C)\right)+240 a b \left(C \left(a^2+b^2\right)+A b^2\right) \sin (2 (c+d x))+5 b^2 \left(24 a^2 C+4 A b^2+5 b^2 C\right) \sin (3 (c+d x))+30 \left(8 a^4 C+12 a^2 b^2 (4 A+3 C)+b^4 (6 A+5 C)\right) \sin (c+d x)+30 a b^3 C \sin (4 (c+d x))+3 b^4 C \sin (5 (c+d x))}{240 d}","\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a b \left(6 a^2 C+40 A b^2+29 b^2 C\right) \sin (c+d x) \cos (c+d x)}{30 d}+\frac{\left(3 a^2 C+b^2 (5 A+4 C)\right) \sin (c+d x) (a+b \cos (c+d x))^2}{15 d}+\frac{1}{2} a b x \left(4 a^2 (2 A+C)+b^2 (4 A+3 C)\right)+\frac{\left(6 a^4 C+a^2 b^2 (85 A+56 C)+2 b^4 (5 A+4 C)\right) \sin (c+d x)}{15 d}+\frac{a C \sin (c+d x) (a+b \cos (c+d x))^3}{5 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}",1,"(120*a*b*(4*a^2*(2*A + C) + b^2*(4*A + 3*C))*(c + d*x) - 240*a^4*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 240*a^4*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 30*(8*a^4*C + 12*a^2*b^2*(4*A + 3*C) + b^4*(6*A + 5*C))*Sin[c + d*x] + 240*a*b*(A*b^2 + (a^2 + b^2)*C)*Sin[2*(c + d*x)] + 5*b^2*(4*A*b^2 + 24*a^2*C + 5*b^2*C)*Sin[3*(c + d*x)] + 30*a*b^3*C*Sin[4*(c + d*x)] + 3*b^4*C*Sin[5*(c + d*x)])/(240*d)","A",1
552,1,274,229,1.3569509,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{\frac{96 a^4 A \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{96 a^4 A \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}-384 a^3 A b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+384 a^3 A b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+96 a b \left(4 a^2 C+4 A b^2+3 b^2 C\right) \sin (c+d x)+24 b^2 \left(C \left(6 a^2+b^2\right)+A b^2\right) \sin (2 (c+d x))+12 (c+d x) \left(8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right)+32 a b^3 C \sin (3 (c+d x))+3 b^4 C \sin (4 (c+d x))}{96 d}","\frac{4 a^3 A b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a b \left(a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right) \sin (c+d x)}{6 d}-\frac{b^2 \left(a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right)-\frac{b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}-\frac{a b (12 A-7 C) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^4}{d}",1,"(12*(8*a^4*C + 24*a^2*b^2*(2*A + C) + b^4*(4*A + 3*C))*(c + d*x) - 384*a^3*A*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 384*a^3*A*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (96*a^4*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (96*a^4*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 96*a*b*(4*A*b^2 + 4*a^2*C + 3*b^2*C)*Sin[c + d*x] + 24*b^2*(A*b^2 + (6*a^2 + b^2)*C)*Sin[2*(c + d*x)] + 32*a*b^3*C*Sin[3*(c + d*x)] + 3*b^4*C*Sin[4*(c + d*x)])/(96*d)","A",1
553,1,323,219,2.9916741,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{\frac{3 a^4 A}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{3 a^4 A}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{48 a^3 A b \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{48 a^3 A b \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+24 a b (c+d x) \left(C \left(2 a^2+b^2\right)+2 A b^2\right)+3 b^2 \left(3 C \left(8 a^2+b^2\right)+4 A b^2\right) \sin (c+d x)-6 a^2 \left(a^2 (A+2 C)+12 A b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 a^2 \left(a^2 (A+2 C)+12 A b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+12 a b^3 C \sin (2 (c+d x))+b^4 C \sin (3 (c+d x))}{12 d}","-\frac{b^2 \left(a^2 (39 A-34 C)-2 b^2 (3 A+2 C)\right) \sin (c+d x)}{6 d}+\frac{a^2 \left(a^2 (A+2 C)+12 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+2 a b x \left(C \left(2 a^2+b^2\right)+2 A b^2\right)-\frac{a b^3 (9 A-4 C) \sin (c+d x) \cos (c+d x)}{3 d}-\frac{b^2 (15 A-2 C) \sin (c+d x) (a+b \cos (c+d x))^2}{6 d}+\frac{2 A b \tan (c+d x) (a+b \cos (c+d x))^3}{d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^4}{2 d}",1,"(24*a*b*(2*A*b^2 + (2*a^2 + b^2)*C)*(c + d*x) - 6*a^2*(12*A*b^2 + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*a^2*(12*A*b^2 + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (3*a^4*A)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (48*a^3*A*b*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (3*a^4*A)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (48*a^3*A*b*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 3*b^2*(4*A*b^2 + 3*(8*a^2 + b^2)*C)*Sin[c + d*x] + 12*a*b^3*C*Sin[2*(c + d*x)] + b^4*C*Sin[3*(c + d*x)])/(12*d)","A",1
554,1,412,251,6.2276436,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\frac{2 a^4 A \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2 a^4 A \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{a^3 A (a+12 b)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{a^3 A (a+12 b)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+6 b^2 (c+d x) \left(C \left(12 a^2+b^2\right)+2 A b^2\right)+\frac{4 a^2 \left(a^2 (2 A+3 C)+18 A b^2\right) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 a^2 \left(a^2 (2 A+3 C)+18 A b^2\right) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}-24 a b \left(a^2 (A+2 C)+2 A b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+24 a b \left(a^2 (A+2 C)+2 A b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+48 a b^3 C \sin (c+d x)+3 b^4 C \sin (2 (c+d x))}{12 d}","-\frac{2 a b \left(a^2 (2 A+3 C)+b^2 (11 A-6 C)\right) \sin (c+d x)}{3 d}+\frac{2 a b \left(a^2 (A+2 C)+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^2 \left(a^2 (4 A+6 C)+3 b^2 (6 A-C)\right) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{\left(a^2 (2 A+3 C)+6 A b^2\right) \tan (c+d x) (a+b \cos (c+d x))^2}{3 d}+\frac{1}{2} b^2 x \left(C \left(12 a^2+b^2\right)+2 A b^2\right)+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^4}{3 d}+\frac{2 A b \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^3}{3 d}",1,"(6*b^2*(2*A*b^2 + (12*a^2 + b^2)*C)*(c + d*x) - 24*a*b*(2*A*b^2 + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 24*a*b*(2*A*b^2 + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a^3*A*(a + 12*b))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (2*a^4*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (4*a^2*(18*A*b^2 + a^2*(2*A + 3*C))*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (2*a^4*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - (a^3*A*(a + 12*b))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*a^2*(18*A*b^2 + a^2*(2*A + 3*C))*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 48*a*b^3*C*Sin[c + d*x] + 3*b^4*C*Sin[2*(c + d*x)])/(12*d)","A",1
555,1,612,246,6.344886,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{a^4 A}{16 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}-\frac{a^4 A}{16 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}+\frac{4 \left(2 a^3 A b \sin \left(\frac{1}{2} (c+d x)\right)+3 a^3 b C \sin \left(\frac{1}{2} (c+d x)\right)+3 a A b^3 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{4 \left(2 a^3 A b \sin \left(\frac{1}{2} (c+d x)\right)+3 a^3 b C \sin \left(\frac{1}{2} (c+d x)\right)+3 a A b^3 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{2 a^3 A b \sin \left(\frac{1}{2} (c+d x)\right)}{3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2 a^3 A b \sin \left(\frac{1}{2} (c+d x)\right)}{3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{\left(-3 a^4 A-4 a^4 C-24 a^2 A b^2-48 a^2 b^2 C-8 A b^4\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}+\frac{\left(3 a^4 A+4 a^4 C+24 a^2 A b^2+48 a^2 b^2 C+8 A b^4\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}+\frac{9 a^4 A+12 a^4 C+16 a^3 A b+72 a^2 A b^2}{48 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{-9 a^4 A-12 a^4 C-16 a^3 A b-72 a^2 A b^2}{48 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{4 a b^3 C (c+d x)}{d}+\frac{b^4 C \sin (c+d x)}{d}","-\frac{b^2 \left(3 a^2 (3 A+4 C)+2 b^2 (13 A-12 C)\right) \sin (c+d x)}{24 d}+\frac{a b \left(a^2 (23 A+36 C)+12 A b^2\right) \tan (c+d x)}{12 d}+\frac{\left(a^2 (3 A+4 C)+4 A b^2\right) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{8 d}+\frac{\left(a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+8 A b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}+\frac{A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}+4 a b^3 C x",1,"(4*a*b^3*C*(c + d*x))/d + ((-3*a^4*A - 24*a^2*A*b^2 - 8*A*b^4 - 4*a^4*C - 48*a^2*b^2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(8*d) + ((3*a^4*A + 24*a^2*A*b^2 + 8*A*b^4 + 4*a^4*C + 48*a^2*b^2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(8*d) + (a^4*A)/(16*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4) + (9*a^4*A + 16*a^3*A*b + 72*a^2*A*b^2 + 12*a^4*C)/(48*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (2*a^3*A*b*Sin[(c + d*x)/2])/(3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) - (a^4*A)/(16*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4) + (2*a^3*A*b*Sin[(c + d*x)/2])/(3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (-9*a^4*A - 16*a^3*A*b - 72*a^2*A*b^2 - 12*a^4*C)/(48*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (4*(2*a^3*A*b*Sin[(c + d*x)/2] + 3*a*A*b^3*Sin[(c + d*x)/2] + 3*a^3*b*C*Sin[(c + d*x)/2]))/(3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (4*(2*a^3*A*b*Sin[(c + d*x)/2] + 3*a*A*b^3*Sin[(c + d*x)/2] + 3*a^3*b*C*Sin[(c + d*x)/2]))/(3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (b^4*C*Sin[c + d*x])/d","B",1
556,1,169,250,1.1075048,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{6 a^4 A \tan ^5(c+d x)+10 a^2 \left(a^2 (2 A+C)+6 A b^2\right) \tan ^3(c+d x)+15 a b \left(a^2 (3 A+4 C)+4 b^2 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))+15 \tan (c+d x) \left(2 a^3 A b \sec ^3(c+d x)+a b \left(a^2 (3 A+4 C)+4 A b^2\right) \sec (c+d x)+2 \left(a^4 (A+C)+6 a^2 b^2 (A+C)+A b^4\right)\right)+30 b^4 C d x}{30 d}","\frac{a b \left(a^2 (3 A+4 C)+4 b^2 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a b \left(a^2 (29 A+40 C)+6 A b^2\right) \tan (c+d x) \sec (c+d x)}{30 d}+\frac{\left(a^2 (4 A+5 C)+3 A b^2\right) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{15 d}+\frac{\left(2 a^4 (4 A+5 C)+a^2 b^2 (56 A+85 C)+6 A b^4\right) \tan (c+d x)}{15 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^4}{5 d}+\frac{A b \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{5 d}+b^4 C x",1,"(30*b^4*C*d*x + 15*a*b*(4*b^2*(A + 2*C) + a^2*(3*A + 4*C))*ArcTanh[Sin[c + d*x]] + 15*(2*(A*b^4 + a^4*(A + C) + 6*a^2*b^2*(A + C)) + a*b*(4*A*b^2 + a^2*(3*A + 4*C))*Sec[c + d*x] + 2*a^3*A*b*Sec[c + d*x]^3)*Tan[c + d*x] + 10*a^2*(6*A*b^2 + a^2*(2*A + C))*Tan[c + d*x]^3 + 6*a^4*A*Tan[c + d*x]^5)/(30*d)","A",1
557,1,204,307,4.7858756,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","\frac{15 \left(a^4 (5 A+6 C)+12 a^2 b^2 (3 A+4 C)+8 b^4 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(40 a^4 A \sec ^5(c+d x)+64 a b \left(5 \left(a^2 (2 A+C)+A b^2\right) \tan ^2(c+d x)+15 \left(a^2+b^2\right) (A+C)+3 a^2 A \tan ^4(c+d x)\right)+10 a^2 \left(a^2 (5 A+6 C)+36 A b^2\right) \sec ^3(c+d x)+15 \left(a^4 (5 A+6 C)+12 a^2 b^2 (3 A+4 C)+8 A b^4\right) \sec (c+d x)\right)}{240 d}","\frac{4 a b \left(2 a^2 (4 A+5 C)+5 b^2 (2 A+3 C)\right) \tan (c+d x)}{15 d}+\frac{a b \left(a^2 (39 A+50 C)+4 A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{60 d}+\frac{\left(5 a^2 (5 A+6 C)+12 A b^2\right) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{120 d}+\frac{\left(a^4 (5 A+6 C)+12 a^2 b^2 (3 A+4 C)+8 b^4 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\left(15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)+24 A b^4\right) \tan (c+d x) \sec (c+d x)}{240 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^4}{6 d}+\frac{2 A b \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{15 d}",1,"(15*(8*b^4*(A + 2*C) + 12*a^2*b^2*(3*A + 4*C) + a^4*(5*A + 6*C))*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*(8*A*b^4 + 12*a^2*b^2*(3*A + 4*C) + a^4*(5*A + 6*C))*Sec[c + d*x] + 10*a^2*(36*A*b^2 + a^2*(5*A + 6*C))*Sec[c + d*x]^3 + 40*a^4*A*Sec[c + d*x]^5 + 64*a*b*(15*(a^2 + b^2)*(A + C) + 5*(A*b^2 + a^2*(2*A + C))*Tan[c + d*x]^2 + 3*a^2*A*Tan[c + d*x]^4)))/(240*d)","A",1
558,1,233,355,2.1694781,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^8(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^8,x]","\frac{105 a b \left(a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(60 a^4 A \tan ^6(c+d x)+280 a^3 A b \sec ^5(c+d x)+84 a^2 \left(a^2 (3 A+C)+6 A b^2\right) \tan ^4(c+d x)+70 a b \left(a^2 (5 A+6 C)+6 A b^2\right) \sec ^3(c+d x)+105 a b \left(a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right) \sec (c+d x)+140 \left(a^4 (3 A+2 C)+6 a^2 b^2 (2 A+C)+A b^4\right) \tan ^2(c+d x)+420 \left(a^4+6 a^2 b^2+b^4\right) (A+C)\right)}{420 d}","\frac{a b \left(a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a b \left(a^2 (103 A+126 C)+6 A b^2\right) \tan (c+d x) \sec ^3(c+d x)}{210 d}+\frac{a b \left(a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right) \tan (c+d x) \sec (c+d x)}{4 d}+\frac{\left(a^2 (6 A+7 C)+2 A b^2\right) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{35 d}+\frac{\left(8 a^4 (6 A+7 C)+84 a^2 b^2 (4 A+5 C)+35 b^4 (2 A+3 C)\right) \tan (c+d x)}{105 d}+\frac{\left(4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)+4 A b^4\right) \tan (c+d x) \sec ^2(c+d x)}{105 d}+\frac{A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}+\frac{2 A b \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{21 d}",1,"(105*a*b*(2*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(420*(a^4 + 6*a^2*b^2 + b^4)*(A + C) + 105*a*b*(2*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*Sec[c + d*x] + 70*a*b*(6*A*b^2 + a^2*(5*A + 6*C))*Sec[c + d*x]^3 + 280*a^3*A*b*Sec[c + d*x]^5 + 140*(A*b^4 + 6*a^2*b^2*(2*A + C) + a^4*(3*A + 2*C))*Tan[c + d*x]^2 + 84*a^2*(6*A*b^2 + a^2*(3*A + C))*Tan[c + d*x]^4 + 60*a^4*A*Tan[c + d*x]^6))/(420*d)","A",1
559,1,139,183,0.5757716,"\int (a+b \cos (c+d x))^3 \left(a^2-b^2 \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(a^2 - b^2*Cos[c + d*x]^2),x]","-\frac{-120 a b^2 \left(2 a^2-3 b^2\right) \sin (2 (c+d x))+10 b^3 \left(8 a^2+5 b^2\right) \sin (3 (c+d x))-60 a \left(8 a^4+8 a^2 b^2-9 b^4\right) (c+d x)+60 b \left(-24 a^4+12 a^2 b^2+5 b^4\right) \sin (c+d x)+45 a b^4 \sin (4 (c+d x))+6 b^5 \sin (5 (c+d x))}{480 d}","\frac{b \left(23 a^2-16 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^2}{60 d}+\frac{a b^2 \left(106 a^2-71 b^2\right) \sin (c+d x) \cos (c+d x)}{120 d}+\frac{b \left(83 a^4-32 a^2 b^2-16 b^4\right) \sin (c+d x)}{30 d}+\frac{1}{8} a x \left(8 a^4+8 a^2 b^2-9 b^4\right)-\frac{b \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}+\frac{a b \sin (c+d x) (a+b \cos (c+d x))^3}{20 d}",1,"-1/480*(-60*a*(8*a^4 + 8*a^2*b^2 - 9*b^4)*(c + d*x) + 60*b*(-24*a^4 + 12*a^2*b^2 + 5*b^4)*Sin[c + d*x] - 120*a*b^2*(2*a^2 - 3*b^2)*Sin[2*(c + d*x)] + 10*b^3*(8*a^2 + 5*b^2)*Sin[3*(c + d*x)] + 45*a*b^4*Sin[4*(c + d*x)] + 6*b^5*Sin[5*(c + d*x)])/d","A",1
560,1,89,129,0.223754,"\int (a+b \cos (c+d x))^2 \left(a^2-b^2 \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(a^2 - b^2*Cos[c + d*x]^2),x]","-\frac{-96 a^4 d x-48 a b \left(4 a^2-3 b^2\right) \sin (c+d x)+16 a b^3 \sin (3 (c+d x))+24 b^4 \sin (2 (c+d x))+3 b^4 \sin (4 (c+d x))+36 b^4 c+36 b^4 d x}{96 d}","\frac{1}{8} x \left(8 a^4-3 b^4\right)+\frac{a b \left(13 a^2-8 b^2\right) \sin (c+d x)}{6 d}+\frac{b^2 \left(14 a^2-9 b^2\right) \sin (c+d x) \cos (c+d x)}{24 d}-\frac{b \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac{a b \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}",1,"-1/96*(36*b^4*c - 96*a^4*d*x + 36*b^4*d*x - 48*a*b*(4*a^2 - 3*b^2)*Sin[c + d*x] + 24*b^4*Sin[2*(c + d*x)] + 16*a*b^3*Sin[3*(c + d*x)] + 3*b^4*Sin[4*(c + d*x)])/d","A",1
561,1,75,92,0.1655674,"\int (a+b \cos (c+d x)) \left(a^2-b^2 \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])*(a^2 - b^2*Cos[c + d*x]^2),x]","-\frac{-12 a^3 d x+\left(9 b^3-12 a^2 b\right) \sin (c+d x)+3 a b^2 \sin (2 (c+d x))+6 a b^2 c+6 a b^2 d x+b^3 \sin (3 (c+d x))}{12 d}","\frac{2 b \left(2 a^2-b^2\right) \sin (c+d x)}{3 d}+\frac{1}{2} a x \left(2 a^2-b^2\right)+\frac{a b^2 \sin (c+d x) \cos (c+d x)}{6 d}-\frac{b \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}",1,"-1/12*(6*a*b^2*c - 12*a^3*d*x + 6*a*b^2*d*x + (-12*a^2*b + 9*b^3)*Sin[c + d*x] + 3*a*b^2*Sin[2*(c + d*x)] + b^3*Sin[3*(c + d*x)])/d","A",1
562,1,194,233,0.6455424,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{24 b^2 \left(C \left(a^2+b^2\right)+A b^2\right) \sin (2 (c+d x))-24 a b \left(4 a^2 C+4 A b^2+3 b^2 C\right) \sin (c+d x)+12 (c+d x) \left(8 a^4 C+4 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right)+\frac{192 a^3 \left(a^2 C+A b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}-8 a b^3 C \sin (3 (c+d x))+3 b^4 C \sin (4 (c+d x))}{96 b^5 d}","-\frac{a \left(3 a^2 C+3 A b^2+2 b^2 C\right) \sin (c+d x)}{3 b^4 d}+\frac{\left(4 a^2 C+b^2 (4 A+3 C)\right) \sin (c+d x) \cos (c+d x)}{8 b^3 d}+\frac{x \left(8 a^4 C+4 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right)}{8 b^5}-\frac{2 a^3 \left(a^2 C+A b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d \sqrt{a-b} \sqrt{a+b}}-\frac{a C \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d}+\frac{C \sin (c+d x) \cos ^3(c+d x)}{4 b d}",1,"(12*(8*a^4*C + 4*a^2*b^2*(2*A + C) + b^4*(4*A + 3*C))*(c + d*x) + (192*a^3*(A*b^2 + a^2*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] - 24*a*b*(4*A*b^2 + 4*a^2*C + 3*b^2*C)*Sin[c + d*x] + 24*b^2*(A*b^2 + (a^2 + b^2)*C)*Sin[2*(c + d*x)] - 8*a*b^3*C*Sin[3*(c + d*x)] + 3*b^4*C*Sin[4*(c + d*x)])/(96*b^5*d)","A",1
563,1,152,177,0.4461292,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{-6 a (c+d x) \left(C \left(2 a^2+b^2\right)+2 A b^2\right)+3 b \left(4 a^2 C+4 A b^2+3 b^2 C\right) \sin (c+d x)-\frac{24 a^2 \left(a^2 C+A b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}-3 a b^2 C \sin (2 (c+d x))+b^3 C \sin (3 (c+d x))}{12 b^4 d}","\frac{2 a^2 \left(a^2 C+A b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d \sqrt{a-b} \sqrt{a+b}}-\frac{a x \left(C \left(2 a^2+b^2\right)+2 A b^2\right)}{2 b^4}+\frac{\left(3 a^2 C+b^2 (3 A+2 C)\right) \sin (c+d x)}{3 b^3 d}-\frac{a C \sin (c+d x) \cos (c+d x)}{2 b^2 d}+\frac{C \sin (c+d x) \cos ^2(c+d x)}{3 b d}",1,"(-6*a*(2*A*b^2 + (2*a^2 + b^2)*C)*(c + d*x) - (24*a^2*(A*b^2 + a^2*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + 3*b*(4*A*b^2 + 4*a^2*C + 3*b^2*C)*Sin[c + d*x] - 3*a*b^2*C*Sin[2*(c + d*x)] + b^3*C*Sin[3*(c + d*x)])/(12*b^4*d)","A",1
564,1,117,128,0.4170189,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{2 (c+d x) \left(C \left(2 a^2+b^2\right)+2 A b^2\right)+\frac{8 a \left(a^2 C+A b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}-4 a b C \sin (c+d x)+b^2 C \sin (2 (c+d x))}{4 b^3 d}","-\frac{2 a \left(a^2 C+A b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d \sqrt{a-b} \sqrt{a+b}}+\frac{x \left(2 a^2 C+b^2 (2 A+C)\right)}{2 b^3}-\frac{a C \sin (c+d x)}{b^2 d}+\frac{C \sin (c+d x) \cos (c+d x)}{2 b d}",1,"(2*(2*A*b^2 + (2*a^2 + b^2)*C)*(c + d*x) + (8*a*(A*b^2 + a^2*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] - 4*a*b*C*Sin[c + d*x] + b^2*C*Sin[2*(c + d*x)])/(4*b^3*d)","A",1
565,1,82,86,0.2107098,"\int \frac{A+C \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]),x]","\frac{-\frac{2 \left(a^2 C+A b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}-a C (c+d x)+b C \sin (c+d x)}{b^2 d}","\frac{2 \left(a^2 C+A b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{a C x}{b^2}+\frac{C \sin (c+d x)}{b d}",1,"(-(a*C*(c + d*x)) - (2*(A*b^2 + a^2*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + b*C*Sin[c + d*x])/(b^2*d)","A",1
566,1,234,88,0.5620618,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x]),x]","\frac{2 \left(A+C \cos ^2(c+d x)\right) \left(\sqrt{-\left(\left(a^2-b^2\right) (\cos (c)-i \sin (c))^2\right)} \left(a C d x-A b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+A b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+2 (\sin (c)+i \cos (c)) \left(a^2 C+A b^2\right) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (b \cos (c)-a)+b \sin (c)\right)}{\sqrt{-\left(\left(a^2-b^2\right) (\cos (c)-i \sin (c))^2\right)}}\right)\right)}{a b d \sqrt{\left(b^2-a^2\right) (\cos (2 c)-i \sin (2 c))} (2 A+C \cos (2 (c+d x))+C)}","-\frac{2 \left(a^2 C+A b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a b d \sqrt{a-b} \sqrt{a+b}}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a d}+\frac{C x}{b}",1,"(2*(A + C*Cos[c + d*x]^2)*((a*C*d*x - A*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + A*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*Sqrt[-((a^2 - b^2)*(Cos[c] - I*Sin[c])^2)] + 2*(A*b^2 + a^2*C)*ArcTan[((I*Cos[c] + Sin[c])*(b*Sin[c] + (-a + b*Cos[c])*Tan[(d*x)/2]))/Sqrt[-((a^2 - b^2)*(Cos[c] - I*Sin[c])^2)]]*(I*Cos[c] + Sin[c])))/(a*b*d*(2*A + C + C*Cos[2*(c + d*x)])*Sqrt[(-a^2 + b^2)*(Cos[2*c] - I*Sin[2*c])])","C",1
567,1,306,95,2.2533718,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x]),x]","\frac{2 \cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right) \left(-\frac{2 i (\cos (c)-i \sin (c)) \left(a^2 C+A b^2\right) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (b \cos (c)-a)+b \sin (c)\right)}{\sqrt{-\left(\left(a^2-b^2\right) (\cos (c)-i \sin (c))^2\right)}}\right)}{\sqrt{\left(b^2-a^2\right) (\cos (c)-i \sin (c))^2}}+\frac{a A \sin \left(\frac{d x}{2}\right)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{a A \sin \left(\frac{d x}{2}\right)}{\left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+A b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-A b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{a^2 d (2 A+C \cos (2 (c+d x))+C)}","\frac{2 \left(a^2 C+A b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{A b \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{A \tan (c+d x)}{a d}",1,"(2*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*(A*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - A*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - ((2*I)*(A*b^2 + a^2*C)*ArcTan[((I*Cos[c] + Sin[c])*(b*Sin[c] + (-a + b*Cos[c])*Tan[(d*x)/2]))/Sqrt[-((a^2 - b^2)*(Cos[c] - I*Sin[c])^2)]]*(Cos[c] - I*Sin[c]))/Sqrt[(-a^2 + b^2)*(Cos[c] - I*Sin[c])^2] + (a*A*Sin[(d*x)/2])/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (a*A*Sin[(d*x)/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(a^2*d*(2*A + C + C*Cos[2*(c + d*x)]))","C",0
568,1,399,137,2.1570058,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x]),x]","\frac{\cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right) \left(-2 \left(a^2 (A+2 C)+2 A b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \left(a^2 (A+2 C)+2 A b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{8 b (\sin (c)+i \cos (c)) \left(a^2 C+A b^2\right) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (b \cos (c)-a)+b \sin (c)\right)}{\sqrt{-\left(\left(a^2-b^2\right) (\cos (c)-i \sin (c))^2\right)}}\right)}{\sqrt{\left(b^2-a^2\right) (\cos (c)-i \sin (c))^2}}+\frac{a^2 A}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{a^2 A}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{4 a A b \sin \left(\frac{d x}{2}\right)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{4 a A b \sin \left(\frac{d x}{2}\right)}{\left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}\right)}{2 a^3 d (2 A+C \cos (2 (c+d x))+C)}","-\frac{A b \tan (c+d x)}{a^2 d}-\frac{2 b \left(a^2 C+A b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d \sqrt{a-b} \sqrt{a+b}}+\frac{\left(a^2 (A+2 C)+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{A \tan (c+d x) \sec (c+d x)}{2 a d}",1,"(Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*(-2*(2*A*b^2 + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(2*A*b^2 + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (8*b*(A*b^2 + a^2*C)*ArcTan[((I*Cos[c] + Sin[c])*(b*Sin[c] + (-a + b*Cos[c])*Tan[(d*x)/2]))/Sqrt[-((a^2 - b^2)*(Cos[c] - I*Sin[c])^2)]]*(I*Cos[c] + Sin[c]))/Sqrt[(-a^2 + b^2)*(Cos[c] - I*Sin[c])^2] + (a^2*A)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 - (4*a*A*b*Sin[(d*x)/2])/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - (a^2*A)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - (4*a*A*b*Sin[(d*x)/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(2*a^3*d*(2*A + C + C*Cos[2*(c + d*x)]))","C",0
569,1,413,184,2.9028762,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + b*Cos[c + d*x]),x]","\frac{\frac{2 a^3 A \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2 a^3 A \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{4 a \left(a^2 (2 A+3 C)+3 A b^2\right) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 a \left(a^2 (2 A+3 C)+3 A b^2\right) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}-\frac{24 b^2 \left(a^2 C+A b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+6 b \left(a^2 (A+2 C)+2 A b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-6 b \left(a^2 (A+2 C)+2 A b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{a^2 A (a-3 b)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{a^2 A (a-3 b)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}}{12 a^4 d}","-\frac{A b \tan (c+d x) \sec (c+d x)}{2 a^2 d}+\frac{2 b^2 \left(a^2 C+A b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d \sqrt{a-b} \sqrt{a+b}}-\frac{b \left(a^2 (A+2 C)+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{\left(a^2 (2 A+3 C)+3 A b^2\right) \tan (c+d x)}{3 a^3 d}+\frac{A \tan (c+d x) \sec ^2(c+d x)}{3 a d}",1,"((-24*b^2*(A*b^2 + a^2*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + 6*b*(2*A*b^2 + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 6*b*(2*A*b^2 + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a^2*A*(a - 3*b))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (2*a^3*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (4*a*(3*A*b^2 + a^2*(2*A + 3*C))*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (2*a^3*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - (a^2*A*(a - 3*b))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*a*(3*A*b^2 + a^2*(2*A + 3*C))*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(12*a^4*d)","B",1
570,1,215,332,1.1044649,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{-12 a (c+d x) \left(C \left(4 a^2+b^2\right)+2 A b^2\right)+3 b \left(3 C \left(4 a^2+b^2\right)+4 A b^2\right) \sin (c+d x)+\frac{24 a^2 \left(4 a^4 C+a^2 b^2 (2 A-5 C)-3 A b^4\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}+\frac{12 a^3 b \left(a^2 C+A b^2\right) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}-6 a b^2 C \sin (2 (c+d x))+b^3 C \sin (3 (c+d x))}{12 b^5 d}","-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^3(c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\left(4 a^2 C+3 A b^2-b^2 C\right) \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d \left(a^2-b^2\right)}-\frac{a x \left(C \left(4 a^2+b^2\right)+2 A b^2\right)}{b^5}-\frac{a \left(2 a^2 C+A b^2-b^2 C\right) \sin (c+d x) \cos (c+d x)}{b^3 d \left(a^2-b^2\right)}+\frac{\left(12 a^4 C+a^2 b^2 (6 A-7 C)-b^4 (3 A+2 C)\right) \sin (c+d x)}{3 b^4 d \left(a^2-b^2\right)}+\frac{2 a^2 \left(4 a^4 C+2 a^2 A b^2-5 a^2 b^2 C-3 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{3/2} (a+b)^{3/2}}",1,"(-12*a*(2*A*b^2 + (4*a^2 + b^2)*C)*(c + d*x) + (24*a^2*(-3*A*b^4 + a^2*b^2*(2*A - 5*C) + 4*a^4*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + 3*b*(4*A*b^2 + 3*(4*a^2 + b^2)*C)*Sin[c + d*x] + (12*a^3*b*(A*b^2 + a^2*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])) - 6*a*b^2*C*Sin[2*(c + d*x)] + b^3*C*Sin[3*(c + d*x)])/(12*b^5*d)","A",1
571,1,178,262,0.9867144,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{2 (c+d x) \left(C \left(6 a^2+b^2\right)+2 A b^2\right)-\frac{4 a^2 b \left(a^2 C+A b^2\right) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}-\frac{8 a \left(3 a^4 C+a^2 b^2 (A-4 C)-2 A b^4\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}-8 a b C \sin (c+d x)+b^2 C \sin (2 (c+d x))}{4 b^4 d}","-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^2(c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\left(3 a^2 C+2 A b^2-b^2 C\right) \sin (c+d x) \cos (c+d x)}{2 b^2 d \left(a^2-b^2\right)}+\frac{x \left(C \left(6 a^2+b^2\right)+2 A b^2\right)}{2 b^4}-\frac{a \left(3 a^2 C+A b^2-2 b^2 C\right) \sin (c+d x)}{b^3 d \left(a^2-b^2\right)}-\frac{2 a \left(3 a^4 C+a^2 A b^2-4 a^2 b^2 C-2 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{3/2} (a+b)^{3/2}}",1,"(2*(2*A*b^2 + (6*a^2 + b^2)*C)*(c + d*x) - (8*a*(-2*A*b^4 + a^2*b^2*(A - 4*C) + 3*a^4*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) - 8*a*b*C*Sin[c + d*x] - (4*a^2*b*(A*b^2 + a^2*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])) + b^2*C*Sin[2*(c + d*x)])/(4*b^4*d)","A",1
572,1,136,144,1.0122344,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{\frac{a b \left(a^2 C+A b^2\right) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}-\frac{2 \left(-2 a^4 C+3 a^2 b^2 C+A b^4\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}-2 a C (c+d x)+b C \sin (c+d x)}{b^3 d}","\frac{a \left(a^2 C+A b^2\right) \sin (c+d x)}{b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{2 \left(-2 a^4 C+3 a^2 b^2 C+A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{2 a C x}{b^3}+\frac{C \sin (c+d x)}{b^2 d}",1,"(-2*a*C*(c + d*x) - (2*(A*b^4 - 2*a^4*C + 3*a^2*b^2*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + b*C*Sin[c + d*x] + (a*b*(A*b^2 + a^2*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])))/(b^3*d)","A",1
573,1,123,126,0.6798538,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2,x]","\frac{-\frac{b \left(a^2 C+A b^2\right) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}-\frac{2 a \left(C \left(a^2-2 b^2\right)-A b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}+C (c+d x)}{b^2 d}","\frac{2 a \left(a^2 (-C)+A b^2+2 b^2 C\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{C x}{b^2}",1,"(C*(c + d*x) - (2*a*(-(A*b^2) + (a^2 - 2*b^2)*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) - (b*(A*b^2 + a^2*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])))/(b^2*d)","A",1
574,1,306,134,1.8506595,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^2,x]","\frac{2 \cos (c+d x) (A \sec (c+d x)+C \cos (c+d x)) \left(\frac{a \left(a^2 C+A b^2\right) (b \sin (d x)-a \sin (c))}{b (a-b) (a+b) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) (a+b \cos (c+d x))}+\frac{2 b (\sin (c)+i \cos (c)) \left(a^2 (2 A+C)-A b^2\right) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (b \cos (c)-a)+b \sin (c)\right)}{\sqrt{-\left(\left(a^2-b^2\right) (\cos (c)-i \sin (c))^2\right)}}\right)}{\left(a^2-b^2\right) \sqrt{\left(b^2-a^2\right) (\cos (c)-i \sin (c))^2}}-A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{a^2 d (2 A+C \cos (2 (c+d x))+C)}","-\frac{2 b \left(2 a^2 A+a^2 C-A b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^2 d}",1,"(2*Cos[c + d*x]*(C*Cos[c + d*x] + A*Sec[c + d*x])*(-(A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]) + A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*b*(-(A*b^2) + a^2*(2*A + C))*ArcTan[((I*Cos[c] + Sin[c])*(b*Sin[c] + (-a + b*Cos[c])*Tan[(d*x)/2]))/Sqrt[-((a^2 - b^2)*(Cos[c] - I*Sin[c])^2)]]*(I*Cos[c] + Sin[c]))/((a^2 - b^2)*Sqrt[(-a^2 + b^2)*(Cos[c] - I*Sin[c])^2]) + (a*(A*b^2 + a^2*C)*(-(a*Sin[c]) + b*Sin[d*x]))/((a - b)*b*(a + b)*(a + b*Cos[c + d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2]))))/(a^2*d*(2*A + C + C*Cos[2*(c + d*x)]))","C",0
575,1,219,180,1.7867718,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^2,x]","\frac{2 \cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right) \left(-\frac{a b \left(a^2 C+A b^2\right) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}+\frac{2 \left(a^4 C+3 a^2 A b^2-2 A b^4\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}+a A \tan (c+d x)+2 A b \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{a^3 d (2 A+C \cos (2 (c+d x))+C)}","-\frac{2 A b \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{\left(2 A b^2-a^2 (A-C)\right) \tan (c+d x)}{a^2 d \left(a^2-b^2\right)}+\frac{\left(a^2 C+A b^2\right) \tan (c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{2 \left(a^4 C+3 a^2 A b^2-2 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{3/2} (a+b)^{3/2}}",1,"(2*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*((2*(3*a^2*A*b^2 - 2*A*b^4 + a^4*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + 2*A*b*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - (a*b*(A*b^2 + a^2*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])) + a*A*Tan[c + d*x]))/(a^3*d*(2*A + C + C*Cos[2*(c + d*x)]))","A",1
576,1,712,265,6.3186331,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^2,x]","-\frac{4 A b \sin \left(\frac{1}{2} (c+d x)\right) \cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right)}{a^3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (2 A+C \cos (2 c+2 d x)+C)}-\frac{4 A b \sin \left(\frac{1}{2} (c+d x)\right) \cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right)}{a^3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) (2 A+C \cos (2 c+2 d x)+C)}+\frac{A \cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right)}{2 a^2 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2 (2 A+C \cos (2 c+2 d x)+C)}-\frac{A \cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right)}{2 a^2 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 (2 A+C \cos (2 c+2 d x)+C)}+\frac{\left(a^2 (-A)-2 a^2 C-6 A b^2\right) \cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a^4 d (2 A+C \cos (2 c+2 d x)+C)}+\frac{\left(a^2 A+2 a^2 C+6 A b^2\right) \cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^4 d (2 A+C \cos (2 c+2 d x)+C)}+\frac{4 b \left(2 a^4 C+4 a^2 A b^2-a^2 b^2 C-3 A b^4\right) \cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{a^4 d \left(a^2-b^2\right) \sqrt{b^2-a^2} (2 A+C \cos (2 c+2 d x)+C)}+\frac{2 \cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right) \left(a^2 b^2 C \sin (c+d x)+A b^4 \sin (c+d x)\right)}{a^3 d (a-b) (a+b) (a+b \cos (c+d x)) (2 A+C \cos (2 c+2 d x)+C)}","-\frac{\left(3 A b^2-a^2 (A-2 C)\right) \tan (c+d x) \sec (c+d x)}{2 a^2 d \left(a^2-b^2\right)}+\frac{\left(a^2 C+A b^2\right) \tan (c+d x) \sec (c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\left(a^2 (A+2 C)+6 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{2 b \left(2 a^4 C+4 a^2 A b^2-a^2 b^2 C-3 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{b \left(3 A b^2-a^2 (2 A-C)\right) \tan (c+d x)}{a^3 d \left(a^2-b^2\right)}",1,"(4*b*(4*a^2*A*b^2 - 3*A*b^4 + 2*a^4*C - a^2*b^2*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]]*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2))/(a^4*(a^2 - b^2)*Sqrt[-a^2 + b^2]*d*(2*A + C + C*Cos[2*c + 2*d*x])) + ((-(a^2*A) - 6*A*b^2 - 2*a^2*C)*Cos[c + d*x]^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(C + A*Sec[c + d*x]^2))/(a^4*d*(2*A + C + C*Cos[2*c + 2*d*x])) + ((a^2*A + 6*A*b^2 + 2*a^2*C)*Cos[c + d*x]^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(C + A*Sec[c + d*x]^2))/(a^4*d*(2*A + C + C*Cos[2*c + 2*d*x])) + (A*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2))/(2*a^2*d*(2*A + C + C*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) - (4*A*b*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*Sin[(c + d*x)/2])/(a^3*d*(2*A + C + C*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - (A*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2))/(2*a^2*d*(2*A + C + C*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) - (4*A*b*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*Sin[(c + d*x)/2])/(a^3*d*(2*A + C + C*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (2*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*(A*b^4*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(a^3*(a - b)*(a + b)*d*(a + b*Cos[c + d*x])*(2*A + C + C*Cos[2*c + 2*d*x]))","B",1
577,1,593,335,6.2795099,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + b*Cos[c + d*x])^2,x]","\frac{A (a-6 b)}{12 a^3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{A (a-6 b)}{12 a^3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{A \sin \left(\frac{1}{2} (c+d x)\right)}{6 a^2 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{A \sin \left(\frac{1}{2} (c+d x)\right)}{6 a^2 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{\left(a^2 A b+2 a^2 b C+4 A b^3\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a^5 d}+\frac{\left(-a^2 A b-2 a^2 b C-4 A b^3\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^5 d}+\frac{2 a^2 A \sin \left(\frac{1}{2} (c+d x)\right)+3 a^2 C \sin \left(\frac{1}{2} (c+d x)\right)+9 A b^2 \sin \left(\frac{1}{2} (c+d x)\right)}{3 a^4 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{2 a^2 A \sin \left(\frac{1}{2} (c+d x)\right)+3 a^2 C \sin \left(\frac{1}{2} (c+d x)\right)+9 A b^2 \sin \left(\frac{1}{2} (c+d x)\right)}{3 a^4 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{-a^2 b^3 C \sin (c+d x)-A b^5 \sin (c+d x)}{a^4 d (a-b) (a+b) (a+b \cos (c+d x))}-\frac{2 b^2 \left(3 a^4 C+5 a^2 A b^2-2 a^2 b^2 C-4 A b^4\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{a^5 d \left(a^2-b^2\right) \sqrt{b^2-a^2}}","-\frac{\left(4 A b^2-a^2 (A-3 C)\right) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d \left(a^2-b^2\right)}+\frac{\left(a^2 C+A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{b \left(a^2 (A+2 C)+4 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{\left(-\left(a^4 (2 A+3 C)\right)-a^2 b^2 (7 A-6 C)+12 A b^4\right) \tan (c+d x)}{3 a^4 d \left(a^2-b^2\right)}+\frac{b \left(2 A b^2-a^2 (A-C)\right) \tan (c+d x) \sec (c+d x)}{a^3 d \left(a^2-b^2\right)}+\frac{2 b^2 \left(3 a^4 C+5 a^2 A b^2-2 a^2 b^2 C-4 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{3/2} (a+b)^{3/2}}",1,"(-2*b^2*(5*a^2*A*b^2 - 4*A*b^4 + 3*a^4*C - 2*a^2*b^2*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(a^5*(a^2 - b^2)*Sqrt[-a^2 + b^2]*d) + ((a^2*A*b + 4*A*b^3 + 2*a^2*b*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(a^5*d) + ((-(a^2*A*b) - 4*A*b^3 - 2*a^2*b*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(a^5*d) + (A*(a - 6*b))/(12*a^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (A*Sin[(c + d*x)/2])/(6*a^2*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + (A*Sin[(c + d*x)/2])/(6*a^2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - (A*(a - 6*b))/(12*a^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (2*a^2*A*Sin[(c + d*x)/2] + 9*A*b^2*Sin[(c + d*x)/2] + 3*a^2*C*Sin[(c + d*x)/2])/(3*a^4*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (2*a^2*A*Sin[(c + d*x)/2] + 9*A*b^2*Sin[(c + d*x)/2] + 3*a^2*C*Sin[(c + d*x)/2])/(3*a^4*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (-(A*b^5*Sin[c + d*x]) - a^2*b^3*C*Sin[c + d*x])/(a^4*(a - b)*(a + b)*d*(a + b*Cos[c + d*x]))","A",0
578,1,256,372,2.3759577,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{2 (c+d x) \left(C \left(12 a^2+b^2\right)+2 A b^2\right)+\frac{2 a^2 b \left(-7 a^4 C+a^2 b^2 (10 C-3 A)+6 A b^4\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}+\frac{2 a^3 b \left(a^2 C+A b^2\right) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}+\frac{4 a \left(12 a^6 C+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+6 A b^6\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}-12 a b C \sin (c+d x)+b^2 C \sin (2 (c+d x))}{4 b^5 d}","-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{x \left(C \left(12 a^2+b^2\right)+2 A b^2\right)}{2 b^5}-\frac{a \left(12 a^4 C+a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)\right) \sin (c+d x)}{2 b^4 d \left(a^2-b^2\right)^2}+\frac{\left(-4 a^4 C+7 a^2 b^2 C+3 A b^4\right) \sin (c+d x) \cos ^2(c+d x)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(6 a^4 C+a^2 b^2 (A-10 C)-b^4 (4 A-C)\right) \sin (c+d x) \cos (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2}-\frac{a \left(12 a^6 C+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+6 A b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{5/2} (a+b)^{5/2}}",1,"(2*(2*A*b^2 + (12*a^2 + b^2)*C)*(c + d*x) + (4*a*(6*A*b^6 + a^4*b^2*(2*A - 29*C) - 5*a^2*b^4*(A - 4*C) + 12*a^6*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) - 12*a*b*C*Sin[c + d*x] + (2*a^3*b*(A*b^2 + a^2*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^2) + (2*a^2*b*(6*A*b^4 - 7*a^4*C + a^2*b^2*(-3*A + 10*C))*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])) + b^2*C*Sin[2*(c + d*x)])/(4*b^5*d)","A",1
579,1,214,262,1.629818,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{-\frac{a^2 b \left(a^2 C+A b^2\right) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}+\frac{a b \left(5 a^4 C+a^2 b^2 (A-8 C)-4 A b^4\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}-\frac{2 \left(6 a^6 C-15 a^4 b^2 C+a^2 b^4 (A+12 C)+2 A b^6\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}-6 a C (c+d x)+2 b C \sin (c+d x)}{2 b^4 d}","-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^2(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\left(3 a^2 C+A b^2-2 b^2 C\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right)}-\frac{a \left(-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(6 a^6 C-15 a^4 b^2 C+a^2 b^4 (A+12 C)+2 A b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{3 a C x}{b^4}",1,"(-6*a*C*(c + d*x) - (2*(2*A*b^6 + 6*a^6*C - 15*a^4*b^2*C + a^2*b^4*(A + 12*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + 2*b*C*Sin[c + d*x] - (a^2*b*(A*b^2 + a^2*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^2) + (a*b*(-4*A*b^4 + a^2*b^2*(A - 8*C) + 5*a^4*C)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])))/(2*b^4*d)","A",1
580,1,194,203,1.2890905,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{a b \left(a^2 C+A b^2\right) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}+\frac{b \left(-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}+\frac{2 a \left(C \left(2 a^4-5 a^2 b^2+6 b^4\right)+3 A b^4\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}+2 C (c+d x)}{2 b^3 d}","\frac{a \left(a^2 C+A b^2\right) \sin (c+d x)}{2 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\left(-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right) \sin (c+d x)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{a \left(C \left(2 a^4-5 a^2 b^2+6 b^4\right)+3 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{C x}{b^3}",1,"(2*C*(c + d*x) + (2*a*(3*A*b^4 + (2*a^4 - 5*a^2*b^2 + 6*b^4)*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (a*b*(A*b^2 + a^2*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^2) + (b*(2*A*b^4 - 3*a^4*C + a^2*b^2*(A + 6*C))*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])))/(2*b^3*d)","A",1
581,1,170,177,0.8115408,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{a \left(C \left(a^2-4 b^2\right)-3 A b^2\right) \sin (c+d x)}{b (a-b)^2 (a+b)^2 (a+b \cos (c+d x))}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{b (b-a) (a+b) (a+b \cos (c+d x))^2}-\frac{2 \left(a^2 (2 A+C)+b^2 (A+2 C)\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}}{2 d}","\frac{\left(a^2 (2 A+C)+b^2 (A+2 C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{a \left(a^2 (-C)+3 A b^2+4 b^2 C\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"((-2*(a^2*(2*A + C) + b^2*(A + 2*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(b*(-a + b)*(a + b)*(a + b*Cos[c + d*x])^2) + (a*(-3*A*b^2 + (a^2 - 4*b^2)*C)*Sin[c + d*x])/((a - b)^2*b*(a + b)^2*(a + b*Cos[c + d*x])))/(2*d)","A",1
582,1,409,211,3.5738285,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^3,x]","\frac{\cos (c+d x) (A \sec (c+d x)+C \cos (c+d x)) \left(\frac{4 b (\sin (c)+i \cos (c)) \left(3 a^4 (2 A+C)-5 a^2 A b^2+2 A b^4\right) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (b \cos (c)-a)+b \sin (c)\right)}{\sqrt{-\left(\left(a^2-b^2\right) (\cos (c)-i \sin (c))^2\right)}}\right)}{\left(a^2-b^2\right)^2 \sqrt{\left(b^2-a^2\right) (\cos (c)-i \sin (c))^2}}-\frac{a \sec (c) \left(b \left(b \left(a b \left(a^2 (4 A+3 C)-A b^2\right) \sin (2 c+d x)-\left(a^4 C+a^2 b^2 (5 A+2 C)-2 A b^4\right) \sin (c+2 d x)\right)-a \sin (d x) \left(4 a^4 C+a^2 b^2 (16 A+5 C)-7 A b^4\right)\right)+\left(2 a^2+b^2\right) \sin (c) \left(a^4 C+a^2 b^2 (5 A+2 C)-2 A b^4\right)\right)}{b \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-4 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{2 a^3 d (2 A+C \cos (2 (c+d x))+C)}","\frac{A \tanh ^{-1}(\sin (c+d x))}{a^3 d}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{\left(a^4 (-C)-a^2 b^2 (5 A+2 C)+2 A b^4\right) \sin (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{b \left(-3 a^4 (2 A+C)+5 a^2 A b^2-2 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}",1,"(Cos[c + d*x]*(C*Cos[c + d*x] + A*Sec[c + d*x])*(-4*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (4*b*(-5*a^2*A*b^2 + 2*A*b^4 + 3*a^4*(2*A + C))*ArcTan[((I*Cos[c] + Sin[c])*(b*Sin[c] + (-a + b*Cos[c])*Tan[(d*x)/2]))/Sqrt[-((a^2 - b^2)*(Cos[c] - I*Sin[c])^2)]]*(I*Cos[c] + Sin[c]))/((a^2 - b^2)^2*Sqrt[(-a^2 + b^2)*(Cos[c] - I*Sin[c])^2]) - (a*Sec[c]*((2*a^2 + b^2)*(-2*A*b^4 + a^4*C + a^2*b^2*(5*A + 2*C))*Sin[c] + b*(-(a*(-7*A*b^4 + 4*a^4*C + a^2*b^2*(16*A + 5*C))*Sin[d*x]) + b*(a*b*(-(A*b^2) + a^2*(4*A + 3*C))*Sin[2*c + d*x] - (-2*A*b^4 + a^4*C + a^2*b^2*(5*A + 2*C))*Sin[c + 2*d*x]))))/(b*(a^2 - b^2)^2*(a + b*Cos[c + d*x])^2)))/(2*a^3*d*(2*A + C + C*Cos[2*(c + d*x)]))","C",0
583,1,649,275,6.3323703,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^3,x]","\frac{6 A b \cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a^4 d (2 A+C \cos (2 c+2 d x)+C)}-\frac{6 A b \cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^4 d (2 A+C \cos (2 c+2 d x)+C)}+\frac{2 A \sin \left(\frac{1}{2} (c+d x)\right) \cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right)}{a^3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (2 A+C \cos (2 c+2 d x)+C)}+\frac{2 A \sin \left(\frac{1}{2} (c+d x)\right) \cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right)}{a^3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) (2 A+C \cos (2 c+2 d x)+C)}+\frac{\cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right) \left(-a^2 b C \sin (c+d x)-A b^3 \sin (c+d x)\right)}{a^2 d (a-b) (a+b) (a+b \cos (c+d x))^2 (2 A+C \cos (2 c+2 d x)+C)}-\frac{2 \left(2 a^6 C+12 a^4 A b^2+a^4 b^2 C-15 a^2 A b^4+6 A b^6\right) \cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{a^4 d \left(a^2-b^2\right)^2 \sqrt{b^2-a^2} (2 A+C \cos (2 c+2 d x)+C)}+\frac{\cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right) \left(-3 a^4 b C \sin (c+d x)-7 a^2 A b^3 \sin (c+d x)+4 A b^5 \sin (c+d x)\right)}{a^3 d (a-b)^2 (a+b)^2 (a+b \cos (c+d x)) (2 A+C \cos (2 c+2 d x)+C)}","-\frac{3 A b \tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac{\left(a^2 C+A b^2\right) \tan (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{\left(-2 a^4 C-a^2 b^2 (6 A+C)+3 A b^4\right) \tan (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\left(-2 a^6 C-a^4 b^2 (12 A+C)+15 a^2 A b^4-6 A b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\left(-\left(a^4 (2 A-3 C)\right)+11 a^2 A b^2-6 A b^4\right) \tan (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}",1,"(-2*(12*a^4*A*b^2 - 15*a^2*A*b^4 + 6*A*b^6 + 2*a^6*C + a^4*b^2*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]]*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2))/(a^4*(a^2 - b^2)^2*Sqrt[-a^2 + b^2]*d*(2*A + C + C*Cos[2*c + 2*d*x])) + (6*A*b*Cos[c + d*x]^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(C + A*Sec[c + d*x]^2))/(a^4*d*(2*A + C + C*Cos[2*c + 2*d*x])) - (6*A*b*Cos[c + d*x]^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(C + A*Sec[c + d*x]^2))/(a^4*d*(2*A + C + C*Cos[2*c + 2*d*x])) + (2*A*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*Sin[(c + d*x)/2])/(a^3*d*(2*A + C + C*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (2*A*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*Sin[(c + d*x)/2])/(a^3*d*(2*A + C + C*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*(-(A*b^3*Sin[c + d*x]) - a^2*b*C*Sin[c + d*x]))/(a^2*(a - b)*(a + b)*d*(a + b*Cos[c + d*x])^2*(2*A + C + C*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*(-7*a^2*A*b^3*Sin[c + d*x] + 4*A*b^5*Sin[c + d*x] - 3*a^4*b*C*Sin[c + d*x]))/(a^3*(a - b)^2*(a + b)^2*d*(a + b*Cos[c + d*x])*(2*A + C + C*Cos[2*c + 2*d*x]))","B",1
584,1,856,378,6.3886891,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^3,x]","\frac{\left(-A a^2-2 C a^2-12 A b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(A \sec ^2(c+d x)+C\right) \cos ^2(c+d x)}{a^5 d (2 A+C+C \cos (2 c+2 d x))}+\frac{\left(A a^2+2 C a^2+12 A b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(A \sec ^2(c+d x)+C\right) \cos ^2(c+d x)}{a^5 d (2 A+C+C \cos (2 c+2 d x))}+\frac{2 b \left(6 C a^6+20 A b^2 a^4-5 b^2 C a^4-29 A b^4 a^2+2 b^4 C a^2+12 A b^6\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right) \left(A \sec ^2(c+d x)+C\right) \cos ^2(c+d x)}{a^5 \left(a^2-b^2\right)^2 \sqrt{b^2-a^2} d (2 A+C+C \cos (2 c+2 d x))}-\frac{6 A b \left(A \sec ^2(c+d x)+C\right) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^2(c+d x)}{a^4 d (2 A+C+C \cos (2 c+2 d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{\left(A \sec ^2(c+d x)+C\right) \left(A \sin (c+d x) b^4+a^2 C \sin (c+d x) b^2\right) \cos ^2(c+d x)}{a^3 (a-b) (a+b) d (a+b \cos (c+d x))^2 (2 A+C+C \cos (2 c+2 d x))}+\frac{\left(A \sec ^2(c+d x)+C\right) \left(-6 A \sin (c+d x) b^6+9 a^2 A \sin (c+d x) b^4-2 a^2 C \sin (c+d x) b^4+5 a^4 C \sin (c+d x) b^2\right) \cos ^2(c+d x)}{a^4 (a-b)^2 (a+b)^2 d (a+b \cos (c+d x)) (2 A+C+C \cos (2 c+2 d x))}-\frac{6 A b \left(A \sec ^2(c+d x)+C\right) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^2(c+d x)}{a^4 d (2 A+C+C \cos (2 c+2 d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{A \left(A \sec ^2(c+d x)+C\right) \cos ^2(c+d x)}{2 a^3 d (2 A+C+C \cos (2 c+2 d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{A \left(A \sec ^2(c+d x)+C\right) \cos ^2(c+d x)}{2 a^3 d (2 A+C+C \cos (2 c+2 d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}","\frac{\left(a^2 C+A b^2\right) \tan (c+d x) \sec (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\left(a^2 (A+2 C)+12 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}-\frac{b \left(a^4 (6 A-5 C)-a^2 b^2 (21 A-2 C)+12 A b^4\right) \tan (c+d x)}{2 a^4 d \left(a^2-b^2\right)^2}+\frac{\left(3 a^4 C+7 a^2 A b^2-4 A b^4\right) \tan (c+d x) \sec (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(a^4 (A-4 C)-a^2 b^2 (10 A-C)+6 A b^4\right) \tan (c+d x) \sec (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}-\frac{b \left(6 a^6 C+5 a^4 b^2 (4 A-C)-a^2 b^4 (29 A-2 C)+12 A b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{5/2} (a+b)^{5/2}}",1,"(2*b*(20*a^4*A*b^2 - 29*a^2*A*b^4 + 12*A*b^6 + 6*a^6*C - 5*a^4*b^2*C + 2*a^2*b^4*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]]*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2))/(a^5*(a^2 - b^2)^2*Sqrt[-a^2 + b^2]*d*(2*A + C + C*Cos[2*c + 2*d*x])) + ((-(a^2*A) - 12*A*b^2 - 2*a^2*C)*Cos[c + d*x]^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(C + A*Sec[c + d*x]^2))/(a^5*d*(2*A + C + C*Cos[2*c + 2*d*x])) + ((a^2*A + 12*A*b^2 + 2*a^2*C)*Cos[c + d*x]^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(C + A*Sec[c + d*x]^2))/(a^5*d*(2*A + C + C*Cos[2*c + 2*d*x])) + (A*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2))/(2*a^3*d*(2*A + C + C*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) - (6*A*b*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*Sin[(c + d*x)/2])/(a^4*d*(2*A + C + C*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - (A*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2))/(2*a^3*d*(2*A + C + C*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) - (6*A*b*Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*Sin[(c + d*x)/2])/(a^4*d*(2*A + C + C*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*(A*b^4*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(a^3*(a - b)*(a + b)*d*(a + b*Cos[c + d*x])^2*(2*A + C + C*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*(9*a^2*A*b^4*Sin[c + d*x] - 6*A*b^6*Sin[c + d*x] + 5*a^4*b^2*C*Sin[c + d*x] - 2*a^2*b^4*C*Sin[c + d*x]))/(a^4*(a - b)^2*(a + b)^2*d*(a + b*Cos[c + d*x])*(2*A + C + C*Cos[2*c + 2*d*x]))","B",0
585,1,1452,514,6.6775983,"\int \frac{\cos ^4(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^4,x]","\frac{a \left(20 C a^8+2 A b^2 a^6-69 b^2 C a^6-7 A b^4 a^4+84 b^4 C a^4+8 A b^6 a^2-40 b^6 C a^2-8 A b^8\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{b^6 \left(a^2-b^2\right)^3 \sqrt{b^2-a^2} d}-\frac{960 C (c+d x) a^{11}+2880 b C (c+d x) \cos (c+d x) a^{10}-960 b C \sin (c+d x) a^{10}+96 A b^2 (c+d x) a^9-1392 b^2 C (c+d x) a^9+1440 b^2 C (c+d x) \cos (2 (c+d x)) a^9-1200 b^2 C \sin (2 (c+d x)) a^9+288 A b^3 (c+d x) \cos (c+d x) a^8-7776 b^3 C (c+d x) \cos (c+d x) a^8+240 b^3 C (c+d x) \cos (3 (c+d x)) a^8-96 A b^3 \sin (c+d x) a^8+2232 b^3 C \sin (c+d x) a^8-440 b^3 C \sin (3 (c+d x)) a^8-144 A b^4 (c+d x) a^7-1512 b^4 C (c+d x) a^7+144 A b^4 (c+d x) \cos (2 (c+d x)) a^7-4248 b^4 C (c+d x) \cos (2 (c+d x)) a^7-120 A b^4 \sin (2 (c+d x)) a^7+3300 b^4 C \sin (2 (c+d x)) a^7-30 b^4 C \sin (4 (c+d x)) a^7-792 A b^5 (c+d x) \cos (c+d x) a^6+6084 b^5 C (c+d x) \cos (c+d x) a^6+24 A b^5 (c+d x) \cos (3 (c+d x)) a^6-708 b^5 C (c+d x) \cos (3 (c+d x)) a^6+228 A b^5 \sin (c+d x) a^6-1086 b^5 C \sin (c+d x) a^6-44 A b^5 \sin (3 (c+d x)) a^6+1253 b^5 C \sin (3 (c+d x)) a^6+3 b^5 C \sin (5 (c+d x)) a^6-144 A b^6 (c+d x) a^5+3288 b^6 C (c+d x) a^5-432 A b^6 (c+d x) \cos (2 (c+d x)) a^5+4104 b^6 C (c+d x) \cos (2 (c+d x)) a^5+360 A b^6 \sin (2 (c+d x)) a^5-2772 b^6 C \sin (2 (c+d x)) a^5+90 b^6 C \sin (4 (c+d x)) a^5+648 A b^7 (c+d x) \cos (c+d x) a^4-396 b^7 C (c+d x) \cos (c+d x) a^4-72 A b^7 (c+d x) \cos (3 (c+d x)) a^4+684 b^7 C (c+d x) \cos (3 (c+d x)) a^4-288 A b^7 \sin (c+d x) a^4-750 b^7 C \sin (c+d x) a^4+128 A b^7 \sin (3 (c+d x)) a^4-1143 b^7 C \sin (3 (c+d x)) a^4-9 b^7 C \sin (5 (c+d x)) a^4+336 A b^8 (c+d x) a^3-1272 b^8 C (c+d x) a^3+432 A b^8 (c+d x) \cos (2 (c+d x)) a^3-1224 b^8 C (c+d x) \cos (2 (c+d x)) a^3-480 A b^8 \sin (2 (c+d x)) a^3+372 b^8 C \sin (2 (c+d x)) a^3-90 b^8 C \sin (4 (c+d x)) a^3-72 A b^9 (c+d x) \cos (c+d x) a^2-756 b^9 C (c+d x) \cos (c+d x) a^2+72 A b^9 (c+d x) \cos (3 (c+d x)) a^2-204 b^9 C (c+d x) \cos (3 (c+d x)) a^2-144 A b^9 \sin (c+d x) a^2+270 b^9 C \sin (c+d x) a^2-144 A b^9 \sin (3 (c+d x)) a^2+279 b^9 C \sin (3 (c+d x)) a^2+9 b^9 C \sin (5 (c+d x)) a^2-144 A b^{10} (c+d x) a-72 b^{10} C (c+d x) a-144 A b^{10} (c+d x) \cos (2 (c+d x)) a-72 b^{10} C (c+d x) \cos (2 (c+d x)) a+60 b^{10} C \sin (2 (c+d x)) a+30 b^{10} C \sin (4 (c+d x)) a-72 A b^{11} (c+d x) \cos (c+d x)-36 b^{11} C (c+d x) \cos (c+d x)-24 A b^{11} (c+d x) \cos (3 (c+d x))-12 b^{11} C (c+d x) \cos (3 (c+d x))-6 b^{11} C \sin (c+d x)-9 b^{11} C \sin (3 (c+d x))-3 b^{11} C \sin (5 (c+d x))}{96 b^6 \left(b^2-a^2\right)^3 d (a+b \cos (c+d x))^3}","-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^4(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{x \left(C \left(20 a^2+b^2\right)+2 A b^2\right)}{2 b^6}+\frac{\left(-5 a^4 C+a^2 b^2 (A+10 C)+4 A b^4\right) \sin (c+d x) \cos ^3(c+d x)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\left(10 a^6 C+a^4 b^2 (A-27 C)-a^2 b^4 (2 A-23 C)+b^6 (6 A-C)\right) \sin (c+d x) \cos (c+d x)}{2 b^4 d \left(a^2-b^2\right)^3}-\frac{a \left(60 a^6 C+a^4 b^2 (6 A-167 C)-a^2 b^4 (17 A-146 C)+2 b^6 (13 A-12 C)\right) \sin (c+d x)}{6 b^5 d \left(a^2-b^2\right)^3}-\frac{\left(20 a^6 C+a^4 b^2 (2 A-53 C)+a^2 b^4 (A+48 C)+12 A b^6\right) \sin (c+d x) \cos ^2(c+d x)}{6 b^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{\left(-20 a^9 C-a^7 b^2 (2 A-69 C)+7 a^5 b^4 (A-12 C)-8 a^3 b^6 (A-5 C)+8 a A b^8\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^6 d \sqrt{a-b} \sqrt{a+b} \left(a^2-b^2\right)^3}",1,"(a*(2*a^6*A*b^2 - 7*a^4*A*b^4 + 8*a^2*A*b^6 - 8*A*b^8 + 20*a^8*C - 69*a^6*b^2*C + 84*a^4*b^4*C - 40*a^2*b^6*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(b^6*(a^2 - b^2)^3*Sqrt[-a^2 + b^2]*d) - (96*a^9*A*b^2*(c + d*x) - 144*a^7*A*b^4*(c + d*x) - 144*a^5*A*b^6*(c + d*x) + 336*a^3*A*b^8*(c + d*x) - 144*a*A*b^10*(c + d*x) + 960*a^11*C*(c + d*x) - 1392*a^9*b^2*C*(c + d*x) - 1512*a^7*b^4*C*(c + d*x) + 3288*a^5*b^6*C*(c + d*x) - 1272*a^3*b^8*C*(c + d*x) - 72*a*b^10*C*(c + d*x) + 288*a^8*A*b^3*(c + d*x)*Cos[c + d*x] - 792*a^6*A*b^5*(c + d*x)*Cos[c + d*x] + 648*a^4*A*b^7*(c + d*x)*Cos[c + d*x] - 72*a^2*A*b^9*(c + d*x)*Cos[c + d*x] - 72*A*b^11*(c + d*x)*Cos[c + d*x] + 2880*a^10*b*C*(c + d*x)*Cos[c + d*x] - 7776*a^8*b^3*C*(c + d*x)*Cos[c + d*x] + 6084*a^6*b^5*C*(c + d*x)*Cos[c + d*x] - 396*a^4*b^7*C*(c + d*x)*Cos[c + d*x] - 756*a^2*b^9*C*(c + d*x)*Cos[c + d*x] - 36*b^11*C*(c + d*x)*Cos[c + d*x] + 144*a^7*A*b^4*(c + d*x)*Cos[2*(c + d*x)] - 432*a^5*A*b^6*(c + d*x)*Cos[2*(c + d*x)] + 432*a^3*A*b^8*(c + d*x)*Cos[2*(c + d*x)] - 144*a*A*b^10*(c + d*x)*Cos[2*(c + d*x)] + 1440*a^9*b^2*C*(c + d*x)*Cos[2*(c + d*x)] - 4248*a^7*b^4*C*(c + d*x)*Cos[2*(c + d*x)] + 4104*a^5*b^6*C*(c + d*x)*Cos[2*(c + d*x)] - 1224*a^3*b^8*C*(c + d*x)*Cos[2*(c + d*x)] - 72*a*b^10*C*(c + d*x)*Cos[2*(c + d*x)] + 24*a^6*A*b^5*(c + d*x)*Cos[3*(c + d*x)] - 72*a^4*A*b^7*(c + d*x)*Cos[3*(c + d*x)] + 72*a^2*A*b^9*(c + d*x)*Cos[3*(c + d*x)] - 24*A*b^11*(c + d*x)*Cos[3*(c + d*x)] + 240*a^8*b^3*C*(c + d*x)*Cos[3*(c + d*x)] - 708*a^6*b^5*C*(c + d*x)*Cos[3*(c + d*x)] + 684*a^4*b^7*C*(c + d*x)*Cos[3*(c + d*x)] - 204*a^2*b^9*C*(c + d*x)*Cos[3*(c + d*x)] - 12*b^11*C*(c + d*x)*Cos[3*(c + d*x)] - 96*a^8*A*b^3*Sin[c + d*x] + 228*a^6*A*b^5*Sin[c + d*x] - 288*a^4*A*b^7*Sin[c + d*x] - 144*a^2*A*b^9*Sin[c + d*x] - 960*a^10*b*C*Sin[c + d*x] + 2232*a^8*b^3*C*Sin[c + d*x] - 1086*a^6*b^5*C*Sin[c + d*x] - 750*a^4*b^7*C*Sin[c + d*x] + 270*a^2*b^9*C*Sin[c + d*x] - 6*b^11*C*Sin[c + d*x] - 120*a^7*A*b^4*Sin[2*(c + d*x)] + 360*a^5*A*b^6*Sin[2*(c + d*x)] - 480*a^3*A*b^8*Sin[2*(c + d*x)] - 1200*a^9*b^2*C*Sin[2*(c + d*x)] + 3300*a^7*b^4*C*Sin[2*(c + d*x)] - 2772*a^5*b^6*C*Sin[2*(c + d*x)] + 372*a^3*b^8*C*Sin[2*(c + d*x)] + 60*a*b^10*C*Sin[2*(c + d*x)] - 44*a^6*A*b^5*Sin[3*(c + d*x)] + 128*a^4*A*b^7*Sin[3*(c + d*x)] - 144*a^2*A*b^9*Sin[3*(c + d*x)] - 440*a^8*b^3*C*Sin[3*(c + d*x)] + 1253*a^6*b^5*C*Sin[3*(c + d*x)] - 1143*a^4*b^7*C*Sin[3*(c + d*x)] + 279*a^2*b^9*C*Sin[3*(c + d*x)] - 9*b^11*C*Sin[3*(c + d*x)] - 30*a^7*b^4*C*Sin[4*(c + d*x)] + 90*a^5*b^6*C*Sin[4*(c + d*x)] - 90*a^3*b^8*C*Sin[4*(c + d*x)] + 30*a*b^10*C*Sin[4*(c + d*x)] + 3*a^6*b^5*C*Sin[5*(c + d*x)] - 9*a^4*b^7*C*Sin[5*(c + d*x)] + 9*a^2*b^9*C*Sin[5*(c + d*x)] - 3*b^11*C*Sin[5*(c + d*x)])/(96*b^6*(-a^2 + b^2)^3*d*(a + b*Cos[c + d*x])^3)","B",1
586,1,849,369,3.914505,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^4,x]","\frac{\frac{24 \left(8 C a^8-28 b^2 C a^6+35 b^4 C a^4-b^6 (3 A+20 C) a^2-2 A b^8\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{7/2}}+\frac{-96 c C a^{10}-96 C d x a^{10}+96 b C \sin (c+d x) a^9+144 b^2 c C a^8+144 b^2 C d x a^8+120 b^2 C \sin (2 (c+d x)) a^8-24 b^3 c C \cos (3 (c+d x)) a^7-24 b^3 C d x \cos (3 (c+d x)) a^7-228 b^3 C \sin (c+d x) a^7+44 b^3 C \sin (3 (c+d x)) a^7+144 b^4 c C a^6+144 b^4 C d x a^6-336 b^4 C \sin (2 (c+d x)) a^6+3 b^4 C \sin (4 (c+d x)) a^6+72 b^5 c C \cos (3 (c+d x)) a^5+72 b^5 C d x \cos (3 (c+d x)) a^5+18 A b^5 \sin (c+d x) a^5+135 b^5 C \sin (c+d x) a^5+2 A b^5 \sin (3 (c+d x)) a^5-125 b^5 C \sin (3 (c+d x)) a^5-336 b^6 c C a^4-336 b^6 C d x a^4+6 A b^6 \sin (2 (c+d x)) a^4+300 b^6 C \sin (2 (c+d x)) a^4-9 b^6 C \sin (4 (c+d x)) a^4-72 b^7 c C \cos (3 (c+d x)) a^3-72 b^7 C d x \cos (3 (c+d x)) a^3+39 A b^7 \sin (c+d x) a^3+90 b^7 C \sin (c+d x) a^3-5 A b^7 \sin (3 (c+d x)) a^3+114 b^7 C \sin (3 (c+d x)) a^3+144 b^8 c C a^2+144 b^8 C d x a^2-144 b^2 \left(a^2-b^2\right)^3 C (c+d x) \cos (2 (c+d x)) a^2+54 A b^8 \sin (2 (c+d x)) a^2-18 b^8 C \sin (2 (c+d x)) a^2+9 b^8 C \sin (4 (c+d x)) a^2-72 b \left(a^2-b^2\right)^3 \left(4 a^2+b^2\right) C (c+d x) \cos (c+d x) a+24 b^9 c C \cos (3 (c+d x)) a+24 b^9 C d x \cos (3 (c+d x)) a+18 A b^9 \sin (c+d x) a-18 b^9 C \sin (c+d x) a+18 A b^9 \sin (3 (c+d x)) a-18 b^9 C \sin (3 (c+d x)) a-6 b^{10} C \sin (2 (c+d x))-3 b^{10} C \sin (4 (c+d x))}{\left(a^2-b^2\right)^3 (a+b \cos (c+d x))^3}}{24 b^5 d}","-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}-\frac{\left(5 A b^4-C \left(12 a^4-23 a^2 b^2+6 b^4\right)\right) \sin (c+d x)}{6 b^4 d \left(a^2-b^2\right)^2}+\frac{\left(-4 a^4 C+a^2 b^2 (2 A+9 C)+3 A b^4\right) \sin (c+d x) \cos ^2(c+d x)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{a \left(4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)+2 A b^6\right) \sin (c+d x)}{2 b^4 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{\left(-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 C)+2 A b^8\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{4 a C x}{b^5}",1,"((24*(-2*A*b^8 + 8*a^8*C - 28*a^6*b^2*C + 35*a^4*b^4*C - a^2*b^6*(3*A + 20*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) + (-96*a^10*c*C + 144*a^8*b^2*c*C + 144*a^6*b^4*c*C - 336*a^4*b^6*c*C + 144*a^2*b^8*c*C - 96*a^10*C*d*x + 144*a^8*b^2*C*d*x + 144*a^6*b^4*C*d*x - 336*a^4*b^6*C*d*x + 144*a^2*b^8*C*d*x - 72*a*b*(a^2 - b^2)^3*(4*a^2 + b^2)*C*(c + d*x)*Cos[c + d*x] - 144*a^2*b^2*(a^2 - b^2)^3*C*(c + d*x)*Cos[2*(c + d*x)] - 24*a^7*b^3*c*C*Cos[3*(c + d*x)] + 72*a^5*b^5*c*C*Cos[3*(c + d*x)] - 72*a^3*b^7*c*C*Cos[3*(c + d*x)] + 24*a*b^9*c*C*Cos[3*(c + d*x)] - 24*a^7*b^3*C*d*x*Cos[3*(c + d*x)] + 72*a^5*b^5*C*d*x*Cos[3*(c + d*x)] - 72*a^3*b^7*C*d*x*Cos[3*(c + d*x)] + 24*a*b^9*C*d*x*Cos[3*(c + d*x)] + 18*a^5*A*b^5*Sin[c + d*x] + 39*a^3*A*b^7*Sin[c + d*x] + 18*a*A*b^9*Sin[c + d*x] + 96*a^9*b*C*Sin[c + d*x] - 228*a^7*b^3*C*Sin[c + d*x] + 135*a^5*b^5*C*Sin[c + d*x] + 90*a^3*b^7*C*Sin[c + d*x] - 18*a*b^9*C*Sin[c + d*x] + 6*a^4*A*b^6*Sin[2*(c + d*x)] + 54*a^2*A*b^8*Sin[2*(c + d*x)] + 120*a^8*b^2*C*Sin[2*(c + d*x)] - 336*a^6*b^4*C*Sin[2*(c + d*x)] + 300*a^4*b^6*C*Sin[2*(c + d*x)] - 18*a^2*b^8*C*Sin[2*(c + d*x)] - 6*b^10*C*Sin[2*(c + d*x)] + 2*a^5*A*b^5*Sin[3*(c + d*x)] - 5*a^3*A*b^7*Sin[3*(c + d*x)] + 18*a*A*b^9*Sin[3*(c + d*x)] + 44*a^7*b^3*C*Sin[3*(c + d*x)] - 125*a^5*b^5*C*Sin[3*(c + d*x)] + 114*a^3*b^7*C*Sin[3*(c + d*x)] - 18*a*b^9*C*Sin[3*(c + d*x)] + 3*a^6*b^4*C*Sin[4*(c + d*x)] - 9*a^4*b^6*C*Sin[4*(c + d*x)] + 9*a^2*b^8*C*Sin[4*(c + d*x)] - 3*b^10*C*Sin[4*(c + d*x)])/((a^2 - b^2)^3*(a + b*Cos[c + d*x])^3))/(24*b^5*d)","B",1
587,1,723,304,6.3199343,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^4,x]","-\frac{\frac{24 a \left(2 a^6 C-7 a^4 b^2 C-a^2 b^4 (A-8 C)-4 b^6 (A+2 C)\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{7/2}}+\frac{-24 a^9 c C-24 a^9 C d x+24 a^8 b C \sin (c+d x)+30 a^7 b^2 C \sin (2 (c+d x))+36 a^7 b^2 c C+36 a^7 b^2 C d x-57 a^6 b^3 C \sin (c+d x)+11 a^6 b^3 C \sin (3 (c+d x))-6 a^6 b^3 c C \cos (3 (c+d x))-6 a^6 b^3 C d x \cos (3 (c+d x))-6 a^5 A b^4 \sin (2 (c+d x))-90 a^5 b^4 C \sin (2 (c+d x))+36 a^5 b^4 c C+36 a^5 b^4 C d x+51 a^4 A b^5 \sin (c+d x)-a^4 A b^5 \sin (3 (c+d x))+72 a^4 b^5 C \sin (c+d x)-32 a^4 b^5 C \sin (3 (c+d x))+18 a^4 b^5 c C \cos (3 (c+d x))+18 a^4 b^5 C d x \cos (3 (c+d x))+54 a^3 A b^6 \sin (2 (c+d x))+120 a^3 b^6 C \sin (2 (c+d x))-84 a^3 b^6 c C-84 a^3 b^6 C d x+18 a^2 A b^7 \sin (c+d x)+10 a^2 A b^7 \sin (3 (c+d x))+36 a^2 b^7 C \sin (c+d x)+36 a^2 b^7 C \sin (3 (c+d x))-18 a^2 b^7 c C \cos (3 (c+d x))-18 a^2 b^7 C d x \cos (3 (c+d x))-36 a b^2 C \left(a^2-b^2\right)^3 (c+d x) \cos (2 (c+d x))+18 b C \left(b^2-a^2\right)^3 \left(4 a^2+b^2\right) (c+d x) \cos (c+d x)+12 a A b^8 \sin (2 (c+d x))+36 a b^8 c C+36 a b^8 C d x+6 A b^9 \sin (c+d x)+6 A b^9 \sin (3 (c+d x))+6 b^9 c C \cos (3 (c+d x))+6 b^9 C d x \cos (3 (c+d x))}{\left(a^2-b^2\right)^3 (a+b \cos (c+d x))^3}}{24 b^4 d}","-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}-\frac{a \left(-3 a^4 C+a^2 b^2 (3 A+8 C)+2 A b^4\right) \sin (c+d x)}{6 b^3 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{a \left(-2 a^6 C+7 a^4 b^2 C+a^2 b^4 (A-8 C)+4 b^6 (A+2 C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left(9 a^6 C-a^4 b^2 (3 A+28 C)+2 a^2 b^4 (7 A+17 C)+4 A b^6\right) \sin (c+d x)}{6 b^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{C x}{b^4}",1,"-1/24*((24*a*(-(a^2*b^4*(A - 8*C)) + 2*a^6*C - 7*a^4*b^2*C - 4*b^6*(A + 2*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) + (-24*a^9*c*C + 36*a^7*b^2*c*C + 36*a^5*b^4*c*C - 84*a^3*b^6*c*C + 36*a*b^8*c*C - 24*a^9*C*d*x + 36*a^7*b^2*C*d*x + 36*a^5*b^4*C*d*x - 84*a^3*b^6*C*d*x + 36*a*b^8*C*d*x + 18*b*(-a^2 + b^2)^3*(4*a^2 + b^2)*C*(c + d*x)*Cos[c + d*x] - 36*a*b^2*(a^2 - b^2)^3*C*(c + d*x)*Cos[2*(c + d*x)] - 6*a^6*b^3*c*C*Cos[3*(c + d*x)] + 18*a^4*b^5*c*C*Cos[3*(c + d*x)] - 18*a^2*b^7*c*C*Cos[3*(c + d*x)] + 6*b^9*c*C*Cos[3*(c + d*x)] - 6*a^6*b^3*C*d*x*Cos[3*(c + d*x)] + 18*a^4*b^5*C*d*x*Cos[3*(c + d*x)] - 18*a^2*b^7*C*d*x*Cos[3*(c + d*x)] + 6*b^9*C*d*x*Cos[3*(c + d*x)] + 51*a^4*A*b^5*Sin[c + d*x] + 18*a^2*A*b^7*Sin[c + d*x] + 6*A*b^9*Sin[c + d*x] + 24*a^8*b*C*Sin[c + d*x] - 57*a^6*b^3*C*Sin[c + d*x] + 72*a^4*b^5*C*Sin[c + d*x] + 36*a^2*b^7*C*Sin[c + d*x] - 6*a^5*A*b^4*Sin[2*(c + d*x)] + 54*a^3*A*b^6*Sin[2*(c + d*x)] + 12*a*A*b^8*Sin[2*(c + d*x)] + 30*a^7*b^2*C*Sin[2*(c + d*x)] - 90*a^5*b^4*C*Sin[2*(c + d*x)] + 120*a^3*b^6*C*Sin[2*(c + d*x)] - a^4*A*b^5*Sin[3*(c + d*x)] + 10*a^2*A*b^7*Sin[3*(c + d*x)] + 6*A*b^9*Sin[3*(c + d*x)] + 11*a^6*b^3*C*Sin[3*(c + d*x)] - 32*a^4*b^5*C*Sin[3*(c + d*x)] + 36*a^2*b^7*C*Sin[3*(c + d*x)])/((a^2 - b^2)^3*(a + b*Cos[c + d*x])^3))/(b^4*d)","B",1
588,1,224,261,1.2179078,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^4,x]","\frac{\frac{24 b \left(a^2 (4 A+3 C)+b^2 (A+2 C)\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+\frac{2 \sin (c+d x) \left(6 b \left(a^4 (2 A+C)+9 a^2 b^2 (A+C)-A b^4\right) \cos (c+d x)+a \left(2 a^4 (6 A+5 C)+a^2 b^2 (22 A+17 C)+\left(2 a^4 C+a^2 b^2 (2 A-5 C)+b^4 (13 A+18 C)\right) \cos (2 (c+d x))+b^4 (11 A+18 C)\right)\right)}{(a+b \cos (c+d x))^3}}{24 d \left(a^2-b^2\right)^3}","-\frac{b \left(a^2 (4 A+3 C)+b^2 (A+2 C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac{a \left(a^2 C+A b^2\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{a \left(2 a^4 C+a^2 b^2 (2 A-5 C)+b^4 (13 A+18 C)\right) \sin (c+d x)}{6 b^2 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{\left(-4 a^4 C+a^2 b^2 (2 A+9 C)+3 A b^4\right) \sin (c+d x)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}",1,"((24*b*(b^2*(A + 2*C) + a^2*(4*A + 3*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (2*(6*b*(-(A*b^4) + 9*a^2*b^2*(A + C) + a^4*(2*A + C))*Cos[c + d*x] + a*(2*a^4*(6*A + 5*C) + a^2*b^2*(22*A + 17*C) + b^4*(11*A + 18*C) + (a^2*b^2*(2*A - 5*C) + 2*a^4*C + b^4*(13*A + 18*C))*Cos[2*(c + d*x)]))*Sin[c + d*x])/(a + b*Cos[c + d*x])^3)/(24*(a^2 - b^2)^3*d)","A",1
589,1,224,252,1.1549382,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4,x]","\frac{\frac{2 \sin (c+d x) \left(6 a \left(a^4 C-9 a^2 b^2 (A+C)-b^4 (A+2 C)\right) \cos (c+d x)-b \left(a^4 (36 A+25 C)+a^2 b^2 (A+14 C)+\left(a^4 (-C)+a^2 b^2 (11 A+10 C)+2 b^4 (2 A+3 C)\right) \cos (2 (c+d x))+2 b^4 (4 A+3 C)\right)\right)}{(a+b \cos (c+d x))^3}-\frac{24 a \left(a^2 (2 A+C)+b^2 (3 A+4 C)\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}}{24 d \left(a^2-b^2\right)^3}","\frac{a \left(a^2 (2 A+C)+b^2 (3 A+4 C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{a \left(a^2 (-C)+5 A b^2+6 b^2 C\right) \sin (c+d x)}{6 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{\left(a^4 C-a^2 b^2 (11 A+10 C)-2 b^4 (2 A+3 C)\right) \sin (c+d x)}{6 b d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}",1,"((-24*a*(a^2*(2*A + C) + b^2*(3*A + 4*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (2*(6*a*(a^4*C - 9*a^2*b^2*(A + C) - b^4*(A + 2*C))*Cos[c + d*x] - b*(2*b^4*(4*A + 3*C) + a^2*b^2*(A + 14*C) + a^4*(36*A + 25*C) + (-(a^4*C) + 2*b^4*(2*A + 3*C) + a^2*b^2*(11*A + 10*C))*Cos[2*(c + d*x)]))*Sin[c + d*x])/(a + b*Cos[c + d*x])^3)/(24*(a^2 - b^2)^3*d)","A",1
590,1,498,301,5.3806409,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^4} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^4,x]","\frac{\cos (c+d x) (A \sec (c+d x)+C \cos (c+d x)) \left(-\frac{6 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a^4}+\frac{6 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^4}+\frac{2 \sec (c) \left(a^2 C+A b^2\right) (b \sin (d x)-a \sin (c))}{a b \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{a b \tan (c) \left(A b^2-a^2 (6 A+5 C)\right)+\sec (c) \sin (d x) \left(2 a^4 C+a^2 b^2 (8 A+3 C)-3 A b^4\right)}{a^2 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{6 b (\sin (c)+i \cos (c)) \left(4 a^6 (2 A+C)+a^4 b^2 (C-8 A)+7 a^2 A b^4-2 A b^6\right) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (b \cos (c)-a)+b \sin (c)\right)}{\sqrt{-\left(\left(a^2-b^2\right) (\cos (c)-i \sin (c))^2\right)}}\right)}{a^4 \left(a^2-b^2\right)^3 \sqrt{-\left(\left(a^2-b^2\right) (\cos (c)-i \sin (c))^2\right)}}+\frac{\sec (c) \sin (d x) \left(2 a^6 C+13 a^4 b^2 (2 A+C)-17 a^2 A b^4+6 A b^6\right)-3 a b \tan (c) \left(a^4 (6 A+4 C)+a^2 b^2 (C-2 A)+A b^4\right)}{\left(a^3-a b^2\right)^3 (a+b \cos (c+d x))}\right)}{3 d (2 A+C \cos (2 (c+d x))+C)}","\frac{A \tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}-\frac{\left(-2 a^4 C-a^2 b^2 (8 A+3 C)+3 A b^4\right) \sin (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{b \left(4 a^6 (2 A+C)-a^4 b^2 (8 A-C)+7 a^2 A b^4-2 A b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left(-2 a^6 C-13 a^4 b^2 (2 A+C)+17 a^2 A b^4-6 A b^6\right) \sin (c+d x)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}",1,"(Cos[c + d*x]*(C*Cos[c + d*x] + A*Sec[c + d*x])*((-6*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/a^4 + (6*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/a^4 + (6*b*(7*a^2*A*b^4 - 2*A*b^6 + a^4*b^2*(-8*A + C) + 4*a^6*(2*A + C))*ArcTan[((I*Cos[c] + Sin[c])*(b*Sin[c] + (-a + b*Cos[c])*Tan[(d*x)/2]))/Sqrt[-((a^2 - b^2)*(Cos[c] - I*Sin[c])^2)]]*(I*Cos[c] + Sin[c]))/(a^4*(a^2 - b^2)^3*Sqrt[-((a^2 - b^2)*(Cos[c] - I*Sin[c])^2)]) + (2*(A*b^2 + a^2*C)*Sec[c]*(-(a*Sin[c]) + b*Sin[d*x]))/(a*b*(a^2 - b^2)*(a + b*Cos[c + d*x])^3) + ((-17*a^2*A*b^4 + 6*A*b^6 + 2*a^6*C + 13*a^4*b^2*(2*A + C))*Sec[c]*Sin[d*x] - 3*a*b*(A*b^4 + a^2*b^2*(-2*A + C) + a^4*(6*A + 4*C))*Tan[c])/((a^3 - a*b^2)^3*(a + b*Cos[c + d*x])) + ((-3*A*b^4 + 2*a^4*C + a^2*b^2*(8*A + 3*C))*Sec[c]*Sin[d*x] + a*b*(A*b^2 - a^2*(6*A + 5*C))*Tan[c])/(a^2*(a^2 - b^2)^2*(a + b*Cos[c + d*x])^2)))/(3*d*(2*A + C + C*Cos[2*(c + d*x)]))","C",1
591,1,515,376,2.9998297,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^4,x]","\frac{\cos (c+d x) \left(A \sec ^2(c+d x)+C\right) \left(\frac{24 \left(2 a^8 C+a^6 b^2 (20 A+3 C)-35 a^4 A b^4+28 a^2 A b^6-8 A b^8\right) \cos (c+d x) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{7/2}}+\frac{a \sin (c+d x) \left(24 a^9 A-36 a^7 A b^2-54 a^7 b^2 C+6 a^6 A b^3 \cos (3 (c+d x))-11 a^6 b^3 C \cos (3 (c+d x))-246 a^5 A b^4-6 a^5 b^4 C-65 a^4 A b^5 \cos (3 (c+d x))-4 a^4 b^5 C \cos (3 (c+d x))+318 a^3 A b^6+68 a^2 A b^7 \cos (3 (c+d x))+6 a b^2 \left(a^6 (6 A-9 C)-a^4 b^2 (53 A+C)+57 a^2 A b^4-20 A b^6\right) \cos (2 (c+d x))-b \left(-72 a^8 (A-C)+a^6 b^2 (438 A+13 C)-5 a^4 b^4 (61 A-4 C)-28 a^2 A b^6+72 A b^8\right) \cos (c+d x)-120 a A b^8-24 A b^9 \cos (3 (c+d x))\right)}{\left(a^2-b^2\right)^3 (a+b \cos (c+d x))^3}+96 A b \cos (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-96 A b \cos (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{12 a^5 d (2 A+C \cos (2 (c+d x))+C)}","-\frac{4 A b \tanh ^{-1}(\sin (c+d x))}{a^5 d}+\frac{\left(a^2 C+A b^2\right) \tan (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}-\frac{\left(-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right) \tan (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\left(a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)+68 a^2 A b^4-24 A b^6\right) \tan (c+d x)}{6 a^4 d \left(a^2-b^2\right)^3}-\frac{\left(-2 a^6 C-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-4 A b^6\right) \tan (c+d x)}{2 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{\left(-2 a^8 C-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6+8 A b^8\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{7/2} (a+b)^{7/2}}",1,"(Cos[c + d*x]*(C + A*Sec[c + d*x]^2)*((24*(-35*a^4*A*b^4 + 28*a^2*A*b^6 - 8*A*b^8 + 2*a^8*C + a^6*b^2*(20*A + 3*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]]*Cos[c + d*x])/(-a^2 + b^2)^(7/2) + 96*A*b*Cos[c + d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 96*A*b*Cos[c + d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a*(24*a^9*A - 36*a^7*A*b^2 - 246*a^5*A*b^4 + 318*a^3*A*b^6 - 120*a*A*b^8 - 54*a^7*b^2*C - 6*a^5*b^4*C - b*(-28*a^2*A*b^6 + 72*A*b^8 - 5*a^4*b^4*(61*A - 4*C) - 72*a^8*(A - C) + a^6*b^2*(438*A + 13*C))*Cos[c + d*x] + 6*a*b^2*(57*a^2*A*b^4 - 20*A*b^6 + a^6*(6*A - 9*C) - a^4*b^2*(53*A + C))*Cos[2*(c + d*x)] + 6*a^6*A*b^3*Cos[3*(c + d*x)] - 65*a^4*A*b^5*Cos[3*(c + d*x)] + 68*a^2*A*b^7*Cos[3*(c + d*x)] - 24*A*b^9*Cos[3*(c + d*x)] - 11*a^6*b^3*C*Cos[3*(c + d*x)] - 4*a^4*b^5*C*Cos[3*(c + d*x)])*Sin[c + d*x])/((a^2 - b^2)^3*(a + b*Cos[c + d*x])^3)))/(12*a^5*d*(2*A + C + C*Cos[2*(c + d*x)]))","A",0
592,1,740,522,5.5859318,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^4,x]","\frac{\left(A \sec ^2(c+d x)+C\right) \left(-48 \left(a^2 (A+2 C)+20 A b^2\right) \cos ^2(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+48 \left(a^2 (A+2 C)+20 A b^2\right) \cos ^2(c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{96 b \left(-8 a^8 C+8 a^6 b^2 (C-5 A)+7 a^4 b^4 (12 A-C)+a^2 b^6 (2 C-69 A)+20 A b^8\right) \cos ^2(c+d x) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{7/2}}+\frac{2 a \sin (c+d x) \left(24 a^{10} A-324 a^8 A b^2+144 a^8 b^2 C-138 a^7 A b^3 \cos (3 (c+d x))+120 a^7 b^3 C \cos (3 (c+d x))-24 a^6 A b^4 \cos (4 (c+d x))+1116 a^6 A b^4+26 a^6 b^4 C \cos (4 (c+d x))-50 a^6 b^4 C+738 a^5 A b^5 \cos (3 (c+d x))-90 a^5 b^5 C \cos (3 (c+d x))+146 a^4 A b^6 \cos (4 (c+d x))-830 a^4 A b^6-17 a^4 b^6 C \cos (4 (c+d x))-7 a^4 b^6 C-840 a^3 A b^7 \cos (3 (c+d x))+30 a^3 b^7 C \cos (3 (c+d x))-167 a^2 A b^8 \cos (4 (c+d x))-61 a^2 A b^8+6 a^2 b^8 C \cos (4 (c+d x))+18 a^2 b^8 C-6 a b \left(20 a^8 A+3 a^6 b^2 (3 A-20 C)+3 a^4 b^4 (15 C-103 A)+5 a^2 b^6 (80 A-3 C)-150 A b^8\right) \cos (c+d x)+12 b^2 \left(-3 a^8 (7 A-4 C)+a^6 b^2 (85 A-2 C)-a^4 b^4 (55 A+2 C)+a^2 b^6 (2 C-19 A)+20 A b^8\right) \cos (2 (c+d x))+300 a A b^9 \cos (3 (c+d x))+60 A b^{10} \cos (4 (c+d x))+180 A b^{10}\right)}{\left(a^2-b^2\right)^3 (a+b \cos (c+d x))^3}\right)}{48 a^6 d (2 A+C \cos (2 (c+d x))+C)}","\frac{\left(a^2 C+A b^2\right) \tan (c+d x) \sec (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{\left(a^2 (A+2 C)+20 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}-\frac{\left(-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right) \tan (c+d x) \sec (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{\left(-\left(a^6 (A-6 C)\right)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right) \tan (c+d x) \sec (c+d x)}{2 a^4 d \left(a^2-b^2\right)^3}+\frac{\left(-8 a^8 b C-8 a^6 b^3 (5 A-C)+7 a^4 b^5 (12 A-C)-a^2 b^7 (69 A-2 C)+20 A b^9\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^6 d \sqrt{a-b} \sqrt{a+b} \left(a^2-b^2\right)^3}+\frac{b \left(-\left(a^6 (24 A-26 C)\right)+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+60 A b^6\right) \tan (c+d x)}{6 a^5 d \left(a^2-b^2\right)^3}+\frac{\left(12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right) \tan (c+d x) \sec (c+d x)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}",1,"((C + A*Sec[c + d*x]^2)*((96*b*(20*A*b^8 + 7*a^4*b^4*(12*A - C) - 8*a^8*C + 8*a^6*b^2*(-5*A + C) + a^2*b^6*(-69*A + 2*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]]*Cos[c + d*x]^2)/(-a^2 + b^2)^(7/2) - 48*(20*A*b^2 + a^2*(A + 2*C))*Cos[c + d*x]^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 48*(20*A*b^2 + a^2*(A + 2*C))*Cos[c + d*x]^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*a*(24*a^10*A - 324*a^8*A*b^2 + 1116*a^6*A*b^4 - 830*a^4*A*b^6 - 61*a^2*A*b^8 + 180*A*b^10 + 144*a^8*b^2*C - 50*a^6*b^4*C - 7*a^4*b^6*C + 18*a^2*b^8*C - 6*a*b*(20*a^8*A - 150*A*b^8 + 3*a^6*b^2*(3*A - 20*C) + 5*a^2*b^6*(80*A - 3*C) + 3*a^4*b^4*(-103*A + 15*C))*Cos[c + d*x] + 12*b^2*(20*A*b^8 - 3*a^8*(7*A - 4*C) + a^6*b^2*(85*A - 2*C) + a^2*b^6*(-19*A + 2*C) - a^4*b^4*(55*A + 2*C))*Cos[2*(c + d*x)] - 138*a^7*A*b^3*Cos[3*(c + d*x)] + 738*a^5*A*b^5*Cos[3*(c + d*x)] - 840*a^3*A*b^7*Cos[3*(c + d*x)] + 300*a*A*b^9*Cos[3*(c + d*x)] + 120*a^7*b^3*C*Cos[3*(c + d*x)] - 90*a^5*b^5*C*Cos[3*(c + d*x)] + 30*a^3*b^7*C*Cos[3*(c + d*x)] - 24*a^6*A*b^4*Cos[4*(c + d*x)] + 146*a^4*A*b^6*Cos[4*(c + d*x)] - 167*a^2*A*b^8*Cos[4*(c + d*x)] + 60*A*b^10*Cos[4*(c + d*x)] + 26*a^6*b^4*C*Cos[4*(c + d*x)] - 17*a^4*b^6*C*Cos[4*(c + d*x)] + 6*a^2*b^8*C*Cos[4*(c + d*x)])*Sin[c + d*x])/((a^2 - b^2)^3*(a + b*Cos[c + d*x])^3)))/(48*a^6*d*(2*A + C + C*Cos[2*(c + d*x)]))","A",0
593,1,168,193,0.9921322,"\int \frac{\cos ^3(c+d x) \left(1-\cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{-96 a^4 c-96 a^4 d x-24 a^2 b^2 \sin (2 (c+d x))+24 a b \left(4 a^2-b^2\right) \sin (c+d x)+48 a^2 b^2 c+48 a^2 b^2 d x+192 a^3 \sqrt{b^2-a^2} \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)+8 a b^3 \sin (3 (c+d x))-3 b^4 \sin (4 (c+d x))+12 b^4 c+12 b^4 d x}{96 b^5 d}","\frac{2 a^3 \sqrt{a-b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d}+\frac{a \left(3 a^2-b^2\right) \sin (c+d x)}{3 b^4 d}-\frac{\left(4 a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{8 b^3 d}-\frac{x \left(8 a^4-4 a^2 b^2-b^4\right)}{8 b^5}+\frac{a \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{4 b d}",1,"(-96*a^4*c + 48*a^2*b^2*c + 12*b^4*c - 96*a^4*d*x + 48*a^2*b^2*d*x + 12*b^4*d*x + 192*a^3*Sqrt[-a^2 + b^2]*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]] + 24*a*b*(4*a^2 - b^2)*Sin[c + d*x] - 24*a^2*b^2*Sin[2*(c + d*x)] + 8*a*b^3*Sin[3*(c + d*x)] - 3*b^4*Sin[4*(c + d*x)])/(96*b^5*d)","A",1
594,1,125,150,0.4348595,"\int \frac{\cos ^2(c+d x) \left(1-\cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","-\frac{-6 a \left(2 a^2-b^2\right) (c+d x)+24 a^2 \sqrt{b^2-a^2} \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)-3 a b^2 \sin (2 (c+d x))+3 b (2 a-b) (2 a+b) \sin (c+d x)+b^3 \sin (3 (c+d x))}{12 b^4 d}","-\frac{2 a^2 \sqrt{a-b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d}+\frac{a x \left(2 a^2-b^2\right)}{2 b^4}-\frac{\left(3 a^2-b^2\right) \sin (c+d x)}{3 b^3 d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 b^2 d}-\frac{\sin (c+d x) \cos ^2(c+d x)}{3 b d}",1,"-1/12*(-6*a*(2*a^2 - b^2)*(c + d*x) + 24*a^2*Sqrt[-a^2 + b^2]*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]] + 3*(2*a - b)*b*(2*a + b)*Sin[c + d*x] - 3*a*b^2*Sin[2*(c + d*x)] + b^3*Sin[3*(c + d*x)])/(b^4*d)","A",1
595,1,98,109,0.3043255,"\int \frac{\cos (c+d x) \left(1-\cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{-2 \left(2 a^2-b^2\right) (c+d x)+8 a \sqrt{b^2-a^2} \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)+4 a b \sin (c+d x)+b^2 (-\sin (2 (c+d x)))}{4 b^3 d}","-\frac{x \left(2 a^2-b^2\right)}{2 b^3}+\frac{2 a \sqrt{a-b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d}+\frac{a \sin (c+d x)}{b^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{2 b d}",1,"(-2*(2*a^2 - b^2)*(c + d*x) + 8*a*Sqrt[-a^2 + b^2]*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]] + 4*a*b*Sin[c + d*x] - b^2*Sin[2*(c + d*x)])/(4*b^3*d)","A",1
596,1,69,73,0.1390384,"\int \frac{1-\cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x]),x]","\frac{-2 \sqrt{b^2-a^2} \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)+a (c+d x)-b \sin (c+d x)}{b^2 d}","-\frac{2 \sqrt{a-b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d}+\frac{a x}{b^2}-\frac{\sin (c+d x)}{b d}",1,"(a*(c + d*x) - 2*Sqrt[-a^2 + b^2]*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]] - b*Sin[c + d*x])/(b^2*d)","A",1
597,1,115,76,0.1420365,"\int \frac{\left(1-\cos ^2(c+d x)\right) \sec (c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((1 - Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x]),x]","-\frac{-2 \sqrt{b^2-a^2} \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)+a c+a d x+b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a b d}","\frac{2 \sqrt{a-b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a b d}+\frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{x}{b}",1,"-((a*c + a*d*x - 2*Sqrt[-a^2 + b^2]*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]] + b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(a*b*d))","A",1
598,1,112,82,0.2796069,"\int \frac{\left(1-\cos ^2(c+d x)\right) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((1 - Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x]),x]","\frac{-2 \sqrt{b^2-a^2} \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)+a \tan (c+d x)+b \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{a^2 d}","-\frac{2 \sqrt{a-b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d}-\frac{b \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{\tan (c+d x)}{a d}",1,"(-2*Sqrt[-a^2 + b^2]*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]] + b*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + a*Tan[c + d*x])/(a^2*d)","A",1
599,1,236,117,1.1012675,"\int \frac{\left(1-\cos ^2(c+d x)\right) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((1 - Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x]),x]","\frac{8 b \sqrt{b^2-a^2} \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)+\frac{a^2}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{a^2}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+2 a^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 a^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-4 a b \tan (c+d x)-4 b^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 a^3 d}","\frac{2 b \sqrt{a-b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d}-\frac{b \tan (c+d x)}{a^2 d}-\frac{\left(a^2-2 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{\tan (c+d x) \sec (c+d x)}{2 a d}",1,"(8*b*Sqrt[-a^2 + b^2]*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]] + 2*a^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 4*b^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 2*a^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 4*b^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + a^2/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 - a^2/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - 4*a*b*Tan[c + d*x])/(4*a^3*d)","B",1
600,1,256,155,2.5627828,"\int \frac{\left(1-\cos ^2(c+d x)\right) \sec ^4(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((1 - Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + b*Cos[c + d*x]),x]","-\frac{24 b^2 \sqrt{b^2-a^2} \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)+\frac{1}{2} \sec ^3(c+d x) \left(4 a \sin (c+d x) \left(\left(a^2-3 b^2\right) \cos (2 (c+d x))-a^2+3 a b \cos (c+d x)-3 b^2\right)+9 b \left(a^2-2 b^2\right) \cos (c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+3 b \left(a^2-2 b^2\right) \cos (3 (c+d x)) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{12 a^4 d}","-\frac{2 b^2 \sqrt{a-b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d}-\frac{b \tan (c+d x) \sec (c+d x)}{2 a^2 d}+\frac{b \left(a^2-2 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{\left(a^2-3 b^2\right) \tan (c+d x)}{3 a^3 d}+\frac{\tan (c+d x) \sec ^2(c+d x)}{3 a d}",1,"-1/12*(24*b^2*Sqrt[-a^2 + b^2]*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]] + (Sec[c + d*x]^3*(9*b*(a^2 - 2*b^2)*Cos[c + d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 3*b*(a^2 - 2*b^2)*Cos[3*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 4*a*(-a^2 - 3*b^2 + 3*a*b*Cos[c + d*x] + (a^2 - 3*b^2)*Cos[2*(c + d*x)])*Sin[c + d*x]))/2)/(a^4*d)","A",1
601,1,271,237,3.8757238,"\int \frac{\cos ^4(c+d x) \left(1-\cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^4*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{\frac{-960 a^5 c-960 a^5 d x+240 a^3 b^2 \sin (2 (c+d x))+288 a^3 b^2 c+288 a^3 b^2 d x-40 a^2 b^3 \sin (3 (c+d x))+24 a^2 b \left(40 a^2-7 b^2\right) \sin (c+d x)+24 b \left(-40 a^4+12 a^2 b^2+b^4\right) (c+d x) \cos (c+d x)-32 a b^4 \sin (2 (c+d x))+10 a b^4 \sin (4 (c+d x))+24 a b^4 c+24 a b^4 d x-3 b^5 \sin (3 (c+d x))-3 b^5 \sin (5 (c+d x))}{a+b \cos (c+d x)}-\frac{384 a^3 \left(5 a^2-4 b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}}{192 b^6 d}","\frac{a \left(15 a^2-2 b^2\right) \sin (c+d x)}{3 b^5 d}-\frac{\left(20 a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{8 b^4 d}-\frac{x \left(40 a^4-12 a^2 b^2-b^4\right)}{8 b^6}+\frac{2 a^3 \left(5 a^2-4 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^6 d \sqrt{a-b} \sqrt{a+b}}+\frac{5 a \sin (c+d x) \cos ^2(c+d x)}{3 b^3 d}+\frac{\sin (c+d x) \cos ^4(c+d x)}{b d (a+b \cos (c+d x))}-\frac{5 \sin (c+d x) \cos ^3(c+d x)}{4 b^2 d}",1,"((-384*a^3*(5*a^2 - 4*b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (-960*a^5*c + 288*a^3*b^2*c + 24*a*b^4*c - 960*a^5*d*x + 288*a^3*b^2*d*x + 24*a*b^4*d*x + 24*b*(-40*a^4 + 12*a^2*b^2 + b^4)*(c + d*x)*Cos[c + d*x] + 24*a^2*b*(40*a^2 - 7*b^2)*Sin[c + d*x] + 240*a^3*b^2*Sin[2*(c + d*x)] - 32*a*b^4*Sin[2*(c + d*x)] - 40*a^2*b^3*Sin[3*(c + d*x)] - 3*b^5*Sin[3*(c + d*x)] + 10*a*b^4*Sin[4*(c + d*x)] - 3*b^5*Sin[5*(c + d*x)])/(a + b*Cos[c + d*x]))/(192*b^6*d)","A",1
602,1,217,189,2.380299,"\int \frac{\cos ^3(c+d x) \left(1-\cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{\frac{48 a^2 \left(4 a^2-3 b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+\frac{96 a^4 c+96 a^4 d x-24 a^2 b^2 \sin (2 (c+d x))+12 a b \left(b^2-8 a^2\right) \sin (c+d x)+24 a b \left(4 a^2-b^2\right) (c+d x) \cos (c+d x)-24 a^2 b^2 c-24 a^2 b^2 d x+4 a b^3 \sin (3 (c+d x))+2 b^4 \sin (2 (c+d x))-b^4 \sin (4 (c+d x))}{a+b \cos (c+d x)}}{24 b^5 d}","-\frac{2 a^2 \left(4 a^2-3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d \sqrt{a-b} \sqrt{a+b}}+\frac{a x \left(4 a^2-b^2\right)}{b^5}-\frac{\left(12 a^2-b^2\right) \sin (c+d x)}{3 b^4 d}+\frac{2 a \sin (c+d x) \cos (c+d x)}{b^3 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{b d (a+b \cos (c+d x))}-\frac{4 \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d}",1,"((48*a^2*(4*a^2 - 3*b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (96*a^4*c - 24*a^2*b^2*c + 96*a^4*d*x - 24*a^2*b^2*d*x + 24*a*b*(4*a^2 - b^2)*(c + d*x)*Cos[c + d*x] + 12*a*b*(-8*a^2 + b^2)*Sin[c + d*x] - 24*a^2*b^2*Sin[2*(c + d*x)] + 2*b^4*Sin[2*(c + d*x)] + 4*a*b^3*Sin[3*(c + d*x)] - b^4*Sin[4*(c + d*x)])/(a + b*Cos[c + d*x]))/(24*b^5*d)","A",1
603,1,131,154,0.3099586,"\int \frac{\cos ^2(c+d x) \left(1-\cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{2 \left(b^2-6 a^2\right) (c+d x)-\frac{8 a \left(3 a^2-2 b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+\frac{4 a^2 b \sin (c+d x)}{a+b \cos (c+d x)}+8 a b \sin (c+d x)-b^2 \sin (2 (c+d x))}{4 b^4 d}","\frac{2 a \left(3 a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d \sqrt{a-b} \sqrt{a+b}}-\frac{x \left(6 a^2-b^2\right)}{2 b^4}+\frac{3 a \sin (c+d x)}{b^3 d}+\frac{\sin (c+d x) \cos ^2(c+d x)}{b d (a+b \cos (c+d x))}-\frac{3 \sin (c+d x) \cos (c+d x)}{2 b^2 d}",1,"(2*(-6*a^2 + b^2)*(c + d*x) - (8*a*(3*a^2 - 2*b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + 8*a*b*Sin[c + d*x] + (4*a^2*b*Sin[c + d*x])/(a + b*Cos[c + d*x]) - b^2*Sin[2*(c + d*x)])/(4*b^4*d)","A",1
604,1,132,112,1.0713857,"\int \frac{\cos (c+d x) \left(1-\cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{\frac{4 a^2 c+4 a^2 d x-4 a b \sin (c+d x)+4 a b (c+d x) \cos (c+d x)-b^2 \sin (2 (c+d x))}{a+b \cos (c+d x)}+\frac{4 \left(2 a^2-b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}}{2 b^3 d}","-\frac{2 \left(2 a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d \sqrt{a-b} \sqrt{a+b}}+\frac{2 a x}{b^3}-\frac{a \sin (c+d x)}{b^2 d (a+b \cos (c+d x))}-\frac{\sin (c+d x)}{b^2 d}",1,"((4*(2*a^2 - b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (4*a^2*c + 4*a^2*d*x + 4*a*b*(c + d*x)*Cos[c + d*x] - 4*a*b*Sin[c + d*x] - b^2*Sin[2*(c + d*x)])/(a + b*Cos[c + d*x]))/(2*b^3*d)","A",1
605,1,80,85,0.2462221,"\int \frac{1-\cos ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2,x]","-\frac{\frac{2 a \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}-\frac{b \sin (c+d x)}{a+b \cos (c+d x)}+c+d x}{b^2 d}","\frac{2 a \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d \sqrt{a-b} \sqrt{a+b}}+\frac{\sin (c+d x)}{b d (a+b \cos (c+d x))}-\frac{x}{b^2}",1,"-((c + d*x + (2*a*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] - (b*Sin[c + d*x])/(a + b*Cos[c + d*x]))/(b^2*d))","A",1
606,1,123,94,0.2014205,"\int \frac{\left(1-\cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((1 - Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^2,x]","\frac{\frac{2 b \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}-\frac{a \sin (c+d x)}{a+b \cos (c+d x)}-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^2 d}","-\frac{2 b \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d \sqrt{a-b} \sqrt{a+b}}+\frac{\tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{\sin (c+d x)}{a d (a+b \cos (c+d x))}",1,"((2*b*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] - Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - (a*Sin[c + d*x])/(a + b*Cos[c + d*x]))/(a^2*d)","A",1
607,1,143,118,0.6594726,"\int \frac{\left(1-\cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((1 - Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^2,x]","\frac{\frac{2 \left(a^2-2 b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+\frac{a b \sin (c+d x)}{a+b \cos (c+d x)}+a \tan (c+d x)+2 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^3 d}","-\frac{2 b \tanh ^{-1}(\sin (c+d x))}{a^3 d}+\frac{2 \tan (c+d x)}{a^2 d}-\frac{2 \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d \sqrt{a-b} \sqrt{a+b}}-\frac{\tan (c+d x)}{a d (a+b \cos (c+d x))}",1,"((2*(a^2 - 2*b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + 2*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 2*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a*b*Sin[c + d*x])/(a + b*Cos[c + d*x]) + a*Tan[c + d*x])/(a^3*d)","A",1
608,1,271,160,3.48055,"\int \frac{\left(1-\cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((1 - Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^2,x]","\frac{\frac{8 b \left(3 b^2-2 a^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+\frac{a^2}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{a^2}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+2 a^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 a^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-\frac{4 a b^2 \sin (c+d x)}{a+b \cos (c+d x)}-8 a b \tan (c+d x)-12 b^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 a^4 d}","-\frac{3 b \tan (c+d x)}{a^3 d}+\frac{3 \tan (c+d x) \sec (c+d x)}{2 a^2 d}+\frac{2 b \left(2 a^2-3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d \sqrt{a-b} \sqrt{a+b}}-\frac{\left(a^2-6 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{\tan (c+d x) \sec (c+d x)}{a d (a+b \cos (c+d x))}",1,"((8*b*(-2*a^2 + 3*b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + 2*a^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 12*b^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 2*a^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 12*b^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + a^2/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 - a^2/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - (4*a*b^2*Sin[c + d*x])/(a + b*Cos[c + d*x]) - 8*a*b*Tan[c + d*x])/(4*a^4*d)","A",1
609,1,475,195,6.2137984,"\int \frac{\left(1-\cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((1 - Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + b*Cos[c + d*x])^2,x]","\frac{b^3 \sin (c+d x)}{a^4 d (a+b \cos (c+d x))}+\frac{a-6 b}{12 a^3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{6 b-a}{12 a^3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{6 a^2 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{6 a^2 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{\left(4 b^3-a^2 b\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a^5 d}+\frac{\left(a^2 b-4 b^3\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^5 d}+\frac{2 b^2 \left(3 a^2-4 b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{a^5 d \sqrt{b^2-a^2}}+\frac{9 b^2 \sin \left(\frac{1}{2} (c+d x)\right)-a^2 \sin \left(\frac{1}{2} (c+d x)\right)}{3 a^4 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{9 b^2 \sin \left(\frac{1}{2} (c+d x)\right)-a^2 \sin \left(\frac{1}{2} (c+d x)\right)}{3 a^4 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{2 b \tan (c+d x) \sec (c+d x)}{a^3 d}+\frac{4 \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d}-\frac{2 b^2 \left(3 a^2-4 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d \sqrt{a-b} \sqrt{a+b}}+\frac{b \left(a^2-4 b^2\right) \tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{\left(a^2-12 b^2\right) \tan (c+d x)}{3 a^4 d}-\frac{\tan (c+d x) \sec ^2(c+d x)}{a d (a+b \cos (c+d x))}",1,"(2*b^2*(3*a^2 - 4*b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(a^5*Sqrt[-a^2 + b^2]*d) + ((-(a^2*b) + 4*b^3)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(a^5*d) + ((a^2*b - 4*b^3)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(a^5*d) + (a - 6*b)/(12*a^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + Sin[(c + d*x)/2]/(6*a^2*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + Sin[(c + d*x)/2]/(6*a^2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (-a + 6*b)/(12*a^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (-(a^2*Sin[(c + d*x)/2]) + 9*b^2*Sin[(c + d*x)/2])/(3*a^4*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (-(a^2*Sin[(c + d*x)/2]) + 9*b^2*Sin[(c + d*x)/2])/(3*a^4*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (b^3*Sin[c + d*x])/(a^4*d*(a + b*Cos[c + d*x]))","B",1
610,1,979,326,7.2714735,"\int \frac{\cos ^4(c+d x) \left(1-\cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^4*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{-\frac{12 \left(-48 a (c+d x)-\frac{6 \left(16 a^6-40 b^2 a^4+30 b^4 a^2-5 b^6\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}+16 b \sin (c+d x)+\frac{a b \left(40 a^4-72 b^2 a^2+29 b^4\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}-\frac{b \left(8 a^4-8 b^2 a^2+b^4\right) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}\right)}{b^4}+12 \left(\frac{b \left(-4 a^2-3 b \cos (c+d x) a+b^2\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))^2}-\frac{2 \left(2 a^2+b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}\right)+\frac{6 \left(\frac{\left(a \left(2 a^2-5 b^2\right) \cos (c+d x)-b \left(2 a^2+b^2\right)\right) \sin (c+d x)}{(a+b \cos (c+d x))^2}-\frac{6 b^2 \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}\right)}{(a-b)^2 (a+b)^2}+\frac{\frac{12 \left(640 a^8-1792 b^2 a^6+1680 b^4 a^4-560 b^6 a^2+35 b^8\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}+\frac{3840 c a^9+3840 d x a^9-3840 b \sin (c+d x) a^8-6912 b^2 c a^7-6912 b^2 d x a^7-2880 b^2 \sin (2 (c+d x)) a^7+7872 b^3 \sin (c+d x) a^6-320 b^3 \sin (3 (c+d x)) a^6+1728 b^4 c a^5+1728 b^4 d x a^5+6304 b^4 \sin (2 (c+d x)) a^5+40 b^4 \sin (4 (c+d x)) a^5-4256 b^5 \sin (c+d x) a^4+696 b^5 \sin (3 (c+d x)) a^4-8 b^5 \sin (5 (c+d x)) a^4+1920 b^6 c a^3+1920 b^6 d x a^3-4022 b^6 \sin (2 (c+d x)) a^3-80 b^6 \sin (4 (c+d x)) a^3+172 b^7 \sin (c+d x) a^2-432 b^7 \sin (3 (c+d x)) a^2+16 b^7 \sin (5 (c+d x)) a^2-576 b^8 c a-576 b^8 d x a+192 b^2 \left(10 a^2-3 b^2\right) \left(a^2-b^2\right)^2 (c+d x) \cos (2 (c+d x)) a+607 b^8 \sin (2 (c+d x)) a+40 b^8 \sin (4 (c+d x)) a+768 b \left(10 a^2-3 b^2\right) \left(a^3-a b^2\right)^2 (c+d x) \cos (c+d x)+70 b^9 \sin (c+d x)+56 b^9 \sin (3 (c+d x))-8 b^9 \sin (5 (c+d x))}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}}{b^6}}{384 d}","\frac{\left(5 a^2-4 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{2 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{a x \left(20 a^2-3 b^2\right)}{2 b^6}+\frac{a \left(10 a^2-9 b^2\right) \sin (c+d x) \cos (c+d x)}{2 b^4 d \left(a^2-b^2\right)}-\frac{\left(20 a^2-17 b^2\right) \sin (c+d x) \cos ^2(c+d x)}{6 b^3 d \left(a^2-b^2\right)}-\frac{a^2 \left(20 a^4-33 a^2 b^2+12 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^6 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\left(60 a^4-59 a^2 b^2+2 b^4\right) \sin (c+d x)}{6 b^5 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \cos ^4(c+d x)}{2 b d (a+b \cos (c+d x))^2}",1,"((-12*(-48*a*(c + d*x) - (6*(16*a^6 - 40*a^4*b^2 + 30*a^2*b^4 - 5*b^6)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + 16*b*Sin[c + d*x] - (b*(8*a^4 - 8*a^2*b^2 + b^4)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^2) + (a*b*(40*a^4 - 72*a^2*b^2 + 29*b^4)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x]))))/b^4 + 12*((-2*(2*a^2 + b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (b*(-4*a^2 + b^2 - 3*a*b*Cos[c + d*x])*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])^2)) + (6*((-6*b^2*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + ((-(b*(2*a^2 + b^2)) + a*(2*a^2 - 5*b^2)*Cos[c + d*x])*Sin[c + d*x])/(a + b*Cos[c + d*x])^2))/((a - b)^2*(a + b)^2) + ((12*(640*a^8 - 1792*a^6*b^2 + 1680*a^4*b^4 - 560*a^2*b^6 + 35*b^8)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (3840*a^9*c - 6912*a^7*b^2*c + 1728*a^5*b^4*c + 1920*a^3*b^6*c - 576*a*b^8*c + 3840*a^9*d*x - 6912*a^7*b^2*d*x + 1728*a^5*b^4*d*x + 1920*a^3*b^6*d*x - 576*a*b^8*d*x + 768*b*(10*a^2 - 3*b^2)*(a^3 - a*b^2)^2*(c + d*x)*Cos[c + d*x] + 192*a*b^2*(10*a^2 - 3*b^2)*(a^2 - b^2)^2*(c + d*x)*Cos[2*(c + d*x)] - 3840*a^8*b*Sin[c + d*x] + 7872*a^6*b^3*Sin[c + d*x] - 4256*a^4*b^5*Sin[c + d*x] + 172*a^2*b^7*Sin[c + d*x] + 70*b^9*Sin[c + d*x] - 2880*a^7*b^2*Sin[2*(c + d*x)] + 6304*a^5*b^4*Sin[2*(c + d*x)] - 4022*a^3*b^6*Sin[2*(c + d*x)] + 607*a*b^8*Sin[2*(c + d*x)] - 320*a^6*b^3*Sin[3*(c + d*x)] + 696*a^4*b^5*Sin[3*(c + d*x)] - 432*a^2*b^7*Sin[3*(c + d*x)] + 56*b^9*Sin[3*(c + d*x)] + 40*a^5*b^4*Sin[4*(c + d*x)] - 80*a^3*b^6*Sin[4*(c + d*x)] + 40*a*b^8*Sin[4*(c + d*x)] - 8*a^4*b^5*Sin[5*(c + d*x)] + 16*a^2*b^7*Sin[5*(c + d*x)] - 8*b^9*Sin[5*(c + d*x)])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2))/b^6)/(384*d)","B",1
611,1,374,268,5.0560422,"\int \frac{\cos ^3(c+d x) \left(1-\cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^3*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{-96 a^6 c-96 a^6 d x+96 a^5 b \sin (c+d x)+72 a^4 b^2 \sin (2 (c+d x))+56 a^4 b^2 c+56 a^4 b^2 d x-80 a^3 b^3 \sin (c+d x)+8 a^3 b^3 \sin (3 (c+d x))-70 a^2 b^4 \sin (2 (c+d x))-a^2 b^4 \sin (4 (c+d x))+44 a^2 b^4 c+44 a^2 b^4 d x-16 a b \left(12 a^4-13 a^2 b^2+b^4\right) (c+d x) \cos (c+d x)-4 b^2 \left(12 a^4-13 a^2 b^2+b^4\right) (c+d x) \cos (2 (c+d x))-8 a b^5 \sin (c+d x)-8 a b^5 \sin (3 (c+d x))+2 b^6 \sin (2 (c+d x))+b^6 \sin (4 (c+d x))-4 b^6 c-4 b^6 d x}{(a+b \cos (c+d x))^2}-\frac{16 a \left(12 a^4-19 a^2 b^2+6 b^4\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}}{16 b^5 d (a-b) (a+b)}","\frac{\left(4 a^2-3 b^2\right) \sin (c+d x) \cos ^2(c+d x)}{2 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{x \left(12 a^2-b^2\right)}{2 b^5}+\frac{a \left(12 a^2-11 b^2\right) \sin (c+d x)}{2 b^4 d \left(a^2-b^2\right)}-\frac{\left(6 a^2-5 b^2\right) \sin (c+d x) \cos (c+d x)}{2 b^3 d \left(a^2-b^2\right)}+\frac{a \left(12 a^4-19 a^2 b^2+6 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\sin (c+d x) \cos ^3(c+d x)}{2 b d (a+b \cos (c+d x))^2}",1,"((-16*a*(12*a^4 - 19*a^2*b^2 + 6*b^4)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (-96*a^6*c + 56*a^4*b^2*c + 44*a^2*b^4*c - 4*b^6*c - 96*a^6*d*x + 56*a^4*b^2*d*x + 44*a^2*b^4*d*x - 4*b^6*d*x - 16*a*b*(12*a^4 - 13*a^2*b^2 + b^4)*(c + d*x)*Cos[c + d*x] - 4*b^2*(12*a^4 - 13*a^2*b^2 + b^4)*(c + d*x)*Cos[2*(c + d*x)] + 96*a^5*b*Sin[c + d*x] - 80*a^3*b^3*Sin[c + d*x] - 8*a*b^5*Sin[c + d*x] + 72*a^4*b^2*Sin[2*(c + d*x)] - 70*a^2*b^4*Sin[2*(c + d*x)] + 2*b^6*Sin[2*(c + d*x)] + 8*a^3*b^3*Sin[3*(c + d*x)] - 8*a*b^5*Sin[3*(c + d*x)] - a^2*b^4*Sin[4*(c + d*x)] + b^6*Sin[4*(c + d*x)])/(a + b*Cos[c + d*x])^2)/(16*(a - b)*b^5*(a + b)*d)","A",1
612,1,159,182,1.0680417,"\int \frac{\cos ^2(c+d x) \left(1-\cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{a b \left(4 b^2-5 a^2\right) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}+\frac{a^2 b \sin (c+d x)}{(a+b \cos (c+d x))^2}-\frac{2 \left(6 a^4-9 a^2 b^2+2 b^4\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}+6 a (c+d x)-2 b \sin (c+d x)}{2 b^4 d}","-\frac{a \left(3 a^2-2 b^2\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\left(6 a^4-9 a^2 b^2+2 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{3 a x}{b^4}+\frac{\sin (c+d x) \cos ^2(c+d x)}{2 b d (a+b \cos (c+d x))^2}-\frac{3 \sin (c+d x)}{2 b^3 d}",1,"(6*a*(c + d*x) - (2*(6*a^4 - 9*a^2*b^2 + 2*b^4)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) - 2*b*Sin[c + d*x] + (a^2*b*Sin[c + d*x])/(a + b*Cos[c + d*x])^2 + (a*b*(-5*a^2 + 4*b^2)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])))/(2*b^4*d)","A",1
613,1,291,149,1.591908,"\int \frac{\cos (c+d x) \left(1-\cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*(1 - Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{\frac{\sin (c+d x) \left(b \left(a^2+2 b^2\right) \cos (c+d x)+a \left(2 a^2+b^2\right)\right)}{(a+b \cos (c+d x))^2}+\frac{6 a b \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}}{(a-b)^2 (a+b)^2}-\frac{\frac{a b \left(4 a^2-3 b^2\right) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}-\frac{3 b \left(4 a^4-7 a^2 b^2+2 b^4\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}+\frac{2 a \left(8 a^4-20 a^2 b^2+15 b^4\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}+8 (c+d x)}{b^3}}{8 d}","\frac{\left(3 a^2-2 b^2\right) \sin (c+d x)}{2 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{a \left(2 a^2-3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{a \sin (c+d x)}{2 b^2 d (a+b \cos (c+d x))^2}-\frac{x}{b^3}",1,"(-((8*(c + d*x) + (2*a*(8*a^4 - 20*a^2*b^2 + 15*b^4)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (a*b*(4*a^2 - 3*b^2)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^2) - (3*b*(4*a^4 - 7*a^2*b^2 + 2*b^4)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])))/b^3) + ((6*a*b*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + ((a*(2*a^2 + b^2) + b*(a^2 + 2*b^2)*Cos[c + d*x])*Sin[c + d*x])/(a + b*Cos[c + d*x])^2)/((a - b)^2*(a + b)^2))/(8*d)","A",1
614,1,94,117,0.2882616,"\int \frac{1-\cos ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(1 - Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3,x]","-\frac{\frac{2 \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+\frac{\sin (c+d x) (a \cos (c+d x)+b)}{(a+b \cos (c+d x))^2}}{2 d (a-b) (a+b)}","-\frac{a \sin (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\sin (c+d x)}{2 b d (a+b \cos (c+d x))^2}",1,"-1/2*((2*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + ((b + a*Cos[c + d*x])*Sin[c + d*x])/(a + b*Cos[c + d*x])^2)/((a - b)*(a + b)*d)","A",1
615,1,180,155,1.0607992,"\int \frac{\left(1-\cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[((1 - Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{2 b \left(2 b^2-3 a^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}-\frac{a \sin (c+d x) \left(2 a^3+b \left(a^2-2 b^2\right) \cos (c+d x)-3 a b^2\right)}{(a-b) (a+b) (a+b \cos (c+d x))^2}-2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^3 d}","\frac{\tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{\left(a^2-2 b^2\right) \sin (c+d x)}{2 a^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{b \left(3 a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\sin (c+d x)}{2 a d (a+b \cos (c+d x))^2}",1,"((2*b*(-3*a^2 + 2*b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) - 2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - (a*(2*a^3 - 3*a*b^2 + b*(a^2 - 2*b^2)*Cos[c + d*x])*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^2))/(2*a^3*d)","A",1
616,1,200,204,2.9330548,"\int \frac{\left(1-\cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[((1 - Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^3,x]","\frac{-\frac{2 \left(2 a^4-9 a^2 b^2+6 b^4\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}+\frac{a b \sin (c+d x) \left(4 a^3+b \left(3 a^2-4 b^2\right) \cos (c+d x)-5 a b^2\right)}{(a-b) (a+b) (a+b \cos (c+d x))^2}+2 a \tan (c+d x)+6 b \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{2 a^4 d}","-\frac{3 b \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{\left(2 a^2-3 b^2\right) \tan (c+d x)}{2 a^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\left(2 a^4-9 a^2 b^2+6 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left(5 a^2-6 b^2\right) \tan (c+d x)}{2 a^3 d \left(a^2-b^2\right)}-\frac{\tan (c+d x)}{2 a d (a+b \cos (c+d x))^2}",1,"((-2*(2*a^4 - 9*a^2*b^2 + 6*b^4)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + 6*b*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (a*b*(4*a^3 - 5*a*b^2 + b*(3*a^2 - 4*b^2)*Cos[c + d*x])*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^2) + 2*a*Tan[c + d*x])/(2*a^4*d)","A",1
617,1,414,271,6.2129279,"\int \frac{\left(1-\cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[((1 - Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^3,x]","-\frac{3 b \sin \left(\frac{1}{2} (c+d x)\right)}{a^4 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{3 b \sin \left(\frac{1}{2} (c+d x)\right)}{a^4 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{b^2 \sin (c+d x)}{2 a^3 d (a+b \cos (c+d x))^2}+\frac{1}{4 a^3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{1}{4 a^3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{\left(a^2-12 b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^5 d}+\frac{\left(12 b^2-a^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^5 d}+\frac{6 b^4 \sin (c+d x)-5 a^2 b^2 \sin (c+d x)}{2 a^4 d (a-b) (a+b) (a+b \cos (c+d x))}-\frac{b \left(6 a^4-19 a^2 b^2+12 b^4\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{a^5 d \left(a^2-b^2\right) \sqrt{b^2-a^2}}","-\frac{\left(3 a^2-4 b^2\right) \tan (c+d x) \sec (c+d x)}{2 a^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\left(a^2-12 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}-\frac{b \left(11 a^2-12 b^2\right) \tan (c+d x)}{2 a^4 d \left(a^2-b^2\right)}+\frac{\left(5 a^2-6 b^2\right) \tan (c+d x) \sec (c+d x)}{2 a^3 d \left(a^2-b^2\right)}+\frac{b \left(6 a^4-19 a^2 b^2+12 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^2}",1,"-((b*(6*a^4 - 19*a^2*b^2 + 12*b^4)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(a^5*(a^2 - b^2)*Sqrt[-a^2 + b^2]*d)) + ((a^2 - 12*b^2)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(2*a^5*d) + ((-a^2 + 12*b^2)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(2*a^5*d) + 1/(4*a^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) - (3*b*Sin[(c + d*x)/2])/(a^4*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - 1/(4*a^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) - (3*b*Sin[(c + d*x)/2])/(a^4*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) - (b^2*Sin[c + d*x])/(2*a^3*d*(a + b*Cos[c + d*x])^2) + (-5*a^2*b^2*Sin[c + d*x] + 6*b^4*Sin[c + d*x])/(2*a^4*(a - b)*(a + b)*d*(a + b*Cos[c + d*x]))","A",1
618,1,563,335,6.259414,"\int \frac{\left(1-\cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[((1 - Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + b*Cos[c + d*x])^3,x]","\frac{b^3 \sin (c+d x)}{2 a^4 d (a+b \cos (c+d x))^2}+\frac{a-9 b}{12 a^4 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{9 b-a}{12 a^4 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{6 a^3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{6 a^3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{\left(20 b^3-3 a^2 b\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^6 d}+\frac{\left(3 a^2 b-20 b^3\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^6 d}+\frac{18 b^2 \sin \left(\frac{1}{2} (c+d x)\right)-a^2 \sin \left(\frac{1}{2} (c+d x)\right)}{3 a^5 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{18 b^2 \sin \left(\frac{1}{2} (c+d x)\right)-a^2 \sin \left(\frac{1}{2} (c+d x)\right)}{3 a^5 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{7 a^2 b^3 \sin (c+d x)-8 b^5 \sin (c+d x)}{2 a^5 d (a-b) (a+b) (a+b \cos (c+d x))}+\frac{b^2 \left(12 a^4-33 a^2 b^2+20 b^4\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{a^6 d \left(a^2-b^2\right) \sqrt{b^2-a^2}}","-\frac{\left(4 a^2-5 b^2\right) \tan (c+d x) \sec ^2(c+d x)}{2 a^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{b \left(3 a^2-20 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}-\frac{b \left(9 a^2-10 b^2\right) \tan (c+d x) \sec (c+d x)}{2 a^4 d \left(a^2-b^2\right)}+\frac{\left(17 a^2-20 b^2\right) \tan (c+d x) \sec ^2(c+d x)}{6 a^3 d \left(a^2-b^2\right)}-\frac{b^2 \left(12 a^4-33 a^2 b^2+20 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^6 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\left(2 a^4-59 a^2 b^2+60 b^4\right) \tan (c+d x)}{6 a^5 d \left(a^2-b^2\right)}-\frac{\tan (c+d x) \sec ^2(c+d x)}{2 a d (a+b \cos (c+d x))^2}",1,"(b^2*(12*a^4 - 33*a^2*b^2 + 20*b^4)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(a^6*(a^2 - b^2)*Sqrt[-a^2 + b^2]*d) + ((-3*a^2*b + 20*b^3)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(2*a^6*d) + ((3*a^2*b - 20*b^3)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(2*a^6*d) + (a - 9*b)/(12*a^4*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + Sin[(c + d*x)/2]/(6*a^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + Sin[(c + d*x)/2]/(6*a^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (-a + 9*b)/(12*a^4*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (-(a^2*Sin[(c + d*x)/2]) + 18*b^2*Sin[(c + d*x)/2])/(3*a^5*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (-(a^2*Sin[(c + d*x)/2]) + 18*b^2*Sin[(c + d*x)/2])/(3*a^5*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (b^3*Sin[c + d*x])/(2*a^4*d*(a + b*Cos[c + d*x])^2) + (7*a^2*b^3*Sin[c + d*x] - 8*b^5*Sin[c + d*x])/(2*a^5*(a - b)*(a + b)*d*(a + b*Cos[c + d*x]))","A",1
619,1,28,16,0.0091279,"\int \frac{a^2-b^2 \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]),x]","a x-\frac{b \sin (c) \cos (d x)}{d}-\frac{b \cos (c) \sin (d x)}{d}","a x-\frac{b \sin (c+d x)}{d}",1,"a*x - (b*Cos[d*x]*Sin[c])/d - (b*Cos[c]*Sin[d*x])/d","A",1
620,1,53,54,0.0690808,"\int \frac{a^2-b^2 \cos ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2,x]","-\frac{4 a \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{d \sqrt{b^2-a^2}}-x","\frac{4 a \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d \sqrt{a-b} \sqrt{a+b}}-x",1,"-x - (4*a*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(Sqrt[-a^2 + b^2]*d)","A",1
621,1,91,93,0.2492782,"\int \frac{a^2-b^2 \cos ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3,x]","\frac{2 \left(\frac{\left(a^2+b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}-\frac{a b \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}\right)}{d}","\frac{2 \left(a^2+b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{3/2} (a+b)^{3/2}}-\frac{2 a b \sin (c+d x)}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"(2*(((a^2 + b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) - (a*b*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x]))))/d","A",1
622,1,116,140,0.6222381,"\int \frac{a^2-b^2 \cos ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Integrate[(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4,x]","-\frac{\frac{2 a \left(a^2+2 b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}+\frac{b \sin (c+d x) \left(3 a^3+b \left(2 a^2+b^2\right) \cos (c+d x)\right)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))^2}}{d}","\frac{2 a \left(a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{b \left(2 a^2+b^2\right) \sin (c+d x)}{d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{a b \sin (c+d x)}{d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"-(((2*a*(a^2 + 2*b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (b*(3*a^3 + b*(2*a^2 + b^2)*Cos[c + d*x])*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])^2))/d)","A",1
623,1,269,364,1.3195304,"\int \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","\frac{b (a+b \cos (c+d x)) \left(2 a \left(32 a^2 C+84 A b^2+57 b^2 C\right) \sin (c+d x)+b \left(\left(-24 a^2 C+252 A b^2+266 b^2 C\right) \sin (2 (c+d x))+5 b C (2 a \sin (3 (c+d x))+7 b \sin (4 (c+d x)))\right)\right)+8 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(a b^2 \left(-4 a^2 C+147 A b^2+111 b^2 C\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-\left(16 a^4 C+6 a^2 b^2 (7 A+4 C)-21 b^4 (9 A+7 C)\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{1260 b^4 d \sqrt{a+b \cos (c+d x)}}","\frac{4 a \left(a^2-b^2\right) \left(8 a^2 C+21 A b^2+18 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(24 a^2 C+7 b^2 (9 A+7 C)\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^3 d}-\frac{4 a \left(8 a^2 C+21 A b^2+18 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^3 d}-\frac{2 \left(16 a^4 C+6 a^2 b^2 (7 A+4 C)-21 b^4 (9 A+7 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{21 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}",1,"(8*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(a*b^2*(147*A*b^2 - 4*a^2*C + 111*b^2*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - (16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x])*(2*a*(84*A*b^2 + 32*a^2*C + 57*b^2*C)*Sin[c + d*x] + b*((252*A*b^2 - 24*a^2*C + 266*b^2*C)*Sin[2*(c + d*x)] + 5*b*C*(2*a*Sin[3*(c + d*x)] + 7*b*Sin[4*(c + d*x)]))))/(1260*b^4*d*Sqrt[a + b*Cos[c + d*x]])","A",1
624,1,216,291,0.8503904,"\int \cos (c+d x) \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","\frac{2 b \sin (c+d x) (a+b \cos (c+d x)) \left(-8 a^2 C+6 a b C \cos (c+d x)+70 A b^2+15 b^2 C \cos (2 (c+d x))+65 b^2 C\right)+4 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b \left(2 a^2 b C+35 A b^3+25 b^3 C\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+a \left(8 a^2 C+35 A b^2+19 b^2 C\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{210 b^3 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(8 a^2 C+5 b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b^2 d}-\frac{2 \left(a^2-b^2\right) \left(8 a^2 C+35 A b^2+25 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 a \left(8 a^2 C+35 A b^2+19 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}",1,"(4*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b*(35*A*b^3 + 2*a^2*b*C + 25*b^3*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + a*(35*A*b^2 + 8*a^2*C + 19*b^2*C)*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + 2*b*(a + b*Cos[c + d*x])*(70*A*b^2 - 8*a^2*C + 65*b^2*C + 6*a*b*C*Cos[c + d*x] + 15*b^2*C*Cos[2*(c + d*x)])*Sin[c + d*x])/(210*b^3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
625,1,181,218,0.9377752,"\int \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","\frac{-2 (a+b) \left(2 a^2 C-15 A b^2-9 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+b C \sin (c+d x) \left(2 a^2+8 a b \cos (c+d x)+3 b^2 \cos (2 (c+d x))+3 b^2\right)+4 a C \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(2 a^2 C-3 b^2 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{4 a C \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}-\frac{4 a C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b d}",1,"(-2*(a + b)*(-15*A*b^2 + 2*a^2*C - 9*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + 4*a*(a^2 - b^2)*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + b*C*(2*a^2 + 3*b^2 + 8*a*b*Cos[c + d*x] + 3*b^2*Cos[2*(c + d*x)])*Sin[c + d*x])/(15*b^2*d*Sqrt[a + b*Cos[c + d*x]])","A",1
626,1,371,231,2.3903602,"\int \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{\frac{4 b (3 A+C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 a (6 A+C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 i C \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{b^2 \sqrt{-\frac{1}{a+b}}}+4 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{6 d}","-\frac{2 \left(a^2 C-b^2 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 a A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{2 a C \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"((4*b*(3*A + C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*a*(6*A + C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + ((2*I)*C*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(b^2*Sqrt[-(a + b)^(-1)]) + 4*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(6*d)","C",1
627,1,374,205,2.2456737,"\int \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{\frac{2 b (A+2 C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i (A-2 C) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}+4 A \tan (c+d x) \sqrt{a+b \cos (c+d x)}+\frac{8 a C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{4 d}","-\frac{(A-2 C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{A \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{d}+\frac{a A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}",1,"((8*a*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*b*(A + 2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(A - 2*C)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b*Sqrt[-(a + b)^(-1)]) + 4*A*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*d)","C",1
628,1,406,277,3.3734705,"\int \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{\frac{2 \left(8 a^2 (A+2 C)-3 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a \sqrt{a+b \cos (c+d x)}}-\frac{2 i A \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a^2 \sqrt{-\frac{1}{a+b}}}+\frac{8 b (A+4 C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{4 A \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)} (2 a+b \cos (c+d x))}{a}}{16 d}","-\frac{\left(A b^2-4 a^2 (A+2 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{a+b \cos (c+d x)}}+\frac{b (3 A+8 C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{A b \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a d}-\frac{A b \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{A \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d}",1,"((8*b*(A + 4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(-3*A*b^2 + 8*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(a*Sqrt[a + b*Cos[c + d*x]]) - ((2*I)*A*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a^2*Sqrt[-(a + b)^(-1)]) + (4*A*Sqrt[a + b*Cos[c + d*x]]*(2*a + b*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/a)/(16*d)","C",1
629,1,601,365,6.5186695,"\int \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\sqrt{a+b \cos (c+d x)} \left(\frac{\sec (c+d x) \left(16 a^2 A \sin (c+d x)+24 a^2 C \sin (c+d x)-3 A b^2 \sin (c+d x)\right)}{24 a^2}+\frac{A b \tan (c+d x) \sec (c+d x)}{12 a}+\frac{1}{3} A \tan (c+d x) \sec ^2(c+d x)\right)}{d}-\frac{b \left(\frac{2 \left(-8 a^2 A-24 a^2 C-9 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \left(16 a^2 A+24 a^2 C-3 A b^2\right) \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{b \cos (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left(2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right)}-\frac{8 a A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}\right)}{96 a^2 d}","-\frac{\left(3 A b^2-8 a^2 (2 A+3 C)\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 a^2 d}-\frac{\left(A b^2-8 a^2 (2 A+3 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a d \sqrt{a+b \cos (c+d x)}}+\frac{\left(3 A b^2-8 a^2 (2 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{b \left(4 a^2 (A+2 C)+A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{A b \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{12 a d}",1,"-1/96*(b*((-8*a*A*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(-8*a^2*A - 9*A*b^2 - 24*a^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(16*a^2*A - 3*A*b^2 + 24*a^2*C)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2))))/(a^2*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]*(16*a^2*A*Sin[c + d*x] - 3*A*b^2*Sin[c + d*x] + 24*a^2*C*Sin[c + d*x]))/(24*a^2) + (A*b*Sec[c + d*x]*Tan[c + d*x])/(12*a) + (A*Sec[c + d*x]^2*Tan[c + d*x])/3))/d","C",1
630,1,331,443,1.7517272,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2),x]","\frac{b (a+b \cos (c+d x)) \left(b \left(16 a \left(-3 a^2 C+132 A b^2+136 b^2 C\right) \sin (2 (c+d x))+5 b \left(\left(4 a^2 C+132 A b^2+171 b^2 C\right) \sin (3 (c+d x))+7 b C (8 a \sin (4 (c+d x))+3 b \sin (5 (c+d x)))\right)\right)+2 \left(64 a^4 C+6 a^2 b^2 (44 A+27 C)+5 b^4 (506 A+435 C)\right) \sin (c+d x)\right)+16 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b \left(-4 a^4 b C+3 a^2 b^3 (187 A+141 C)+25 b^5 (11 A+9 C)\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-2 a \left(8 a^4 C+3 a^2 b^2 (11 A+6 C)-b^4 (451 A+348 C)\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{9240 b^4 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(8 a^2 C+3 b^2 (11 A+9 C)\right) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{231 b^3 d}-\frac{4 a \left(8 a^2 C+33 A b^2+34 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{1155 b^3 d}+\frac{2 \left(a^2-b^2\right) \left(16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{1155 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{4 a \left(8 a^4 C+3 a^2 b^2 (11 A+6 C)-b^4 (451 A+348 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{1155 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 \left(16 a^4 C+6 a^2 b^2 (11 A+8 C)-25 b^4 (11 A+9 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1155 b^3 d}-\frac{4 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{33 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}",1,"(16*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b*(-4*a^4*b*C + 25*b^5*(11*A + 9*C) + 3*a^2*b^3*(187*A + 141*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - 2*a*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x])*(2*(64*a^4*C + 6*a^2*b^2*(44*A + 27*C) + 5*b^4*(506*A + 435*C))*Sin[c + d*x] + b*(16*a*(132*A*b^2 - 3*a^2*C + 136*b^2*C)*Sin[2*(c + d*x)] + 5*b*((132*A*b^2 + 4*a^2*C + 171*b^2*C)*Sin[3*(c + d*x)] + 7*b*C*(8*a*Sin[4*(c + d*x)] + 3*b*Sin[5*(c + d*x)])))))/(9240*b^4*d*Sqrt[a + b*Cos[c + d*x]])","A",1
631,1,269,356,1.3721713,"\int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2),x]","\frac{b (a+b \cos (c+d x)) \left(b \left(2 \left(6 a^2 C+126 A b^2+133 b^2 C\right) \sin (2 (c+d x))+5 b C (20 a \sin (3 (c+d x))+7 b \sin (4 (c+d x)))\right)-4 a \left(8 a^2 C-252 A b^2-201 b^2 C\right) \sin (c+d x)\right)+8 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(2 a b^2 \left(C \left(a^2+93 b^2\right)+126 A b^2\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(8 a^4 C+3 a^2 b^2 (21 A+11 C)+21 b^4 (9 A+7 C)\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{1260 b^3 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(8 a^2 C+7 b^2 (9 A+7 C)\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}+\frac{2 a \left(8 a^2 C+63 A b^2+39 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^2 d}-\frac{2 a \left(a^2-b^2\right) \left(8 a^2 C+63 A b^2+39 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(8 a^4 C+3 a^2 b^2 (21 A+11 C)+21 b^4 (9 A+7 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}",1,"(8*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(2*a*b^2*(126*A*b^2 + (a^2 + 93*b^2)*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x])*(-4*a*(-252*A*b^2 + 8*a^2*C - 201*b^2*C)*Sin[c + d*x] + b*(2*(126*A*b^2 + 6*a^2*C + 133*b^2*C)*Sin[2*(c + d*x)] + 5*b*C*(20*a*Sin[3*(c + d*x)] + 7*b*Sin[4*(c + d*x)]))))/(1260*b^3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
632,1,224,285,0.8581687,"\int (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2),x]","\frac{2 b \sin (c+d x) (a+b \cos (c+d x)) \left(6 a^2 C+48 a b C \cos (c+d x)+70 A b^2+15 b^2 C \cos (2 (c+d x))+65 b^2 C\right)+4 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b^2 \left(3 a^2 (35 A+17 C)+5 b^2 (7 A+5 C)\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-2 a \left(3 a^2 C-70 A b^2-41 b^2 C\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{210 b^2 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(6 a^2 C-5 b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b d}-\frac{2 \left(a^2-b^2\right) \left(-6 a^2 C+35 A b^2+25 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{4 a \left(-3 a^2 C+70 A b^2+41 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}-\frac{4 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b d}",1,"(4*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(5*b^2*(7*A + 5*C) + 3*a^2*(35*A + 17*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - 2*a*(-70*A*b^2 + 3*a^2*C - 41*b^2*C)*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + 2*b*(a + b*Cos[c + d*x])*(70*A*b^2 + 6*a^2*C + 65*b^2*C + 48*a*b*C*Cos[c + d*x] + 15*b^2*C*Cos[2*(c + d*x)])*Sin[c + d*x])/(210*b^2*d*Sqrt[a + b*Cos[c + d*x]])","A",1
633,1,421,281,3.354068,"\int (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{\frac{2 \left(a^2 (10 A+C)+b^2 (5 A+3 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 i \left(C \left(a^2+3 b^2\right)+5 A b^2\right) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b^2 \sqrt{-\frac{1}{a+b}}}+\frac{8 a b (5 A+2 C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+4 C \sin (c+d x) \sqrt{a+b \cos (c+d x)} (2 a+b \cos (c+d x))}{10 d}","\frac{2 a \left(5 A b^2-C \left(a^2-b^2\right)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{5 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 C+b^2 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{5 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 a^2 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 a C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}",1,"((8*a*b*(5*A + 2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(a^2*(10*A + C) + b^2*(5*A + 3*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + ((2*I)*(5*A*b^2 + (a^2 + 3*b^2)*C)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b^2*Sqrt[-(a + b)^(-1)]) + 4*C*Sqrt[a + b*Cos[c + d*x]]*(2*a + b*Cos[c + d*x])*Sin[c + d*x])/(10*d)","C",1
634,1,406,270,3.5174561,"\int (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{\frac{8 \left(C \left(3 a^2+b^2\right)+3 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+4 \tan (c+d x) \sqrt{a+b \cos (c+d x)} (3 a A+2 b C \cos (c+d x))+\frac{2 a b (15 A+8 C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i (3 A-8 C) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{b \sqrt{-\frac{1}{a+b}}}}{12 d}","\frac{\left(a^2 (3 A-2 C)+2 b^2 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}-\frac{b (3 A-2 C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}-\frac{a (3 A-8 C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}+\frac{3 a A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}",1,"((8*(3*A*b^2 + (3*a^2 + b^2)*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*a*b*(15*A + 8*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(3*A - 8*C)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(b*Sqrt[-(a + b)^(-1)]) + 4*Sqrt[a + b*Cos[c + d*x]]*(3*a*A + 2*b*C*Cos[c + d*x])*Tan[c + d*x])/(12*d)","C",1
635,1,411,276,4.748808,"\int (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{\frac{2 \left(8 a^2 (A+2 C)+b^2 (A+8 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{8 a b (A+8 C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i (5 A-8 C) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a \sqrt{-\frac{1}{a+b}}}+4 A \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)} (2 a+5 b \cos (c+d x))}{16 d}","\frac{\left(4 a^2 (A+2 C)+3 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{a b (7 A+8 C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}-\frac{b (5 A-8 C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{3 A b \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}",1,"((8*a*b*(A + 8*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(8*a^2*(A + 2*C) + b^2*(A + 8*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(5*A - 8*C)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*Sqrt[-(a + b)^(-1)]) + 4*A*Sqrt[a + b*Cos[c + d*x]]*(2*a + 5*b*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(16*d)","C",1
636,1,607,365,6.6117583,"\int (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\sqrt{a+b \cos (c+d x)} \left(\frac{\sec (c+d x) \left(16 a^2 A \sin (c+d x)+24 a^2 C \sin (c+d x)+3 A b^2 \sin (c+d x)\right)}{24 a}+\frac{1}{3} a A \tan (c+d x) \sec ^2(c+d x)+\frac{7}{12} A b \tan (c+d x) \sec (c+d x)\right)}{d}-\frac{b \left(\frac{2 \left(-56 a^2 A-120 a^2 C+9 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \left(16 a^2 A+24 a^2 C+3 A b^2\right) \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{b \cos (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left(2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right)}+\frac{2 (-28 a A b-96 a b C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}\right)}{96 a d}","\frac{\left(8 a^2 (2 A+3 C)+3 A b^2\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 a d}+\frac{\left(8 a^2 (2 A+3 C)+b^2 (17 A+48 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(8 a^2 (2 A+3 C)+3 A b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{b \left(A b^2-12 a^2 (A+2 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 a d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}+\frac{A b \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}",1,"-1/96*(b*((2*(-28*a*A*b - 96*a*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(-56*a^2*A + 9*A*b^2 - 120*a^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(16*a^2*A + 3*A*b^2 + 24*a^2*C)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2))))/(a*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]*(16*a^2*A*Sin[c + d*x] + 3*A*b^2*Sin[c + d*x] + 24*a^2*C*Sin[c + d*x]))/(24*a) + (7*A*b*Sec[c + d*x]*Tan[c + d*x])/12 + (a*A*Sec[c + d*x]^2*Tan[c + d*x])/3))/d","C",1
637,1,696,436,6.7995482,"\int (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{\sqrt{a+b \cos (c+d x)} \left(\frac{\sec (c+d x) \left(52 a^2 A b \sin (c+d x)+80 a^2 b C \sin (c+d x)-3 A b^3 \sin (c+d x)\right)}{64 a^2}+\frac{\sec ^2(c+d x) \left(12 a^2 A \sin (c+d x)+16 a^2 C \sin (c+d x)+A b^2 \sin (c+d x)\right)}{32 a}+\frac{1}{4} a A \tan (c+d x) \sec ^3(c+d x)+\frac{3}{8} A b \tan (c+d x) \sec ^2(c+d x)\right)}{d}+\frac{\frac{2 \left(48 a^3 A b+64 a^3 b C+4 a A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \left(-52 a^2 A b^2-80 a^2 b^2 C+3 A b^4\right) \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{b \cos (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left(2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right)}+\frac{2 \left(96 a^4 A+128 a^4 C-4 a^2 A b^2+16 a^2 b^2 C+9 A b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{256 a^2 d}","-\frac{b \left(3 A b^2-4 a^2 (13 A+20 C)\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{64 a^2 d}-\frac{b \left(A b^2-4 a^2 (19 A+28 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{64 a d \sqrt{a+b \cos (c+d x)}}+\frac{b \left(3 A b^2-4 a^2 (13 A+20 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{64 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(4 a^2 (3 A+4 C)+A b^2\right) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{32 a d}+\frac{\left(16 a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+3 A b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{64 a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}+\frac{A b \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{8 d}",1,"((2*(48*a^3*A*b + 4*a*A*b^3 + 64*a^3*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(96*a^4*A - 4*a^2*A*b^2 + 9*A*b^4 + 128*a^4*C + 16*a^2*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(-52*a^2*A*b^2 + 3*A*b^4 - 80*a^2*b^2*C)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2)))/(256*a^2*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]^2*(12*a^2*A*Sin[c + d*x] + A*b^2*Sin[c + d*x] + 16*a^2*C*Sin[c + d*x]))/(32*a) + (Sec[c + d*x]*(52*a^2*A*b*Sin[c + d*x] - 3*A*b^3*Sin[c + d*x] + 80*a^2*b*C*Sin[c + d*x]))/(64*a^2) + (3*A*b*Sec[c + d*x]^2*Tan[c + d*x])/8 + (a*A*Sec[c + d*x]^3*Tan[c + d*x])/4))/d","C",1
638,1,395,523,2.5948397,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2),x]","\frac{b (a+b \cos (c+d x)) \left(4 a \left(960 a^4 C+10 a^2 b^2 (572 A+331 C)+3 b^4 (71214 A+60793 C)\right) \sin (c+d x)+b \left(5 b \left(2 a \left(60 a^2 C+10868 A b^2+13939 b^2 C\right) \sin (3 (c+d x))+7 b \left(\left(636 a^2 C+572 A b^2+880 b^2 C\right) \sin (4 (c+d x))+9 b C (54 a \sin (5 (c+d x))+11 b \sin (6 (c+d x)))\right)\right)+\left(-1440 a^4 C+120 a^2 b^2 (1430 A+1457 C)+77 b^4 (1976 A+1897 C)\right) \sin (2 (c+d x))\right)\right)+32 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(a b^2 \left(-60 a^4 C+5 a^2 b^2 (4433 A+3337 C)+3 b^4 (12441 A+10277 C)\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-\left(240 a^6 C+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)-1617 b^6 (13 A+11 C)\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{720720 b^4 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(24 a^2 C+11 b^2 (13 A+11 C)\right) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{1287 b^3 d}-\frac{4 a \left(24 a^2 C+143 A b^2+166 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{9009 b^3 d}+\frac{4 a \left(a^2-b^2\right) \left(120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{45045 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(240 a^4 C+10 a^2 b^2 (143 A+124 C)-539 b^4 (13 A+11 C)\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{45045 b^3 d}-\frac{4 a \left(120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{45045 b^3 d}-\frac{2 \left(240 a^6 C+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)-1617 b^6 (13 A+11 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{45045 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{143 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}",1,"(32*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(a*b^2*(-60*a^4*C + 5*a^2*b^2*(4433*A + 3337*C) + 3*b^4*(12441*A + 10277*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - (240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x])*(4*a*(960*a^4*C + 10*a^2*b^2*(572*A + 331*C) + 3*b^4*(71214*A + 60793*C))*Sin[c + d*x] + b*((-1440*a^4*C + 120*a^2*b^2*(1430*A + 1457*C) + 77*b^4*(1976*A + 1897*C))*Sin[2*(c + d*x)] + 5*b*(2*a*(10868*A*b^2 + 60*a^2*C + 13939*b^2*C)*Sin[3*(c + d*x)] + 7*b*((572*A*b^2 + 636*a^2*C + 880*b^2*C)*Sin[4*(c + d*x)] + 9*b*C*(54*a*Sin[5*(c + d*x)] + 11*b*Sin[6*(c + d*x)]))))))/(720720*b^4*d*Sqrt[a + b*Cos[c + d*x]])","A",1
639,1,328,435,1.6667601,"\int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2),x]","\frac{b (a+b \cos (c+d x)) \left(b \left(4 a \left(6 a^2 C+594 A b^2+619 b^2 C\right) \sin (2 (c+d x))+b \left(\left(452 a^2 C+396 A b^2+513 b^2 C\right) \sin (3 (c+d x))+7 b C (46 a \sin (4 (c+d x))+9 b \sin (5 (c+d x)))\right)\right)+\left(-64 a^4 C+12 a^2 b^2 (396 A+311 C)+6 b^4 (506 A+435 C)\right) \sin (c+d x)\right)+16 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b \left(2 a^4 b C+3 a^2 b^3 (297 A+221 C)+15 b^5 (11 A+9 C)\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+a \left(8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{5544 b^3 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(8 a^2 C+9 b^2 (11 A+9 C)\right) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}+\frac{2 a \left(8 a^2 C+99 A b^2+67 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{693 b^2 d}+\frac{2 \left(8 a^4 C+3 a^2 b^2 (33 A+19 C)+15 b^4 (11 A+9 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{693 b^2 d}-\frac{2 \left(a^2-b^2\right) \left(8 a^4 C+3 a^2 b^2 (33 A+19 C)+15 b^4 (11 A+9 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{693 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 a \left(8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{693 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}",1,"(16*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b*(2*a^4*b*C + 15*b^5*(11*A + 9*C) + 3*a^2*b^3*(297*A + 221*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + a*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x])*((-64*a^4*C + 12*a^2*b^2*(396*A + 311*C) + 6*b^4*(506*A + 435*C))*Sin[c + d*x] + b*(4*a*(594*A*b^2 + 6*a^2*C + 619*b^2*C)*Sin[2*(c + d*x)] + b*((396*A*b^2 + 452*a^2*C + 513*b^2*C)*Sin[3*(c + d*x)] + 7*b*C*(46*a*Sin[4*(c + d*x)] + 9*b*Sin[5*(c + d*x)])))))/(5544*b^3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
640,1,274,350,1.34847,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2),x]","\frac{b (a+b \cos (c+d x)) \left(2 a \left(20 a^2 C+924 A b^2+747 b^2 C\right) \sin (c+d x)+b \left(\left(300 a^2 C+252 A b^2+266 b^2 C\right) \sin (2 (c+d x))+5 b C (38 a \sin (3 (c+d x))+7 b \sin (4 (c+d x)))\right)\right)+8 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(a b^2 \left(5 a^2 (63 A+31 C)+3 b^2 (119 A+87 C)\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(-10 a^4 C+3 a^2 b^2 (161 A+93 C)+21 b^4 (9 A+7 C)\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{1260 b^2 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(10 a^2 C-7 b^2 (9 A+7 C)\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b d}+\frac{4 a \left(-5 a^2 C+84 A b^2+57 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b d}-\frac{4 a \left(a^2-b^2\right) \left(-5 a^2 C+84 A b^2+57 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(10 a^4 C-3 a^2 b^2 (161 A+93 C)-21 b^4 (9 A+7 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}-\frac{4 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b d}",1,"(8*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(a*b^2*(5*a^2*(63*A + 31*C) + 3*b^2*(119*A + 87*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (-10*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(161*A + 93*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x])*(2*a*(924*A*b^2 + 20*a^2*C + 747*b^2*C)*Sin[c + d*x] + b*((252*A*b^2 + 300*a^2*C + 266*b^2*C)*Sin[2*(c + d*x)] + 5*b*C*(38*a*Sin[3*(c + d*x)] + 7*b*Sin[4*(c + d*x)]))))/(1260*b^2*d*Sqrt[a + b*Cos[c + d*x]])","A",1
641,1,468,342,3.0823891,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{2 \sin (c+d x) \sqrt{a+b \cos (c+d x)} \left(18 a^2 C+18 a b C \cos (c+d x)+14 A b^2+3 b^2 C \cos (2 (c+d x))+13 b^2 C\right)+\frac{4 b \left(9 a^2 (7 A+3 C)+b^2 (7 A+5 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 a \left(3 a^2 (14 A+C)+b^2 (49 A+29 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 i \left(3 a^2 C+49 A b^2+29 b^2 C\right) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{b^2 \sqrt{-\frac{1}{a+b}}}}{42 d}","\frac{2 a^3 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(3 a^2 C+b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{21 d}+\frac{2 a \left(3 a^2 C+49 A b^2+29 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{21 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \left(-3 a^4 C+2 a^2 b^2 (7 A-C)+b^4 (7 A+5 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{21 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{7 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}",1,"((4*b*(9*a^2*(7*A + 3*C) + b^2*(7*A + 5*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*a*(3*a^2*(14*A + C) + b^2*(49*A + 29*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + ((2*I)*(49*A*b^2 + 3*a^2*C + 29*b^2*C)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(b^2*Sqrt[-(a + b)^(-1)]) + 2*Sqrt[a + b*Cos[c + d*x]]*(14*A*b^2 + 18*a^2*C + 13*b^2*C + 18*a*b*C*Cos[c + d*x] + 3*b^2*C*Cos[2*(c + d*x)])*Sin[c + d*x])/(42*d)","C",1
642,1,462,327,3.5133341,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{\frac{8 a \left(15 a^2 C+45 A b^2+17 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 b \left(a^2 (135 A+46 C)+6 b^2 (5 A+3 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+4 \sqrt{a+b \cos (c+d x)} \left(15 a^2 A \tan (c+d x)+22 a b C \sin (c+d x)+3 b^2 C \sin (2 (c+d x))\right)+\frac{2 i \left(a^2 (46 C-15 A)+6 b^2 (5 A+3 C)\right) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\cos (c+d x)+1)}{a-b}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}}{60 d}","\frac{a \left(a^2 (15 A-16 C)+4 b^2 (15 A+4 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(a^2 (15 A-46 C)-6 b^2 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{5 a^2 A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{b (5 A-2 C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac{a b (15 A-16 C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^{5/2}}{d}",1,"((8*a*(45*A*b^2 + 15*a^2*C + 17*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*b*(6*b^2*(5*A + 3*C) + a^2*(135*A + 46*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + ((2*I)*(6*b^2*(5*A + 3*C) + a^2*(-15*A + 46*C))*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Cos[c + d*x]))/(a - b))]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b*Sqrt[-(a + b)^(-1)]) + 4*Sqrt[a + b*Cos[c + d*x]]*(22*a*b*C*Sin[c + d*x] + 3*b^2*C*Sin[2*(c + d*x)] + 15*a^2*A*Tan[c + d*x]))/(60*d)","C",1
643,1,445,329,4.1665957,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{\frac{8 b \left(3 a^2 (A+12 C)+4 b^2 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 a \left(24 a^2 (A+2 C)+7 b^2 (9 A+8 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+4 \sec (c+d x) \sqrt{a+b \cos (c+d x)} \left(6 a^2 A \tan (c+d x)+27 a A b \sin (c+d x)+4 b^2 C \sin (2 (c+d x))\right)-\frac{2 i (27 A-56 C) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{\sqrt{-\frac{1}{a+b}}}}{48 d}","\frac{b \left(a^2 (33 A+16 C)+8 b^2 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{12 d \sqrt{a+b \cos (c+d x)}}+\frac{a \left(4 a^2 (A+2 C)+15 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}-\frac{b^2 (21 A-8 C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{12 d}-\frac{a b (27 A-56 C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{12 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{5 A b \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}",1,"((8*b*(4*b^2*(3*A + C) + 3*a^2*(A + 12*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*a*(24*a^2*(A + 2*C) + 7*b^2*(9*A + 8*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(27*A - 56*C)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/Sqrt[-(a + b)^(-1)] + 4*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*(27*a*A*b*Sin[c + d*x] + 4*b^2*C*Sin[2*(c + d*x)] + 6*a^2*A*Tan[c + d*x]))/(48*d)","C",1
644,1,477,363,6.0050294,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\frac{2 b \left(8 a^2 (13 A+27 C)-3 b^2 (A-16 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+4 \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)} \left(\left(4 a^2 (2 A+3 C)+\frac{33 A b^2}{2}\right) \sin (2 (c+d x))+8 a^2 A \tan (c+d x)+26 a A b \sin (c+d x)\right)-\frac{2 i \left(8 a^2 (2 A+3 C)+3 b^2 (11 A-16 C)\right) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\cos (c+d x)+1)}{a-b}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}+\frac{8 a b^2 (13 A+72 C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{96 d}","\frac{\left(8 a^2 (2 A+3 C)+15 A b^2\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{a \left(8 a^2 (2 A+3 C)+b^2 (59 A+96 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(8 a^2 (2 A+3 C)+3 b^2 (11 A-16 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{5 b \left(4 a^2 (A+2 C)+A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}+\frac{5 A b \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{12 d}",1,"((8*a*b^2*(13*A + 72*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*b*(-3*b^2*(A - 16*C) + 8*a^2*(13*A + 27*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(3*b^2*(11*A - 16*C) + 8*a^2*(2*A + 3*C))*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Cos[c + d*x]))/(a - b))]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b*Sqrt[-(a + b)^(-1)]) + 4*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*(26*a*A*b*Sin[c + d*x] + ((33*A*b^2)/2 + 4*a^2*(2*A + 3*C))*Sin[2*(c + d*x)] + 8*a^2*A*Tan[c + d*x]))/(96*d)","C",1
645,1,704,437,6.873461,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{\sqrt{a+b \cos (c+d x)} \left(\frac{\sec (c+d x) \left(284 a^2 A b \sin (c+d x)+432 a^2 b C \sin (c+d x)+15 A b^3 \sin (c+d x)\right)}{192 a}+\frac{1}{96} \sec ^2(c+d x) \left(36 a^2 A \sin (c+d x)+48 a^2 C \sin (c+d x)+59 A b^2 \sin (c+d x)\right)+\frac{1}{4} a^2 A \tan (c+d x) \sec ^3(c+d x)+\frac{17}{24} a A b \tan (c+d x) \sec ^2(c+d x)\right)}{d}+\frac{\frac{2 \left(144 a^3 A b+192 a^3 b C+236 a A b^3+768 a b^3 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \left(-284 a^2 A b^2-432 a^2 b^2 C-15 A b^4\right) \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{b \cos (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left(2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right)}+\frac{2 \left(288 a^4 A+384 a^4 C+436 a^2 A b^2+1008 a^2 b^2 C-45 A b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{768 a d}","\frac{b \left(4 a^2 (71 A+108 C)+15 A b^2\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{192 a d}+\frac{b \left(4 a^2 (89 A+132 C)+b^2 (133 A+384 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{192 d \sqrt{a+b \cos (c+d x)}}-\frac{b \left(4 a^2 (71 A+108 C)+15 A b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{192 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(4 a^2 (3 A+4 C)+5 A b^2\right) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{32 d}-\frac{\left(-16 a^4 (3 A+4 C)-120 a^2 b^2 (A+2 C)+5 A b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{64 a d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{5/2}}{4 d}+\frac{5 A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{24 d}",1,"((2*(144*a^3*A*b + 236*a*A*b^3 + 192*a^3*b*C + 768*a*b^3*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(288*a^4*A + 436*a^2*A*b^2 - 45*A*b^4 + 384*a^4*C + 1008*a^2*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(-284*a^2*A*b^2 - 15*A*b^4 - 432*a^2*b^2*C)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2)))/(768*a*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]^2*(36*a^2*A*Sin[c + d*x] + 59*A*b^2*Sin[c + d*x] + 48*a^2*C*Sin[c + d*x]))/96 + (Sec[c + d*x]*(284*a^2*A*b*Sin[c + d*x] + 15*A*b^3*Sin[c + d*x] + 432*a^2*b*C*Sin[c + d*x]))/(192*a) + (17*a*A*b*Sec[c + d*x]^2*Tan[c + d*x])/24 + (a^2*A*Sec[c + d*x]^3*Tan[c + d*x])/4))/d","C",1
646,1,212,246,1.1489924,"\int (a+b \cos (c+d x))^{3/2} \left(a^2-b^2 \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(a^2 - b^2*Cos[c + d*x]^2),x]","\frac{-4 \left(41 a^4-66 a^2 b^2+25 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-b \sin (c+d x) \left(-128 a^3+\left(145 b^3-32 a^2 b\right) \cos (c+d x)+78 a b^2 \cos (2 (c+d x))+178 a b^2+15 b^3 \cos (3 (c+d x))\right)+8 a \left(73 a^3+73 a^2 b-41 a b^2-41 b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{210 d \sqrt{a+b \cos (c+d x)}}","\frac{2 b \left(41 a^2-25 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 d}+\frac{4 a \left(73 a^2-41 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 \left(41 a^4-66 a^2 b^2+25 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 d \sqrt{a+b \cos (c+d x)}}-\frac{2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}+\frac{4 a b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d}",1,"(8*a*(73*a^3 + 73*a^2*b - 41*a*b^2 - 41*b^3)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 4*(41*a^4 - 66*a^2*b^2 + 25*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - b*(-128*a^3 + 178*a*b^2 + (-32*a^2*b + 145*b^3)*Cos[c + d*x] + 78*a*b^2*Cos[2*(c + d*x)] + 15*b^3*Cos[3*(c + d*x)])*Sin[c + d*x])/(210*d*Sqrt[a + b*Cos[c + d*x]])","A",1
647,1,178,197,0.8537372,"\int \sqrt{a+b \cos (c+d x)} \left(a^2-b^2 \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(a^2 - b^2*Cos[c + d*x]^2),x]","\frac{-b \sin (c+d x) \left(2 a^2+8 a b \cos (c+d x)+3 b^2 \cos (2 (c+d x))+3 b^2\right)-4 a \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+2 \left(17 a^3+17 a^2 b-9 a b^2-9 b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{a+b \cos (c+d x)}}","-\frac{4 a \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(17 a^2-9 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}+\frac{4 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}",1,"(2*(17*a^3 + 17*a^2*b - 9*a*b^2 - 9*b^3)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 4*a*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - b*(2*a^2 + 3*b^2 + 8*a*b*Cos[c + d*x] + 3*b^2*Cos[2*(c + d*x)])*Sin[c + d*x])/(15*d*Sqrt[a + b*Cos[c + d*x]])","A",1
648,1,272,378,1.4499229,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{8 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b \left(32 a^3 b C+6 a b^3 (7 A+6 C)\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(128 a^4 C+12 a^2 b^2 (14 A+9 C)+21 b^4 (9 A+7 C)\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)-b (a+b \cos (c+d x)) \left(32 a \left(2 C \left(8 a^2+9 b^2\right)+21 A b^2\right) \sin (c+d x)-b \left(2 \left(96 a^2 C+126 A b^2+133 b^2 C\right) \sin (2 (c+d x))+5 b C (7 b \sin (4 (c+d x))-16 a \sin (3 (c+d x)))\right)\right)}{1260 b^5 d \sqrt{a+b \cos (c+d x)}}","-\frac{4 a \left(32 a^2 C+42 A b^2+31 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^4 d}+\frac{2 \left(48 a^2 C+7 b^2 (9 A+7 C)\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^3 d}-\frac{2 a \left(128 a^4 C+4 a^2 b^2 (42 A+19 C)+3 b^4 (49 A+37 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^5 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(128 a^4 C+12 a^2 b^2 (14 A+9 C)+21 b^4 (9 A+7 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^5 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{16 a C \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{63 b^2 d}+\frac{2 C \sin (c+d x) \cos ^3(c+d x) \sqrt{a+b \cos (c+d x)}}{9 b d}",1,"(8*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b*(32*a^3*b*C + 6*a*b^3*(7*A + 6*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (128*a^4*C + 21*b^4*(9*A + 7*C) + 12*a^2*b^2*(14*A + 9*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) - b*(a + b*Cos[c + d*x])*(32*a*(21*A*b^2 + 2*(8*a^2 + 9*b^2)*C)*Sin[c + d*x] - b*(2*(126*A*b^2 + 96*a^2*C + 133*b^2*C)*Sin[2*(c + d*x)] + 5*b*C*(-16*a*Sin[3*(c + d*x)] + 7*b*Sin[4*(c + d*x)]))))/(1260*b^5*d*Sqrt[a + b*Cos[c + d*x]])","A",1
649,1,217,305,0.9071608,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 b \sin (c+d x) (a+b \cos (c+d x)) \left(48 a^2 C-36 a b C \cos (c+d x)+70 A b^2+15 b^2 C \cos (2 (c+d x))+65 b^2 C\right)+4 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b \left(-12 a^2 b C+35 A b^3+25 b^3 C\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-2 a \left(24 a^2 C+35 A b^2+22 b^2 C\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{210 b^4 d \sqrt{a+b \cos (c+d x)}}","-\frac{4 a \left(24 a^2 C+35 A b^2+22 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \left(24 a^2 C+5 b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b^3 d}+\frac{2 \left(48 a^4 C+2 a^2 b^2 (35 A+16 C)+5 b^4 (7 A+5 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{12 a C \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{35 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{7 b d}",1,"(4*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b*(35*A*b^3 - 12*a^2*b*C + 25*b^3*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - 2*a*(35*A*b^2 + 24*a^2*C + 22*b^2*C)*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + 2*b*(a + b*Cos[c + d*x])*(70*A*b^2 + 48*a^2*C + 65*b^2*C - 36*a*b*C*Cos[c + d*x] + 15*b^2*C*Cos[2*(c + d*x)])*Sin[c + d*x])/(210*b^4*d*Sqrt[a + b*Cos[c + d*x]])","A",1
650,1,190,233,0.930216,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{-2 a \left(8 a^2 C+15 A b^2+7 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+2 (a+b) \left(8 a^2 C+15 A b^2+9 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+b C \sin (c+d x) \left(-8 a^2-2 a b \cos (c+d x)+3 b^2 \cos (2 (c+d x))+3 b^2\right)}{15 b^3 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 a \left(8 a^2 C+15 A b^2+7 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(8 a^2 C+3 b^2 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b d}",1,"(2*(a + b)*(15*A*b^2 + 8*a^2*C + 9*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 2*a*(15*A*b^2 + 8*a^2*C + 7*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + b*C*(-8*a^2 + 3*b^2 - 2*a*b*Cos[c + d*x] + 3*b^2*Cos[2*(c + d*x)])*Sin[c + d*x])/(15*b^3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
651,1,148,174,0.6922379,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \left(C \left(2 a^2+b^2\right)+3 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+2 b C \sin (c+d x) (a+b \cos (c+d x))-4 a C (a+b) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(2 a^2 C+b^2 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{a+b \cos (c+d x)}}-\frac{4 a C \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d}",1,"(-4*a*(a + b)*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + 2*(3*A*b^2 + (2*a^2 + b^2)*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + 2*b*C*(a + b*Cos[c + d*x])*Sin[c + d*x])/(3*b^2*d*Sqrt[a + b*Cos[c + d*x]])","A",1
652,0,0,183,8.4878477,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/Sqrt[a + b*Cos[c + d*x]],x]","\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","\frac{2 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{2 a C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/Sqrt[a + b*Cos[c + d*x]], x]","F",-1
653,1,559,214,11.7017351,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 A \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)} \left(A \sec ^2(c+d x)+C\right)}{a d (2 A+C \cos (2 c+2 d x)+C)}+\frac{\cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right) \left(\frac{2 i A b \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{b \cos (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left(2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right)}-\frac{6 A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{8 a C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}\right)}{2 a d (2 A+C \cos (2 c+2 d x)+C)}","\frac{(A+2 C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{a d}-\frac{A \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}",1,"(2*A*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*(C + A*Sec[c + d*x]^2)*Sin[c + d*x])/(a*d*(2*A + C + C*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*((8*a*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - (6*A*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + ((2*I)*A*b*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2))))/(2*a*d*(2*A + C + C*Cos[2*c + 2*d*x]))","C",0
654,1,603,278,6.7586207,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\cos ^2(c+d x) \sqrt{a+b \cos (c+d x)} \left(A \sec ^2(c+d x)+C\right) \left(\frac{A \tan (c+d x) \sec (c+d x)}{a}-\frac{3 A b \tan (c+d x)}{2 a^2}\right)}{d (2 A+C \cos (2 c+2 d x)+C)}+\frac{\cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right) \left(\frac{2 \left(8 a^2 A+16 a^2 C+9 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{6 i A b^2 \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{b \cos (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left(2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right)}+\frac{8 a A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}\right)}{8 a^2 d (2 A+C \cos (2 c+2 d x)+C)}","\frac{\left(4 a^2 (A+2 C)+3 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{3 A b \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a^2 d}+\frac{3 A b \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 a d}",1,"(Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*((8*a*A*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(8*a^2*A + 9*A*b^2 + 16*a^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((6*I)*A*b^2*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2))))/(8*a^2*d*(2*A + C + C*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*(C + A*Sec[c + d*x]^2)*((-3*A*b*Tan[c + d*x])/(2*a^2) + (A*Sec[c + d*x]*Tan[c + d*x])/a))/(d*(2*A + C + C*Cos[2*c + 2*d*x]))","C",0
655,1,604,370,6.7307466,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\sqrt{a+b \cos (c+d x)} \left(-\frac{5 A b \tan (c+d x) \sec (c+d x)}{12 a^2}+\frac{\sec (c+d x) \left(16 a^2 A \sin (c+d x)+24 a^2 C \sin (c+d x)+15 A b^2 \sin (c+d x)\right)}{24 a^3}+\frac{A \tan (c+d x) \sec ^2(c+d x)}{3 a}\right)}{d}-\frac{b \left(\frac{2 \left(40 a^2 A+72 a^2 C+45 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \left(16 a^2 A+24 a^2 C+15 A b^2\right) \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{b \cos (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left(2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right)}+\frac{40 a A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}\right)}{96 a^3 d}","\frac{\left(8 a^2 (2 A+3 C)+5 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{5 A b \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{12 a^2 d}+\frac{\left(8 a^2 (2 A+3 C)+15 A b^2\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 a^3 d}-\frac{\left(8 a^2 (2 A+3 C)+15 A b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{b \left(4 a^2 (A+2 C)+5 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 a^3 d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{3 a d}",1,"-1/96*(b*((40*a*A*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(40*a^2*A + 45*A*b^2 + 72*a^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(16*a^2*A + 15*A*b^2 + 24*a^2*C)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2))))/(a^3*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]*(16*a^2*A*Sin[c + d*x] + 15*A*b^2*Sin[c + d*x] + 24*a^2*C*Sin[c + d*x]))/(24*a^3) - (5*A*b*Sec[c + d*x]*Tan[c + d*x])/(12*a^2) + (A*Sec[c + d*x]^2*Tan[c + d*x])/(3*a)))/d","C",1
656,1,358,473,1.683618,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{b (a-b) (a+b) \left(\left(a^2-b^2\right) \left(348 a^2 C+140 A b^2+115 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))-78 a b C \left(a^2-b^2\right) \sin (2 (c+d x)) (a+b \cos (c+d x))+15 b^2 C \left(a^2-b^2\right) \sin (3 (c+d x)) (a+b \cos (c+d x))+420 a^3 \left(a^2 C+A b^2\right) \sin (c+d x)\right)-4 \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b \left(96 a^4 b C+2 a^2 b^3 (35 A-8 C)+5 b^5 (7 A+5 C)\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+a \left(384 a^4 C+4 a^2 b^2 (70 A-43 C)-b^4 (175 A+107 C)\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{210 b^5 d (a-b) (a+b) \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \cos ^3(c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(8 a^2 C+7 A b^2-b^2 C\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{7 b^2 d \left(a^2-b^2\right)}-\frac{2 a \left(48 a^2 C+35 A b^2-13 b^2 C\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{35 b^3 d \left(a^2-b^2\right)}+\frac{2 \left(192 a^4 C+2 a^2 b^2 (70 A-31 C)-5 b^4 (7 A+5 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b^4 d \left(a^2-b^2\right)}+\frac{2 \left(384 a^4 C+4 a^2 b^2 (70 A+29 C)+5 b^4 (7 A+5 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^5 d \sqrt{a+b \cos (c+d x)}}-\frac{2 a \left(384 a^4 C+4 a^2 b^2 (70 A-43 C)-b^4 (175 A+107 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^5 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(-4*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b*(2*a^2*b^3*(35*A - 8*C) + 96*a^4*b*C + 5*b^5*(7*A + 5*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + a*(4*a^2*b^2*(70*A - 43*C) + 384*a^4*C - b^4*(175*A + 107*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + (a - b)*b*(a + b)*(420*a^3*(A*b^2 + a^2*C)*Sin[c + d*x] + (a^2 - b^2)*(140*A*b^2 + 348*a^2*C + 115*b^2*C)*(a + b*Cos[c + d*x])*Sin[c + d*x] - 78*a*b*(a^2 - b^2)*C*(a + b*Cos[c + d*x])*Sin[2*(c + d*x)] + 15*b^2*(a^2 - b^2)*C*(a + b*Cos[c + d*x])*Sin[3*(c + d*x)]))/(210*(a - b)*b^5*(a + b)*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",1
657,1,289,375,1.6515695,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\frac{10 a^2 b \left(a^2 C+A b^2\right) \sin (c+d x)}{b^2-a^2}+\frac{2 a b^2 \left(C \left(4 a^2+b^2\right)+5 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{(a-b) (a+b)}+\frac{2 \left(16 a^4 C+2 a^2 b^2 (5 A-4 C)-b^4 (5 A+3 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)}{(a-b) (a+b)}+b^2 C \sin (2 (c+d x)) (a+b \cos (c+d x))-6 a b C \sin (c+d x) (a+b \cos (c+d x))}{5 b^4 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \cos ^2(c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(6 a^2 C+5 A b^2-b^2 C\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b^2 d \left(a^2-b^2\right)}-\frac{4 a \left(2 C \left(4 a^2+b^2\right)+5 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{5 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{2 a \left(8 a^2 C+5 A b^2-3 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b^3 d \left(a^2-b^2\right)}+\frac{2 \left(16 a^4 C+2 a^2 b^2 (5 A-4 C)-b^4 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{5 b^4 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"((2*a*b^2*(5*A*b^2 + (4*a^2 + b^2)*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/((a - b)*(a + b)) + (2*(2*a^2*b^2*(5*A - 4*C) + 16*a^4*C - b^4*(5*A + 3*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/((a - b)*(a + b)) + (10*a^2*b*(A*b^2 + a^2*C)*Sin[c + d*x])/(-a^2 + b^2) - 6*a*b*C*(a + b*Cos[c + d*x])*Sin[c + d*x] + b^2*C*(a + b*Cos[c + d*x])*Sin[2*(c + d*x)])/(5*b^4*d*Sqrt[a + b*Cos[c + d*x]])","A",1
658,1,209,256,1.3705759,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 \left(-\left(a^2-b^2\right) \left(C \left(8 a^2+b^2\right)+3 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+a (a+b) \left(8 a^2 C+3 A b^2-5 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+b \sin (c+d x) \left(-4 a^3 C+b C \left(b^2-a^2\right) \cos (c+d x)+a b^2 (C-3 A)\right)\right)}{3 b^3 d (a-b) (a+b) \sqrt{a+b \cos (c+d x)}}","\frac{2 a \left(a^2 C+A b^2\right) \sin (c+d x)}{b^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(C \left(8 a^2+b^2\right)+3 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 a \left(8 a^2 C+3 A b^2-5 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^2 d}",1,"(-2*(a*(a + b)*(3*A*b^2 + 8*a^2*C - 5*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - (a^2 - b^2)*(3*A*b^2 + (8*a^2 + b^2)*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + b*(-4*a^3*C + a*b^2*(-3*A + C) + b*(-a^2 + b^2)*C*Cos[c + d*x])*Sin[c + d*x]))/(3*(a - b)*b^3*(a + b)*d*Sqrt[a + b*Cos[c + d*x]])","A",1
659,1,166,202,0.6961238,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2),x]","\frac{-2 b \left(a^2 C+A b^2\right) \sin (c+d x)+2 (a+b) \left(2 a^2 C+A b^2-b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-4 a C \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b^2 d (a-b) (a+b) \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(2 a^2 C+A b^2-b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{4 a C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b^2 d \sqrt{a+b \cos (c+d x)}}",1,"(2*(a + b)*(A*b^2 + 2*a^2*C - b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 4*a*(a^2 - b^2)*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - 2*b*(A*b^2 + a^2*C)*Sin[c + d*x])/((a - b)*b^2*(a + b)*d*Sqrt[a + b*Cos[c + d*x]])","A",1
660,0,0,259,31.3492183,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^(3/2),x]","\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(a^2 C+A b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a b d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{a+b \cos (c+d x)}}",1,"Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x]","F",-1
661,1,511,296,6.4224097,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right) \left(\frac{4 \tan (c+d x) \left(b \left(a^2 (A-2 C)-3 A b^2\right) \cos (c+d x)+a A \left(a^2-b^2\right)\right)}{\left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\frac{8 a \left(a^2 C+A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 b \left(a^2 (2 C-7 A)+9 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 i \left(3 A b^2-a^2 (A-2 C)\right) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\cos (c+d x)+1)}{a-b}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}}{(a-b) (a+b)}\right)}{2 a^2 d (2 A+C \cos (2 (c+d x))+C)}","-\frac{b \left(3 A b^2-a^2 (A-2 C)\right) \sin (c+d x)}{a^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\left(3 A b^2-a^2 (A-2 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{3 A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x)}{a d \sqrt{a+b \cos (c+d x)}}+\frac{A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}",1,"(Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*(((8*a*(A*b^2 + a^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*b*(9*A*b^2 + a^2*(-7*A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + ((2*I)*(3*A*b^2 - a^2*(A - 2*C))*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Cos[c + d*x]))/(a - b))]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b*Sqrt[-(a + b)^(-1)]))/((a - b)*(a + b)) + (4*(a*A*(a^2 - b^2) + b*(-3*A*b^2 + a^2*(A - 2*C))*Cos[c + d*x])*Tan[c + d*x])/((a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]])))/(2*a^2*d*(2*A + C + C*Cos[2*(c + d*x)]))","C",1
662,1,727,370,6.8719137,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\cos ^2(c+d x) \sqrt{a+b \cos (c+d x)} \left(A \sec ^2(c+d x)+C\right) \left(-\frac{7 A b \tan (c+d x)}{2 a^3}+\frac{A \tan (c+d x) \sec (c+d x)}{a^2}+\frac{4 \left(a^2 b^2 C \sin (c+d x)+A b^4 \sin (c+d x)\right)}{a^3 \left(a^2-b^2\right) (a+b \cos (c+d x))}\right)}{d (2 A+C \cos (2 c+2 d x)+C)}-\frac{\cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right) \left(\frac{2 \left(4 a^3 A b-16 a^3 b C-20 a A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \left(7 a^2 A b^2-8 a^2 b^2 C-15 A b^4\right) \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{b \cos (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left(2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right)}+\frac{2 \left(8 a^4 A+16 a^4 C+29 a^2 A b^2-24 a^2 b^2 C-45 A b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}\right)}{8 a^3 d (b-a) (a+b) (2 A+C \cos (2 c+2 d x)+C)}","-\frac{5 A b \tan (c+d x)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{5 A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{b^2 \left(15 A b^2-a^2 (7 A-8 C)\right) \sin (c+d x)}{4 a^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{b \left(15 A b^2-a^2 (7 A-8 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^3 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(4 a^2 (A+2 C)+15 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^3 d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt{a+b \cos (c+d x)}}",1,"-1/8*(Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*((2*(4*a^3*A*b - 20*a*A*b^3 - 16*a^3*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(8*a^4*A + 29*a^2*A*b^2 - 45*A*b^4 + 16*a^4*C - 24*a^2*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(7*a^2*A*b^2 - 15*A*b^4 - 8*a^2*b^2*C)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2))))/(a^3*(-a + b)*(a + b)*d*(2*A + C + C*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*(C + A*Sec[c + d*x]^2)*((4*(A*b^4*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(a^3*(a^2 - b^2)*(a + b*Cos[c + d*x])) - (7*A*b*Tan[c + d*x])/(2*a^3) + (A*Sec[c + d*x]*Tan[c + d*x])/a^2))/(d*(2*A + C + C*Cos[2*c + 2*d*x]))","C",0
663,1,350,521,3.7092013,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{\frac{2 \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{3/2} \left(\left(128 a^6 C+4 a^4 b^2 (10 A-53 C)+5 a^2 b^4 (11 C-15 A)+3 b^6 (5 A+3 C)\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)-2 a b^2 \left(-16 a^4 C+a^2 b^2 (22 C-5 A)+b^4 (15 A+4 C)\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)}{(a-b)^2 (a+b)}+b \left(-\frac{10 a^2 \left(11 a^4 C+5 a^2 b^2 (A-3 C)-9 A b^4\right) \sin (c+d x) (a+b \cos (c+d x))}{\left(a^2-b^2\right)^2}+\frac{10 a^3 \left(a^2 C+A b^2\right) \sin (c+d x)}{a^2-b^2}-28 a C \sin (c+d x) (a+b \cos (c+d x))^2+3 b C \sin (2 (c+d x)) (a+b \cos (c+d x))^2\right)}{15 b^5 d (a+b \cos (c+d x))^{3/2}}","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{4 \left(-4 a^4 C-a^2 b^2 (A-6 C)+3 A b^4\right) \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{4 a \left(32 a^4 C+a^2 b^2 (10 A-49 C)-b^4 (20 A-7 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^4 d \left(a^2-b^2\right)^2}-\frac{2 a \left(128 a^4 C+4 a^2 b^2 (10 A-29 C)-b^4 (45 A+17 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^5 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(48 a^4 C+a^2 b^2 (15 A-71 C)-b^4 (35 A-3 C)\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^3 d \left(a^2-b^2\right)^2}+\frac{2 \left(128 a^6 C+4 a^4 b^2 (10 A-53 C)-5 a^2 b^4 (15 A-11 C)+3 b^6 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^5 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"((2*((a + b*Cos[c + d*x])/(a + b))^(3/2)*(-2*a*b^2*(-16*a^4*C + b^4*(15*A + 4*C) + a^2*b^2*(-5*A + 22*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (4*a^4*b^2*(10*A - 53*C) + 128*a^6*C + 3*b^6*(5*A + 3*C) + 5*a^2*b^4*(-15*A + 11*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])))/((a - b)^2*(a + b)) + b*((10*a^3*(A*b^2 + a^2*C)*Sin[c + d*x])/(a^2 - b^2) - (10*a^2*(-9*A*b^4 + 5*a^2*b^2*(A - 3*C) + 11*a^4*C)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2 - 28*a*C*(a + b*Cos[c + d*x])^2*Sin[c + d*x] + 3*b*C*(a + b*Cos[c + d*x])^2*Sin[2*(c + d*x)]))/(15*b^5*d*(a + b*Cos[c + d*x])^(3/2))","A",1
664,1,306,392,2.7854756,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(\frac{\left(\frac{a+b \cos (c+d x)}{a+b}\right)^{3/2} \left(b \left(-4 a^4 b C+a^2 b^3 (A+7 C)+b^5 (3 A+C)\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-2 a \left(8 a^4 C+a^2 b^2 (A-14 C)+b^4 (4 C-3 A)\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{(a-b)^2 (a+b)}+\frac{b \sin (c+d x) \left(16 a^6 C+2 a^4 A b^2-25 a^4 b^2 C-10 a^2 A b^4+C \left(b^3-a^2 b\right)^2 \cos (2 (c+d x))+4 a b \left(5 a^4 C+a^2 b^2 (A-8 C)+b^4 (C-3 A)\right) \cos (c+d x)+b^6 C\right)}{2 \left(a^2-b^2\right)^2}\right)}{3 b^4 d (a+b \cos (c+d x))^{3/2}}","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(2 a^2 C+A b^2-b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)}+\frac{2 \left(16 a^4 C+2 a^2 b^2 (A-8 C)-b^4 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^4 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{4 a \left(8 a^4 C+a^2 b^2 (A-14 C)-b^4 (3 A-4 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^4 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{4 a \left(-3 a^4 C+5 a^2 b^2 C+2 A b^4\right) \sin (c+d x)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}",1,"(2*((((a + b*Cos[c + d*x])/(a + b))^(3/2)*(b*(-4*a^4*b*C + b^5*(3*A + C) + a^2*b^3*(A + 7*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - 2*a*(a^2*b^2*(A - 14*C) + 8*a^4*C + b^4*(-3*A + 4*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])))/((a - b)^2*(a + b)) + (b*(2*a^4*A*b^2 - 10*a^2*A*b^4 + 16*a^6*C - 25*a^4*b^2*C + b^6*C + 4*a*b*(a^2*b^2*(A - 8*C) + 5*a^4*C + b^4*(-3*A + C))*Cos[c + d*x] + (-(a^2*b) + b^3)^2*C*Cos[2*(c + d*x)])*Sin[c + d*x])/(2*(a^2 - b^2)^2)))/(3*b^4*d*(a + b*Cos[c + d*x])^(3/2))","A",1
665,1,227,314,2.2015963,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(\frac{\left(\frac{a+b \cos (c+d x)}{a+b}\right)^{3/2} \left(\left(8 a^4 C-a^2 b^2 (A+15 C)+3 b^4 (C-A)\right) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a (a-b) \left(8 a^2 C-A b^2-9 b^2 C\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)}{(a-b)^2}+\frac{b \sin (c+d x) \left(-4 a^5 C+2 a^3 b^2 (A+4 C)+\left(-5 a^4 b C+a^2 b^3 (A+9 C)+3 A b^5\right) \cos (c+d x)+2 a A b^4\right)}{\left(a^2-b^2\right)^2}\right)}{3 b^3 d (a+b \cos (c+d x))^{3/2}}","\frac{2 a \left(a^2 C+A b^2\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 a \left(-8 a^2 C+A b^2+9 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-5 a^4 C+a^2 b^2 (A+9 C)+3 A b^4\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-8 a^4 C+a^2 b^2 (A+15 C)+3 b^4 (A-C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*((((a + b*Cos[c + d*x])/(a + b))^(3/2)*((8*a^4*C + 3*b^4*(-A + C) - a^2*b^2*(A + 15*C))*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*(a - b)*(-(A*b^2) + 8*a^2*C - 9*b^2*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/(a - b)^2 + (b*(2*a*A*b^4 - 4*a^5*C + 2*a^3*b^2*(A + 4*C) + (3*A*b^5 - 5*a^4*b*C + a^2*b^3*(A + 9*C))*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2))/(3*b^3*d*(a + b*Cos[c + d*x])^(3/2))","A",1
666,1,205,298,1.8607517,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(\frac{b \sin (c+d x) \left(a^4 C+2 a b \left(C \left(a^2-3 b^2\right)-2 A b^2\right) \cos (c+d x)-5 a^2 b^2 (A+C)+A b^4\right)}{\left(a^2-b^2\right)^2}+\frac{\left(\frac{a+b \cos (c+d x)}{a+b}\right)^{3/2} \left(\left(2 a b^2 (2 A+3 C)-2 a^3 C\right) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+(a-b) \left(2 a^2 C-A b^2-3 b^2 C\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)}{(a-b)^2}\right)}{3 b^2 d (a+b \cos (c+d x))^{3/2}}","-\frac{4 a \left(a^2 (-C)+2 A b^2+3 b^2 C\right) \sin (c+d x)}{3 b d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(-2 a^2 C+A b^2+3 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{4 a \left(2 A b^2-C \left(a^2-3 b^2\right)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*((((a + b*Cos[c + d*x])/(a + b))^(3/2)*((-2*a^3*C + 2*a*b^2*(2*A + 3*C))*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + (a - b)*(-(A*b^2) + 2*a^2*C - 3*b^2*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/(a - b)^2 + (b*(A*b^4 + a^4*C - 5*a^2*b^2*(A + C) + 2*a*b*(-2*A*b^2 + (a^2 - 3*b^2)*C)*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2))/(3*b^2*d*(a + b*Cos[c + d*x])^(3/2))","A",1
667,0,0,375,38.1955475,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^(5/2),x]","\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(a^2 C+A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(a^4 (-C)-a^2 b^2 (7 A+3 C)+3 A b^4\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^4 (-C)-a^2 b^2 (7 A+3 C)+3 A b^4\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^2 b d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x]","F",-1
668,1,786,416,7.1449418,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2),x]","\frac{\cos ^2(c+d x) \sqrt{a+b \cos (c+d x)} \left(A \sec ^2(c+d x)+C\right) \left(\frac{2 A \tan (c+d x)}{a^3}-\frac{4 \left(a^2 b C \sin (c+d x)+A b^3 \sin (c+d x)\right)}{3 a^2 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{8 \left(2 a^4 b C \sin (c+d x)+5 a^2 A b^3 \sin (c+d x)-3 A b^5 \sin (c+d x)\right)}{3 a^3 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}\right)}{d (2 A+C \cos (2 c+2 d x)+C)}+\frac{\cos ^2(c+d x) \left(A \sec ^2(c+d x)+C\right) \left(\frac{2 \left(12 a^5 C+36 a^3 A b^2+4 a^3 b^2 C-20 a A b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-33 a^4 A b+8 a^4 b C+86 a^2 A b^3-45 A b^5\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \left(-3 a^4 A b+8 a^4 b C+26 a^2 A b^3-15 A b^5\right) \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{b \cos (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left(2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right)}\right)}{6 a^3 d (b-a)^2 (a+b)^2 (2 A+C \cos (2 c+2 d x)+C)}","-\frac{5 A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^3 d \sqrt{a+b \cos (c+d x)}}-\frac{b \left(5 A b^2-a^2 (3 A-2 C)\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{\left(5 A b^2-a^2 (3 A-2 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{b \left(-\left(a^4 (3 A-8 C)\right)+26 a^2 A b^2-15 A b^4\right) \sin (c+d x)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{\left(-\left(a^4 (3 A-8 C)\right)+26 a^2 A b^2-15 A b^4\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}",1,"(Cos[c + d*x]^2*(C + A*Sec[c + d*x]^2)*((2*(36*a^3*A*b^2 - 20*a*A*b^4 + 12*a^5*C + 4*a^3*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(-33*a^4*A*b + 86*a^2*A*b^3 - 45*A*b^5 + 8*a^4*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(-3*a^4*A*b + 26*a^2*A*b^3 - 15*A*b^5 + 8*a^4*b*C)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2))))/(6*a^3*(-a + b)^2*(a + b)^2*d*(2*A + C + C*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*(C + A*Sec[c + d*x]^2)*((-4*(A*b^3*Sin[c + d*x] + a^2*b*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) - (8*(5*a^2*A*b^3*Sin[c + d*x] - 3*A*b^5*Sin[c + d*x] + 2*a^4*b*C*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) + (2*A*Tan[c + d*x])/a^3))/(d*(2*A + C + C*Cos[2*c + 2*d*x]))","C",0
669,1,314,389,2.7629779,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{7/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(7/2),x]","\frac{2 \left(\frac{\left(\frac{a+b \cos (c+d x)}{a+b}\right)^{5/2} \left(2 a (a-b) \left(C \left(a^2-5 b^2\right)-4 A b^2\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(-2 a^4 C+a^2 b^2 (23 A+19 C)+3 b^4 (3 A+5 C)\right) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)}{(a-b)^3}+\frac{b \sin (c+d x) \left(-2 a^6 C+68 a^4 A b^2+48 a^4 b^2 C+13 a^2 A b^4+35 a^2 b^4 C-4 a b \left(3 a^4 C-a^2 b^2 (27 A+25 C)-5 b^4 (A+2 C)\right) \cos (c+d x)+\left(-2 a^4 b^2 C+a^2 b^4 (23 A+19 C)+3 b^6 (3 A+5 C)\right) \cos (2 (c+d x))+15 A b^6+15 b^6 C\right)}{2 \left(b^2-a^2\right)^3}\right)}{15 b^2 d (a+b \cos (c+d x))^{5/2}}","-\frac{4 a \left(a^2 (-C)+4 A b^2+5 b^2 C\right) \sin (c+d x)}{15 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{5 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{5/2}}-\frac{4 a \left(4 A b^2-C \left(a^2-5 b^2\right)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right) \sin (c+d x)}{15 b d \left(a^2-b^2\right)^3 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(2 a^4 C-a^2 b^2 (23 A+19 C)-3 b^4 (3 A+5 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \left(a^2-b^2\right)^3 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*((((a + b*Cos[c + d*x])/(a + b))^(5/2)*((-2*a^4*C + 3*b^4*(3*A + 5*C) + a^2*b^2*(23*A + 19*C))*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + 2*a*(a - b)*(-4*A*b^2 + (a^2 - 5*b^2)*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/(a - b)^3 + (b*(68*a^4*A*b^2 + 13*a^2*A*b^4 + 15*A*b^6 - 2*a^6*C + 48*a^4*b^2*C + 35*a^2*b^4*C + 15*b^6*C - 4*a*b*(3*a^4*C - 5*b^4*(A + 2*C) - a^2*b^2*(27*A + 25*C))*Cos[c + d*x] + (-2*a^4*b^2*C + 3*b^6*(3*A + 5*C) + a^2*b^4*(23*A + 19*C))*Cos[2*(c + d*x)])*Sin[c + d*x])/(2*(-a^2 + b^2)^3)))/(15*b^2*d*(a + b*Cos[c + d*x])^(5/2))","A",1
670,1,134,157,0.5080786,"\int \frac{a^2-b^2 \cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(a^2 - b^2*Cos[c + d*x]^2)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-2 b \sin (c+d x) (a+b \cos (c+d x))+4 a (a+b) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{4 a \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(4*a*(a + b)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + 2*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - 2*b*(a + b*Cos[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
671,1,83,116,0.1677449,"\int \frac{a^2-b^2 \cos ^2(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-2 a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)}{d \sqrt{a+b \cos (c+d x)}}","\frac{4 a \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{2 \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(-2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 2*a*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/(d*Sqrt[a + b*Cos[c + d*x]])","A",1
672,1,134,165,0.3629976,"\int \frac{a^2-b^2 \cos ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2),x]","\frac{-2 \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-4 a b \sin (c+d x)+4 a (a+b) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d (a-b) (a+b) \sqrt{a+b \cos (c+d x)}}","-\frac{4 a b \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{4 a \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}",1,"(4*a*(a + b)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 2*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - 4*a*b*Sin[c + d*x])/((a - b)*(a + b)*d*Sqrt[a + b*Cos[c + d*x]])","A",1
673,1,158,243,1.0111223,"\int \frac{a^2-b^2 \cos ^2(c+d x)}{(a+b \cos (c+d x))^{7/2}} \, dx","Integrate[(a^2 - b^2*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(7/2),x]","\frac{2 \left(\frac{\left(\frac{a+b \cos (c+d x)}{a+b}\right)^{3/2} \left(\left(5 a^2+3 b^2\right) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+2 a (b-a) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)}{(a-b)^2}-\frac{b \sin (c+d x) \left(b \left(5 a^2+3 b^2\right) \cos (c+d x)+a \left(7 a^2+b^2\right)\right)}{\left(a^2-b^2\right)^2}\right)}{3 d (a+b \cos (c+d x))^{3/2}}","-\frac{2 b \left(5 a^2+3 b^2\right) \sin (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{4 a b \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{4 a \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(5 a^2+3 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*((((a + b*Cos[c + d*x])/(a + b))^(3/2)*((5*a^2 + 3*b^2)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + 2*a*(-a + b)*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/(a - b)^2 - (b*(a*(7*a^2 + b^2) + b*(5*a^2 + 3*b^2)*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2))/(3*d*(a + b*Cos[c + d*x])^(3/2))","A",1
674,1,134,196,1.6490538,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} (154 a (36 A+43 C) \cos (c+d x)+770 a C \cos (3 (c+d x))+180 b (11 A+16 C) \cos (2 (c+d x))+8580 A b+315 b C \cos (4 (c+d x))+7965 b C)+1848 a (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+600 b (11 A+9 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{13860 d}","\frac{2 a (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a (9 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{10 b (11 A+9 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 b (11 A+9 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{10 b (11 A+9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{11 d}",1,"(1848*a*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2] + 600*b*(11*A + 9*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(8580*A*b + 7965*b*C + 154*a*(36*A + 43*C)*Cos[c + d*x] + 180*b*(11*A + 16*C)*Cos[2*(c + d*x)] + 770*a*C*Cos[3*(c + d*x)] + 315*b*C*Cos[4*(c + d*x)])*Sin[c + d*x])/(13860*d)","A",1
675,1,119,165,0.9270011,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} (5 (84 a A+18 a C \cos (2 (c+d x))+78 a C+7 b C \cos (3 (c+d x)))+7 b (36 A+43 C) \cos (c+d x))+60 a (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+84 b (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{630 d}","\frac{2 a (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (7 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 b (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b (9 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}",1,"(84*b*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2] + 60*a*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*b*(36*A + 43*C)*Cos[c + d*x] + 5*(84*a*A + 78*a*C + 18*a*C*Cos[2*(c + d*x)] + 7*b*C*Cos[3*(c + d*x)]))*Sin[c + d*x])/(630*d)","A",1
676,1,98,134,0.6912019,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} (42 a C \cos (c+d x)+70 A b+15 b C \cos (2 (c+d x))+65 b C)+42 a (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+10 b (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{105 d}","\frac{2 a (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 b (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 b (7 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}",1,"(42*a*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2] + 10*b*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(70*A*b + 65*b*C + 42*a*C*Cos[c + d*x] + 15*b*C*Cos[2*(c + d*x)])*Sin[c + d*x])/(105*d)","A",1
677,1,79,101,0.421881,"\int \frac{(a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{2 \left(5 a (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (c+d x) \sqrt{\cos (c+d x)} (5 a+3 b \cos (c+d x))+3 b (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d}","\frac{2 a (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(2*(3*b*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2] + 5*a*(3*A + C)*EllipticF[(c + d*x)/2, 2] + C*Sqrt[Cos[c + d*x]]*(5*a + 3*b*Cos[c + d*x])*Sin[c + d*x]))/(15*d)","A",1
678,1,78,95,0.4871443,"\int \frac{(a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{\frac{2 \sin (c+d x) (3 a A+b C \cos (c+d x))}{\sqrt{\cos (c+d x)}}+6 a (C-A) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 b (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}","-\frac{2 a (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 b (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 b C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(6*a*(-A + C)*EllipticE[(c + d*x)/2, 2] + 2*b*(3*A + C)*EllipticF[(c + d*x)/2, 2] + (2*(3*a*A + b*C*Cos[c + d*x])*Sin[c + d*x])/Sqrt[Cos[c + d*x]])/(3*d)","A",1
679,1,76,95,0.7822448,"\int \frac{(a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{\frac{2 A \sin (c+d x) (a+3 b \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)}+2 a (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 b (C-A) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}","\frac{2 a (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A b \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(6*b*(-A + C)*EllipticE[(c + d*x)/2, 2] + 2*a*(A + 3*C)*EllipticF[(c + d*x)/2, 2] + (2*A*(a + 3*b*Cos[c + d*x])*Sin[c + d*x])/Cos[c + d*x]^(3/2))/(3*d)","A",1
680,1,122,132,0.7614798,"\int \frac{(a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{-6 a (3 A+5 C) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+9 a A \sin (2 (c+d x))+6 a A \tan (c+d x)+15 a C \sin (2 (c+d x))+10 b (A+3 C) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+10 A b \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{2 a (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (3 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 b (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A b \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-6*a*(3*A + 5*C)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 10*b*(A + 3*C)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 10*A*b*Sin[c + d*x] + 9*a*A*Sin[2*(c + d*x)] + 15*a*C*Sin[2*(c + d*x)] + 6*a*A*Tan[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",1
681,1,160,165,0.8083492,"\int \frac{(a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{10 a (5 A+7 C) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+25 a A \sin (2 (c+d x))+30 a A \tan (c+d x)+35 a C \sin (2 (c+d x))-42 b (3 A+5 C) \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+42 A b \sin (c+d x)+126 A b \sin (c+d x) \cos ^2(c+d x)+210 b C \sin (c+d x) \cos ^2(c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 a (5 A+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (5 A+7 C) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}-\frac{2 b (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b (3 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 A b \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-42*b*(3*A + 5*C)*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 10*a*(5*A + 7*C)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 42*A*b*Sin[c + d*x] + 126*A*b*Cos[c + d*x]^2*Sin[c + d*x] + 210*b*C*Cos[c + d*x]^2*Sin[c + d*x] + 25*a*A*Sin[2*(c + d*x)] + 35*a*C*Sin[2*(c + d*x)] + 30*a*A*Tan[c + d*x])/(105*d*Cos[c + d*x]^(5/2))","A",1
682,1,187,254,1.5126871,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2),x]","\frac{240 \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(5 \left(36 \left(11 a^2 C+11 A b^2+16 b^2 C\right) \cos (2 (c+d x))+132 a^2 (14 A+13 C)+308 a b C \cos (3 (c+d x))+3 b^2 (572 A+531 C)+63 b^2 C \cos (4 (c+d x))\right)+308 a b (36 A+43 C) \cos (c+d x)\right)+7392 a b (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{27720 d}","\frac{2 \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 \left(4 a^2 C+b^2 (11 A+9 C)\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{2 \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{4 a b (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a b (9 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{8 a b C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}",1,"(7392*a*b*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2] + 240*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2] + 2*Sqrt[Cos[c + d*x]]*(308*a*b*(36*A + 43*C)*Cos[c + d*x] + 5*(132*a^2*(14*A + 13*C) + 3*b^2*(572*A + 531*C) + 36*(11*A*b^2 + 11*a^2*C + 16*b^2*C)*Cos[2*(c + d*x)] + 308*a*b*C*Cos[3*(c + d*x)] + 63*b^2*C*Cos[4*(c + d*x)]))*Sin[c + d*x])/(27720*d)","A",1
683,1,148,205,1.2119557,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2),x]","\frac{84 \left(3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} \left(7 \left(36 a^2 C+36 A b^2+43 b^2 C\right) \cos (c+d x)+5 b (168 a A+36 a C \cos (2 (c+d x))+156 a C+7 b C \cos (3 (c+d x)))\right)+120 a b (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{630 d}","\frac{2 \left(3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(4 a^2 C+b^2 (9 A+7 C)\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{4 a b (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a b (7 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{8 a b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}",1,"(84*(3*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2] + 120*a*b*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*(36*A*b^2 + 36*a^2*C + 43*b^2*C)*Cos[c + d*x] + 5*b*(168*a*A + 156*a*C + 36*a*C*Cos[2*(c + d*x)] + 7*b*C*Cos[3*(c + d*x)]))*Sin[c + d*x])/(630*d)","A",1
684,1,126,171,1.0050473,"\int \frac{(a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{10 \left(7 a^2 (3 A+C)+b^2 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} \left(70 a^2 C+84 a b C \cos (c+d x)+70 A b^2+15 b^2 C \cos (2 (c+d x))+65 b^2 C\right)+84 a b (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{105 d}","\frac{2 \left(7 a^2 (3 A+C)+b^2 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(4 a^2 C+b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{4 a b (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 a b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}",1,"(84*a*b*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2] + 10*(7*a^2*(3*A + C) + b^2*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(70*A*b^2 + 70*a^2*C + 65*b^2*C + 84*a*b*C*Cos[c + d*x] + 15*b^2*C*Cos[2*(c + d*x)])*Sin[c + d*x])/(105*d)","A",1
685,1,119,166,1.0628709,"\int \frac{(a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{-6 \left(5 a^2 (A-C)-b^2 (5 A+3 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{\sin (c+d x) \left(30 a^2 A+20 a b C \cos (c+d x)+3 b^2 C \cos (2 (c+d x))+3 b^2 C\right)}{\sqrt{\cos (c+d x)}}+20 a b (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}","-\frac{2 \left(5 a^2 (A-C)-b^2 (5 A+3 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a b (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a b (3 A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt{\cos (c+d x)}}-\frac{2 b^2 (5 A-C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(-6*(5*a^2*(A - C) - b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2] + 20*a*b*(3*A + C)*EllipticF[(c + d*x)/2, 2] + ((30*a^2*A + 3*b^2*C + 20*a*b*C*Cos[c + d*x] + 3*b^2*C*Cos[2*(c + d*x)])*Sin[c + d*x])/Sqrt[Cos[c + d*x]])/(15*d)","A",1
686,1,108,154,1.4564281,"\int \frac{(a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 \left(a^2 (A+3 C)+b^2 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{2 a^2 A \tan (c+d x)+12 a A b \sin (c+d x)+b^2 C \sin (2 (c+d x))}{\sqrt{\cos (c+d x)}}+12 a b (C-A) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}","\frac{2 \left(a^2 (A+3 C)+b^2 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a b (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{8 a A b \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}-\frac{2 b^2 (A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(12*a*b*(-A + C)*EllipticE[(c + d*x)/2, 2] + 2*(b^2*(3*A + C) + a^2*(A + 3*C))*EllipticF[(c + d*x)/2, 2] + (12*a*A*b*Sin[c + d*x] + b^2*C*Sin[2*(c + d*x)] + 2*a^2*A*Tan[c + d*x])/Sqrt[Cos[c + d*x]])/(3*d)","A",1
687,1,158,169,0.8564956,"\int \frac{(a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{-6 \left(a^2 (3 A+5 C)+5 b^2 (A-C)\right) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+9 a^2 A \sin (2 (c+d x))+6 a^2 A \tan (c+d x)+15 a^2 C \sin (2 (c+d x))+20 a b (A+3 C) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+20 a A b \sin (c+d x)+15 A b^2 \sin (2 (c+d x))}{15 d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{2 \left(a^2 (3 A+5 C)+5 b^2 (A-C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(a^2 (3 A+5 C)+4 A b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{4 a b (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{8 a A b \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-6*(5*b^2*(A - C) + a^2*(3*A + 5*C))*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 20*a*b*(A + 3*C)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 20*a*A*b*Sin[c + d*x] + 9*a^2*A*Sin[2*(c + d*x)] + 15*A*b^2*Sin[2*(c + d*x)] + 15*a^2*C*Sin[2*(c + d*x)] + 6*a^2*A*Tan[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",1
688,1,198,203,1.1865102,"\int \frac{(a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{10 \left(a^2 (5 A+7 C)+7 b^2 (A+3 C)\right) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+25 a^2 A \sin (2 (c+d x))+30 a^2 A \tan (c+d x)+35 a^2 C \sin (2 (c+d x))-84 a b (3 A+5 C) \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+84 a A b \sin (c+d x)+252 a A b \sin (c+d x) \cos ^2(c+d x)+420 a b C \sin (c+d x) \cos ^2(c+d x)+35 A b^2 \sin (2 (c+d x))}{105 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 \left(a^2 (5 A+7 C)+7 b^2 (A+3 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(a^2 (5 A+7 C)+4 A b^2\right) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{4 a b (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a b (3 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{8 a A b \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-84*a*b*(3*A + 5*C)*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 10*(7*b^2*(A + 3*C) + a^2*(5*A + 7*C))*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 84*a*A*b*Sin[c + d*x] + 252*a*A*b*Cos[c + d*x]^2*Sin[c + d*x] + 420*a*b*C*Cos[c + d*x]^2*Sin[c + d*x] + 25*a^2*A*Sin[2*(c + d*x)] + 35*A*b^2*Sin[2*(c + d*x)] + 35*a^2*C*Sin[2*(c + d*x)] + 30*a^2*A*Tan[c + d*x])/(105*d*Cos[c + d*x]^(5/2))","A",1
689,1,215,295,1.6113647,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2),x]","\frac{3696 \left(a^3 (5 A+3 C)+a b^2 (9 A+7 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+80 b \left(33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(154 a \left(12 a^2 C+36 A b^2+43 b^2 C\right) \cos (c+d x)+5 b \left(12 \left(33 a^2 C+11 A b^2+16 b^2 C\right) \cos (2 (c+d x))+1848 a^2 A+1716 a^2 C+154 a b C \cos (3 (c+d x))+572 A b^2+21 b^2 C \cos (4 (c+d x))+531 b^2 C\right)\right)}{9240 d}","\frac{2 b \left(33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 a \left(a^2 (5 A+3 C)+b^2 (9 A+7 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b \left(8 a^2 C+3 b^2 (11 A+9 C)\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{231 d}+\frac{2 a \left(8 a^2 C+99 A b^2+77 b^2 C\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{165 d}+\frac{2 b \left(33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}+\frac{4 a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{33 d}",1,"(3696*(a^3*(5*A + 3*C) + a*b^2*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2] + 80*b*(33*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2] + 2*Sqrt[Cos[c + d*x]]*(154*a*(36*A*b^2 + 12*a^2*C + 43*b^2*C)*Cos[c + d*x] + 5*b*(1848*a^2*A + 572*A*b^2 + 1716*a^2*C + 531*b^2*C + 12*(11*A*b^2 + 33*a^2*C + 16*b^2*C)*Cos[2*(c + d*x)] + 154*a*b*C*Cos[3*(c + d*x)] + 21*b^2*C*Cos[4*(c + d*x)]))*Sin[c + d*x])/(9240*d)","A",1
690,1,181,245,1.5804632,"\int \frac{(a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{84 \left(9 a^2 b (5 A+3 C)+b^3 (9 A+7 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+60 a \left(7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} \left(5 \left(84 a^3 C+252 a A b^2+54 a b^2 C \cos (2 (c+d x))+234 a b^2 C+7 b^3 C \cos (3 (c+d x))\right)+7 b \left(108 a^2 C+36 A b^2+43 b^2 C\right) \cos (c+d x)\right)}{630 d}","\frac{2 a \left(7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 b \left(9 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b \left(24 a^2 C+7 b^2 (9 A+7 C)\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{315 d}+\frac{2 a \left(8 a^2 C+63 A b^2+45 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{63 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac{4 a C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{21 d}",1,"(84*(9*a^2*b*(5*A + 3*C) + b^3*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2] + 60*a*(7*a^2*(3*A + C) + 3*b^2*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*b*(36*A*b^2 + 108*a^2*C + 43*b^2*C)*Cos[c + d*x] + 5*(252*a*A*b^2 + 84*a^3*C + 234*a*b^2*C + 54*a*b^2*C*Cos[2*(c + d*x)] + 7*b^3*C*Cos[3*(c + d*x)]))*Sin[c + d*x])/(630*d)","A",1
691,1,172,244,1.8007325,"\int \frac{(a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{-84 \left(5 a^3 (A-C)-3 a b^2 (5 A+3 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+20 b \left(21 a^2 (3 A+C)+b^2 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{\sin (c+d x) \left(3 \left(140 a^3 A+42 a b^2 C \cos (2 (c+d x))+42 a b^2 C+5 b^3 C \cos (3 (c+d x))\right)+5 b \left(84 a^2 C+28 A b^2+29 b^2 C\right) \cos (c+d x)\right)}{\sqrt{\cos (c+d x)}}}{210 d}","\frac{2 b \left(21 a^2 (3 A+C)+b^2 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a \left(5 a^2 (A-C)-3 b^2 (5 A+3 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{2 b \left(6 a^2 (7 A-3 C)-b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}-\frac{2 a b^2 (35 A-11 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}-\frac{2 b (7 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt{\cos (c+d x)}}",1,"(-84*(5*a^3*(A - C) - 3*a*b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2] + 20*b*(21*a^2*(3*A + C) + b^2*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2] + ((5*b*(28*A*b^2 + 84*a^2*C + 29*b^2*C)*Cos[c + d*x] + 3*(140*a^3*A + 42*a*b^2*C + 42*a*b^2*C*Cos[2*(c + d*x)] + 5*b^3*C*Cos[3*(c + d*x)]))*Sin[c + d*x])/Sqrt[Cos[c + d*x]])/(210*d)","A",1
692,1,150,218,1.7519206,"\int \frac{(a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{10 \left(a^3 (A+3 C)+3 a b^2 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \left(3 b^3 (5 A+3 C)-45 a^2 b (A-C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{6 \sin (c+d x) \left(15 a^2 A b+b^3 C \cos ^2(c+d x)\right)+5 a \left(2 a^2 A \tan (c+d x)+3 b^2 C \sin (2 (c+d x))\right)}{\sqrt{\cos (c+d x)}}}{15 d}","\frac{2 a \left(a^2 (A+3 C)+3 b^2 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 b \left(15 a^2 (A-C)-b^2 (5 A+3 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{2 a b^2 (5 A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 A b \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt{\cos (c+d x)}}-\frac{2 b^3 (35 A-3 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 d}",1,"(2*(-45*a^2*b*(A - C) + 3*b^3*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2] + 10*(3*a*b^2*(3*A + C) + a^3*(A + 3*C))*EllipticF[(c + d*x)/2, 2] + (6*(15*a^2*A*b + b^3*C*Cos[c + d*x]^2)*Sin[c + d*x] + 5*a*(3*b^2*C*Sin[2*(c + d*x)] + 2*a^2*A*Tan[c + d*x]))/Sqrt[Cos[c + d*x]])/(15*d)","A",1
693,1,196,229,1.4032301,"\int \frac{(a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{9 a^3 A \sin (2 (c+d x))+6 a^3 A \tan (c+d x)+15 a^3 C \sin (2 (c+d x))+10 b \left(3 a^2 (A+3 C)+b^2 (3 A+C)\right) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 a \left(a^2 (3 A+5 C)+15 b^2 (A-C)\right) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+30 a^2 A b \sin (c+d x)+45 a A b^2 \sin (2 (c+d x))+10 b^3 C \sin (c+d x) \cos ^2(c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 b \left(3 a^2 (A+3 C)+b^2 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a \left(a^2 (3 A+5 C)+15 b^2 (A-C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \left(a^2 (3 A+5 C)+8 A b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{4 A b \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 b^3 (9 A-5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}",1,"(-6*a*(15*b^2*(A - C) + a^2*(3*A + 5*C))*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 10*b*(b^2*(3*A + C) + 3*a^2*(A + 3*C))*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 30*a^2*A*b*Sin[c + d*x] + 10*b^3*C*Cos[c + d*x]^2*Sin[c + d*x] + 9*a^3*A*Sin[2*(c + d*x)] + 45*a*A*b^2*Sin[2*(c + d*x)] + 15*a^3*C*Sin[2*(c + d*x)] + 6*a^3*A*Tan[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",1
694,1,241,243,1.5759571,"\int \frac{(a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{25 a^3 A \sin (2 (c+d x))+30 a^3 A \tan (c+d x)+35 a^3 C \sin (2 (c+d x))+10 a \left(a^2 (5 A+7 C)+21 b^2 (A+3 C)\right) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-42 b \left(3 a^2 (3 A+5 C)+5 b^2 (A-C)\right) \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+126 a^2 A b \sin (c+d x)+378 a^2 A b \sin (c+d x) \cos ^2(c+d x)+630 a^2 b C \sin (c+d x) \cos ^2(c+d x)+105 a A b^2 \sin (2 (c+d x))+210 A b^3 \sin (c+d x) \cos ^2(c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 a \left(a^2 (5 A+7 C)+21 b^2 (A+3 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 b \left(3 a^2 (3 A+5 C)+5 b^2 (A-C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \left(5 a^2 (5 A+7 C)+24 A b^2\right) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{6 b \left(7 a^2 (3 A+5 C)+8 A b^2\right) \sin (c+d x)}{35 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{12 A b \sin (c+d x) (a+b \cos (c+d x))^2}{35 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-42*b*(5*b^2*(A - C) + 3*a^2*(3*A + 5*C))*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 10*a*(21*b^2*(A + 3*C) + a^2*(5*A + 7*C))*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 126*a^2*A*b*Sin[c + d*x] + 378*a^2*A*b*Cos[c + d*x]^2*Sin[c + d*x] + 210*A*b^3*Cos[c + d*x]^2*Sin[c + d*x] + 630*a^2*b*C*Cos[c + d*x]^2*Sin[c + d*x] + 25*a^3*A*Sin[2*(c + d*x)] + 105*a*A*b^2*Sin[2*(c + d*x)] + 35*a^3*C*Sin[2*(c + d*x)] + 30*a^3*A*Tan[c + d*x])/(105*d*Cos[c + d*x]^(5/2))","A",1
695,1,250,293,5.1006564,"\int \frac{(a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{-14 \left(a^3 (7 A+9 C)+9 a b^2 (3 A+5 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{70 a^3 A \sin (c+d x)}{3 \cos ^{\frac{9}{2}}(c+d x)}+10 \left(3 a^2 b (5 A+7 C)+7 b^3 (A+3 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{14 a \left(a^2 (7 A+9 C)+27 A b^2\right) \sin (c+d x)}{3 \cos ^{\frac{5}{2}}(c+d x)}+\frac{10 b \left(3 a^2 (5 A+7 C)+7 A b^2\right) \sin (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)}+\frac{14 a \left(a^2 (7 A+9 C)+9 b^2 (3 A+5 C)\right) \sin (c+d x)}{\sqrt{\cos (c+d x)}}+\frac{90 a^2 A b \sin (c+d x)}{\cos ^{\frac{7}{2}}(c+d x)}}{105 d}","\frac{2 b \left(3 a^2 (5 A+7 C)+7 b^2 (A+3 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a \left(a^2 (7 A+9 C)+9 b^2 (3 A+5 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b \left(9 a^2 (5 A+7 C)+8 A b^2\right) \sin (c+d x)}{63 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left(7 a^2 (7 A+9 C)+24 A b^2\right) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a \left(a^2 (7 A+9 C)+9 b^2 (3 A+5 C)\right) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{4 A b \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(-14*(9*a*b^2*(3*A + 5*C) + a^3*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2] + 10*(7*b^3*(A + 3*C) + 3*a^2*b*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2] + (70*a^3*A*Sin[c + d*x])/(3*Cos[c + d*x]^(9/2)) + (90*a^2*A*b*Sin[c + d*x])/Cos[c + d*x]^(7/2) + (14*a*(27*A*b^2 + a^2*(7*A + 9*C))*Sin[c + d*x])/(3*Cos[c + d*x]^(5/2)) + (10*b*(7*A*b^2 + 3*a^2*(5*A + 7*C))*Sin[c + d*x])/Cos[c + d*x]^(3/2) + (14*a*(9*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sin[c + d*x])/Sqrt[Cos[c + d*x]])/(105*d)","A",1
696,1,281,382,2.9479412,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2),x]","\frac{24960 a b \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+7392 \left(39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(5 b \left(77 \left(312 a^2 b C+52 A b^3+89 b^3 C\right) \cos (3 (c+d x))+3744 a \left(11 a^2 C+11 A b^2+16 b^2 C\right) \cos (2 (c+d x))+312 a \left(44 a^2 (14 A+13 C)+b^2 (572 A+531 C)\right)+6552 a b^2 C \cos (4 (c+d x))+693 b^3 C \cos (5 (c+d x))\right)+154 \left(936 a^4 C+156 a^2 b^2 (36 A+43 C)+b^4 (1118 A+1171 C)\right) \cos (c+d x)\right)}{720720 d}","\frac{8 a b \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 \left(48 a^2 C+11 b^2 (13 A+11 C)\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{1287 d}+\frac{4 a b \left(96 a^2 C+1573 A b^2+1259 b^2 C\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{9009 d}+\frac{8 a b \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 \left(39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{195 d}+\frac{2 \left(192 a^4 C+11 a^2 b^2 (637 A+491 C)+77 b^4 (13 A+11 C)\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{6435 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}+\frac{16 a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{143 d}",1,"(7392*(39*a^4*(5*A + 3*C) + 78*a^2*b^2*(9*A + 7*C) + 7*b^4*(13*A + 11*C))*EllipticE[(c + d*x)/2, 2] + 24960*a*b*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2] + 2*Sqrt[Cos[c + d*x]]*(154*(936*a^4*C + 156*a^2*b^2*(36*A + 43*C) + b^4*(1118*A + 1171*C))*Cos[c + d*x] + 5*b*(312*a*(44*a^2*(14*A + 13*C) + b^2*(572*A + 531*C)) + 3744*a*(11*A*b^2 + 11*a^2*C + 16*b^2*C)*Cos[2*(c + d*x)] + 77*(52*A*b^3 + 312*a^2*b*C + 89*b^3*C)*Cos[3*(c + d*x)] + 6552*a*b^2*C*Cos[4*(c + d*x)] + 693*b^3*C*Cos[5*(c + d*x)]))*Sin[c + d*x])/(720720*d)","A",1
697,1,243,329,2.1247031,"\int \frac{(a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{14784 \left(3 a^3 b (5 A+3 C)+a b^3 (9 A+7 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+240 \left(77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(616 a b \left(36 a^2 C+36 A b^2+43 b^2 C\right) \cos (c+d x)+5 \left(1848 a^4 C+792 a^2 b^2 (14 A+13 C)+36 \left(66 a^2 b^2 C+11 A b^4+16 b^4 C\right) \cos (2 (c+d x))+616 a b^3 C \cos (3 (c+d x))+3 b^4 (572 A+531 C)+63 b^4 C \cos (4 (c+d x))\right)\right)}{27720 d}","\frac{8 a b \left(3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a b \left(96 a^2 C+891 A b^2+673 b^2 C\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3465 d}+\frac{2 \left(16 a^2 C+3 b^2 (11 A+9 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{231 d}+\frac{2 \left(77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 \left(64 a^4 C+9 a^2 b^2 (143 A+101 C)+15 b^4 (11 A+9 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{693 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}+\frac{16 a C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3}{99 d}",1,"(14784*(3*a^3*b*(5*A + 3*C) + a*b^3*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2] + 240*(77*a^4*(3*A + C) + 66*a^2*b^2*(7*A + 5*C) + 5*b^4*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2] + 2*Sqrt[Cos[c + d*x]]*(616*a*b*(36*A*b^2 + 36*a^2*C + 43*b^2*C)*Cos[c + d*x] + 5*(1848*a^4*C + 792*a^2*b^2*(14*A + 13*C) + 3*b^4*(572*A + 531*C) + 36*(11*A*b^4 + 66*a^2*b^2*C + 16*b^4*C)*Cos[2*(c + d*x)] + 616*a*b^3*C*Cos[3*(c + d*x)] + 63*b^4*C*Cos[4*(c + d*x)]))*Sin[c + d*x])/(27720*d)","A",1
698,1,216,320,3.9247879,"\int \frac{(a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{40 \left(7 a^3 b (3 A+C)+a b^3 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-14 \left(15 a^4 (A-C)-18 a^2 b^2 (5 A+3 C)-b^4 (9 A+7 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{1}{12} \sqrt{\cos (c+d x)} \left(5 \left(504 a^4 A \tan (c+d x)+72 a b^3 C \sin (3 (c+d x))+7 b^4 C \sin (4 (c+d x))\right)+120 a b \left(28 a^2 C+28 A b^2+23 b^2 C\right) \sin (c+d x)+14 \left(108 a^2 b^2 C+18 A b^4+19 b^4 C\right) \sin (2 (c+d x))\right)}{105 d}","\frac{8 a b \left(7 a^2 (3 A+C)+b^2 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 b^2 \left(3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{315 d}-\frac{4 a b \left(a^2 (63 A-31 C)-6 b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{63 d}-\frac{2 \left(15 a^4 (A-C)-18 a^2 b^2 (5 A+3 C)-b^4 (9 A+7 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}-\frac{2 b (9 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}-\frac{2 a b (21 A-5 C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{21 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt{\cos (c+d x)}}",1,"(-14*(15*a^4*(A - C) - 18*a^2*b^2*(5*A + 3*C) - b^4*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2] + 40*(7*a^3*b*(3*A + C) + a*b^3*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2] + (Sqrt[Cos[c + d*x]]*(120*a*b*(28*A*b^2 + 28*a^2*C + 23*b^2*C)*Sin[c + d*x] + 14*(18*A*b^4 + 108*a^2*b^2*C + 19*b^4*C)*Sin[2*(c + d*x)] + 5*(72*a*b^3*C*Sin[3*(c + d*x)] + 7*b^4*C*Sin[4*(c + d*x)] + 504*a^4*A*Tan[c + d*x])))/12)/(105*d)","A",1
699,1,206,300,2.7839688,"\int \frac{(a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{-168 \left(5 a^3 b (A-C)-a b^3 (5 A+3 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+10 \left(7 a^4 (A+3 C)+42 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+105 \sqrt{\cos (c+d x)} \left(\frac{2}{3} a^4 A \tan (c+d x) \sec (c+d x)+8 a^3 A b \tan (c+d x)+\left(4 a^2 b^2 C+\frac{2 A b^4}{3}+\frac{23 b^4 C}{42}\right) \sin (c+d x)+\frac{4}{5} a b^3 C \sin (2 (c+d x))+\frac{1}{14} b^4 C \sin (3 (c+d x))\right)}{105 d}","-\frac{8 a b \left(5 a^2 (A-C)-b^2 (5 A+3 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{2 b^2 \left(3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 \left(7 a^4 (A+3 C)+42 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a b^3 (175 A-27 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d}-\frac{2 b^2 (21 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \sqrt{\cos (c+d x)}}",1,"(-168*(5*a^3*b*(A - C) - a*b^3*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2] + 10*(42*a^2*b^2*(3*A + C) + 7*a^4*(A + 3*C) + b^4*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2] + 105*Sqrt[Cos[c + d*x]]*(((2*A*b^4)/3 + 4*a^2*b^2*C + (23*b^4*C)/42)*Sin[c + d*x] + (4*a*b^3*C*Sin[2*(c + d*x)])/5 + (b^4*C*Sin[3*(c + d*x)])/14 + 8*a^3*A*b*Tan[c + d*x] + (2*a^4*A*Sec[c + d*x]*Tan[c + d*x])/3))/(105*d)","A",1
700,1,233,321,1.6539992,"\int \frac{(a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{9 a^4 A \sin (2 (c+d x))+6 a^4 A \tan (c+d x)+15 a^4 C \sin (2 (c+d x))+40 a^3 A b \sin (c+d x)+40 a b \left(a^2 (A+3 C)+b^2 (3 A+C)\right) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+90 a^2 A b^2 \sin (2 (c+d x))-6 \left(a^4 (3 A+5 C)+30 a^2 b^2 (A-C)-b^4 (5 A+3 C)\right) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+40 a b^3 C \sin (c+d x) \cos ^2(c+d x)+6 b^4 C \sin (c+d x) \cos ^3(c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{8 a b \left(a^2 (A+3 C)+b^2 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 b^2 \left(3 a^2 (3 A+5 C)+b^2 (59 A-3 C)\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{2 \left(a^2 (3 A+5 C)+16 A b^2\right) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \sqrt{\cos (c+d x)}}-\frac{4 a b \left(3 a^2 (3 A+5 C)+2 b^2 (33 A-5 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}-\frac{2 \left(a^4 (3 A+5 C)+30 a^2 b^2 (A-C)-b^4 (5 A+3 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{15 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-6*(30*a^2*b^2*(A - C) - b^4*(5*A + 3*C) + a^4*(3*A + 5*C))*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 40*a*b*(b^2*(3*A + C) + a^2*(A + 3*C))*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 40*a^3*A*b*Sin[c + d*x] + 40*a*b^3*C*Cos[c + d*x]^2*Sin[c + d*x] + 6*b^4*C*Cos[c + d*x]^3*Sin[c + d*x] + 9*a^4*A*Sin[2*(c + d*x)] + 90*a^2*A*b^2*Sin[2*(c + d*x)] + 15*a^4*C*Sin[2*(c + d*x)] + 6*a^4*A*Tan[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",1
701,1,276,316,2.6103821,"\int \frac{(a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{25 a^4 A \sin (2 (c+d x))+30 a^4 A \tan (c+d x)+35 a^4 C \sin (2 (c+d x))+168 a^3 A b \sin (c+d x)+504 a^3 A b \sin (c+d x) \cos ^2(c+d x)+840 a^3 b C \sin (c+d x) \cos ^2(c+d x)-168 a b \left(a^2 (3 A+5 C)+5 b^2 (A-C)\right) \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+210 a^2 A b^2 \sin (2 (c+d x))+10 \left(a^4 (5 A+7 C)+42 a^2 b^2 (A+3 C)+7 b^4 (3 A+C)\right) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+840 a A b^3 \sin (c+d x) \cos ^2(c+d x)+70 b^4 C \sin (c+d x) \cos ^3(c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x)}","-\frac{8 a b \left(a^2 (3 A+5 C)+5 b^2 (A-C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(5 a^2 (5 A+7 C)+48 A b^2\right) \sin (c+d x) (a+b \cos (c+d x))^2}{105 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b^2 \left(5 a^2 (5 A+7 C)+b^2 (87 A-35 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{4 a b \left(a^2 (101 A+175 C)+96 A b^2\right) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)}}+\frac{2 \left(a^4 (5 A+7 C)+42 a^2 b^2 (A+3 C)+7 b^4 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{35 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-168*a*b*(5*b^2*(A - C) + a^2*(3*A + 5*C))*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 10*(7*b^4*(3*A + C) + 42*a^2*b^2*(A + 3*C) + a^4*(5*A + 7*C))*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 168*a^3*A*b*Sin[c + d*x] + 504*a^3*A*b*Cos[c + d*x]^2*Sin[c + d*x] + 840*a*A*b^3*Cos[c + d*x]^2*Sin[c + d*x] + 840*a^3*b*C*Cos[c + d*x]^2*Sin[c + d*x] + 70*b^4*C*Cos[c + d*x]^3*Sin[c + d*x] + 25*a^4*A*Sin[2*(c + d*x)] + 210*a^2*A*b^2*Sin[2*(c + d*x)] + 35*a^4*C*Sin[2*(c + d*x)] + 30*a^4*A*Tan[c + d*x])/(105*d*Cos[c + d*x]^(5/2))","A",1
702,1,268,325,5.3892818,"\int \frac{(a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{2 \left(\frac{35 a^4 A \sin (c+d x)}{\cos ^{\frac{9}{2}}(c+d x)}+60 \left(a^3 b (5 A+7 C)+7 a b^3 (A+3 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{180 a^3 A b \sin (c+d x)}{\cos ^{\frac{7}{2}}(c+d x)}+\frac{7 a^2 \left(a^2 (7 A+9 C)+54 A b^2\right) \sin (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)}+\frac{60 a b \left(a^2 (5 A+7 C)+7 A b^2\right) \sin (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)}-21 \left(a^4 (7 A+9 C)+18 a^2 b^2 (3 A+5 C)+15 b^4 (A-C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{21 \left(a^4 (7 A+9 C)+18 a^2 b^2 (3 A+5 C)+15 A b^4\right) \sin (c+d x)}{\sqrt{\cos (c+d x)}}\right)}{315 d}","\frac{8 a b \left(a^2 (5 A+7 C)+7 b^2 (A+3 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(7 a^2 (7 A+9 C)+48 A b^2\right) \sin (c+d x) (a+b \cos (c+d x))^2}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a b \left(a^2 (101 A+147 C)+32 A b^2\right) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \left(a^4 (7 A+9 C)+18 a^2 b^2 (3 A+5 C)+15 b^4 (A-C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(21 a^4 (7 A+9 C)+7 a^2 b^2 (155 A+261 C)+192 A b^4\right) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{63 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(2*(-21*(15*b^4*(A - C) + 18*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2] + 60*(7*a*b^3*(A + 3*C) + a^3*b*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2] + (35*a^4*A*Sin[c + d*x])/Cos[c + d*x]^(9/2) + (180*a^3*A*b*Sin[c + d*x])/Cos[c + d*x]^(7/2) + (7*a^2*(54*A*b^2 + a^2*(7*A + 9*C))*Sin[c + d*x])/Cos[c + d*x]^(5/2) + (60*a*b*(7*A*b^2 + a^2*(5*A + 7*C))*Sin[c + d*x])/Cos[c + d*x]^(3/2) + (21*(15*A*b^4 + 18*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(315*d)","A",1
703,1,284,377,4.6021429,"\int \frac{(a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","\frac{-616 \left(a^3 b (7 A+9 C)+3 a b^3 (3 A+5 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+10 \left(5 a^4 (9 A+11 C)+66 a^2 b^2 (5 A+7 C)+77 b^4 (A+3 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{45 \left(14 a^4 A \tan (c+d x)+\left(a^4 (9 A+11 C)+66 a^2 A b^2\right) \sin (2 (c+d x))\right)+2 \sin (c+d x) \left(1540 a^3 A b+924 a b \left(a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right) \cos ^4(c+d x)+308 a b \left(a^2 (7 A+9 C)+9 A b^2\right) \cos ^2(c+d x)+15 \left(5 a^4 (9 A+11 C)+66 a^2 b^2 (5 A+7 C)+77 A b^4\right) \cos ^3(c+d x)\right)}{3 \cos ^{\frac{9}{2}}(c+d x)}}{1155 d}","-\frac{8 a b \left(a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(3 a^2 (9 A+11 C)+16 A b^2\right) \sin (c+d x) (a+b \cos (c+d x))^2}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 a b \left(a^2 (673 A+891 C)+96 A b^2\right) \sin (c+d x)}{3465 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{8 a b \left(a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 \left(5 a^4 (9 A+11 C)+66 a^2 b^2 (5 A+7 C)+77 b^4 (A+3 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 \left(15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)+64 A b^4\right) \sin (c+d x)}{693 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{99 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(-616*(3*a*b^3*(3*A + 5*C) + a^3*b*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2] + 10*(77*b^4*(A + 3*C) + 66*a^2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*EllipticF[(c + d*x)/2, 2] + (2*(1540*a^3*A*b + 308*a*b*(9*A*b^2 + a^2*(7*A + 9*C))*Cos[c + d*x]^2 + 15*(77*A*b^4 + 66*a^2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*Cos[c + d*x]^3 + 924*a*b*(3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Cos[c + d*x]^4)*Sin[c + d*x] + 45*((66*a^2*A*b^2 + a^4*(9*A + 11*C))*Sin[2*(c + d*x)] + 14*a^4*A*Tan[c + d*x]))/(3*Cos[c + d*x]^(9/2)))/(1155*d)","A",1
704,1,360,299,2.6492112,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(7/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{6 \left(\frac{8 a \left(7 a^2 C+7 A b^2+6 b^2 C\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{\left(35 a^4 C+a^2 b^2 (35 A+13 C)+7 b^4 (9 A+7 C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{7 \left(15 a^4 C+3 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}\right)+\sin (c+d x) \sqrt{\cos (c+d x)} \left(7 b \left(36 a^2 C+36 A b^2+43 b^2 C\right) \cos (c+d x)-5 \left(84 a^3 C+84 a A b^2+18 a b^2 C \cos (2 (c+d x))+78 a b^2 C-7 b^3 C \cos (3 (c+d x))\right)\right)}{630 b^4 d}","-\frac{2 a \left(7 a^2 C+7 A b^2+5 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 b^4 d}+\frac{2 \left(9 a^2 C+b^2 (9 A+7 C)\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 b^3 d}+\frac{2 a^4 \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^6 d (a+b)}-\frac{2 a \left(21 a^4 C+7 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b^6 d}+\frac{2 \left(15 a^4 C+3 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 b^5 d}-\frac{2 a C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 b^2 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 b d}",1,"(Sqrt[Cos[c + d*x]]*(7*b*(36*A*b^2 + 36*a^2*C + 43*b^2*C)*Cos[c + d*x] - 5*(84*a*A*b^2 + 84*a^3*C + 78*a*b^2*C + 18*a*b^2*C*Cos[2*(c + d*x)] - 7*b^3*C*Cos[3*(c + d*x)]))*Sin[c + d*x] + 6*(((35*a^4*C + 7*b^4*(9*A + 7*C) + a^2*b^2*(35*A + 13*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*a*(7*A*b^2 + 7*a^2*C + 6*b^2*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (7*(15*a^4*C + 3*a^2*b^2*(5*A + 3*C) + b^4*(9*A + 7*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2])))/(630*b^4*d)","A",1
705,1,291,239,2.2582884,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{-\frac{2 a \left(35 a^2 C+35 A b^2+13 b^2 C\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(70 a^2 C-42 a b C \cos (c+d x)+70 A b^2+15 b^2 C \cos (2 (c+d x))+65 b^2 C\right)+\frac{4 \left(-28 a^2 C+35 A b^2+25 b^2 C\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}-\frac{42 \left(5 a^2 C+5 A b^2+3 b^2 C\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{b^2 \sqrt{\sin ^2(c+d x)}}}{210 b^3 d}","-\frac{2 a \left(5 a^2 C+5 A b^2+3 b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^4 d}+\frac{2 \left(7 a^2 C+b^2 (7 A+5 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 b^3 d}+\frac{2 \left(21 a^4 C+7 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b^5 d}-\frac{2 a^3 \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a+b)}-\frac{2 a C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b^2 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 b d}",1,"((-2*a*(35*A*b^2 + 35*a^2*C + 13*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (4*(35*A*b^2 - 28*a^2*C + 25*b^2*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + 2*Sqrt[Cos[c + d*x]]*(70*A*b^2 + 70*a^2*C + 65*b^2*C - 42*a*b*C*Cos[c + d*x] + 15*b^2*C*Cos[2*(c + d*x)])*Sin[c + d*x] - (42*(5*A*b^2 + 5*a^2*C + 3*b^2*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(b^2*Sqrt[Sin[c + d*x]^2]))/(210*b^3*d)","A",1
706,1,244,181,2.2413971,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{\frac{2 \left(5 a^2 C+15 A b^2+9 b^2 C\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{6 \left(5 a^2 C+5 A b^2+3 b^2 C\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}+4 C \sin (c+d x) \sqrt{\cos (c+d x)} (3 b \cos (c+d x)-5 a)+8 a C \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{30 b^2 d}","-\frac{2 a \left(C \left(3 a^2+b^2\right)+3 A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d}+\frac{2 a^2 \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a+b)}+\frac{2 \left(5 a^2 C+b^2 (5 A+3 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^3 d}-\frac{2 a C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b d}",1,"((2*(15*A*b^2 + 5*a^2*C + 9*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + 8*a*C*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)) + 4*C*Sqrt[Cos[c + d*x]]*(-5*a + 3*b*Cos[c + d*x])*Sin[c + d*x] + (6*(5*A*b^2 + 5*a^2*C + 3*b^2*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/(30*b^2*d)","A",1
707,1,198,130,1.7774119,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{-\frac{6 C \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{b^2 \sqrt{\sin ^2(c+d x)}}+\frac{4 (3 A+C) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}-\frac{2 a C \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+4 C \sin (c+d x) \sqrt{\cos (c+d x)}}{6 b d}","\frac{2 \left(3 a^2 C+b^2 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^3 d}-\frac{2 a \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a+b)}-\frac{2 a C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d}",1,"((-2*a*C*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (4*(3*A + C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + 4*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x] - (6*C*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(b^2*Sqrt[Sin[c + d*x]^2]))/(6*b*d)","A",1
708,1,127,85,0.8520621,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])),x]","\frac{\frac{C \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}+\frac{(2 A+C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{d}","\frac{2 \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}-\frac{2 a C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(((2*A + C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (C*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/d","A",1
709,1,205,112,2.8375078,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])),x]","-\frac{\frac{2 A \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}+\frac{4 a (A-C) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{b (a+b)}+\frac{6 A b \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}-\frac{4 A \sin (c+d x)}{\sqrt{\cos (c+d x)}}}{2 a d}","-\frac{2 \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d (a+b)}-\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{2 A \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}+\frac{2 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"-1/2*((6*A*b*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (4*a*(A - C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(b*(a + b)) - (4*A*Sin[c + d*x])/Sqrt[Cos[c + d*x]] + (2*A*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/(a*d)","A",1
710,1,219,140,5.7886366,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])),x]","\frac{\frac{2 \left(2 a^2 (A+3 C)+9 A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{6 A \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a \sqrt{\sin ^2(c+d x)}}+\frac{4 A \sin (c+d x) (a-3 b \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)}+8 a A \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{6 a^2 d}","\frac{2 \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a+b)}+\frac{2 A b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{2 A b \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}+\frac{2 A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{2 A \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}",1,"((2*(9*A*b^2 + 2*a^2*(A + 3*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + 8*a*A*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)) + (4*A*(a - 3*b*Cos[c + d*x])*Sin[c + d*x])/Cos[c + d*x]^(3/2) + (6*A*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*Sqrt[Sin[c + d*x]^2]))/(6*a^2*d)","A",1
711,1,295,206,4.3204339,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])),x]","-\frac{\frac{\left(6 a^3 (3 A+5 C)+40 a A b^2\right) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{b}+\frac{2 \left(a^2 b (19 A+45 C)+45 A b^3\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}-\frac{2 \left(3 \left(a^2 (3 A+5 C)+5 A b^2\right) \sin (2 (c+d x))+6 a^2 A \tan (c+d x)-10 a A b \sin (c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)}+\frac{6 \left(a^2 (3 A+5 C)+5 A b^2\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}}{30 a^3 d}","-\frac{2 A b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{2 A b \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \left(a^2 (3 A+5 C)+5 A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^3 d}-\frac{2 b \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a+b)}+\frac{2 \left(a^2 (3 A+5 C)+5 A b^2\right) \sin (c+d x)}{5 a^3 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x)}{5 a d \cos ^{\frac{5}{2}}(c+d x)}",1,"-1/30*((2*(45*A*b^3 + a^2*b*(19*A + 45*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + ((40*a*A*b^2 + 6*a^3*(3*A + 5*C))*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)))/b + (6*(5*A*b^2 + a^2*(3*A + 5*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]) - (2*(-10*a*A*b*Sin[c + d*x] + 3*(5*A*b^2 + a^2*(3*A + 5*C))*Sin[2*(c + d*x)] + 6*a^2*A*Tan[c + d*x]))/Cos[c + d*x]^(3/2))/(a^3*d)","A",1
712,1,338,270,3.9655042,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*(a + b*Cos[c + d*x])),x]","\frac{\frac{8 \left(a^3 (22 A+35 C)+35 a A b^2\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{21 \left(a^2 (3 A+5 C)+5 A b^2\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a \sqrt{\sin ^2(c+d x)}}+\frac{\left(10 a^4 (5 A+7 C)+7 a^2 b^2 (19 A+45 C)+315 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{5 \left(\left(a^3 (5 A+7 C)+7 a A b^2\right) \sin (2 (c+d x))+6 a^3 A \tan (c+d x)\right)-42 b \sin (c+d x) \left(\left(a^2 (3 A+5 C)+5 A b^2\right) \cos ^2(c+d x)+a^2 A\right)}{\cos ^{\frac{5}{2}}(c+d x)}}{105 a^4 d}","-\frac{2 A b \sin (c+d x)}{5 a^2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 b \left(a^2 (3 A+5 C)+5 A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^4 d}+\frac{2 b^2 \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d (a+b)}-\frac{2 b \left(a^2 (3 A+5 C)+5 A b^2\right) \sin (c+d x)}{5 a^4 d \sqrt{\cos (c+d x)}}+\frac{2 \left(a^2 (5 A+7 C)+7 A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a^3 d}+\frac{2 \left(a^2 (5 A+7 C)+7 A b^2\right) \sin (c+d x)}{21 a^3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x)}{7 a d \cos ^{\frac{7}{2}}(c+d x)}",1,"(((315*A*b^4 + 10*a^4*(5*A + 7*C) + 7*a^2*b^2*(19*A + 45*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(35*a*A*b^2 + a^3*(22*A + 35*C))*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (21*(5*A*b^2 + a^2*(3*A + 5*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*Sqrt[Sin[c + d*x]^2]) + (-42*b*(a^2*A + (5*A*b^2 + a^2*(3*A + 5*C))*Cos[c + d*x]^2)*Sin[c + d*x] + 5*((7*a*A*b^2 + a^3*(5*A + 7*C))*Sin[2*(c + d*x)] + 6*a^3*A*Tan[c + d*x]))/Cos[c + d*x]^(5/2))/(105*a^4*d)","A",1
713,1,427,344,4.2935431,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{11}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(11/2)*(a + b*Cos[c + d*x])),x]","\frac{\frac{2 \left(70 a^4 A \tan (c+d x)+7 \left(a^4 (7 A+9 C)+9 a^2 A b^2\right) \sin (2 (c+d x))+6 \sin (c+d x) \left(-15 a^3 A b-5 a b \left(a^2 (5 A+7 C)+7 A b^2\right) \cos ^2(c+d x)+7 \left(a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right) \cos ^3(c+d x)\right)\right)}{\cos ^{\frac{7}{2}}(c+d x)}-3 \left(\frac{4 \left(7 a^5 (7 A+9 C)+4 a^3 b^2 (22 A+35 C)+140 a A b^4\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{b (a+b)}+\frac{2 \left(a^4 b (99 A+133 C)+7 a^2 b^3 (19 A+45 C)+315 A b^5\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{14 \left(a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}\right)}{630 a^5 d}","-\frac{2 A b \sin (c+d x)}{7 a^2 d \cos ^{\frac{7}{2}}(c+d x)}-\frac{2 b^3 \left(a^2 C+A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^5 d (a+b)}-\frac{2 b \left(a^2 (5 A+7 C)+7 A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a^4 d}-\frac{2 b \left(a^2 (5 A+7 C)+7 A b^2\right) \sin (c+d x)}{21 a^4 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(a^2 (7 A+9 C)+9 A b^2\right) \sin (c+d x)}{45 a^3 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 \left(a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 a^5 d}+\frac{2 \left(a^4 (7 A+9 C)+3 a^2 b^2 (3 A+5 C)+15 A b^4\right) \sin (c+d x)}{15 a^5 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x)}{9 a d \cos ^{\frac{9}{2}}(c+d x)}",1,"(-3*((2*(315*A*b^5 + 7*a^2*b^3*(19*A + 45*C) + a^4*b*(99*A + 133*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (4*(140*a*A*b^4 + 7*a^5*(7*A + 9*C) + 4*a^3*b^2*(22*A + 35*C))*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(b*(a + b)) + (14*(15*A*b^4 + 3*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2])) + (2*(6*(-15*a^3*A*b - 5*a*b*(7*A*b^2 + a^2*(5*A + 7*C))*Cos[c + d*x]^2 + 7*(15*A*b^4 + 3*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*Cos[c + d*x]^3)*Sin[c + d*x] + 7*(9*a^2*A*b^2 + a^4*(7*A + 9*C))*Sin[2*(c + d*x)] + 70*a^4*A*Tan[c + d*x]))/Cos[c + d*x]^(7/2))/(630*a^5*d)","A",1
714,1,354,370,4.156071,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{4 \sqrt{\cos (c+d x)} \left(-\frac{15 a^2 \left(a^2 C+A b^2\right) \sin (c+d x)}{\left(a^2-b^2\right) (a+b \cos (c+d x))}-20 a C \sin (c+d x)+3 b C \sin (2 (c+d x))\right)+\frac{\frac{8 a \left(C \left(14 a^2+b^2\right)+15 A b^2\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{2 \left(35 a^4 C+a^2 b^2 (15 A-32 C)-6 b^4 (5 A+3 C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{6 \left(35 a^4 C+3 a^2 b^2 (5 A-8 C)-2 b^4 (5 A+3 C)\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}}{(a-b) (a+b)}}{60 b^3 d}","-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\left(7 a^2 C+5 A b^2-2 b^2 C\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b^2 d \left(a^2-b^2\right)}-\frac{a \left(7 a^2 C+3 A b^2-4 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)}+\frac{\left(35 a^4 C+3 a^2 b^2 (5 A-8 C)-2 b^4 (5 A+3 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^4 d \left(a^2-b^2\right)}-\frac{a \left(21 a^4 C+a^2 b^2 (9 A-20 C)-4 b^4 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^5 d \left(a^2-b^2\right)}-\frac{a^2 \left(-7 a^4 C-3 a^2 b^2 (A-3 C)+5 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a-b) (a+b)^2}",1,"(((2*(a^2*b^2*(15*A - 32*C) + 35*a^4*C - 6*b^4*(5*A + 3*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*a*(15*A*b^2 + (14*a^2 + b^2)*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (6*(3*a^2*b^2*(5*A - 8*C) + 35*a^4*C - 2*b^4*(5*A + 3*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b)) + 4*Sqrt[Cos[c + d*x]]*(-20*a*C*Sin[c + d*x] - (15*a^2*(A*b^2 + a^2*C)*Sin[c + d*x])/((a^2 - b^2)*(a + b*Cos[c + d*x])) + 3*b*C*Sin[2*(c + d*x)]))/(60*b^3*d)","A",1
715,1,301,292,2.78908,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)} \left(\frac{3 a \left(a^2 C+A b^2\right)}{\left(a^2-b^2\right) (a+b \cos (c+d x))}+2 C\right)-\frac{\frac{2 a \left(5 a^2 C-3 A b^2-8 b^2 C\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{8 \left(C \left(2 a^2+b^2\right)+3 A b^2\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{6 \left(5 a^2 C+A b^2-4 b^2 C\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{b^2 \sqrt{\sin ^2(c+d x)}}}{(a-b) (a+b)}}{12 b^2 d}","-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\left(5 a^2 C+3 A b^2-2 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \left(a^2-b^2\right)}-\frac{a \left(5 a^2 C+A b^2-4 b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}+\frac{\left(15 a^4 C+a^2 b^2 (3 A-16 C)-2 b^4 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d \left(a^2-b^2\right)}+\frac{a \left(-5 a^4 C-a^2 b^2 (A-7 C)+3 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a-b) (a+b)^2}",1,"(4*Sqrt[Cos[c + d*x]]*(2*C + (3*a*(A*b^2 + a^2*C))/((a^2 - b^2)*(a + b*Cos[c + d*x])))*Sin[c + d*x] - ((2*a*(-3*A*b^2 + 5*a^2*C - 8*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(3*A*b^2 + (2*a^2 + b^2)*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (6*(A*b^2 + 5*a^2*C - 4*b^2*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b)))/(12*b^2*d)","A",1
716,1,280,217,3.2040788,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","-\frac{\frac{4 \left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{\left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\frac{2 \left(C \left(a^2-2 b^2\right)-A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{2 \left(3 a^2 C+A b^2-2 b^2 C\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}+\frac{8 a (A+C) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}}{(b-a) (a+b)}}{4 b d}","\frac{\left(3 a^2 C+A b^2-2 b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{a \left(-3 a^2 C+A b^2+4 b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}-\frac{\left(-3 a^4 C+a^2 b^2 (A+5 C)+A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}",1,"-1/4*((4*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*(a + b*Cos[c + d*x])) + ((2*(-(A*b^2) + (a^2 - 2*b^2)*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*a*(A + C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (2*(A*b^2 + 3*a^2*C - 2*b^2*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/((-a + b)*(a + b)))/(b*d)","A",1
717,1,271,214,2.4621602,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2),x]","\frac{\frac{4 \left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{\left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\frac{2 \left(a^2 (4 A+C)-3 A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}-\frac{8 a (A+C) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}}{(a-b) (a+b)}}{4 a d}","-\frac{\left(a^2 (-C)+A b^2+2 b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}-\frac{\left(a^2 C+A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d \left(a^2-b^2\right)}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\left(a^4 C-3 a^2 b^2 (A+C)+A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b^2 d (a-b) (a+b)^2}",1,"((4*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*(a + b*Cos[c + d*x])) + ((2*(-3*A*b^2 + a^2*(4*A + C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) - (8*a*(A + C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) - (2*(A*b^2 + a^2*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b)))/(4*a*d)","A",1
718,1,306,270,4.105577,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2),x]","\frac{4 \sqrt{\cos (c+d x)} \left(\frac{\left(a^2 b C+A b^3\right) \sin (c+d x)}{\left(b^2-a^2\right) (a+b \cos (c+d x))}+2 A \tan (c+d x)\right)-\frac{\frac{\left(8 a A b^2-4 a^3 (A-C)\right) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{b}-\frac{2 \left(a^2 b (10 A+C)-9 A b^3\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}-\frac{2 \left(a^2 (2 A-C)-3 A b^2\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}}{(b-a) (a+b)}}{4 a^2 d}","\frac{\left(a^2 C+A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d \left(a^2-b^2\right)}+\frac{\left(3 A b^2-a^2 (2 A-C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}-\frac{\left(3 A b^2-a^2 (2 A-C)\right) \sin (c+d x)}{a^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}+\frac{\left(a^4 (-C)-a^2 b^2 (5 A+C)+3 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 b d (a-b) (a+b)^2}",1,"(-(((-2*(-9*A*b^3 + a^2*b*(10*A + C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + ((8*a*A*b^2 - 4*a^3*(A - C))*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)))/b - (2*(-3*A*b^2 + a^2*(2*A - C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/((-a + b)*(a + b))) + 4*Sqrt[Cos[c + d*x]]*(((A*b^3 + a^2*b*C)*Sin[c + d*x])/((-a^2 + b^2)*(a + b*Cos[c + d*x])) + 2*A*Tan[c + d*x]))/(4*a^2*d)","A",1
719,1,332,336,5.3620069,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2),x]","\frac{4 \sqrt{\cos (c+d x)} \left(\frac{3 b^2 \left(a^2 C+A b^2\right) \sin (c+d x)}{\left(a^2-b^2\right) (a+b \cos (c+d x))}+2 A \tan (c+d x) (a \sec (c+d x)-6 b)\right)+\frac{\frac{8 \left(a^3 (7 A-3 C)-10 a A b^2\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{6 \left(a^2 (4 A-C)-5 A b^2\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a \sqrt{\sin ^2(c+d x)}}+\frac{2 \left(4 a^4 (A+3 C)+a^2 b^2 (44 A-9 C)-45 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{(a-b) (a+b)}}{12 a^3 d}","-\frac{\left(5 A b^2-a^2 (2 A-3 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d \left(a^2-b^2\right)}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{\left(5 A b^2-a^2 (2 A-3 C)\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}-\frac{b \left(5 A b^2-a^2 (4 A-C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \left(a^2-b^2\right)}+\frac{b \left(5 A b^2-a^2 (4 A-C)\right) \sin (c+d x)}{a^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{\left(-3 a^4 C-a^2 b^2 (7 A-C)+5 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}",1,"(((2*(-45*A*b^4 + a^2*b^2*(44*A - 9*C) + 4*a^4*(A + 3*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(-10*a*A*b^2 + a^3*(7*A - 3*C))*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (6*(-5*A*b^2 + a^2*(4*A - C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b)) + 4*Sqrt[Cos[c + d*x]]*((3*b^2*(A*b^2 + a^2*C)*Sin[c + d*x])/((a^2 - b^2)*(a + b*Cos[c + d*x])) + 2*A*(-6*b + a*Sec[c + d*x])*Tan[c + d*x]))/(12*a^3*d)","A",1
720,1,405,427,7.0079565,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])^2),x]","\frac{4 \sqrt{\cos (c+d x)} \left(2 \tan (c+d x) \left(3 a^2 A \sec ^2(c+d x)+3 a^2 (3 A+5 C)-10 a A b \sec (c+d x)+45 A b^2\right)+\frac{15 \left(a^2 b^3 C+A b^5\right) \sin (c+d x)}{\left(b^2-a^2\right) (a+b \cos (c+d x))}\right)-\frac{-\frac{8 \left(3 a^5 (3 A+5 C)+2 a^3 b^2 (23 A-15 C)-70 a A b^4\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{b (a+b)}-\frac{2 \left(2 a^4 b (29 A+75 C)+a^2 b^3 (272 A-135 C)-315 A b^5\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}-\frac{6 \left(2 a^4 (3 A+5 C)+3 a^2 b^2 (8 A-5 C)-35 A b^4\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}}{(b-a) (a+b)}}{60 a^4 d}","\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{\left(7 A b^2-a^2 (2 A-5 C)\right) \sin (c+d x)}{5 a^2 d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x)}+\frac{\left(-2 a^4 (3 A+5 C)-3 a^2 b^2 (8 A-5 C)+35 A b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^4 d \left(a^2-b^2\right)}+\frac{b \left(-5 a^4 C-3 a^2 b^2 (3 A-C)+7 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d (a-b) (a+b)^2}-\frac{\left(-2 a^4 (3 A+5 C)-3 a^2 b^2 (8 A-5 C)+35 A b^4\right) \sin (c+d x)}{5 a^4 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}+\frac{b \left(7 A b^2-a^2 (4 A-3 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^3 d \left(a^2-b^2\right)}+\frac{b \left(7 A b^2-a^2 (4 A-3 C)\right) \sin (c+d x)}{3 a^3 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}",1,"(-(((-2*(-315*A*b^5 + a^2*b^3*(272*A - 135*C) + 2*a^4*b*(29*A + 75*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) - (8*(-70*a*A*b^4 + 2*a^3*b^2*(23*A - 15*C) + 3*a^5*(3*A + 5*C))*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(b*(a + b)) - (6*(-35*A*b^4 + 3*a^2*b^2*(8*A - 5*C) + 2*a^4*(3*A + 5*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/((-a + b)*(a + b))) + 4*Sqrt[Cos[c + d*x]]*((15*(A*b^5 + a^2*b^3*C)*Sin[c + d*x])/((-a^2 + b^2)*(a + b*Cos[c + d*x])) + 2*(45*A*b^2 + 3*a^2*(3*A + 5*C) - 10*a*A*b*Sec[c + d*x] + 3*a^2*A*Sec[c + d*x]^2)*Tan[c + d*x]))/(60*a^4*d)","A",1
721,1,428,433,4.3857184,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)} \left(35 a^6 C+3 a^4 A b^2-57 a^4 b^2 C-21 a^2 A b^4+4 C \left(b^3-a^2 b\right)^2 \cos (2 (c+d x))+a b \left(49 a^4 C+a^2 b^2 (9 A-83 C)+b^4 (16 C-27 A)\right) \cos (c+d x)+4 b^6 C\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{\frac{2 \left(35 a^5 C+a^3 b^2 (3 A-73 C)+a b^4 (15 A+56 C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}-\frac{16 \left(-7 a^4 C+a^2 b^2 (3 A+14 C)+2 b^4 (3 A+C)\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{6 \left(35 a^4 C+a^2 b^2 (3 A-65 C)+3 b^4 (8 C-3 A)\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{b^2 \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{48 b^3 d}","-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{a \left(35 a^4 C+a^2 b^2 (3 A-65 C)-3 b^4 (3 A-8 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d \left(a^2-b^2\right)^2}+\frac{\left(-7 a^4 C+a^2 b^2 (A+13 C)+5 A b^4\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(35 a^4 C+a^2 b^2 (3 A-61 C)-b^4 (21 A-8 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{12 b^3 d \left(a^2-b^2\right)^2}+\frac{\left(105 a^6 C+a^4 b^2 (9 A-223 C)-a^2 b^4 (15 A-128 C)+8 b^6 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 b^5 d \left(a^2-b^2\right)^2}-\frac{a \left(35 a^6 C+a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+15 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^5 d (a-b)^2 (a+b)^3}",1,"((4*Sqrt[Cos[c + d*x]]*(3*a^4*A*b^2 - 21*a^2*A*b^4 + 35*a^6*C - 57*a^4*b^2*C + 4*b^6*C + a*b*(a^2*b^2*(9*A - 83*C) + 49*a^4*C + b^4*(-27*A + 16*C))*Cos[c + d*x] + 4*(-(a^2*b) + b^3)^2*C*Cos[2*(c + d*x)])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) - ((2*(a^3*b^2*(3*A - 73*C) + 35*a^5*C + a*b^4*(15*A + 56*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) - (16*(-7*a^4*C + 2*b^4*(3*A + C) + a^2*b^2*(3*A + 14*C))*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (6*(a^2*b^2*(3*A - 65*C) + 35*a^4*C + 3*b^4*(-3*A + 8*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(48*b^3*d)","A",1
722,1,368,345,4.6190503,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{\frac{8 a \left(C \left(a^2-4 b^2\right)-3 A b^2\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{\left(5 a^4 C+a^2 b^2 (5 A-7 C)+b^4 (A+8 C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{\left(15 a^4 C-a^2 b^2 (A+29 C)+b^4 (8 C-5 A)\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(-5 a^5 C+a^3 b^2 (3 A+11 C)+\left(-7 a^4 b C+a^2 b^3 (A+13 C)+5 A b^5\right) \cos (c+d x)+3 a A b^4\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}}{8 b^2 d}","-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{a \left(15 a^4 C-a^2 b^2 (A+33 C)+b^4 (7 A+24 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d \left(a^2-b^2\right)^2}+\frac{\left(-5 a^4 C+a^2 b^2 (3 A+11 C)+3 A b^4\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\left(-15 a^4 C+a^2 b^2 (A+29 C)+b^4 (5 A-8 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d \left(a^2-b^2\right)^2}+\frac{\left(15 a^6 C-a^4 b^2 (A+38 C)+5 a^2 b^4 (2 A+7 C)+3 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d (a-b)^2 (a+b)^3}",1,"((2*Sqrt[Cos[c + d*x]]*(3*a*A*b^4 - 5*a^5*C + a^3*b^2*(3*A + 11*C) + (5*A*b^5 - 7*a^4*b*C + a^2*b^3*(A + 13*C))*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) + (((a^2*b^2*(5*A - 7*C) + 5*a^4*C + b^4*(A + 8*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*a*(-3*A*b^2 + (a^2 - 4*b^2)*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + ((15*a^4*C + b^4*(-5*A + 8*C) - a^2*b^2*(A + 29*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(8*b^2*d)","A",1
723,1,364,348,3.6822917,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^5 C-7 a^3 b^2 (A+C)+b \left(3 a^4 C-a^2 b^2 (5 A+9 C)-A b^4\right) \cos (c+d x)+a A b^4\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{-\frac{8 \left(a^3 (2 A+C)+a b^2 (A+2 C)\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{\left(a^4 C+a^2 b^2 (9 A+5 C)-3 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{\left(3 a^4 C-a^2 b^2 (5 A+9 C)-A b^4\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{8 a b d}","-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\left(-3 a^4 C+a^2 b^2 (5 A+9 C)+A b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^2 d \left(a^2-b^2\right)^2}-\frac{\left(-3 a^4 C+a^2 b^2 (5 A+9 C)+A b^4\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 a b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(3 a^4 C+a^2 b^2 (3 A-5 C)+b^4 (3 A+8 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d \left(a^2-b^2\right)^2}+\frac{\left(-3 a^6 C-3 a^4 b^2 (A-2 C)-5 a^2 b^4 (2 A+3 C)+A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^3 d (a-b)^2 (a+b)^3}",1,"((2*Sqrt[Cos[c + d*x]]*(a*A*b^4 + a^5*C - 7*a^3*b^2*(A + C) + b*(-(A*b^4) + 3*a^4*C - a^2*b^2*(5*A + 9*C))*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) - (((-3*A*b^4 + a^4*C + a^2*b^2*(9*A + 5*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) - (8*(a^3*(2*A + C) + a*b^2*(A + 2*C))*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + ((-(A*b^4) + 3*a^4*C - a^2*b^2*(5*A + 9*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(8*a*b*d)","A",1
724,1,366,345,4.4305515,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3),x]","\frac{\frac{\frac{16 \left(a A b^2-a^3 (4 A+3 C)\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{2 \left(a^4 (16 A+5 C)+a^2 b^2 (C-19 A)+9 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}-\frac{2 \left(a^4 C+a^2 b^2 (9 A+5 C)-3 A b^4\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}+\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)} \left(3 a^5 C+a^3 b^2 (11 A+3 C)+b \left(a^4 C+a^2 b^2 (9 A+5 C)-3 A b^4\right) \cos (c+d x)-5 a A b^4\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}}{16 a^2 d}","\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\left(a^4 C-7 a^2 b^2 (A+C)+A b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^2 d \left(a^2-b^2\right)^2}+\frac{\left(a^4 (-C)-a^2 b^2 (9 A+5 C)+3 A b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b d \left(a^2-b^2\right)^2}-\frac{\left(a^4 (-C)-a^2 b^2 (9 A+5 C)+3 A b^4\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(a^6 (-C)+5 a^4 b^2 (3 A+2 C)-3 a^2 b^4 (2 A-C)+3 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b^2 d (a-b)^2 (a+b)^3}",1,"((4*Sqrt[Cos[c + d*x]]*(-5*a*A*b^4 + 3*a^5*C + a^3*b^2*(11*A + 3*C) + b*(-3*A*b^4 + a^4*C + a^2*b^2*(9*A + 5*C))*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) + ((2*(9*A*b^4 + a^2*b^2*(-19*A + C) + a^4*(16*A + 5*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (16*(a*A*b^2 - a^3*(4*A + 3*C))*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) - (2*(-3*A*b^4 + a^4*C + a^2*b^2*(9*A + 5*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(16*a^2*d)","A",1
725,1,425,417,5.2939875,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3),x]","\frac{\frac{\sqrt{\cos (c+d x)} \left(16 A \left(a^3-a b^2\right)^2 \tan (c+d x)+b^2 \left(a^4 (8 A-5 C)-a^2 b^2 (29 A+C)+15 A b^4\right) \sin (2 (c+d x))+2 a b \left(a^4 (16 A-7 C)+a^2 b^2 (C-47 A)+25 A b^4\right) \sin (c+d x)\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{\frac{8 \left(2 a^5 (A-C)-a^3 b^2 (10 A+C)+5 a A b^4\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{b (a+b)}+\frac{\left(a^4 b (56 A+9 C)-a^2 b^3 (95 A+3 C)+45 A b^5\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{\left(a^4 (8 A-5 C)-a^2 b^2 (29 A+C)+15 A b^4\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{8 a^3 d}","\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}-\frac{\left(-3 a^4 C-a^2 b^2 (11 A+3 C)+5 A b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b d \left(a^2-b^2\right)^2}-\frac{\left(-3 a^4 C-a^2 b^2 (11 A+3 C)+5 A b^4\right) \sin (c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}-\frac{\left(a^4 (8 A-5 C)-a^2 b^2 (29 A+C)+15 A b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}+\frac{\left(a^4 (8 A-5 C)-a^2 b^2 (29 A+C)+15 A b^4\right) \sin (c+d x)}{4 a^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}-\frac{\left(3 a^6 C+5 a^4 b^2 (7 A+2 C)-a^2 b^4 (38 A+C)+15 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 b d (a-b)^2 (a+b)^3}",1,"(-((((45*A*b^5 - a^2*b^3*(95*A + 3*C) + a^4*b*(56*A + 9*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(5*a*A*b^4 + 2*a^5*(A - C) - a^3*b^2*(10*A + C))*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(b*(a + b)) + ((15*A*b^4 + a^4*(8*A - 5*C) - a^2*b^2*(29*A + C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2)) + (Sqrt[Cos[c + d*x]]*(2*a*b*(25*A*b^4 + a^4*(16*A - 7*C) + a^2*b^2*(-47*A + C))*Sin[c + d*x] + b^2*(15*A*b^4 + a^4*(8*A - 5*C) - a^2*b^2*(29*A + C))*Sin[2*(c + d*x)] + 16*A*(a^3 - a*b^2)^2*Tan[c + d*x]))/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2))/(8*a^3*d)","A",1
726,1,543,494,7.3349796,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^3),x]","\frac{\sqrt{\cos (c+d x)} \left(-\frac{6 A b \tan (c+d x)}{a^4}+\frac{2 A \tan (c+d x) \sec (c+d x)}{3 a^3}+\frac{9 a^4 b^2 C \sin (c+d x)+17 a^2 A b^4 \sin (c+d x)-3 a^2 b^4 C \sin (c+d x)-11 A b^6 \sin (c+d x)}{4 a^4 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{a^2 b^2 C \sin (c+d x)+A b^4 \sin (c+d x)}{2 a^3 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}\right)}{d}+\frac{\frac{\left(160 a^5 A b-96 a^5 b C-512 a^3 A b^3+24 a^3 b^3 C+280 a A b^5\right) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{b}+\frac{2 \left(72 a^4 A b^2-27 a^4 b^2 C-195 a^2 A b^4+9 a^2 b^4 C+105 A b^6\right) \sin (c+d x) \cos (2 (c+d x)) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{1-\cos ^2(c+d x)} \left(2 \cos ^2(c+d x)-1\right)}+\frac{2 \left(16 a^6 A+48 a^6 C+328 a^4 A b^2-57 a^4 b^2 C-641 a^2 A b^4+27 a^2 b^4 C+315 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{48 a^4 d (a-b)^2 (a+b)^2}","\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac{b \left(3 a^4 (8 A-3 C)-a^2 b^2 (65 A-3 C)+35 A b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d \left(a^2-b^2\right)^2}-\frac{\left(-5 a^4 C-a^2 b^2 (13 A+C)+7 A b^4\right) \sin (c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{b \left(3 a^4 (8 A-3 C)-a^2 b^2 (65 A-3 C)+35 A b^4\right) \sin (c+d x)}{4 a^4 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}+\frac{\left(15 a^6 C+3 a^4 b^2 (21 A-2 C)-a^2 b^4 (86 A-3 C)+35 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}+\frac{\left(a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)+35 A b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 a^3 d \left(a^2-b^2\right)^2}+\frac{\left(a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)+35 A b^4\right) \sin (c+d x)}{12 a^3 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x)}",1,"((2*(16*a^6*A + 328*a^4*A*b^2 - 641*a^2*A*b^4 + 315*A*b^6 + 48*a^6*C - 57*a^4*b^2*C + 27*a^2*b^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + ((160*a^5*A*b - 512*a^3*A*b^3 + 280*a*A*b^5 - 96*a^5*b*C + 24*a^3*b^3*C)*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)))/b + (2*(72*a^4*A*b^2 - 195*a^2*A*b^4 + 105*A*b^6 - 27*a^4*b^2*C + 9*a^2*b^4*C)*Cos[2*(c + d*x)]*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[1 - Cos[c + d*x]^2]*(-1 + 2*Cos[c + d*x]^2)))/(48*a^4*(a - b)^2*(a + b)^2*d) + (Sqrt[Cos[c + d*x]]*((A*b^4*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x])/(2*a^3*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) + (17*a^2*A*b^4*Sin[c + d*x] - 11*A*b^6*Sin[c + d*x] + 9*a^4*b^2*C*Sin[c + d*x] - 3*a^2*b^4*C*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) - (6*A*b*Tan[c + d*x])/a^4 + (2*A*Sec[c + d*x]*Tan[c + d*x])/(3*a^3)))/d","A",1
727,1,1220,553,6.3319882,"\int \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","\frac{-\frac{4 a \left(-C a^2+24 A b^2+16 b^2 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a (48 a A b+28 a C b) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-3 C a^2+24 A b^2+16 b^2 C\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{48 b d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{a C \sin (c+d x)}{12 b}+\frac{1}{6} C \sin (2 (c+d x))\right)}{d}","-\frac{\left(3 a^2 C-8 b^2 (3 A+2 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 b^2 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(3 a^2 C-2 a b C-8 b^2 (3 A+2 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^2 d}+\frac{(a-b) \sqrt{a+b} \left(3 a^2 C-8 b^2 (3 A+2 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^2 d}-\frac{a \sqrt{a+b} \left(C \left(a^2+4 b^2\right)+8 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^3 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 b d}-\frac{a C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}",1,"((-4*a*(24*A*b^2 - a^2*C + 16*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(48*a*A*b + 28*a*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(24*A*b^2 - 3*a^2*C + 16*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(48*b*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((a*C*Sin[c + d*x])/(12*b) + (C*Sin[2*(c + d*x)])/6))/d","C",0
728,1,1169,455,8.8052867,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{C \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{2 d}+\frac{-\frac{4 a (8 a A+3 a C) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a (8 A b+4 C b) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 a C \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{8 d}","\frac{\sqrt{a+b} \left(a^2 C-4 b^2 (2 A+C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}+\frac{\sqrt{a+b} (C (a+2 b)+8 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}+\frac{a C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b d \sqrt{\cos (c+d x)}}-\frac{C (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}",1,"(C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + ((-4*a*(8*a*A + 3*a*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(8*A*b + 4*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*a*C*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(8*d)","C",1
729,1,1166,439,20.1122746,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{2 A \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{-\frac{4 a b C \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a (2 a C-2 a A) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 (b C-2 A b) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{2 d}","-\frac{(2 A-C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} (2 a A-a C-2 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}+\frac{(a-b) \sqrt{a+b} (2 A-C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{a C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d}",1,"(2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + ((-4*a*b*C*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-2*a*A + 2*a*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-2*A*b + b*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(2*d)","C",1
730,1,315,394,8.0856057,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","-\frac{2 \left(\sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right)} \left(-a (a (A+3 C)+b (A-3 C)) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))}{a+b}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+A b \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} (a+b \cos (c+d x))+A b (a+b) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-6 a b C \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))}{a+b}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)-\frac{A \sin (c+d x) (a+b \cos (c+d x))^2}{\cos ^{\frac{3}{2}}(c+d x)}\right)}{3 a d \sqrt{a+b \cos (c+d x)}}","\frac{2 A b (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d}-\frac{2 \sqrt{a+b} (A b-a (A+3 C)) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}",1,"(-2*(-((A*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/Cos[c + d*x]^(3/2)) + Sqrt[Cos[(c + d*x)/2]^2]*(A*b*(a + b)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - a*(b*(A - 3*C) + a*(A + 3*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - 6*a*b*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + A*b*(a + b*Cos[c + d*x])*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Tan[(c + d*x)/2])))/(3*a*d*Sqrt[a + b*Cos[c + d*x]])","A",0
731,1,1288,345,6.3567535,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{5} A \tan (c+d x) \sec ^2(c+d x)+\frac{2 \left(9 A \sin (c+d x) a^2+15 C \sin (c+d x) a^2-2 A b^2 \sin (c+d x)\right) \sec (c+d x)}{15 a^2}+\frac{2 A b \tan (c+d x) \sec (c+d x)}{15 a}\right)}{d}-\frac{-\frac{4 a \left(2 a^2 A b-2 A b^3\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(9 A a^3+15 C a^3-2 A b^2 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-2 A b^3+9 a^2 A b+15 a^2 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{15 a^2 d}","-\frac{2 (a-b) \sqrt{a+b} (9 a A+15 a C+2 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^2 d}-\frac{2 (a-b) \sqrt{a+b} \left(2 A b^2-3 a^2 (3 A+5 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d}+\frac{2 A b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"-1/15*((-4*a*(2*a^2*A*b - 2*A*b^3)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(9*a^3*A - 2*a*A*b^2 + 15*a^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(9*a^2*A*b - 2*A*b^3 + 15*a^2*b*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a^2*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]*(9*a^2*A*Sin[c + d*x] - 2*A*b^2*Sin[c + d*x] + 15*a^2*C*Sin[c + d*x]))/(15*a^2) + (2*A*b*Sec[c + d*x]*Tan[c + d*x])/(15*a) + (2*A*Sec[c + d*x]^2*Tan[c + d*x])/5))/d","C",1
732,1,1373,415,6.4450234,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{-\frac{4 a \left(25 A a^4+35 C a^4-17 A b^2 a^2-35 b^2 C a^2-8 A b^4\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-19 A b a^3-35 b C a^3-8 A b^3 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-8 A b^4-19 a^2 A b^2-35 a^2 C b^2\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{105 a^3 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{7} A \tan (c+d x) \sec ^3(c+d x)+\frac{2 \left(25 A \sin (c+d x) a^2+35 C \sin (c+d x) a^2-4 A b^2 \sin (c+d x)\right) \sec ^2(c+d x)}{105 a^2}+\frac{2 A b \tan (c+d x) \sec ^2(c+d x)}{35 a}+\frac{2 \left(8 A \sin (c+d x) b^3+19 a^2 A \sin (c+d x) b+35 a^2 C \sin (c+d x) b\right) \sec (c+d x)}{105 a^3}\right)}{d}","-\frac{2 \left(4 A b^2-5 a^2 (5 A+7 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b (a-b) \sqrt{a+b} \left(a^2 (19 A+35 C)+8 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d}+\frac{2 (a-b) \sqrt{a+b} \left(5 a^2 (5 A+7 C)+6 a A b+8 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d}+\frac{2 A b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"((-4*a*(25*a^4*A - 17*a^2*A*b^2 - 8*A*b^4 + 35*a^4*C - 35*a^2*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-19*a^3*A*b - 8*a*A*b^3 - 35*a^3*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-19*a^2*A*b^2 - 8*A*b^4 - 35*a^2*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(105*a^3*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^2*(25*a^2*A*Sin[c + d*x] - 4*A*b^2*Sin[c + d*x] + 35*a^2*C*Sin[c + d*x]))/(105*a^2) + (2*Sec[c + d*x]*(19*a^2*A*b*Sin[c + d*x] + 8*A*b^3*Sin[c + d*x] + 35*a^2*b*C*Sin[c + d*x]))/(105*a^3) + (2*A*b*Sec[c + d*x]^2*Tan[c + d*x])/(35*a) + (2*A*Sec[c + d*x]^3*Tan[c + d*x])/7))/d","C",0
733,1,1270,638,6.3823193,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{\left(C a^2+16 A b^2+14 b^2 C\right) \sin (c+d x)}{32 b}+\frac{3}{16} a C \sin (2 (c+d x))+\frac{1}{16} b C \sin (3 (c+d x))\right)}{d}-\frac{-\frac{4 a \left(C a^3-112 A b^2 a-76 b^2 C a\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-64 A b^3-48 C b^3-128 a^2 A b-76 a^2 C b\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(3 C a^3-80 A b^2 a-52 b^2 C a\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{128 b d}","-\frac{\left(3 a^2 C-4 b^2 (4 A+3 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{32 b d}+\frac{a \left(-3 a^2 C+80 A b^2+52 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{64 b^2 d \sqrt{\cos (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(-3 a^2 C+80 A b^2+52 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d}-\frac{\sqrt{a+b} \left(3 a^4 C+24 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^3 d}-\frac{\sqrt{a+b} \left(3 a^3 C-2 a^2 b C-4 a b^2 (20 A+13 C)-8 b^3 (4 A+3 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}-\frac{a C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{8 b d}",1,"-1/128*((-4*a*(-112*a*A*b^2 + a^3*C - 76*a*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-128*a^2*A*b - 64*A*b^3 - 76*a^2*b*C - 48*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-80*a*A*b^2 + 3*a^3*C - 52*a*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(b*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(((16*A*b^2 + a^2*C + 14*b^2*C)*Sin[c + d*x])/(32*b) + (3*a*C*Sin[2*(c + d*x)])/16 + (b*C*Sin[3*(c + d*x)])/16))/d","C",0
734,1,1221,553,6.418807,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{-\frac{4 a \left(48 A a^2+17 C a^2+24 A b^2+16 b^2 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a (96 a A b+52 a C b) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(3 C a^2+24 A b^2+16 b^2 C\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{48 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{7}{12} a C \sin (c+d x)+\frac{1}{6} b C \sin (2 (c+d x))\right)}{d}","\frac{\left(3 a^2 C+8 b^2 (3 A+2 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(3 a^2 C+48 a A b+14 a b C+24 A b^2+16 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b d}-\frac{(a-b) \sqrt{a+b} \left(3 a^2 C+8 b^2 (3 A+2 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b d}-\frac{a \sqrt{a+b} \left(a^2 (-C)+24 A b^2+12 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^2 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac{a C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}",1,"((-4*a*(48*a^2*A + 24*A*b^2 + 17*a^2*C + 16*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(96*a*A*b + 52*a*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(24*A*b^2 + 3*a^2*C + 16*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(48*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((7*a*C*Sin[c + d*x])/12 + (b*C*Sin[2*(c + d*x)])/6))/d","C",0
735,1,1209,509,6.4366125,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{\frac{4 a (-8 a A b-7 a C b) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+4 a \left(8 A a^2-8 C a^2-8 A b^2-4 b^2 C\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)-2 (8 a A b-5 a b C) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{8 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{1}{2} b C \sin (c+d x)+2 a A \tan (c+d x)\right)}{d}","-\frac{\sqrt{a+b} \left(3 a^2 C+8 A b^2+4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}-\frac{b (4 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}-\frac{a (8 A-5 C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} (8 a A-5 a C-16 A b-2 b C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}+\frac{(a-b) \sqrt{a+b} (8 A-5 C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt{\cos (c+d x)}}",1,"((4*a*(-8*a*A*b - 7*a*b*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) + 4*a*(8*a^2*A - 8*A*b^2 - 8*a^2*C - 4*b^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) - 2*(8*a*A*b - 5*a*b*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(8*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((b*C*Sin[c + d*x])/2 + 2*a*A*Tan[c + d*x]))/d","C",1
736,1,1219,500,6.4132577,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{-\frac{4 a \left(2 A a^2+6 C a^2-2 A b^2+3 b^2 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a (12 a b C-8 a A b) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(3 b^2 C-8 A b^2\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{6 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{8}{3} A b \tan (c+d x)+\frac{2}{3} a A \sec (c+d x) \tan (c+d x)\right)}{d}","\frac{\sqrt{a+b} \left(2 a^2 (A+3 C)-a (8 A b-3 b C)+6 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}-\frac{b (8 A-3 C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{b (a-b) \sqrt{a+b} (8 A-3 C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{3 a C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}",1,"((-4*a*(2*a^2*A - 2*A*b^2 + 6*a^2*C + 3*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-8*a*A*b + 12*a*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-8*A*b^2 + 3*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(6*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((8*A*b*Tan[c + d*x])/3 + (2*a*A*Sec[c + d*x]*Tan[c + d*x])/3))/d","C",1
737,1,1296,465,6.5045198,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{5} a A \tan (c+d x) \sec ^2(c+d x)+\frac{2 \left(3 A \sin (c+d x) a^2+5 C \sin (c+d x) a^2+A b^2 \sin (c+d x)\right) \sec (c+d x)}{5 a}+\frac{4}{5} A b \tan (c+d x) \sec (c+d x)\right)}{d}-\frac{-\frac{4 a \left(A b^3-a^2 A b-5 a^2 C b\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(3 A a^3+5 C a^3+A b^2 a-5 b^2 C a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(A b^3+3 a^2 A b+5 a^2 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{5 a d}","-\frac{2 \sqrt{a+b} \left(a^2 (3 A+5 C)-2 a b (2 A+5 C)+A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a d}+\frac{2 (a-b) \sqrt{a+b} \left(a^2 (3 A+5 C)+A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^2 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}",1,"-1/5*((-4*a*(-(a^2*A*b) + A*b^3 - 5*a^2*b*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(3*a^3*A + a*A*b^2 + 5*a^3*C - 5*a*b^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(3*a^2*A*b + A*b^3 + 5*a^2*b*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]*(3*a^2*A*Sin[c + d*x] + A*b^2*Sin[c + d*x] + 5*a^2*C*Sin[c + d*x]))/(5*a) + (4*A*b*Sec[c + d*x]*Tan[c + d*x])/5 + (2*a*A*Sec[c + d*x]^2*Tan[c + d*x])/5))/d","C",0
738,1,1371,418,6.5303649,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{-\frac{4 a \left(25 A a^4+35 C a^4-31 A b^2 a^2-35 b^2 C a^2+6 A b^4\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-82 A b a^3-140 b C a^3+6 A b^3 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(6 A b^4-82 a^2 A b^2-140 a^2 C b^2\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{105 a^2 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{7} a A \tan (c+d x) \sec ^3(c+d x)+\frac{2 \left(25 A \sin (c+d x) a^2+35 C \sin (c+d x) a^2+3 A b^2 \sin (c+d x)\right) \sec ^2(c+d x)}{105 a}+\frac{16}{35} A b \tan (c+d x) \sec ^2(c+d x)+\frac{4 \left(-3 A \sin (c+d x) b^3+41 a^2 A \sin (c+d x) b+70 a^2 C \sin (c+d x) b\right) \sec (c+d x)}{105 a^2}\right)}{d}","\frac{2 \left(5 a^2 (5 A+7 C)+3 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(25 a^2 A+35 a^2 C-57 a A b-105 a b C-6 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d}-\frac{4 b (a-b) \sqrt{a+b} \left(3 A b^2-a^2 (41 A+70 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{6 A b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 d \cos ^{\frac{5}{2}}(c+d x)}",1,"((-4*a*(25*a^4*A - 31*a^2*A*b^2 + 6*A*b^4 + 35*a^4*C - 35*a^2*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-82*a^3*A*b + 6*a*A*b^3 - 140*a^3*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-82*a^2*A*b^2 + 6*A*b^4 - 140*a^2*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(105*a^2*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^2*(25*a^2*A*Sin[c + d*x] + 3*A*b^2*Sin[c + d*x] + 35*a^2*C*Sin[c + d*x]))/(105*a) + (4*Sec[c + d*x]*(41*a^2*A*b*Sin[c + d*x] - 3*A*b^3*Sin[c + d*x] + 70*a^2*b*C*Sin[c + d*x]))/(105*a^2) + (16*A*b*Sec[c + d*x]^2*Tan[c + d*x])/35 + (2*a*A*Sec[c + d*x]^3*Tan[c + d*x])/7))/d","C",1
739,1,1485,502,6.7014133,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{9} a A \tan (c+d x) \sec ^4(c+d x)+\frac{2 \left(49 A \sin (c+d x) a^2+63 C \sin (c+d x) a^2+3 A b^2 \sin (c+d x)\right) \sec ^3(c+d x)}{315 a}+\frac{20}{63} A b \tan (c+d x) \sec ^3(c+d x)+\frac{4 \left(-2 A \sin (c+d x) b^3+44 a^2 A \sin (c+d x) b+63 a^2 C \sin (c+d x) b\right) \sec ^2(c+d x)}{315 a^2}+\frac{2 \left(147 A \sin (c+d x) a^4+189 C \sin (c+d x) a^4+33 A b^2 \sin (c+d x) a^2+63 b^2 C \sin (c+d x) a^2+8 A b^4 \sin (c+d x)\right) \sec (c+d x)}{315 a^3}\right)}{d}-\frac{-\frac{4 a \left(8 A b^5+31 a^2 A b^3+63 a^2 C b^3-39 a^4 A b-63 a^4 C b\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(147 A a^5+189 C a^5+33 A b^2 a^3+63 b^2 C a^3+8 A b^4 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(8 A b^5+33 a^2 A b^3+63 a^2 C b^3+147 a^4 A b+189 a^4 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{315 a^3 d}","-\frac{4 b \left(2 A b^2-a^2 (44 A+63 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(7 a^2 (7 A+9 C)+3 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(21 a^4 (7 A+9 C)+3 a^2 b^2 (11 A+21 C)+8 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^4 d}+\frac{2 (a-b) \sqrt{a+b} \left(-21 a^3 (7 A+9 C)+a^2 (39 A b+63 b C)+6 a A b^2+8 A b^3\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{21 d \cos ^{\frac{7}{2}}(c+d x)}",1,"-1/315*((-4*a*(-39*a^4*A*b + 31*a^2*A*b^3 + 8*A*b^5 - 63*a^4*b*C + 63*a^2*b^3*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(147*a^5*A + 33*a^3*A*b^2 + 8*a*A*b^4 + 189*a^5*C + 63*a^3*b^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(147*a^4*A*b + 33*a^2*A*b^3 + 8*A*b^5 + 189*a^4*b*C + 63*a^2*b^3*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a^3*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^3*(49*a^2*A*Sin[c + d*x] + 3*A*b^2*Sin[c + d*x] + 63*a^2*C*Sin[c + d*x]))/(315*a) + (4*Sec[c + d*x]^2*(44*a^2*A*b*Sin[c + d*x] - 2*A*b^3*Sin[c + d*x] + 63*a^2*b*C*Sin[c + d*x]))/(315*a^2) + (2*Sec[c + d*x]*(147*a^4*A*Sin[c + d*x] + 33*a^2*A*b^2*Sin[c + d*x] + 8*A*b^4*Sin[c + d*x] + 189*a^4*C*Sin[c + d*x] + 63*a^2*b^2*C*Sin[c + d*x]))/(315*a^3) + (20*A*b*Sec[c + d*x]^3*Tan[c + d*x])/63 + (2*a*A*Sec[c + d*x]^4*Tan[c + d*x])/9))/d","C",0
740,1,1341,746,6.5445638,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{1}{40} C \sin (4 (c+d x)) b^2+\frac{21}{160} a C \sin (3 (c+d x)) b+\frac{1}{480} \left(93 C a^2+80 A b^2+88 b^2 C\right) \sin (2 (c+d x))+\frac{a \left(15 C a^2+1040 A b^2+898 b^2 C\right) \sin (c+d x)}{960 b}\right)}{d}-\frac{-\frac{4 a \left(15 C a^4-4720 A b^2 a^2-3236 b^2 C a^2-1280 A b^4-1024 b^4 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-3840 A b a^3-2292 b C a^3-6080 A b^3 a-4624 b^3 C a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(45 C a^4-2640 A b^2 a^2-1692 b^2 C a^2-1280 A b^4-1024 b^4 C\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{3840 b d}","-\frac{\left(15 a^2 C-16 b^2 (5 A+4 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{240 b d}+\frac{a \left(-15 a^2 C+240 A b^2+172 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{320 b d}-\frac{\left(45 a^4 C-12 a^2 b^2 (220 A+141 C)-256 b^4 (5 A+4 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d \sqrt{\cos (c+d x)}}+\frac{(a-b) \sqrt{a+b} \left(45 a^4 C-12 a^2 b^2 (220 A+141 C)-256 b^4 (5 A+4 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{1920 a b^2 d}-\frac{a \sqrt{a+b} \left(3 a^4 C+40 a^2 b^2 (2 A+C)+80 b^4 (4 A+3 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{128 b^3 d}-\frac{\sqrt{a+b} \left(45 a^4 C-30 a^3 b C-12 a^2 b^2 (220 A+141 C)-8 a b^3 (260 A+193 C)-256 b^4 (5 A+4 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{1920 b^2 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}-\frac{3 a C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{40 b d}",1,"-1/3840*((-4*a*(-4720*a^2*A*b^2 - 1280*A*b^4 + 15*a^4*C - 3236*a^2*b^2*C - 1024*b^4*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-3840*a^3*A*b - 6080*a*A*b^3 - 2292*a^3*b*C - 4624*a*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-2640*a^2*A*b^2 - 1280*A*b^4 + 45*a^4*C - 1692*a^2*b^2*C - 1024*b^4*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(b*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((a*(1040*A*b^2 + 15*a^2*C + 898*b^2*C)*Sin[c + d*x])/(960*b) + ((80*A*b^2 + 93*a^2*C + 88*b^2*C)*Sin[2*(c + d*x)])/480 + (21*a*b*C*Sin[3*(c + d*x)])/160 + (b^2*C*Sin[4*(c + d*x)])/40))/d","C",0
741,1,1275,635,6.5924395,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{-\frac{4 a \left(384 A a^3+133 C a^3+528 A b^2 a+356 b^2 C a\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(192 A b^3+144 C b^3+1152 a^2 A b+644 a^2 C b\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(15 C a^3+432 A b^2 a+284 b^2 C a\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{384 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{1}{16} C \sin (3 (c+d x)) b^2+\frac{17}{48} a C \sin (2 (c+d x)) b+\frac{1}{96} \left(59 C a^2+48 A b^2+42 b^2 C\right) \sin (c+d x)\right)}{d}","\frac{\left(5 a^2 C+4 b^2 (4 A+3 C)\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{32 d}+\frac{a \left(15 a^2 C+432 A b^2+284 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{192 b d \sqrt{\cos (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(15 a^2 C+432 A b^2+284 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b d}+\frac{\sqrt{a+b} \left(5 a^4 C-120 a^2 b^2 (2 A+C)-16 b^4 (4 A+3 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d}+\frac{\sqrt{a+b} \left(15 a^3 C+2 a^2 b (192 A+59 C)+4 a b^2 (108 A+71 C)+24 b^3 (4 A+3 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}+\frac{5 a C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{24 d}",1,"((-4*a*(384*a^3*A + 528*a*A*b^2 + 133*a^3*C + 356*a*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(1152*a^2*A*b + 192*A*b^3 + 644*a^2*b*C + 144*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(432*a*A*b^2 + 15*a^3*C + 284*a*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(384*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(((48*A*b^2 + 59*a^2*C + 42*b^2*C)*Sin[c + d*x])/96 + (17*a*b*C*Sin[2*(c + d*x)])/48 + (b^2*C*Sin[3*(c + d*x)])/16))/d","C",0
742,1,1262,609,6.656708,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{\frac{4 a \left(-24 A b^3-16 C b^3-96 a^2 A b-59 a^2 C b\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+4 a \left(48 A a^3-48 C a^3-144 A b^2 a-76 b^2 C a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)-2 \left(-24 A b^3-16 C b^3+48 a^2 A b-33 a^2 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{48 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(2 A \tan (c+d x) a^2+\frac{13}{12} b C \sin (c+d x) a+\frac{1}{6} b^2 C \sin (2 (c+d x))\right)}{d}","-\frac{\left(a^2 (48 A-33 C)-8 b^2 (3 A+2 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(a^2 (48 A-33 C)-2 a b (72 A+13 C)-8 b^2 (3 A+2 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 d}+\frac{(a-b) \sqrt{a+b} \left(a^2 (48 A-33 C)-8 b^2 (3 A+2 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a d}-\frac{5 a \sqrt{a+b} \left(C \left(a^2+4 b^2\right)+8 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b d}-\frac{b (6 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}-\frac{a b (8 A-3 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt{\cos (c+d x)}}",1,"((4*a*(-96*a^2*A*b - 24*A*b^3 - 59*a^2*b*C - 16*b^3*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) + 4*a*(48*a^3*A - 144*a*A*b^2 - 48*a^3*C - 76*a*b^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) - 2*(48*a^2*A*b - 24*A*b^3 - 33*a^2*b*C - 16*b^3*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(48*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((13*a*b*C*Sin[c + d*x])/12 + (b^2*C*Sin[2*(c + d*x)])/6 + 2*a^2*A*Tan[c + d*x]))/d","C",0
743,1,1256,567,6.570098,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{-\frac{4 a \left(8 A a^3+24 C a^3+16 A b^2 a+33 b^2 C a\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(24 A b^3+12 C b^3-56 a^2 A b+72 a^2 C b\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(27 a b^2 C-56 a A b^2\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{24 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{3} A \sec (c+d x) \tan (c+d x) a^2+\frac{14}{3} A b \tan (c+d x) a+\frac{1}{2} b^2 C \sin (c+d x)\right)}{d}","\frac{\sqrt{a+b} \left(8 a^2 (A+3 C)-a (56 A b-27 b C)+6 b^2 (12 A+C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{12 d}-\frac{\sqrt{a+b} \left(15 a^2 C+8 A b^2+4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}-\frac{b^2 (8 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}-\frac{a b (56 A-27 C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{12 d \sqrt{\cos (c+d x)}}+\frac{b (a-b) \sqrt{a+b} (56 A-27 C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{12 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \sqrt{\cos (c+d x)}}",1,"((-4*a*(8*a^3*A + 16*a*A*b^2 + 24*a^3*C + 33*a*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-56*a^2*A*b + 24*A*b^3 + 72*a^2*b*C + 12*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-56*a*A*b^2 + 27*a*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(24*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((b^2*C*Sin[c + d*x])/2 + (14*a*A*b*Tan[c + d*x])/3 + (2*a^2*A*Sec[c + d*x]*Tan[c + d*x])/3))/d","C",1
744,1,1309,606,6.6102827,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{\frac{4 a \left(16 A b^3-15 C b^3-16 a^2 A b-60 a^2 C b\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+4 a \left(18 A a^3+30 C a^3+46 A b^2 a-90 b^2 C a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)-2 \left(46 A b^3-15 C b^3+18 a^2 A b+30 a^2 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{30 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{5} a^2 A \tan (c+d x) \sec ^2(c+d x)+\frac{2}{15} \left(9 A \sin (c+d x) a^2+15 C \sin (c+d x) a^2+23 A b^2 \sin (c+d x)\right) \sec (c+d x)+\frac{22}{15} a A b \tan (c+d x) \sec (c+d x)\right)}{d}","\frac{2 \left(a^2 (3 A+5 C)+5 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}-\frac{\left(6 a^2 (3 A+5 C)+b^2 (46 A-15 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}+\frac{(a-b) \sqrt{a+b} \left(6 a^2 (3 A+5 C)+b^2 (46 A-15 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d}+\frac{\sqrt{a+b} \left(-6 a^3 (3 A+5 C)+a^2 (34 A b+90 b C)-a b^2 (46 A-15 C)+30 A b^3\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{5 a b C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}",1,"((4*a*(-16*a^2*A*b + 16*A*b^3 - 60*a^2*b*C - 15*b^3*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) + 4*a*(18*a^3*A + 46*a*A*b^2 + 30*a^3*C - 90*a*b^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) - 2*(18*a^2*A*b + 46*A*b^3 + 30*a^2*b*C - 15*b^3*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(30*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]*(9*a^2*A*Sin[c + d*x] + 23*A*b^2*Sin[c + d*x] + 15*a^2*C*Sin[c + d*x]))/15 + (22*a*A*b*Sec[c + d*x]*Tan[c + d*x])/15 + (2*a^2*A*Sec[c + d*x]^2*Tan[c + d*x])/5))/d","C",0
745,1,1378,540,6.6683028,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{-\frac{4 a \left(5 A a^4+7 C a^4-2 A b^2 a^2+14 b^2 C a^2-3 A b^4\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-29 A b a^3-49 b C a^3-3 A b^3 a+21 b^3 C a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-3 A b^4-29 a^2 A b^2-49 a^2 C b^2\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{21 a d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{7} a^2 A \tan (c+d x) \sec ^3(c+d x)+\frac{2}{21} \left(5 A \sin (c+d x) a^2+7 C \sin (c+d x) a^2+9 A b^2 \sin (c+d x)\right) \sec ^2(c+d x)+\frac{6}{7} a A b \tan (c+d x) \sec ^2(c+d x)+\frac{2 \left(3 A \sin (c+d x) b^3+29 a^2 A \sin (c+d x) b+49 a^2 C \sin (c+d x) b\right) \sec (c+d x)}{21 a}\right)}{d}","\frac{2 \left(a^2 (5 A+7 C)+3 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b (a-b) \sqrt{a+b} \left(a^2 (29 A+49 C)+3 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{21 a^2 d}-\frac{2 \sqrt{a+b} \left(-\left(a^3 (5 A+7 C)\right)+a^2 b (29 A+49 C)-9 a b^2 (3 A+7 C)+3 A b^3\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{21 a d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{7 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 b^2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}",1,"((-4*a*(5*a^4*A - 2*a^2*A*b^2 - 3*A*b^4 + 7*a^4*C + 14*a^2*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-29*a^3*A*b - 3*a*A*b^3 - 49*a^3*b*C + 21*a*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-29*a^2*A*b^2 - 3*A*b^4 - 49*a^2*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(21*a*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^2*(5*a^2*A*Sin[c + d*x] + 9*A*b^2*Sin[c + d*x] + 7*a^2*C*Sin[c + d*x]))/21 + (2*Sec[c + d*x]*(29*a^2*A*b*Sin[c + d*x] + 3*A*b^3*Sin[c + d*x] + 49*a^2*b*C*Sin[c + d*x]))/(21*a) + (6*a*A*b*Sec[c + d*x]^2*Tan[c + d*x])/7 + (2*a^2*A*Sec[c + d*x]^3*Tan[c + d*x])/7))/d","C",0
746,1,1485,504,6.7687375,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{9} a^2 A \tan (c+d x) \sec ^4(c+d x)+\frac{2}{315} \left(49 A \sin (c+d x) a^2+63 C \sin (c+d x) a^2+75 A b^2 \sin (c+d x)\right) \sec ^3(c+d x)+\frac{38}{63} a A b \tan (c+d x) \sec ^3(c+d x)+\frac{2 \left(5 A \sin (c+d x) b^3+163 a^2 A \sin (c+d x) b+231 a^2 C \sin (c+d x) b\right) \sec ^2(c+d x)}{315 a}+\frac{2 \left(147 A \sin (c+d x) a^4+189 C \sin (c+d x) a^4+279 A b^2 \sin (c+d x) a^2+483 b^2 C \sin (c+d x) a^2-10 A b^4 \sin (c+d x)\right) \sec (c+d x)}{315 a^2}\right)}{d}-\frac{-\frac{4 a \left(-10 A b^5+124 a^2 A b^3+168 a^2 C b^3-114 a^4 A b-168 a^4 C b\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(147 A a^5+189 C a^5+279 A b^2 a^3+483 b^2 C a^3-10 A b^4 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-10 A b^5+279 a^2 A b^3+483 a^2 C b^3+147 a^4 A b+189 a^4 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{315 a^2 d}","\frac{2 b \left(a^2 (163 A+231 C)+5 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(7 a^2 (7 A+9 C)+15 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 (a-b) \sqrt{a+b} \left(21 a^3 (7 A+9 C)-6 a^2 b (19 A+28 C)+15 a b^2 (11 A+21 C)+10 A b^3\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^2 d}-\frac{2 (a-b) \sqrt{a+b} \left(-21 a^4 (7 A+9 C)-3 a^2 b^2 (93 A+161 C)+10 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{63 d \cos ^{\frac{7}{2}}(c+d x)}",1,"-1/315*((-4*a*(-114*a^4*A*b + 124*a^2*A*b^3 - 10*A*b^5 - 168*a^4*b*C + 168*a^2*b^3*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(147*a^5*A + 279*a^3*A*b^2 - 10*a*A*b^4 + 189*a^5*C + 483*a^3*b^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(147*a^4*A*b + 279*a^2*A*b^3 - 10*A*b^5 + 189*a^4*b*C + 483*a^2*b^3*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a^2*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^3*(49*a^2*A*Sin[c + d*x] + 75*A*b^2*Sin[c + d*x] + 63*a^2*C*Sin[c + d*x]))/315 + (2*Sec[c + d*x]^2*(163*a^2*A*b*Sin[c + d*x] + 5*A*b^3*Sin[c + d*x] + 231*a^2*b*C*Sin[c + d*x]))/(315*a) + (2*Sec[c + d*x]*(147*a^4*A*Sin[c + d*x] + 279*a^2*A*b^2*Sin[c + d*x] - 10*A*b^4*Sin[c + d*x] + 189*a^4*C*Sin[c + d*x] + 483*a^2*b^2*C*Sin[c + d*x]))/(315*a^2) + (38*a*A*b*Sec[c + d*x]^3*Tan[c + d*x])/63 + (2*a^2*A*Sec[c + d*x]^4*Tan[c + d*x])/9))/d","C",0
747,1,1591,587,6.9359916,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","\frac{-\frac{4 a \left(135 A a^6+165 C a^6-78 A b^2 a^4-66 b^2 C a^4-49 A b^4 a^2-99 b^4 C a^2-8 A b^6\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-741 A b a^5-957 b C a^5-51 A b^3 a^3-99 b^3 C a^3-8 A b^5 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-8 A b^6-51 a^2 A b^4-99 a^2 C b^4-741 a^4 A b^2-957 a^4 C b^2\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{693 a^3 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{11} a^2 A \tan (c+d x) \sec ^5(c+d x)+\frac{2}{693} \left(81 A \sin (c+d x) a^2+99 C \sin (c+d x) a^2+113 A b^2 \sin (c+d x)\right) \sec ^4(c+d x)+\frac{46}{99} a A b \tan (c+d x) \sec ^4(c+d x)+\frac{2 \left(3 A \sin (c+d x) b^3+229 a^2 A \sin (c+d x) b+297 a^2 C \sin (c+d x) b\right) \sec ^3(c+d x)}{693 a}+\frac{2 \left(135 A \sin (c+d x) a^4+165 C \sin (c+d x) a^4+205 A b^2 \sin (c+d x) a^2+297 b^2 C \sin (c+d x) a^2-4 A b^4 \sin (c+d x)\right) \sec ^2(c+d x)}{693 a^2}+\frac{2 \left(8 A \sin (c+d x) b^5+51 a^2 A \sin (c+d x) b^3+99 a^2 C \sin (c+d x) b^3+741 a^4 A \sin (c+d x) b+957 a^4 C \sin (c+d x) b\right) \sec (c+d x)}{693 a^3}\right)}{d}","\frac{2 b \left(a^2 (229 A+297 C)+3 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{693 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(3 a^2 (9 A+11 C)+5 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{231 d \cos ^{\frac{7}{2}}(c+d x)}-\frac{2 \left(-15 a^4 (9 A+11 C)-a^2 b^2 (205 A+297 C)+4 A b^4\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{693 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b (a-b) \sqrt{a+b} \left(a^4 (741 A+957 C)+3 a^2 b^2 (17 A+33 C)+8 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{693 a^4 d}+\frac{2 (a-b) \sqrt{a+b} \left(15 a^4 (9 A+11 C)-6 a^3 b (101 A+132 C)+3 a^2 b^2 (19 A+33 C)+6 a A b^3+8 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{693 a^3 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{10 A b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{99 d \cos ^{\frac{9}{2}}(c+d x)}",1,"((-4*a*(135*a^6*A - 78*a^4*A*b^2 - 49*a^2*A*b^4 - 8*A*b^6 + 165*a^6*C - 66*a^4*b^2*C - 99*a^2*b^4*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-741*a^5*A*b - 51*a^3*A*b^3 - 8*a*A*b^5 - 957*a^5*b*C - 99*a^3*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-741*a^4*A*b^2 - 51*a^2*A*b^4 - 8*A*b^6 - 957*a^4*b^2*C - 99*a^2*b^4*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(693*a^3*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^4*(81*a^2*A*Sin[c + d*x] + 113*A*b^2*Sin[c + d*x] + 99*a^2*C*Sin[c + d*x]))/693 + (2*Sec[c + d*x]^3*(229*a^2*A*b*Sin[c + d*x] + 3*A*b^3*Sin[c + d*x] + 297*a^2*b*C*Sin[c + d*x]))/(693*a) + (2*Sec[c + d*x]^2*(135*a^4*A*Sin[c + d*x] + 205*a^2*A*b^2*Sin[c + d*x] - 4*A*b^4*Sin[c + d*x] + 165*a^4*C*Sin[c + d*x] + 297*a^2*b^2*C*Sin[c + d*x]))/(693*a^2) + (2*Sec[c + d*x]*(741*a^4*A*b*Sin[c + d*x] + 51*a^2*A*b^3*Sin[c + d*x] + 8*A*b^5*Sin[c + d*x] + 957*a^4*b*C*Sin[c + d*x] + 99*a^2*b^3*C*Sin[c + d*x]))/(693*a^3) + (46*a*A*b*Sec[c + d*x]^4*Tan[c + d*x])/99 + (2*a^2*A*Sec[c + d*x]^5*Tan[c + d*x])/11))/d","C",0
748,1,1216,554,13.4160285,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{-\frac{4 a \left(5 C a^2+24 A b^2+16 b^2 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-16 a^2 b C \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(15 C a^2+24 A b^2+16 b^2 C\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{48 b^2 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{C \sin (2 (c+d x))}{6 b}-\frac{5 a C \sin (c+d x)}{12 b^2}\right)}{d}","\frac{a \sqrt{a+b} \left(5 a^2 C+8 A b^2+4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^4 d}+\frac{\left(15 a^2 C+8 b^2 (3 A+2 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 b^3 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(15 a^2 C-10 a b C+8 b^2 (3 A+2 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^3 d}-\frac{(a-b) \sqrt{a+b} \left(15 a^2 C+8 b^2 (3 A+2 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^3 d}-\frac{5 a C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{12 b^2 d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d}",1,"((-4*a*(24*A*b^2 + 5*a^2*C + 16*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 16*a^2*b*C*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(24*A*b^2 + 15*a^2*C + 16*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(48*b^2*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((-5*a*C*Sin[c + d*x])/(12*b^2) + (C*Sin[2*(c + d*x)])/(6*b)))/d","C",0
749,1,1169,455,11.94732,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{C \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{2 b d}-\frac{-\frac{4 a^2 C \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a (-8 A b-4 C b) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+6 a C \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{8 b d}","-\frac{\sqrt{a+b} \left(3 a^2 C+4 b^2 (2 A+C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d}-\frac{3 a C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b^2 d \sqrt{\cos (c+d x)}}-\frac{C (3 a-2 b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}+\frac{3 C (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 b d}",1,"(C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b*d) - ((-4*a^2*C*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-8*A*b - 4*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 6*a*C*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(8*b*d)","C",1
750,1,340,393,13.9841739,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{8 \cos \left(\frac{1}{2} (c+d x)\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} \sqrt{\cos (c+d x)} \left(2 A b \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+C \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} (a+b \cos (c+d x))+C (a+b) \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-2 a C \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{b d (\cos (c+d x)+1)^{5/2} \sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \cos (c+d x)}}","\frac{\sqrt{a+b} (a C+2 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d}+\frac{a C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}-\frac{C (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d}",1,"(8*Cos[(c + d*x)/2]*(Cos[(c + d*x)/2]^2)^(3/2)*Sqrt[Cos[c + d*x]]*((a + b)*C*Cos[(c + d*x)/2]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*A*b*Cos[(c + d*x)/2]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 2*a*C*Cos[(c + d*x)/2]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*(a + b*Cos[c + d*x])*Sin[(c + d*x)/2]))/(b*d*(1 + Cos[c + d*x])^(5/2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2])","A",0
751,1,351,343,12.7446502,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{\cos (c+d x) \left(\frac{2 a (A-C) \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)}{\sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}}}-\frac{2 A (a+b) \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)}{\sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}}}-2 a A \tan \left(\frac{1}{2} (c+d x)\right)+\frac{4 a C \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)}{\sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}}}+A b \tan \left(\frac{1}{2} (c+d x)\right)-A b \sin \left(\frac{3}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right)\right)+2 A \sin (c+d x) (a+b \cos (c+d x))}{a d \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}","\frac{2 A (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d}-\frac{2 A \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d}",1,"(2*A*(a + b*Cos[c + d*x])*Sin[c + d*x] + Cos[c + d*x]*((-2*A*(a + b)*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (2*a*(A - C)*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (4*a*C*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - A*b*Sec[(c + d*x)/2]*Sin[(3*(c + d*x))/2] - 2*a*A*Tan[(c + d*x)/2] + A*b*Tan[(c + d*x)/2]))/(a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",1
752,1,349,283,11.8437467,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{2 \left(\frac{8 \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \cos ^2\left(\frac{1}{2} (c+d x)\right)^{7/2} \sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} \left(a (a (A+3 C)-2 A b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+A b \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))+2 A b (a+b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{(\cos (c+d x)+1)^{3/2}}+A \sin (c+d x) (a-2 b \cos (c+d x)) (a+b \cos (c+d x))\right)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}","-\frac{4 A b (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d}+\frac{2 \sqrt{a+b} (a (A+3 C)+2 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 a d \cos ^{\frac{3}{2}}(c+d x)}",1,"(2*(A*(a - 2*b*Cos[c + d*x])*(a + b*Cos[c + d*x])*Sin[c + d*x] + (8*(Cos[(c + d*x)/2]^2)^(7/2)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*(2*A*b*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*(-2*A*b + a*(A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + A*b*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(1 + Cos[c + d*x])^(3/2)))/(3*a^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]])","A",1
753,1,1298,354,6.4374109,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 A \tan (c+d x) \sec ^2(c+d x)}{5 a}+\frac{2 \left(9 A \sin (c+d x) a^2+15 C \sin (c+d x) a^2+8 A b^2 \sin (c+d x)\right) \sec (c+d x)}{15 a^3}-\frac{8 A b \tan (c+d x) \sec (c+d x)}{15 a^2}\right)}{d}-\frac{-\frac{4 a \left(8 A b^3+7 a^2 A b+15 a^2 C b\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(9 A a^3+15 C a^3+8 A b^2 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(8 A b^3+9 a^2 A b+15 a^2 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{15 a^3 d}","-\frac{8 A b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(3 a^2 (3 A+5 C)+8 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^4 d}+\frac{2 \sqrt{a+b} \left(-3 a^2 (3 A+5 C)+2 a A b-8 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 a d \cos ^{\frac{5}{2}}(c+d x)}",1,"-1/15*((-4*a*(7*a^2*A*b + 8*A*b^3 + 15*a^2*b*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(9*a^3*A + 8*a*A*b^2 + 15*a^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(9*a^2*A*b + 8*A*b^3 + 15*a^2*b*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a^3*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]*(9*a^2*A*Sin[c + d*x] + 8*A*b^2*Sin[c + d*x] + 15*a^2*C*Sin[c + d*x]))/(15*a^3) - (8*A*b*Sec[c + d*x]*Tan[c + d*x])/(15*a^2) + (2*A*Sec[c + d*x]^2*Tan[c + d*x])/(5*a)))/d","C",1
754,1,1376,429,6.482467,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{9}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{-\frac{4 a \left(25 A a^4+35 C a^4+32 A b^2 a^2+70 b^2 C a^2+48 A b^4\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(44 A b a^3+70 b C a^3+48 A b^3 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(48 A b^4+44 a^2 A b^2+70 a^2 C b^2\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{105 a^4 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 A \tan (c+d x) \sec ^3(c+d x)}{7 a}+\frac{2 \left(25 A \sin (c+d x) a^2+35 C \sin (c+d x) a^2+24 A b^2 \sin (c+d x)\right) \sec ^2(c+d x)}{105 a^3}-\frac{12 A b \tan (c+d x) \sec ^2(c+d x)}{35 a^2}-\frac{4 \left(24 A \sin (c+d x) b^3+22 a^2 A \sin (c+d x) b+35 a^2 C \sin (c+d x) b\right) \sec (c+d x)}{105 a^4}\right)}{d}","-\frac{12 A b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 a^2 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{4 b (a-b) \sqrt{a+b} \left(a^2 (22 A+35 C)+24 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^5 d}+\frac{2 \left(5 a^2 (5 A+7 C)+24 A b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \sqrt{a+b} \left(-5 a^3 (5 A+7 C)-a^2 (44 A b+70 b C)+12 a A b^2-48 A b^3\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 a d \cos ^{\frac{7}{2}}(c+d x)}",1,"((-4*a*(25*a^4*A + 32*a^2*A*b^2 + 48*A*b^4 + 35*a^4*C + 70*a^2*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(44*a^3*A*b + 48*a*A*b^3 + 70*a^3*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(44*a^2*A*b^2 + 48*A*b^4 + 70*a^2*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(105*a^4*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^2*(25*a^2*A*Sin[c + d*x] + 24*A*b^2*Sin[c + d*x] + 35*a^2*C*Sin[c + d*x]))/(105*a^3) - (4*Sec[c + d*x]*(22*a^2*A*b*Sin[c + d*x] + 24*A*b^3*Sin[c + d*x] + 35*a^2*b*C*Sin[c + d*x]))/(105*a^4) - (12*A*b*Sec[c + d*x]^2*Tan[c + d*x])/(35*a^2) + (2*A*Sec[c + d*x]^3*Tan[c + d*x])/(7*a)))/d","C",0
755,1,1276,604,6.5429632,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{C \sin (c+d x)}{2 b^2}-\frac{2 \left(C \sin (c+d x) a^3+A b^2 \sin (c+d x) a\right)}{b^2 \left(b^2-a^2\right) (a+b \cos (c+d x))}\right)}{d}-\frac{-\frac{4 a \left(5 a^3 C-5 a b^2 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(8 A b^3+4 C b^3+4 a^2 C b\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(15 C a^3+8 A b^2 a-7 b^2 C a\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{8 (a-b) b^2 (a+b) d}","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\left(5 a^2 C+4 A b^2-b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 b^2 d \left(a^2-b^2\right)}-\frac{\sqrt{a+b} \left(15 a^2 C+8 A b^2+4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^4 d}-\frac{a \left(15 a^2 C+8 A b^2-7 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{\left(C \left(15 a^2+5 a b-2 b^2\right)+8 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{a+b}}+\frac{\left(15 a^2 C+8 A b^2-7 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{a+b}}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((C*Sin[c + d*x])/(2*b^2) - (2*(a*A*b^2*Sin[c + d*x] + a^3*C*Sin[c + d*x]))/(b^2*(-a^2 + b^2)*(a + b*Cos[c + d*x]))))/d - ((-4*a*(5*a^3*C - 5*a*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(8*A*b^3 + 4*a^2*b*C + 4*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(8*a*A*b^2 + 15*a^3*C - 7*a*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(8*(a - b)*b^2*(a + b)*d)","C",0
756,1,1234,503,6.3943452,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 \sqrt{\cos (c+d x)} \left(C \sin (c+d x) a^2+A b^2 \sin (c+d x)\right)}{b \left(b^2-a^2\right) d \sqrt{a+b \cos (c+d x)}}+\frac{-\frac{4 a \left(a^2 C-b^2 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a (2 a A b+2 a C b) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(3 C a^2+2 A b^2-b^2 C\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{2 (a-b) b (a+b) d}","\frac{\left(3 a^2 C+2 A b^2-b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{b^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{\left(3 a^2 C+2 A b^2-b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b}}+\frac{\left(a C (3 a+b)+2 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b}}+\frac{3 a C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d}",1,"(2*Sqrt[Cos[c + d*x]]*(A*b^2*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(b*(-a^2 + b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((-4*a*(a^2*C - b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(2*a*A*b + 2*a*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(2*A*b^2 + 3*a^2*C - b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(2*(a - b)*b*(a + b)*d)","C",1
757,1,1225,421,6.4052876,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{2 \sqrt{\cos (c+d x)} \left(C \sin (c+d x) a^2+A b^2 \sin (c+d x)\right)}{a \left(a^2-b^2\right) d \sqrt{a+b \cos (c+d x)}}+\frac{-\frac{4 a \left(a^2 A-A b^2\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a (-a A b-a C b) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-C a^2-A b^2\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{a (a-b) (a+b) d}","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 C+A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 b d \sqrt{a+b}}+\frac{2 (A b-a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{a+b}}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d}",1,"(2*Sqrt[Cos[c + d*x]]*(A*b^2*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((-4*a*(a^2*A - A*b^2)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-(a*A*b) - a*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-(A*b^2) - a^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a*(a - b)*(a + b)*d)","C",1
758,1,1269,308,6.5281233,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{-\frac{4 a \left(2 a^2 A b-2 A b^3\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(A a^3-C a^3-2 A b^2 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-2 A b^3+a^2 A b-a^2 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{a^2 (b-a) (a+b) d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 A \tan (c+d x)}{a^2}-\frac{2 \left(A \sin (c+d x) b^3+a^2 C \sin (c+d x) b\right)}{a^2 \left(a^2-b^2\right) (a+b \cos (c+d x))}\right)}{d}","\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{2 (a (A-C)+2 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d \sqrt{a+b}}-\frac{2 \left(2 A b^2-a^2 (A-C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^3 d \sqrt{a+b}}",1,"((-4*a*(2*a^2*A*b - 2*A*b^3)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(a^3*A - 2*a*A*b^2 - a^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(a^2*A*b - 2*A*b^3 - a^2*b*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a^2*(-a + b)*(a + b)*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((-2*(A*b^3*Sin[c + d*x] + a^2*b*C*Sin[c + d*x]))/(a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])) + (2*A*Tan[c + d*x])/a^2))/d","C",1
759,1,1327,392,6.6899235,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{-\frac{4 a \left(A a^4+3 C a^4+7 A b^2 a^2-3 b^2 C a^2-8 A b^4\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(5 A b a^3-3 b C a^3-8 A b^3 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-8 A b^4+5 a^2 A b^2-3 a^2 C b^2\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{3 a^3 (a-b) (a+b) d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \left(A \sin (c+d x) b^4+a^2 C \sin (c+d x) b^2\right)}{a^3 \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{10 A b \tan (c+d x)}{3 a^3}+\frac{2 A \sec (c+d x) \tan (c+d x)}{3 a^2}\right)}{d}","-\frac{2 \left(4 A b^2-a^2 (A-3 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 a^2 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}+\frac{2 b \left(8 A b^2-a^2 (5 A-3 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b}}+\frac{2 \left(a^2 (A+3 C)+6 a A b+8 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b}}",1,"((-4*a*(a^4*A + 7*a^2*A*b^2 - 8*A*b^4 + 3*a^4*C - 3*a^2*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(5*a^3*A*b - 8*a*A*b^3 - 3*a^3*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(5*a^2*A*b^2 - 8*A*b^4 - 3*a^2*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(3*a^3*(a - b)*(a + b)*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*(A*b^4*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(a^3*(a^2 - b^2)*(a + b*Cos[c + d*x])) - (10*A*b*Tan[c + d*x])/(3*a^3) + (2*A*Sec[c + d*x]*Tan[c + d*x])/(3*a^2)))/d","C",1
760,1,1418,494,6.856048,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{-\frac{4 a \left(-16 A b^5+12 a^2 A b^3-10 a^2 C b^3+4 a^4 A b+10 a^4 C b\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(3 A a^5+5 C a^5+8 A b^2 a^3-10 b^2 C a^3-16 A b^4 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-16 A b^5+8 a^2 A b^3-10 a^2 C b^3+3 a^4 A b+5 a^4 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{5 a^4 (b-a) (a+b) d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 A \tan (c+d x) \sec ^2(c+d x)}{5 a^2}+\frac{2 \left(3 A \sin (c+d x) a^2+5 C \sin (c+d x) a^2+11 A b^2 \sin (c+d x)\right) \sec (c+d x)}{5 a^4}-\frac{6 A b \tan (c+d x) \sec (c+d x)}{5 a^3}-\frac{2 \left(A \sin (c+d x) b^5+a^2 C \sin (c+d x) b^3\right)}{a^4 \left(a^2-b^2\right) (a+b \cos (c+d x))}\right)}{d}","-\frac{2 \left(6 A b^2-a^2 (A-5 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 a^2 d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}+\frac{2 b \left(8 A b^2-a^2 (3 A-5 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 a^3 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \left(-\left(a^4 (3 A+5 C)\right)-2 a^2 b^2 (4 A-5 C)+16 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^5 d \sqrt{a+b}}-\frac{2 \left(a^3 (3 A+5 C)+2 a^2 b (2 A+5 C)+12 a A b^2+16 A b^3\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^4 d \sqrt{a+b}}",1,"((-4*a*(4*a^4*A*b + 12*a^2*A*b^3 - 16*A*b^5 + 10*a^4*b*C - 10*a^2*b^3*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(3*a^5*A + 8*a^3*A*b^2 - 16*a*A*b^4 + 5*a^5*C - 10*a^3*b^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(3*a^4*A*b + 8*a^2*A*b^3 - 16*A*b^5 + 5*a^4*b*C - 10*a^2*b^3*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(5*a^4*(-a + b)*(a + b)*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]*(3*a^2*A*Sin[c + d*x] + 11*A*b^2*Sin[c + d*x] + 5*a^2*C*Sin[c + d*x]))/(5*a^4) - (2*(A*b^5*Sin[c + d*x] + a^2*b^3*C*Sin[c + d*x]))/(a^4*(a^2 - b^2)*(a + b*Cos[c + d*x])) - (6*A*b*Sec[c + d*x]*Tan[c + d*x])/(5*a^3) + (2*A*Sec[c + d*x]^2*Tan[c + d*x])/(5*a^2)))/d","C",0
761,1,1366,650,6.6965958,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{4 \left(-3 C \sin (c+d x) a^4+5 b^2 C \sin (c+d x) a^2+2 A b^4 \sin (c+d x)\right)}{3 b^2 \left(b^2-a^2\right)^2 (a+b \cos (c+d x))}-\frac{2 \left(C \sin (c+d x) a^3+A b^2 \sin (c+d x) a\right)}{3 b^2 \left(b^2-a^2\right) (a+b \cos (c+d x))^2}\right)}{d}+\frac{-\frac{4 a \left(5 C a^4+2 A b^2 a^2-8 b^2 C a^2-2 A b^4+3 b^4 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(4 b C a^3-8 A b^3 a-12 b^3 C a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(15 C a^4-26 b^2 C a^2-8 A b^4+3 b^4 C\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{6 (a-b)^2 b^2 (a+b)^2 d}","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(-5 a^4 C+a^2 b^2 (A+9 C)+3 A b^4\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{\left(8 A b^4-C \left(15 a^4-26 a^2 b^2+3 b^4\right)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}+\frac{\left(-15 a^4 C+26 a^2 b^2 C+8 A b^4-3 b^4 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^3 d (a-b) (a+b)^{3/2}}+\frac{\left(15 a^4 C+5 a^3 b C-21 a^2 b^2 C+2 a A b^3-3 a b^3 C-6 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^3 d (a-b) (a+b)^{3/2}}+\frac{5 a C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^4 d}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((-2*(a*A*b^2*Sin[c + d*x] + a^3*C*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (4*(2*A*b^4*Sin[c + d*x] - 3*a^4*C*Sin[c + d*x] + 5*a^2*b^2*C*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)^2*(a + b*Cos[c + d*x]))))/d + ((-4*a*(2*a^2*A*b^2 - 2*A*b^4 + 5*a^4*C - 8*a^2*b^2*C + 3*b^4*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-8*a*A*b^3 + 4*a^3*b*C - 12*a*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-8*A*b^4 + 15*a^4*C - 26*a^2*b^2*C + 3*b^4*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(6*(a - b)^2*b^2*(a + b)^2*d)","C",0
762,1,1388,563,6.6212743,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \left(C \sin (c+d x) a^2+A b^2 \sin (c+d x)\right)}{3 b \left(b^2-a^2\right) (a+b \cos (c+d x))^2}+\frac{2 \left(3 C \sin (c+d x) a^4-3 A b^2 \sin (c+d x) a^2-7 b^2 C \sin (c+d x) a^2-A b^4 \sin (c+d x)\right)}{3 a b \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}\right)}{d}-\frac{-\frac{4 a \left(C a^4+A b^2 a^2-b^2 C a^2-A b^4\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-3 A b a^3-b C a^3-A b^3 a-3 b^3 C a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(3 C a^4-3 A b^2 a^2-7 b^2 C a^2-A b^4\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{3 a (a-b)^2 b (a+b)^2 d}","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(-3 a^4 C+a^2 b^2 (3 A+7 C)+A b^4\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-3 a^4 C+a^2 b^2 (3 A+7 C)+A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 b^2 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{2 \left(-3 a^3 C-a^2 b C+3 a A b^2+6 a b^2 C-A b^3\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^2 d (a-b) (a+b)^{3/2}}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*(A*b^2*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(3*b*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (2*(-3*a^2*A*b^2*Sin[c + d*x] - A*b^4*Sin[c + d*x] + 3*a^4*C*Sin[c + d*x] - 7*a^2*b^2*C*Sin[c + d*x]))/(3*a*b*(a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/d - ((-4*a*(a^2*A*b^2 - A*b^4 + a^4*C - a^2*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-3*a^3*A*b - a*A*b^3 - a^3*b*C - 3*a*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-3*a^2*A*b^2 - A*b^4 + 3*a^4*C - 7*a^2*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(3*a*(a - b)^2*b*(a + b)^2*d)","C",0
763,1,1364,417,6.5443742,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \left(C \sin (c+d x) a^2+A b^2 \sin (c+d x)\right)}{3 a \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{4 \left(-A \sin (c+d x) b^4+3 a^2 A \sin (c+d x) b^2+2 a^2 C \sin (c+d x) b^2\right)}{3 a^2 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}\right)}{d}+\frac{-\frac{4 a \left(3 A a^4+C a^4-5 A b^2 a^2-b^2 C a^2+2 A b^4\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-6 A b a^3-4 b C a^3+2 A b^3 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(2 A b^4-6 a^2 A b^2-4 a^2 C b^2\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{3 a^2 (a-b)^2 (a+b)^2 d}","\frac{4 b \left(A b^2-a^2 (3 A+2 C)\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(-\left(a^2 (3 A+C)\right)+3 a b (A+C)+2 A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{4 b \left(3 a^2 A+2 a^2 C-A b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d (a-b) (a+b)^{3/2}}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*(A*b^2*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(3*a*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) + (4*(3*a^2*A*b^2*Sin[c + d*x] - A*b^4*Sin[c + d*x] + 2*a^2*b^2*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/d + ((-4*a*(3*a^4*A - 5*a^2*A*b^2 + 2*A*b^4 + a^4*C - a^2*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-6*a^3*A*b + 2*a*A*b^3 - 4*a^3*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-6*a^2*A*b^2 + 2*A*b^4 - 4*a^2*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(3*a^2*(a - b)^2*(a + b)^2*d)","C",0
764,1,1421,449,6.8630523,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(5/2)),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(-\frac{2 \left(A \sin (c+d x) b^3+a^2 C \sin (c+d x) b\right)}{3 a^2 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{2 \left(-5 A \sin (c+d x) b^5+9 a^2 A \sin (c+d x) b^3+a^2 C \sin (c+d x) b^3+3 a^4 C \sin (c+d x) b\right)}{3 a^3 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{2 A \tan (c+d x)}{a^3}\right)}{d}-\frac{-\frac{4 a \left(8 A b^5-17 a^2 A b^3-a^2 C b^3+9 a^4 A b+a^4 C b\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(3 A a^5-3 C a^5-15 A b^2 a^3-b^2 C a^3+8 A b^4 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(8 A b^5-15 a^2 A b^3-a^2 C b^3+3 a^4 A b-3 a^4 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{3 a^3 (a-b)^2 (a+b)^2 d}","\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}-\frac{4 \left(a^4 (-C)-a^2 b^2 (4 A+C)+2 A b^4\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(3 a^4 (A-C)-a^2 b^2 (15 A+C)+8 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{2 \left(-3 a^3 (A-C)-a^2 b (9 A+C)+6 a A b^2+8 A b^3\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b} \left(a^2-b^2\right)}",1,"-1/3*((-4*a*(9*a^4*A*b - 17*a^2*A*b^3 + 8*A*b^5 + a^4*b*C - a^2*b^3*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(3*a^5*A - 15*a^3*A*b^2 + 8*a*A*b^4 - 3*a^5*C - a^3*b^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(3*a^4*A*b - 15*a^2*A*b^3 + 8*A*b^5 - 3*a^4*b*C - a^2*b^3*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a^3*(a - b)^2*(a + b)^2*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((-2*(A*b^3*Sin[c + d*x] + a^2*b*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) - (2*(9*a^2*A*b^3*Sin[c + d*x] - 5*A*b^5*Sin[c + d*x] + 3*a^4*b*C*Sin[c + d*x] + a^2*b^3*C*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) + (2*A*Tan[c + d*x])/a^3))/d","C",0
765,1,1471,549,7.0492713,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(5/2)),x]","\frac{-\frac{4 a \left(A a^6+3 C a^6+15 A b^2 a^4-5 b^2 C a^4-32 A b^4 a^2+2 b^4 C a^2+16 A b^6\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(8 A b a^5-6 b C a^5-28 A b^3 a^3+2 b^3 C a^3+16 A b^5 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(16 A b^6-28 a^2 A b^4+2 a^2 C b^4+8 a^4 A b^2-6 a^4 C b^2\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{3 a^4 (a-b)^2 (a+b)^2 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \left(A \sin (c+d x) b^4+a^2 C \sin (c+d x) b^2\right)}{3 a^3 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{4 \left(-4 A \sin (c+d x) b^6+6 a^2 A \sin (c+d x) b^4-a^2 C \sin (c+d x) b^4+3 a^4 C \sin (c+d x) b^2\right)}{3 a^4 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{16 A b \tan (c+d x)}{3 a^4}+\frac{2 A \sec (c+d x) \tan (c+d x)}{3 a^3}\right)}{d}","\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}+\frac{4 \left(2 a^4 C+5 a^2 A b^2-3 A b^4\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}-\frac{4 b \left(a^4 (4 A-3 C)-a^2 b^2 (14 A-C)+8 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^5 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{2 \left(a^4 (A-5 C)-a^2 b^2 (13 A-C)+8 A b^4\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 a^3 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \left(-\left(a^4 (A+3 C)\right)-a^3 (9 A b-3 b C)-2 a^2 b^2 (8 A-C)+12 a A b^3+16 A b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \left(a^2-b^2\right)}",1,"((-4*a*(a^6*A + 15*a^4*A*b^2 - 32*a^2*A*b^4 + 16*A*b^6 + 3*a^6*C - 5*a^4*b^2*C + 2*a^2*b^4*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(8*a^5*A*b - 28*a^3*A*b^3 + 16*a*A*b^5 - 6*a^5*b*C + 2*a^3*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(8*a^4*A*b^2 - 28*a^2*A*b^4 + 16*A*b^6 - 6*a^4*b^2*C + 2*a^2*b^4*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(3*a^4*(a - b)^2*(a + b)^2*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*(A*b^4*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) + (4*(6*a^2*A*b^4*Sin[c + d*x] - 4*A*b^6*Sin[c + d*x] + 3*a^4*b^2*C*Sin[c + d*x] - a^2*b^4*C*Sin[c + d*x]))/(3*a^4*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) - (16*A*b*Tan[c + d*x])/(3*a^4) + (2*A*Sec[c + d*x]*Tan[c + d*x])/(3*a^3)))/d","C",0
766,1,250,318,2.2932398,"\int \cos ^m(c+d x) (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^m*(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2),x]","\frac{\sin (c+d x) \cos ^{m+1}(c+d x) \left(\cos (c+d x) \left(\cos (c+d x) \left(b C \cos (c+d x) \left(-\frac{2 a \, _2F_1\left(\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};\cos ^2(c+d x)\right)}{m+4}-\frac{b \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+5}{2};\frac{m+7}{2};\cos ^2(c+d x)\right)}{m+5}\right)-\frac{\left(a^2 C+A b^2\right) \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\cos ^2(c+d x)\right)}{m+3}\right)-\frac{2 a A b \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{m+2}\right)-\frac{a^2 A \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{m+1}\right)}{d \sqrt{\sin ^2(c+d x)}}","-\frac{\sin (c+d x) \left(a^2 (m+4) (A (m+2)+C (m+1))+b^2 (m+1) (A (m+4)+C (m+3))\right) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) (m+4) \sqrt{\sin ^2(c+d x)}}+\frac{\sin (c+d x) \left(2 a^2 C+b^2 (A (m+4)+C (m+3))\right) \cos ^{m+1}(c+d x)}{d (m+2) (m+4)}-\frac{2 a b (A (m+3)+C (m+2)) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{C \sin (c+d x) \cos ^{m+1}(c+d x) (a+b \cos (c+d x))^2}{d (m+4)}+\frac{2 a b C \sin (c+d x) \cos ^{m+2}(c+d x)}{d (m+3) (m+4)}",1,"(Cos[c + d*x]^(1 + m)*(-((a^2*A*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2])/(1 + m)) + Cos[c + d*x]*((-2*a*A*b*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2])/(2 + m) + Cos[c + d*x]*(-(((A*b^2 + a^2*C)*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, Cos[c + d*x]^2])/(3 + m)) + b*C*Cos[c + d*x]*((-2*a*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, Cos[c + d*x]^2])/(4 + m) - (b*Cos[c + d*x]*Hypergeometric2F1[1/2, (5 + m)/2, (7 + m)/2, Cos[c + d*x]^2])/(5 + m)))))*Sin[c + d*x])/(d*Sqrt[Sin[c + d*x]^2])","A",1
767,1,194,217,1.0048495,"\int \cos ^m(c+d x) (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^m*(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2),x]","\frac{\sqrt{\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+1}(c+d x) \left(\cos (c+d x) \left(C \cos (c+d x) \left(-\frac{a \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\cos ^2(c+d x)\right)}{m+3}-\frac{b \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};\cos ^2(c+d x)\right)}{m+4}\right)-\frac{A b \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{m+2}\right)-\frac{a A \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{m+1}\right)}{d}","-\frac{a (A (m+2)+C (m+1)) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) \sqrt{\sin ^2(c+d x)}}+\frac{a C \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2)}-\frac{b (A (m+3)+C (m+2)) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{b C \sin (c+d x) \cos ^{m+2}(c+d x)}{d (m+3)}",1,"(Cos[c + d*x]^(1 + m)*Csc[c + d*x]*(-((a*A*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2])/(1 + m)) + Cos[c + d*x]*(-((A*b*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2])/(2 + m)) + C*Cos[c + d*x]*(-((a*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, Cos[c + d*x]^2])/(3 + m)) - (b*Cos[c + d*x]*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, Cos[c + d*x]^2])/(4 + m))))*Sqrt[Sin[c + d*x]^2])/d","A",1
768,1,10459,353,28.1470236,"\int \frac{\cos ^m(c+d x) \left(A+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^m*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\text{Result too large to show}","\frac{a \left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{m-1}(c+d x) \cos ^2(c+d x)^{\frac{1-m}{2}} F_1\left(\frac{1}{2};\frac{1-m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{b^2 d \left(a^2-b^2\right)}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^m(c+d x) \cos ^2(c+d x)^{-m/2} F_1\left(\frac{1}{2};-\frac{m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{b d \left(a^2-b^2\right)}+\frac{a C \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{b^2 d (m+1) \sqrt{\sin ^2(c+d x)}}-\frac{C \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{b d (m+2) \sqrt{\sin ^2(c+d x)}}",1,"Result too large to show","B",0
769,1,14082,514,45.5183989,"\int \frac{\cos ^m(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^m*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\text{Result too large to show}","-\frac{\sin (c+d x) \left(a^2 C (m+1)-b^2 (C-A m)\right) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{b^2 d (m+1) \left(a^2-b^2\right) \sqrt{\sin ^2(c+d x)}}+\frac{(m+1) \left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{a b d (m+2) \left(a^2-b^2\right) \sqrt{\sin ^2(c+d x)}}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^{m+1}(c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \left(a^4 (-C) (m+1)+a^2 b^2 (A (-m)+A+C (m+2))+A b^4 m\right) \cos ^{m-1}(c+d x) \cos ^2(c+d x)^{\frac{1-m}{2}} F_1\left(\frac{1}{2};\frac{1-m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{b^2 d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x) \left(a^4 (-C) (m+1)+a^2 b^2 (A (-m)+A+C (m+2))+A b^4 m\right) \cos ^m(c+d x) \cos ^2(c+d x)^{-m/2} F_1\left(\frac{1}{2};-\frac{m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{a b d \left(a^2-b^2\right)^2}",1,"Result too large to show","B",0
770,1,91,105,0.2230821,"\int \cos (c+d x) (a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{-32 (a C+b B) \sin ^3(c+d x)+96 (a C+b B) \sin (c+d x)+24 (a B+b C) \sin (2 (c+d x))+48 a B c+48 a B d x+3 b C \sin (4 (c+d x))+36 b c C+36 b C d x}{96 d}","-\frac{(a C+b B) \sin ^3(c+d x)}{3 d}+\frac{(a C+b B) \sin (c+d x)}{d}+\frac{(4 a B+3 b C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x (4 a B+3 b C)+\frac{b C \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"(48*a*B*c + 36*b*c*C + 48*a*B*d*x + 36*b*C*d*x + 96*(b*B + a*C)*Sin[c + d*x] - 32*(b*B + a*C)*Sin[c + d*x]^3 + 24*(a*B + b*C)*Sin[2*(c + d*x)] + 3*b*C*Sin[4*(c + d*x)])/(96*d)","A",1
771,1,75,84,0.1612979,"\int (a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{3 (4 a B+3 b C) \sin (c+d x)+3 (a C+b B) \sin (2 (c+d x))+6 a c C+6 a C d x+6 b B c+6 b B d x+b C \sin (3 (c+d x))}{12 d}","\frac{(3 a B+2 b C) \sin (c+d x)}{3 d}+\frac{(a C+b B) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (a C+b B)+\frac{b C \sin (c+d x) \cos ^2(c+d x)}{3 d}",1,"(6*b*B*c + 6*a*c*C + 6*b*B*d*x + 6*a*C*d*x + 3*(4*a*B + 3*b*C)*Sin[c + d*x] + 3*(b*B + a*C)*Sin[2*(c + d*x)] + b*C*Sin[3*(c + d*x)])/(12*d)","A",1
772,1,51,52,0.0841869,"\int (a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{4 (a C+b B) \sin (c+d x)+4 a B d x+b C \sin (2 (c+d x))+2 b c C+2 b C d x}{4 d}","\frac{(a C+b B) \sin (c+d x)}{d}+\frac{1}{2} x (2 a B+b C)+\frac{b C \sin (c+d x) \cos (c+d x)}{2 d}",1,"(2*b*c*C + 4*a*B*d*x + 2*b*C*d*x + 4*(b*B + a*C)*Sin[c + d*x] + b*C*Sin[2*(c + d*x)])/(4*d)","A",1
773,1,46,35,0.0261677,"\int (a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+a C x+b B x+\frac{b C \sin (c) \cos (d x)}{d}+\frac{b C \cos (c) \sin (d x)}{d}","x (a C+b B)+\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b C \sin (c+d x)}{d}",1,"b*B*x + a*C*x + (a*B*ArcTanh[Sin[c + d*x]])/d + (b*C*Cos[d*x]*Sin[c])/d + (b*C*Cos[c]*Sin[d*x])/d","A",1
774,1,43,35,0.0130164,"\int (a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{a B \tan (c+d x)}{d}+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b B \tanh ^{-1}(\sin (c+d x))}{d}+b C x","\frac{(a C+b B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a B \tan (c+d x)}{d}+b C x",1,"b*C*x + (b*B*ArcTanh[Sin[c + d*x]])/d + (a*C*ArcTanh[Sin[c + d*x]])/d + (a*B*Tan[c + d*x])/d","A",1
775,1,75,61,0.0202486,"\int (a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{a B \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a B \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a C \tan (c+d x)}{d}+\frac{b B \tan (c+d x)}{d}+\frac{b C \tanh ^{-1}(\sin (c+d x))}{d}","\frac{(a C+b B) \tan (c+d x)}{d}+\frac{(a B+2 b C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a B \tan (c+d x) \sec (c+d x)}{2 d}",1,"(a*B*ArcTanh[Sin[c + d*x]])/(2*d) + (b*C*ArcTanh[Sin[c + d*x]])/d + (b*B*Tan[c + d*x])/d + (a*C*Tan[c + d*x])/d + (a*B*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
776,1,67,93,0.6148835,"\int (a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{3 (a C+b B) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(3 (a C+b B) \sec (c+d x)+2 a B \tan ^2(c+d x)+6 a B+6 b C\right)}{6 d}","\frac{(2 a B+3 b C) \tan (c+d x)}{3 d}+\frac{(a C+b B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(a C+b B) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a B \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"(3*(b*B + a*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(6*a*B + 6*b*C + 3*(b*B + a*C)*Sec[c + d*x] + 2*a*B*Tan[c + d*x]^2))/(6*d)","A",1
777,1,85,114,0.5938708,"\int (a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{3 (3 a B+4 b C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x) \left(8 (a C+b B) (\cos (2 (c+d x))+2) \sec (c+d x)+6 a B \sec ^2(c+d x)+9 a B+12 b C\right)}{24 d}","\frac{(a C+b B) \tan ^3(c+d x)}{3 d}+\frac{(a C+b B) \tan (c+d x)}{d}+\frac{(3 a B+4 b C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(3 a B+4 b C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a B \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"(3*(3*a*B + 4*b*C)*ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*(9*a*B + 12*b*C + 8*(b*B + a*C)*(2 + Cos[2*(c + d*x)])*Sec[c + d*x] + 6*a*B*Sec[c + d*x]^2)*Tan[c + d*x])/(24*d)","A",1
778,1,146,189,0.4631729,"\int \cos (c+d x) (a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{60 (c+d x) \left(4 a^2 B+6 a b C+3 b^2 B\right)+60 \left(6 a^2 C+12 a b B+5 b^2 C\right) \sin (c+d x)+120 \left(a^2 B+2 a b C+b^2 B\right) \sin (2 (c+d x))+10 \left(4 a^2 C+8 a b B+5 b^2 C\right) \sin (3 (c+d x))+15 b (2 a C+b B) \sin (4 (c+d x))+6 b^2 C \sin (5 (c+d x))}{480 d}","\frac{\left(4 a^2 B+6 a b C+3 b^2 B\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(4 a^2 B+6 a b C+3 b^2 B\right)-\frac{\left(5 a (a C+2 b B)+4 b^2 C\right) \sin ^3(c+d x)}{15 d}+\frac{\left(5 a (a C+2 b B)+4 b^2 C\right) \sin (c+d x)}{5 d}+\frac{b (6 a C+5 b B) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{b C \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))}{5 d}",1,"(60*(4*a^2*B + 3*b^2*B + 6*a*b*C)*(c + d*x) + 60*(12*a*b*B + 6*a^2*C + 5*b^2*C)*Sin[c + d*x] + 120*(a^2*B + b^2*B + 2*a*b*C)*Sin[2*(c + d*x)] + 10*(8*a*b*B + 4*a^2*C + 5*b^2*C)*Sin[3*(c + d*x)] + 15*b*(b*B + 2*a*C)*Sin[4*(c + d*x)] + 6*b^2*C*Sin[5*(c + d*x)])/(480*d)","A",1
779,1,118,170,0.4460981,"\int (a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{12 (c+d x) \left(4 a^2 C+8 a b B+3 b^2 C\right)+24 \left(4 a^2 B+6 a b C+3 b^2 B\right) \sin (c+d x)+24 \left(a^2 C+2 a b B+b^2 C\right) \sin (2 (c+d x))+8 b (2 a C+b B) \sin (3 (c+d x))+3 b^2 C \sin (4 (c+d x))}{96 d}","\frac{\left(-2 a^2 C+8 a b B+9 b^2 C\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(4 a^2 C+8 a b B+3 b^2 C\right)+\frac{\left(a^3 (-C)+4 a^2 b B+8 a b^2 C+4 b^3 B\right) \sin (c+d x)}{6 b d}+\frac{(4 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^2}{12 b d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d}",1,"(12*(8*a*b*B + 4*a^2*C + 3*b^2*C)*(c + d*x) + 24*(4*a^2*B + 3*b^2*B + 6*a*b*C)*Sin[c + d*x] + 24*(2*a*b*B + a^2*C + b^2*C)*Sin[2*(c + d*x)] + 8*b*(b*B + 2*a*C)*Sin[3*(c + d*x)] + 3*b^2*C*Sin[4*(c + d*x)])/(96*d)","A",1
780,1,90,107,0.2278144,"\int (a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{6 (c+d x) \left(2 a^2 B+2 a b C+b^2 B\right)+3 \left(4 a^2 C+8 a b B+3 b^2 C\right) \sin (c+d x)+3 b (2 a C+b B) \sin (2 (c+d x))+b^2 C \sin (3 (c+d x))}{12 d}","\frac{2 \left(a^2 C+3 a b B+b^2 C\right) \sin (c+d x)}{3 d}+\frac{1}{2} x \left(2 a^2 B+2 a b C+b^2 B\right)+\frac{b (2 a C+3 b B) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}",1,"(6*(2*a^2*B + b^2*B + 2*a*b*C)*(c + d*x) + 3*(8*a*b*B + 4*a^2*C + 3*b^2*C)*Sin[c + d*x] + 3*b*(b*B + 2*a*C)*Sin[2*(c + d*x)] + b^2*C*Sin[3*(c + d*x)])/(12*d)","A",1
781,1,120,86,0.2319884,"\int (a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{2 (c+d x) \left(2 a^2 C+4 a b B+b^2 C\right)-4 a^2 B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 a^2 B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+4 b (2 a C+b B) \sin (c+d x)+b^2 C \sin (2 (c+d x))}{4 d}","\frac{1}{2} x \left(2 a^2 C+4 a b B+b^2 C\right)+\frac{a^2 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b (3 a C+2 b B) \sin (c+d x)}{2 d}+\frac{b C \sin (c+d x) (a+b \cos (c+d x))}{2 d}",1,"(2*(4*a*b*B + 2*a^2*C + b^2*C)*(c + d*x) - 4*a^2*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*a^2*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 4*b*(b*B + 2*a*C)*Sin[c + d*x] + b^2*C*Sin[2*(c + d*x)])/(4*d)","A",1
782,1,109,60,0.4818458,"\int (a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{a^2 B \tan (c+d x)+b (c+d x) (2 a C+b B)-a (a C+2 b B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+a (a C+2 b B) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+b^2 C \sin (c+d x)}{d}","\frac{a^2 B \tan (c+d x)}{d}+\frac{a (a C+2 b B) \tanh ^{-1}(\sin (c+d x))}{d}+b x (2 a C+b B)+\frac{b^2 C \sin (c+d x)}{d}",1,"(b*(b*B + 2*a*C)*(c + d*x) - a*(2*b*B + a*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + a*(2*b*B + a*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + b^2*C*Sin[c + d*x] + a^2*B*Tan[c + d*x])/d","A",1
783,1,67,80,0.2647839,"\int (a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\left(a^2 B+4 a b C+2 b^2 B\right) \tanh ^{-1}(\sin (c+d x))+a \tan (c+d x) (a B \sec (c+d x)+2 a C+4 b B)+2 b^2 C d x}{2 d}","\frac{\left(a^2 B+4 a b C+2 b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 B \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a (a C+2 b B) \tan (c+d x)}{d}+b^2 C x",1,"(2*b^2*C*d*x + (a^2*B + 2*b^2*B + 4*a*b*C)*ArcTanh[Sin[c + d*x]] + a*(4*b*B + 2*a*C + a*B*Sec[c + d*x])*Tan[c + d*x])/(2*d)","A",1
784,1,92,116,0.4588466,"\int (a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{3 \left(a^2 C+2 a b B+2 b^2 C\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(2 \left(a^2 B \tan ^2(c+d x)+3 a^2 B+6 a b C+3 b^2 B\right)+3 a (a C+2 b B) \sec (c+d x)\right)}{6 d}","\frac{\left(2 a^2 B+6 a b C+3 b^2 B\right) \tan (c+d x)}{3 d}+\frac{\left(a^2 C+2 a b B+2 b^2 C\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 B \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a (a C+2 b B) \tan (c+d x) \sec (c+d x)}{2 d}",1,"(3*(2*a*b*B + a^2*C + 2*b^2*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(3*a*(2*b*B + a*C)*Sec[c + d*x] + 2*(3*a^2*B + 3*b^2*B + 6*a*b*C + a^2*B*Tan[c + d*x]^2)))/(6*d)","A",1
785,1,120,156,0.7408573,"\int (a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{3 \left(3 a^2 B+8 a b C+4 b^2 B\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(3 \left(3 a^2 B+8 a b C+4 b^2 B\right) \sec (c+d x)+24 \left(a^2 C+2 a b B+b^2 C\right)+6 a^2 B \sec ^3(c+d x)+8 a (a C+2 b B) \tan ^2(c+d x)\right)}{24 d}","\frac{\left(2 a^2 C+4 a b B+3 b^2 C\right) \tan (c+d x)}{3 d}+\frac{\left(3 a^2 B+8 a b C+4 b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(3 a^2 B+8 a b C+4 b^2 B\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^2 B \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{a (a C+2 b B) \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"(3*(3*a^2*B + 4*b^2*B + 8*a*b*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(24*(2*a*b*B + a^2*C + b^2*C) + 3*(3*a^2*B + 4*b^2*B + 8*a*b*C)*Sec[c + d*x] + 6*a^2*B*Sec[c + d*x]^3 + 8*a*(2*b*B + a*C)*Tan[c + d*x]^2))/(24*d)","A",1
786,1,176,243,0.6888135,"\int (a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{10 b \left(12 a^2 C+12 a b B+5 b^2 C\right) \sin (3 (c+d x))+60 (c+d x) \left(4 a^3 C+12 a^2 b B+9 a b^2 C+3 b^3 B\right)+60 \left(8 a^3 B+18 a^2 b C+18 a b^2 B+5 b^3 C\right) \sin (c+d x)+120 \left(a^3 C+3 a^2 b B+3 a b^2 C+b^3 B\right) \sin (2 (c+d x))+15 b^2 (3 a C+b B) \sin (4 (c+d x))+6 b^3 C \sin (5 (c+d x))}{480 d}","\frac{\left(-3 a^2 C+15 a b B+16 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^2}{60 b d}+\frac{\left(-6 a^3 C+30 a^2 b B+71 a b^2 C+45 b^3 B\right) \sin (c+d x) \cos (c+d x)}{120 d}+\frac{1}{8} x \left(4 a^3 C+12 a^2 b B+9 a b^2 C+3 b^3 B\right)+\frac{\left(-3 a^4 C+15 a^3 b B+52 a^2 b^2 C+60 a b^3 B+16 b^4 C\right) \sin (c+d x)}{30 b d}+\frac{(5 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^3}{20 b d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^4}{5 b d}",1,"(60*(12*a^2*b*B + 3*b^3*B + 4*a^3*C + 9*a*b^2*C)*(c + d*x) + 60*(8*a^3*B + 18*a*b^2*B + 18*a^2*b*C + 5*b^3*C)*Sin[c + d*x] + 120*(3*a^2*b*B + b^3*B + a^3*C + 3*a*b^2*C)*Sin[2*(c + d*x)] + 10*b*(12*a*b*B + 12*a^2*C + 5*b^2*C)*Sin[3*(c + d*x)] + 15*b^2*(b*B + 3*a*C)*Sin[4*(c + d*x)] + 6*b^3*C*Sin[5*(c + d*x)])/(480*d)","A",1
787,1,140,171,0.4346012,"\int (a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{24 b \left(3 a^2 C+3 a b B+b^2 C\right) \sin (2 (c+d x))+12 (c+d x) \left(8 a^3 B+12 a^2 b C+12 a b^2 B+3 b^3 C\right)+24 \left(4 a^3 C+12 a^2 b B+9 a b^2 C+3 b^3 B\right) \sin (c+d x)+8 b^2 (3 a C+b B) \sin (3 (c+d x))+3 b^3 C \sin (4 (c+d x))}{96 d}","\frac{b \left(6 a^2 C+20 a b B+9 b^2 C\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{\left(3 a^3 C+16 a^2 b B+12 a b^2 C+4 b^3 B\right) \sin (c+d x)}{6 d}+\frac{1}{8} x \left(8 a^3 B+12 a^2 b C+12 a b^2 B+3 b^3 C\right)+\frac{(3 a C+4 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}",1,"(12*(8*a^3*B + 12*a*b^2*B + 12*a^2*b*C + 3*b^3*C)*(c + d*x) + 24*(12*a^2*b*B + 3*b^3*B + 4*a^3*C + 9*a*b^2*C)*Sin[c + d*x] + 24*b*(3*a*b*B + 3*a^2*C + b^2*C)*Sin[2*(c + d*x)] + 8*b^2*(b*B + 3*a*C)*Sin[3*(c + d*x)] + 3*b^3*C*Sin[4*(c + d*x)])/(96*d)","A",1
788,1,159,137,0.4073583,"\int (a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{-12 a^3 B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 a^3 B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+9 b \left(4 a^2 C+4 a b B+b^2 C\right) \sin (c+d x)+6 (c+d x) \left(2 a^3 C+6 a^2 b B+3 a b^2 C+b^3 B\right)+3 b^2 (3 a C+b B) \sin (2 (c+d x))+b^3 C \sin (3 (c+d x))}{12 d}","\frac{a^3 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b \left(8 a^2 C+9 a b B+2 b^2 C\right) \sin (c+d x)}{3 d}+\frac{1}{2} x \left(2 a^3 C+6 a^2 b B+3 a b^2 C+b^3 B\right)+\frac{b^2 (5 a C+3 b B) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{b C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}",1,"(6*(6*a^2*b*B + b^3*B + 2*a^3*C + 3*a*b^2*C)*(c + d*x) - 12*a^3*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*a^3*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 9*b*(4*a*b*B + 4*a^2*C + b^2*C)*Sin[c + d*x] + 3*b^2*(b*B + 3*a*C)*Sin[2*(c + d*x)] + b^3*C*Sin[3*(c + d*x)])/(12*d)","A",1
789,1,217,131,0.6665266,"\int (a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{\frac{4 a^3 B \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 a^3 B \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+2 b (c+d x) \left(6 a^2 C+6 a b B+b^2 C\right)-4 a^2 (a C+3 b B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 a^2 (a C+3 b B) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+4 b^2 (3 a C+b B) \sin (c+d x)+b^3 C \sin (2 (c+d x))}{4 d}","-\frac{b \left(2 a^2 B-3 a b C-b^2 B\right) \sin (c+d x)}{d}+\frac{1}{2} b x \left(6 a^2 C+6 a b B+b^2 C\right)+\frac{a^2 (a C+3 b B) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^2 (2 a B-b C) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a B \tan (c+d x) (a+b \cos (c+d x))^2}{d}",1,"(2*b*(6*a*b*B + 6*a^2*C + b^2*C)*(c + d*x) - 4*a^2*(3*b*B + a*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*a^2*(3*b*B + a*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (4*a^3*B*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (4*a^3*B*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 4*b^2*(b*B + 3*a*C)*Sin[c + d*x] + b^3*C*Sin[2*(c + d*x)])/(4*d)","A",1
790,1,277,124,2.1842018,"\int (a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\frac{a^3 B}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{a^3 B}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-2 a \left(a^2 B+6 a b C+6 b^2 B\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 a \left(a^2 B+6 a b C+6 b^2 B\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{4 a^2 (a C+3 b B) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 a^2 (a C+3 b B) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+4 b^2 (c+d x) (3 a C+b B)+4 b^3 C \sin (c+d x)}{4 d}","\frac{a \left(a^2 B+6 a b C+6 b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 (a C+2 b B) \tan (c+d x)}{d}-\frac{b^2 (a B-2 b C) \sin (c+d x)}{2 d}+b^2 x (3 a C+b B)+\frac{a B \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}",1,"(4*b^2*(b*B + 3*a*C)*(c + d*x) - 2*a*(a^2*B + 6*b^2*B + 6*a*b*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*a*(a^2*B + 6*b^2*B + 6*a*b*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a^3*B)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (4*a^2*(3*b*B + a*C)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (a^3*B)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*a^2*(3*b*B + a*C)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 4*b^3*C*Sin[c + d*x])/(4*d)","B",1
791,1,108,145,0.5933789,"\int (a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{2 a^3 B \tan ^3(c+d x)+3 a \tan (c+d x) \left(2 a^2 B+a (a C+3 b B) \sec (c+d x)+6 a b C+6 b^2 B\right)+3 \left(a^3 C+3 a^2 b B+6 a b^2 C+2 b^3 B\right) \tanh ^{-1}(\sin (c+d x))+6 b^3 C d x}{6 d}","\frac{a \left(2 a^2 B+9 a b C+8 b^2 B\right) \tan (c+d x)}{3 d}+\frac{a^2 (3 a C+5 b B) \tan (c+d x) \sec (c+d x)}{6 d}+\frac{\left(a^3 C+3 a^2 b B+6 a b^2 C+2 b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a B \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}+b^3 C x",1,"(6*b^3*C*d*x + 3*(3*a^2*b*B + 2*b^3*B + a^3*C + 6*a*b^2*C)*ArcTanh[Sin[c + d*x]] + 3*a*(2*a^2*B + 6*b^2*B + 6*a*b*C + a*(3*b*B + a*C)*Sec[c + d*x])*Tan[c + d*x] + 2*a^3*B*Tan[c + d*x]^3)/(6*d)","A",1
792,1,140,188,0.8135786,"\int (a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{3 \left(3 a^3 B+12 a^2 b C+12 a b^2 B+8 b^3 C\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(6 a^3 B \sec ^3(c+d x)+9 a \left(a^2 B+4 a b C+4 b^2 B\right) \sec (c+d x)+8 a^2 (a C+3 b B) \tan ^2(c+d x)+24 \left(a^3 C+3 a^2 b B+3 a b^2 C+b^3 B\right)\right)}{24 d}","\frac{a \left(3 a^2 B+12 a b C+10 b^2 B\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^2 (2 a C+3 b B) \tan (c+d x) \sec ^2(c+d x)}{6 d}+\frac{\left(2 a^3 C+6 a^2 b B+9 a b^2 C+3 b^3 B\right) \tan (c+d x)}{3 d}+\frac{\left(3 a^3 B+12 a^2 b C+12 a b^2 B+8 b^3 C\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a B \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{4 d}",1,"(3*(3*a^3*B + 12*a*b^2*B + 12*a^2*b*C + 8*b^3*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(24*(3*a^2*b*B + b^3*B + a^3*C + 3*a*b^2*C) + 9*a*(a^2*B + 4*b^2*B + 4*a*b*C)*Sec[c + d*x] + 6*a^3*B*Sec[c + d*x]^3 + 8*a^2*(3*b*B + a*C)*Tan[c + d*x]^2))/(24*d)","A",1
793,1,181,236,3.2013059,"\int (a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","\frac{15 \left(3 a^3 C+9 a^2 b B+12 a b^2 C+4 b^3 B\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(30 a^2 (a C+3 b B) \sec ^3(c+d x)+8 \left(3 a^3 B \tan ^4(c+d x)+5 a \left(2 a^2 B+3 a b C+3 b^2 B\right) \tan ^2(c+d x)+15 \left(a^3 B+3 a^2 b C+3 a b^2 B+b^3 C\right)\right)+15 \left(3 a^3 C+9 a^2 b B+12 a b^2 C+4 b^3 B\right) \sec (c+d x)\right)}{120 d}","\frac{a \left(4 a^2 B+15 a b C+12 b^2 B\right) \tan (c+d x) \sec ^2(c+d x)}{15 d}+\frac{a^2 (5 a C+7 b B) \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac{\left(8 a^3 B+30 a^2 b C+30 a b^2 B+15 b^3 C\right) \tan (c+d x)}{15 d}+\frac{\left(3 a^3 C+9 a^2 b B+12 a b^2 C+4 b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(3 a^3 C+9 a^2 b B+12 a b^2 C+4 b^3 B\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a B \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}",1,"(15*(9*a^2*b*B + 4*b^3*B + 3*a^3*C + 12*a*b^2*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*(9*a^2*b*B + 4*b^3*B + 3*a^3*C + 12*a*b^2*C)*Sec[c + d*x] + 30*a^2*(3*b*B + a*C)*Sec[c + d*x]^3 + 8*(15*(a^3*B + 3*a*b^2*B + 3*a^2*b*C + b^3*C) + 5*a*(2*a^2*B + 3*b^2*B + 3*a*b*C)*Tan[c + d*x]^2 + 3*a^3*B*Tan[c + d*x]^4)))/(120*d)","A",1
794,1,152,178,0.4713121,"\int \frac{\cos ^2(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{6 \left(2 a^2+b^2\right) (c+d x) (b B-a C)+3 b \left(4 a^2 C-4 a b B+3 b^2 C\right) \sin (c+d x)-\frac{24 a^3 (a C-b B) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+3 b^2 (b B-a C) \sin (2 (c+d x))+b^3 C \sin (3 (c+d x))}{12 b^4 d}","-\frac{2 a^3 (b B-a C) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d \sqrt{a-b} \sqrt{a+b}}+\frac{x \left(2 a^2+b^2\right) (b B-a C)}{2 b^4}-\frac{\left(-3 a^2 C+3 a b B-2 b^2 C\right) \sin (c+d x)}{3 b^3 d}+\frac{(b B-a C) \sin (c+d x) \cos (c+d x)}{2 b^2 d}+\frac{C \sin (c+d x) \cos ^2(c+d x)}{3 b d}",1,"(6*(2*a^2 + b^2)*(b*B - a*C)*(c + d*x) - (24*a^3*(-(b*B) + a*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + 3*b*(-4*a*b*B + 4*a^2*C + 3*b^2*C)*Sin[c + d*x] + 3*b^2*(b*B - a*C)*Sin[2*(c + d*x)] + b^3*C*Sin[3*(c + d*x)])/(12*b^4*d)","A",1
795,1,121,134,0.3201611,"\int \frac{\cos (c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{2 (c+d x) \left(2 a^2 C-2 a b B+b^2 C\right)+\frac{8 a^2 (a C-b B) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+4 b (b B-a C) \sin (c+d x)+b^2 C \sin (2 (c+d x))}{4 b^3 d}","\frac{2 a^2 (b B-a C) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d \sqrt{a-b} \sqrt{a+b}}-\frac{x \left(-2 a^2 C+2 a b B-b^2 C\right)}{2 b^3}+\frac{(b B-a C) \sin (c+d x)}{b^2 d}+\frac{C \sin (c+d x) \cos (c+d x)}{2 b d}",1,"(2*(-2*a*b*B + 2*a^2*C + b^2*C)*(c + d*x) + (8*a^2*(-(b*B) + a*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + 4*b*(b*B - a*C)*Sin[c + d*x] + b^2*C*Sin[2*(c + d*x)])/(4*b^3*d)","A",1
796,1,85,89,0.2121934,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]),x]","\frac{-\frac{2 a (a C-b B) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+(c+d x) (b B-a C)+b C \sin (c+d x)}{b^2 d}","-\frac{2 a (b B-a C) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d \sqrt{a-b} \sqrt{a+b}}+\frac{x (b B-a C)}{b^2}+\frac{C \sin (c+d x)}{b d}",1,"((b*B - a*C)*(c + d*x) - (2*a*(-(b*B) + a*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + b*C*Sin[c + d*x])/(b^2*d)","A",1
797,1,68,67,0.1207206,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x]),x]","\frac{\frac{2 (a C-b B) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+C (c+d x)}{b d}","\frac{2 (b B-a C) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b d \sqrt{a-b} \sqrt{a+b}}+\frac{C x}{b}",1,"(C*(c + d*x) + (2*(-(b*B) + a*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2])/(b*d)","A",1
798,1,112,76,0.1622335,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x]),x]","\frac{\frac{2 (b B-a C) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+B \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{a d}","\frac{B \tanh ^{-1}(\sin (c+d x))}{a d}-\frac{2 (b B-a C) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a d \sqrt{a-b} \sqrt{a+b}}",1,"((2*(b*B - a*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + B*(-Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]))/(a*d)","A",1
799,1,129,99,0.5468919,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x]),x]","\frac{-\frac{2 b (b B-a C) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+(b B-a C) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+a B \tan (c+d x)}{a^2 d}","\frac{2 b (b B-a C) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{(b B-a C) \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{B \tan (c+d x)}{a d}",1,"((-2*b*(b*B - a*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (b*B - a*C)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + a*B*Tan[c + d*x])/(a^2*d)","A",1
800,1,300,143,1.6321784,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + b*Cos[c + d*x]),x]","\frac{\frac{8 b^2 (b B-a C) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}-2 \left(a^2 B-2 a b C+2 b^2 B\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \left(a^2 B-2 a b C+2 b^2 B\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{a^2 B}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{a^2 B}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{4 a (a C-b B) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 a (a C-b B) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}}{4 a^3 d}","-\frac{2 b^2 (b B-a C) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d \sqrt{a-b} \sqrt{a+b}}-\frac{(b B-a C) \tan (c+d x)}{a^2 d}+\frac{\left(a^2 B-2 a b C+2 b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{B \tan (c+d x) \sec (c+d x)}{2 a d}",1,"((8*b^2*(b*B - a*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] - 2*(a^2*B + 2*b^2*B - 2*a*b*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(a^2*B + 2*b^2*B - 2*a*b*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a^2*B)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (4*a*(-(b*B) + a*C)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (a^2*B)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*a*(-(b*B) + a*C)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(4*a^3*d)","B",1
801,1,184,263,1.0403521,"\int \frac{\cos ^2(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{\frac{4 a^3 b (b B-a C) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}+2 (c+d x) \left(6 a^2 C-4 a b B+b^2 C\right)-\frac{8 a^2 \left(3 a^3 C-2 a^2 b B-4 a b^2 C+3 b^3 B\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}+4 b (b B-2 a C) \sin (c+d x)+b^2 C \sin (2 (c+d x))}{4 b^4 d}","\frac{a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\left(-3 a^2 C+2 a b B+b^2 C\right) \sin (c+d x) \cos (c+d x)}{2 b^2 d \left(a^2-b^2\right)}-\frac{x \left(-6 a^2 C+4 a b B-b^2 C\right)}{2 b^4}+\frac{\left(-3 a^3 C+2 a^2 b B+2 a b^2 C-b^3 B\right) \sin (c+d x)}{b^3 d \left(a^2-b^2\right)}+\frac{2 a^2 \left(-3 a^3 C+2 a^2 b B+4 a b^2 C-3 b^3 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{3/2} (a+b)^{3/2}}",1,"(2*(-4*a*b*B + 6*a^2*C + b^2*C)*(c + d*x) - (8*a^2*(-2*a^2*b*B + 3*b^3*B + 3*a^3*C - 4*a*b^2*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + 4*b*(b*B - 2*a*C)*Sin[c + d*x] + (4*a^3*b*(b*B - a*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])) + b^2*C*Sin[2*(c + d*x)])/(4*b^4*d)","A",1
802,1,147,155,0.8184138,"\int \frac{\cos (c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{\frac{a^2 b (a C-b B) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}+\frac{2 a \left(2 a^3 C-a^2 b B-3 a b^2 C+2 b^3 B\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}+(c+d x) (b B-2 a C)+b C \sin (c+d x)}{b^3 d}","-\frac{a^2 (b B-a C) \sin (c+d x)}{b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{2 a \left(-2 a^3 C+a^2 b B+3 a b^2 C-2 b^3 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{x (b B-2 a C)}{b^3}+\frac{C \sin (c+d x)}{b^2 d}",1,"((b*B - 2*a*C)*(c + d*x) + (2*a*(-(a^2*b*B) + 2*b^3*B + 2*a^3*C - 3*a*b^2*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + b*C*Sin[c + d*x] + (a^2*b*(-(b*B) + a*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])))/(b^3*d)","A",1
803,1,119,122,0.5323062,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2,x]","\frac{-\frac{2 \left(a^3 C-2 a b^2 C+b^3 B\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}+\frac{a b (b B-a C) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}+C (c+d x)}{b^2 d}","-\frac{2 \left(a^3 C-2 a b^2 C+b^3 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{a (b B-a C) \sin (c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{C x}{b^2}",1,"(C*(c + d*x) - (2*(b^3*B + a^3*C - 2*a*b^2*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + (a*b*(b*B - a*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])))/(b^2*d)","A",1
804,1,97,100,0.3394828,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^2,x]","\frac{\frac{2 (a B-b C) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}+\frac{(a C-b B) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}}{d}","\frac{2 (a B-b C) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{3/2} (a+b)^{3/2}}-\frac{(b B-a C) \sin (c+d x)}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"((2*(a*B - b*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + ((-(b*B) + a*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])))/d","A",1
805,1,191,133,0.598192,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^2,x]","\frac{\cos (c+d x) (B \sec (c+d x)+C) \left(\frac{2 \left(a^3 C-2 a^2 b B+b^3 B\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}+\frac{a b (b B-a C) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}-B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{a^2 d (B+C \cos (c+d x))}","\frac{b (b B-a C) \sin (c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{B \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{2 \left(a^3 (-C)+2 a^2 b B-b^3 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}",1,"(Cos[c + d*x]*(C + B*Sec[c + d*x])*((2*(-2*a^2*b*B + b^3*B + a^3*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) - B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a*b*(b*B - a*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x]))))/(a^2*d*(B + C*Cos[c + d*x]))","A",1
806,1,240,189,1.8889892,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^2,x]","\frac{-\frac{2 b \left(2 a^3 C-3 a^2 b B-a b^2 C+2 b^3 B\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}+\frac{a b^2 (a C-b B) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}+a B \tan (c+d x)-a C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+a C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 b B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 b B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^3 d}","-\frac{(2 b B-a C) \tanh ^{-1}(\sin (c+d x))}{a^3 d}+\frac{\left(a^2 B+a b C-2 b^2 B\right) \tan (c+d x)}{a^2 d \left(a^2-b^2\right)}+\frac{b (b B-a C) \tan (c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{2 b \left(-2 a^3 C+3 a^2 b B+a b^2 C-2 b^3 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{3/2} (a+b)^{3/2}}",1,"((-2*b*(-3*a^2*b*B + 2*b^3*B + 2*a^3*C - a*b^2*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + 2*b*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - a*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 2*b*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + a*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a*b^2*(-(b*B) + a*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])) + a*B*Tan[c + d*x])/(a^3*d)","A",1
807,1,734,398,3.4291134,"\int \frac{\cos ^3(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{16 a^2 \left(12 a^5 C-6 a^4 b B-29 a^3 b^2 C+15 a^2 b^3 B+20 a b^4 C-12 b^5 B\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}+\frac{96 a^8 c C+96 a^8 C d x-48 a^7 b B c-48 a^7 b B d x-96 a^7 b C \sin (c+d x)+48 a^6 b^2 B \sin (c+d x)-72 a^6 b^2 C \sin (2 (c+d x))-136 a^6 b^2 c C-136 a^6 b^2 C d x+36 a^5 b^3 B \sin (2 (c+d x))+72 a^5 b^3 B c+72 a^5 b^3 B d x+160 a^5 b^3 C \sin (c+d x)-8 a^5 b^3 C \sin (3 (c+d x))-84 a^4 b^4 B \sin (c+d x)+4 a^4 b^4 B \sin (3 (c+d x))+130 a^4 b^4 C \sin (2 (c+d x))+a^4 b^4 C \sin (4 (c+d x))-12 a^4 b^4 c C-12 a^4 b^4 C d x-64 a^3 b^5 B \sin (2 (c+d x))-32 a^3 b^5 C \sin (c+d x)+16 a^3 b^5 C \sin (3 (c+d x))+8 a^2 b^6 B \sin (c+d x)-8 a^2 b^6 B \sin (3 (c+d x))-48 a^2 b^6 C \sin (2 (c+d x))-2 a^2 b^6 C \sin (4 (c+d x))+48 a^2 b^6 c C+48 a^2 b^6 C d x+16 a b \left(a^2-b^2\right)^2 (c+d x) \left(12 a^2 C-6 a b B+b^2 C\right) \cos (c+d x)+4 \left(b^3-a^2 b\right)^2 (c+d x) \left(12 a^2 C-6 a b B+b^2 C\right) \cos (2 (c+d x))+16 a b^7 B \sin (2 (c+d x))-24 a b^7 B c-24 a b^7 B d x-8 a b^7 C \sin (c+d x)-8 a b^7 C \sin (3 (c+d x))+4 b^8 B \sin (c+d x)+4 b^8 B \sin (3 (c+d x))+2 b^8 C \sin (2 (c+d x))+b^8 C \sin (4 (c+d x))+4 b^8 c C+4 b^8 C d x}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}}{16 b^5 d}","\frac{a (b B-a C) \sin (c+d x) \cos ^3(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{x \left(-12 a^2 C+6 a b B-b^2 C\right)}{2 b^5}+\frac{a \left(-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right) \sin (c+d x) \cos ^2(c+d x)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\left(-6 a^4 C+3 a^3 b B+10 a^2 b^2 C-6 a b^3 B-b^4 C\right) \sin (c+d x) \cos (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2}+\frac{\left(-12 a^5 C+6 a^4 b B+21 a^3 b^2 C-11 a^2 b^3 B-6 a b^4 C+2 b^5 B\right) \sin (c+d x)}{2 b^4 d \left(a^2-b^2\right)^2}+\frac{a^2 \left(-12 a^5 C+6 a^4 b B+29 a^3 b^2 C-15 a^2 b^3 B-20 a b^4 C+12 b^5 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{5/2} (a+b)^{5/2}}",1,"((16*a^2*(-6*a^4*b*B + 15*a^2*b^3*B - 12*b^5*B + 12*a^5*C - 29*a^3*b^2*C + 20*a*b^4*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (-48*a^7*b*B*c + 72*a^5*b^3*B*c - 24*a*b^7*B*c + 96*a^8*c*C - 136*a^6*b^2*c*C - 12*a^4*b^4*c*C + 48*a^2*b^6*c*C + 4*b^8*c*C - 48*a^7*b*B*d*x + 72*a^5*b^3*B*d*x - 24*a*b^7*B*d*x + 96*a^8*C*d*x - 136*a^6*b^2*C*d*x - 12*a^4*b^4*C*d*x + 48*a^2*b^6*C*d*x + 4*b^8*C*d*x + 16*a*b*(a^2 - b^2)^2*(-6*a*b*B + 12*a^2*C + b^2*C)*(c + d*x)*Cos[c + d*x] + 4*(-(a^2*b) + b^3)^2*(-6*a*b*B + 12*a^2*C + b^2*C)*(c + d*x)*Cos[2*(c + d*x)] + 48*a^6*b^2*B*Sin[c + d*x] - 84*a^4*b^4*B*Sin[c + d*x] + 8*a^2*b^6*B*Sin[c + d*x] + 4*b^8*B*Sin[c + d*x] - 96*a^7*b*C*Sin[c + d*x] + 160*a^5*b^3*C*Sin[c + d*x] - 32*a^3*b^5*C*Sin[c + d*x] - 8*a*b^7*C*Sin[c + d*x] + 36*a^5*b^3*B*Sin[2*(c + d*x)] - 64*a^3*b^5*B*Sin[2*(c + d*x)] + 16*a*b^7*B*Sin[2*(c + d*x)] - 72*a^6*b^2*C*Sin[2*(c + d*x)] + 130*a^4*b^4*C*Sin[2*(c + d*x)] - 48*a^2*b^6*C*Sin[2*(c + d*x)] + 2*b^8*C*Sin[2*(c + d*x)] + 4*a^4*b^4*B*Sin[3*(c + d*x)] - 8*a^2*b^6*B*Sin[3*(c + d*x)] + 4*b^8*B*Sin[3*(c + d*x)] - 8*a^5*b^3*C*Sin[3*(c + d*x)] + 16*a^3*b^5*C*Sin[3*(c + d*x)] - 8*a*b^7*C*Sin[3*(c + d*x)] + a^4*b^4*C*Sin[4*(c + d*x)] - 2*a^2*b^6*C*Sin[4*(c + d*x)] + b^8*C*Sin[4*(c + d*x)])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2))/(16*b^5*d)","A",1
808,1,232,280,2.1658789,"\int \frac{\cos ^2(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{a^3 b (b B-a C) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}+\frac{a^2 b \left(5 a^3 C-3 a^2 b B-8 a b^2 C+6 b^3 B\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}-\frac{2 a \left(6 a^5 C-2 a^4 b B-15 a^3 b^2 C+5 a^2 b^3 B+12 a b^4 C-6 b^5 B\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}+2 (c+d x) (b B-3 a C)+2 b C \sin (c+d x)}{2 b^4 d}","\frac{a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{\left(-3 a^2 C+a b B+2 b^2 C\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right)}-\frac{a^2 \left(-3 a^3 C+a^2 b B+6 a b^2 C-4 b^3 B\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{a \left(-6 a^5 C+2 a^4 b B+15 a^3 b^2 C-5 a^2 b^3 B-12 a b^4 C+6 b^5 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{x (b B-3 a C)}{b^4}",1,"(2*(b*B - 3*a*C)*(c + d*x) - (2*a*(-2*a^4*b*B + 5*a^2*b^3*B - 6*b^5*B + 6*a^5*C - 15*a^3*b^2*C + 12*a*b^4*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + 2*b*C*Sin[c + d*x] + (a^3*b*(b*B - a*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^2) + (a^2*b*(-3*a^2*b*B + 6*b^3*B + 5*a^3*C - 8*a*b^2*C)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])))/(2*b^4*d)","A",1
809,1,204,211,1.3698588,"\int \frac{\cos (c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{a^2 b (a C-b B) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}+\frac{a b \left(-3 a^3 C+a^2 b B+6 a b^2 C-4 b^3 B\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}+\frac{2 \left(2 a^5 C-5 a^3 b^2 C-a^2 b^3 B+6 a b^4 C-2 b^5 B\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}+2 C (c+d x)}{2 b^3 d}","-\frac{a^2 (b B-a C) \sin (c+d x)}{2 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{a \left(-3 a^3 C+a^2 b B+6 a b^2 C-4 b^3 B\right) \sin (c+d x)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(-2 a^5 C+5 a^3 b^2 C+a^2 b^3 B-6 a b^4 C+2 b^5 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{C x}{b^3}",1,"(2*C*(c + d*x) + (2*(-(a^2*b^3*B) - 2*b^5*B + 2*a^5*C - 5*a^3*b^2*C + 6*a*b^4*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (a^2*b*(-(b*B) + a*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^2) + (a*b*(a^2*b*B - 4*b^3*B - 3*a^3*C + 6*a*b^2*C)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])))/(2*b^3*d)","A",1
810,1,172,180,0.8482829,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3,x]","\frac{-\frac{2 \left(a^2 C-3 a b B+2 b^2 C\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}+\frac{\left(a^3 C+a^2 b B-4 a b^2 C+2 b^3 B\right) \sin (c+d x)}{b (a-b)^2 (a+b)^2 (a+b \cos (c+d x))}+\frac{a (b B-a C) \sin (c+d x)}{b (a-b) (a+b) (a+b \cos (c+d x))^2}}{2 d}","-\frac{\left(a^2 (-C)+3 a b B-2 b^2 C\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}+\frac{a (b B-a C) \sin (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\left(a^3 C+a^2 b B-4 a b^2 C+2 b^3 B\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}",1,"((-2*(-3*a*b*B + a^2*C + 2*b^2*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (a*(b*B - a*C)*Sin[c + d*x])/((a - b)*b*(a + b)*(a + b*Cos[c + d*x])^2) + ((a^2*b*B + 2*b^3*B + a^3*C - 4*a*b^2*C)*Sin[c + d*x])/((a - b)^2*b*(a + b)^2*(a + b*Cos[c + d*x])))/(2*d)","A",1
811,1,157,164,0.6802072,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{\left(a^2 C-3 a b B+2 b^2 C\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}-\frac{2 \left(2 a^2 B-3 a b C+b^2 B\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}+\frac{(a C-b B) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}}{2 d}","\frac{\left(2 a^2 B-3 a b C+b^2 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\left(a^2 (-C)+3 a b B-2 b^2 C\right) \sin (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{(b B-a C) \sin (c+d x)}{2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"((-2*(2*a^2*B + b^2*B - 3*a*b*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + ((-(b*B) + a*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^2) + ((-3*a*b*B + a^2*C + 2*b^2*C)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])))/(2*d)","A",1
812,1,269,214,1.3030511,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^3,x]","\frac{\cos (c+d x) (B \sec (c+d x)+C) \left(\frac{a^2 b (b B-a C) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}+\frac{a b \left(-3 a^3 C+5 a^2 b B-2 b^3 B\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}-\frac{2 \left(2 a^5 C-6 a^4 b B+a^3 b^2 C+5 a^2 b^3 B-2 b^5 B\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}-2 B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{2 a^3 d (B+C \cos (c+d x))}","\frac{B \tanh ^{-1}(\sin (c+d x))}{a^3 d}+\frac{b (b B-a C) \sin (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{b \left(-3 a^3 C+5 a^2 b B-2 b^3 B\right) \sin (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\left(-2 a^5 C+6 a^4 b B-a^3 b^2 C-5 a^2 b^3 B+2 b^5 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}",1,"(Cos[c + d*x]*(C + B*Sec[c + d*x])*((-2*(-6*a^4*b*B + 5*a^2*b^3*B - 2*b^5*B + 2*a^5*C + a^3*b^2*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) - 2*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a^2*b*(b*B - a*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^2) + (a*b*(5*a^2*b*B - 2*b^3*B - 3*a^3*C)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x]))))/(2*a^3*d*(B + C*Cos[c + d*x]))","A",1
813,1,352,299,5.8879961,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{a^2 b^2 (a C-b B) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}+\frac{a b^2 \left(5 a^3 C-7 a^2 b B-2 a b^2 C+4 b^3 B\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}-\frac{2 b \left(-6 a^5 C+12 a^4 b B+5 a^3 b^2 C-15 a^2 b^3 B-2 a b^4 C+6 b^5 B\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}+2 (3 b B-a C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 (a C-3 b B) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{2 a B \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{2 a B \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}}{2 a^4 d}","-\frac{(3 b B-a C) \tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac{b (b B-a C) \tan (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{b \left(-4 a^3 C+6 a^2 b B+a b^2 C-3 b^3 B\right) \tan (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(2 a^4 B+5 a^3 b C-11 a^2 b^2 B-2 a b^3 C+6 b^4 B\right) \tan (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}+\frac{b \left(-6 a^5 C+12 a^4 b B+5 a^3 b^2 C-15 a^2 b^3 B-2 a b^4 C+6 b^5 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{5/2} (a+b)^{5/2}}",1,"((-2*b*(12*a^4*b*B - 15*a^2*b^3*B + 6*b^5*B - 6*a^5*C + 5*a^3*b^2*C - 2*a*b^4*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + 2*(3*b*B - a*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(-3*b*B + a*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*a*B*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (2*a*B*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + (a^2*b^2*(-(b*B) + a*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^2) + (a*b^2*(-7*a^2*b*B + 4*b^3*B + 5*a^3*C - 2*a*b^2*C)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])))/(2*a^4*d)","A",1
814,1,232,303,1.0021245,"\int \cos (c+d x) \sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{b (a+b \cos (c+d x)) \left(\left(-16 a^2 C+28 a b B+115 b^2 C\right) \sin (c+d x)+3 b (2 (a C+7 b B) \sin (2 (c+d x))+5 b C \sin (3 (c+d x)))\right)+4 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b^2 \left(2 a^2 C+49 a b B+25 b^2 C\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(8 a^3 C-14 a^2 b B+19 a b^2 C+63 b^3 B\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{210 b^3 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(-8 a^2 C+14 a b B-25 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b^2 d}+\frac{2 \left(a^2-b^2\right) \left(-8 a^2 C+14 a b B-25 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-8 a^3 C+14 a^2 b B-19 a b^2 C-63 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (7 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}",1,"(4*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(49*a*b*B + 2*a^2*C + 25*b^2*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (-14*a^2*b*B + 63*b^3*B + 8*a^3*C + 19*a*b^2*C)*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x])*((28*a*b*B - 16*a^2*C + 115*b^2*C)*Sin[c + d*x] + 3*b*(2*(7*b*B + a*C)*Sin[2*(c + d*x)] + 5*b*C*Sin[3*(c + d*x)])))/(210*b^3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
815,1,179,231,0.8835352,"\int \sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(\left(-2 a^2 C+5 a b B+9 b^2 C\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)+b^2 (7 a C+5 b B) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)+2 b \sin (c+d x) (a+b \cos (c+d x)) (a C+5 b B+3 b C \cos (c+d x))}{15 b^2 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(a^2-b^2\right) (5 b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-2 a^2 C+5 a b B+9 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (5 b B-2 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}",1,"(2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(5*b*B + 7*a*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (5*a*b*B - 2*a^2*C + 9*b^2*C)*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + 2*b*(a + b*Cos[c + d*x])*(5*b*B + a*C + 3*b*C*Cos[c + d*x])*Sin[c + d*x])/(15*b^2*d*Sqrt[a + b*Cos[c + d*x]])","A",1
816,1,146,171,0.5820459,"\int \sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{-2 C \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+2 (a+b) (a C+3 b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+2 b C \sin (c+d x) (a+b \cos (c+d x))}{3 b d \sqrt{a+b \cos (c+d x)}}","-\frac{2 C \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 (a C+3 b B) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}",1,"(2*(a + b)*(3*b*B + a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 2*(a^2 - b^2)*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + 2*b*C*(a + b*Cos[c + d*x])*Sin[c + d*x])/(3*b*d*Sqrt[a + b*Cos[c + d*x]])","A",1
817,1,107,178,2.3900956,"\int \sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(B \left(b F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+a \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)+C (a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)}{d \sqrt{a+b \cos (c+d x)}}","\frac{2 b B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 a B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*((a + b)*C*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + B*(b*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + a*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])))/(d*Sqrt[a + b*Cos[c + d*x]])","A",1
818,1,372,213,10.554843,"\int \sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{\frac{2 (4 a C+b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+4 B \tan (c+d x) \sqrt{a+b \cos (c+d x)}-\frac{2 i B \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}+\frac{8 b C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{4 d}","\frac{(a B+2 b C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{(2 a C+b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{B \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{d}-\frac{B \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"((8*b*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(b*B + 4*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*B*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b*Sqrt[-(a + b)^(-1)]) + 4*B*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*d)","C",1
819,1,420,292,4.2643134,"\int \sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\frac{2 \left(8 a^2 B+4 a b C-3 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a \sqrt{a+b \cos (c+d x)}}-\frac{2 i (4 a C+b B) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a^2 b \sqrt{-\frac{1}{a+b}}}+\frac{4 \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)} ((4 a C+b B) \cos (c+d x)+2 a B)}{a}+\frac{8 b B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{16 d}","\frac{\left(4 a^2 B+4 a b C-b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{a+b \cos (c+d x)}}+\frac{(4 a C+b B) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a d}+\frac{(4 a C+3 b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}-\frac{(4 a C+b B) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{B \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d}",1,"((8*b*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(8*a^2*B - 3*b^2*B + 4*a*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(a*Sqrt[a + b*Cos[c + d*x]]) - ((2*I)*(b*B + 4*a*C)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a^2*b*Sqrt[-(a + b)^(-1)]) + (4*Sqrt[a + b*Cos[c + d*x]]*(2*a*B + (b*B + 4*a*C)*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/a)/(16*d)","C",1
820,1,291,378,1.5197018,"\int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{b (a+b \cos (c+d x)) \left(b \left(2 \left(6 a^2 C+144 a b B+133 b^2 C\right) \sin (2 (c+d x))+5 b (2 (10 a C+9 b B) \sin (3 (c+d x))+7 b C \sin (4 (c+d x)))\right)+\left(-32 a^3 C+72 a^2 b B+804 a b^2 C+690 b^3 B\right) \sin (c+d x)\right)+8 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b^2 \left(2 a^3 C+153 a^2 b B+186 a b^2 C+75 b^3 B\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(8 a^4 C-18 a^3 b B+33 a^2 b^2 C+246 a b^3 B+147 b^4 C\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{1260 b^3 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(-8 a^2 C+18 a b B-49 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}-\frac{2 \left(-8 a^3 C+18 a^2 b B-39 a b^2 C-75 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^2 d}+\frac{2 \left(a^2-b^2\right) \left(-8 a^3 C+18 a^2 b B-39 a b^2 C-75 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-8 a^4 C+18 a^3 b B-33 a^2 b^2 C-246 a b^3 B-147 b^4 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}",1,"(8*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(153*a^2*b*B + 75*b^3*B + 2*a^3*C + 186*a*b^2*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*C)*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x])*((72*a^2*b*B + 690*b^3*B - 32*a^3*C + 804*a*b^2*C)*Sin[c + d*x] + b*(2*(144*a*b*B + 6*a^2*C + 133*b^2*C)*Sin[2*(c + d*x)] + 5*b*(2*(9*b*B + 10*a*C)*Sin[3*(c + d*x)] + 7*b*C*Sin[4*(c + d*x)]))))/(1260*b^3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
821,1,233,297,1.0875727,"\int (a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{b (a+b \cos (c+d x)) \left(\left(12 a^2 C+168 a b B+115 b^2 C\right) \sin (c+d x)+3 b (2 (8 a C+7 b B) \sin (2 (c+d x))+5 b C \sin (3 (c+d x)))\right)+4 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b^2 \left(51 a^2 C+84 a b B+25 b^2 C\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(-6 a^3 C+21 a^2 b B+82 a b^2 C+63 b^3 B\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{210 b^2 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(-6 a^2 C+21 a b B+25 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b d}-\frac{2 \left(a^2-b^2\right) \left(-6 a^2 C+21 a b B+25 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-6 a^3 C+21 a^2 b B+82 a b^2 C+63 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}",1,"(4*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(84*a*b*B + 51*a^2*C + 25*b^2*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (21*a^2*b*B + 63*b^3*B - 6*a^3*C + 82*a*b^2*C)*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x])*((168*a*b*B + 12*a^2*C + 115*b^2*C)*Sin[c + d*x] + 3*b*(2*(7*b*B + 8*a*C)*Sin[2*(c + d*x)] + 5*b*C*Sin[3*(c + d*x)])))/(210*b^2*d*Sqrt[a + b*Cos[c + d*x]])","A",1
822,1,203,225,0.7725654,"\int (a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{2 \left(b \left(15 a^2 B+12 a b C+5 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(3 a^2 C+20 a b B+9 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)+b \sin (c+d x) (a+b \cos (c+d x)) (6 a C+5 b B+3 b C \cos (c+d x))\right)}{15 b d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(a^2-b^2\right) (3 a C+5 b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(3 a^2 C+20 a b B+9 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (3 a C+5 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}",1,"(2*(b*(15*a^2*B + 5*b^2*B + 12*a*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (20*a*b*B + 3*a^2*C + 9*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)]) + b*(a + b*Cos[c + d*x])*(5*b*B + 6*a*C + 3*b*C*Cos[c + d*x])*Sin[c + d*x]))/(15*b*d*Sqrt[a + b*Cos[c + d*x]])","A",1
823,1,406,236,2.3966606,"\int (a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{\frac{4 \left(3 a^2 C+6 a b B+b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 \left(6 a^2 B+4 a b C+3 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 i (4 a C+3 b B) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}+4 b C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{6 d}","\frac{2 \left(a^2 (-C)+3 a b B+b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 a^2 B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 (4 a C+3 b B) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 b C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}",1,"((4*(6*a*b*B + 3*a^2*C + b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(6*a^2*B + 3*b^2*B + 4*a*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + ((2*I)*(3*b*B + 4*a*C)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b*Sqrt[-(a + b)^(-1)]) + 4*b*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(6*d)","C",1
824,1,398,232,2.5247326,"\int (a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{\frac{2 \left(4 a^2 C+5 a b B+2 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{8 b (2 a C+b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 i (2 b C-a B) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}+4 a B \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}","\frac{\left(a^2 B+2 a b C+2 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{(a B-2 b C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{a (2 a C+3 b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{a B \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{d}",1,"((8*b*(b*B + 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(5*a*b*B + 4*a^2*C + 2*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + ((2*I)*(-(a*B) + 2*b*C)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b*Sqrt[-(a + b)^(-1)]) + 4*a*B*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*d)","C",1
825,1,422,295,4.9035615,"\int (a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\frac{2 \left(8 a^2 B+20 a b C+b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{8 b (a B+4 b C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+4 \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)} ((4 a C+5 b B) \cos (c+d x)+2 a B)-\frac{2 i (4 a C+5 b B) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}}{16 d}","\frac{\left(4 a^2 C+7 a b B+8 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{\left(4 a^2 B+12 a b C+3 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{(4 a C+5 b B) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{(4 a C+5 b B) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{a B \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d}",1,"((8*b*(a*B + 4*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(8*a^2*B + b^2*B + 20*a*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(5*b*B + 4*a*C)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b*Sqrt[-(a + b)^(-1)]) + 4*Sqrt[a + b*Cos[c + d*x]]*(2*a*B + (5*b*B + 4*a*C)*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(16*d)","C",1
826,1,634,375,6.5905926,"\int (a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{\sqrt{a+b \cos (c+d x)} \left(\frac{\sec (c+d x) \left(16 a^2 B \sin (c+d x)+30 a b C \sin (c+d x)+3 b^2 B \sin (c+d x)\right)}{24 a}+\frac{1}{12} \sec ^2(c+d x) (6 a C \sin (c+d x)+7 b B \sin (c+d x))+\frac{1}{3} a B \tan (c+d x) \sec ^2(c+d x)\right)}{d}+\frac{\frac{2 \left(24 a^2 b C+28 a b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \left(-16 a^2 b B-30 a b^2 C-3 b^3 B\right) \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{b \cos (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left(2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right)}+\frac{2 \left(48 a^3 C+56 a^2 b B+6 a b^2 C-9 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{96 a d}","\frac{\left(16 a^2 B+30 a b C+3 b^2 B\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 a d}+\frac{\left(16 a^2 B+42 a b C+17 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(16 a^2 B+30 a b C+3 b^2 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(8 a^3 C+12 a^2 b B+6 a b^2 C-b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 a d \sqrt{a+b \cos (c+d x)}}+\frac{(6 a C+7 b B) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{12 d}+\frac{a B \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}",1,"((2*(28*a*b^2*B + 24*a^2*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(56*a^2*b*B - 9*b^3*B + 48*a^3*C + 6*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(-16*a^2*b*B - 3*b^3*B - 30*a*b^2*C)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2)))/(96*a*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]^2*(7*b*B*Sin[c + d*x] + 6*a*C*Sin[c + d*x]))/12 + (Sec[c + d*x]*(16*a^2*B*Sin[c + d*x] + 3*b^2*B*Sin[c + d*x] + 30*a*b*C*Sin[c + d*x]))/(24*a) + (a*B*Sec[c + d*x]^2*Tan[c + d*x])/3))/d","C",1
827,1,357,462,2.0957412,"\int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{b (a+b \cos (c+d x)) \left(b \left(5 b \left(\left(452 a^2 C+836 a b B+513 b^2 C\right) \sin (3 (c+d x))+7 b ((46 a C+22 b B) \sin (4 (c+d x))+9 b C \sin (5 (c+d x)))\right)+4 \left(30 a^3 C+1650 a^2 b B+3095 a b^2 C+1463 b^3 B\right) \sin (2 (c+d x))\right)+\left(-320 a^4 C+880 a^3 b B+18660 a^2 b^2 C+32868 a b^3 B+13050 b^4 C\right) \sin (c+d x)\right)+16 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b^2 \left(10 a^4 C+1705 a^3 b B+3315 a^2 b^2 C+2871 a b^3 B+675 b^4 C\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(40 a^5 C-110 a^4 b B+255 a^3 b^2 C+3069 a^2 b^3 B+3705 a b^4 C+1617 b^5 B\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{27720 b^3 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(-8 a^2 C+22 a b B-81 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}-\frac{2 \left(-40 a^3 C+110 a^2 b B-335 a b^2 C-539 b^3 B\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3465 b^2 d}-\frac{2 \left(-40 a^4 C+110 a^3 b B-285 a^2 b^2 C-1254 a b^3 B-675 b^4 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3465 b^2 d}+\frac{2 \left(a^2-b^2\right) \left(-40 a^4 C+110 a^3 b B-285 a^2 b^2 C-1254 a b^3 B-675 b^4 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3465 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-40 a^5 C+110 a^4 b B-255 a^3 b^2 C-3069 a^2 b^3 B-3705 a b^4 C-1617 b^5 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3465 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}",1,"(16*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(1705*a^3*b*B + 2871*a*b^3*B + 10*a^4*C + 3315*a^2*b^2*C + 675*b^4*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (-110*a^4*b*B + 3069*a^2*b^3*B + 1617*b^5*B + 40*a^5*C + 255*a^3*b^2*C + 3705*a*b^4*C)*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x])*((880*a^3*b*B + 32868*a*b^3*B - 320*a^4*C + 18660*a^2*b^2*C + 13050*b^4*C)*Sin[c + d*x] + b*(4*(1650*a^2*b*B + 1463*b^3*B + 30*a^3*C + 3095*a*b^2*C)*Sin[2*(c + d*x)] + 5*b*((836*a*b*B + 452*a^2*C + 513*b^2*C)*Sin[3*(c + d*x)] + 7*b*((22*b*B + 46*a*C)*Sin[4*(c + d*x)] + 9*b*C*Sin[5*(c + d*x)])))))/(27720*b^3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
828,1,291,372,1.5941421,"\int (a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{b (a+b \cos (c+d x)) \left(b \left(\left(300 a^2 C+540 a b B+266 b^2 C\right) \sin (2 (c+d x))+5 b (2 (19 a C+9 b B) \sin (3 (c+d x))+7 b C \sin (4 (c+d x)))\right)+2 \left(20 a^3 C+540 a^2 b B+747 a b^2 C+345 b^3 B\right) \sin (c+d x)\right)+8 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b^2 \left(155 a^3 C+405 a^2 b B+261 a b^2 C+75 b^3 B\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(-10 a^4 C+45 a^3 b B+279 a^2 b^2 C+435 a b^3 B+147 b^4 C\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{1260 b^2 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(-10 a^2 C+45 a b B+49 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b d}+\frac{2 \left(-10 a^3 C+45 a^2 b B+114 a b^2 C+75 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b d}-\frac{2 \left(a^2-b^2\right) \left(-10 a^3 C+45 a^2 b B+114 a b^2 C+75 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-10 a^4 C+45 a^3 b B+279 a^2 b^2 C+435 a b^3 B+147 b^4 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (9 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}",1,"(8*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(405*a^2*b*B + 75*b^3*B + 155*a^3*C + 261*a*b^2*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (45*a^3*b*B + 435*a*b^3*B - 10*a^4*C + 279*a^2*b^2*C + 147*b^4*C)*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x])*(2*(540*a^2*b*B + 345*b^3*B + 20*a^3*C + 747*a*b^2*C)*Sin[c + d*x] + b*((540*a*b*B + 300*a^2*C + 266*b^2*C)*Sin[2*(c + d*x)] + 5*b*(2*(9*b*B + 19*a*C)*Sin[3*(c + d*x)] + 7*b*C*Sin[4*(c + d*x)]))))/(1260*b^2*d*Sqrt[a + b*Cos[c + d*x]])","A",1
829,1,254,288,1.0995211,"\int (a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{b \sin (c+d x) (a+b \cos (c+d x)) \left(90 a^2 C+6 b (15 a C+7 b B) \cos (c+d x)+154 a b B+15 b^2 C \cos (2 (c+d x))+65 b^2 C\right)+2 b \left(105 a^3 B+135 a^2 b C+119 a b^2 B+25 b^3 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+2 \left(15 a^3 C+161 a^2 b B+145 a b^2 C+63 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)}{105 b d \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(15 a^2 C+56 a b B+25 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 d}-\frac{2 \left(a^2-b^2\right) \left(15 a^2 C+56 a b B+25 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(15 a^3 C+161 a^2 b B+145 a b^2 C+63 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}",1,"(2*b*(105*a^3*B + 119*a*b^2*B + 135*a^2*b*C + 25*b^3*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + 2*(161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)]) + b*(a + b*Cos[c + d*x])*(154*a*b*B + 90*a^2*C + 65*b^2*C + 6*b*(7*b*B + 15*a*C)*Cos[c + d*x] + 15*b^2*C*Cos[2*(c + d*x)])*Sin[c + d*x])/(105*b*d*Sqrt[a + b*Cos[c + d*x]])","A",1
830,1,453,292,2.7874022,"\int (a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{\frac{2 i \left(23 a^2 C+35 a b B+9 b^2 C\right) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\cos (c+d x)+1)}{a-b}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}+\frac{4 \left(15 a^3 C+45 a^2 b B+17 a b^2 C+5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 \left(30 a^3 B+23 a^2 b C+35 a b^2 B+9 b^3 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+4 b \sin (c+d x) \sqrt{a+b \cos (c+d x)} (11 a C+5 b B+3 b C \cos (c+d x))}{30 d}","\frac{2 a^3 B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(23 a^2 C+35 a b B+9 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \left(-8 a^3 C+10 a^2 b B+8 a b^2 C+5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{a+b \cos (c+d x)}}+\frac{2 b (8 a C+5 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{2 b C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}",1,"((4*(45*a^2*b*B + 5*b^3*B + 15*a^3*C + 17*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(30*a^3*B + 35*a*b^2*B + 23*a^2*b*C + 9*b^3*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + ((2*I)*(35*a*b*B + 23*a^2*C + 9*b^2*C)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Cos[c + d*x]))/(a - b))]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b*Sqrt[-(a + b)^(-1)]) + 4*b*Sqrt[a + b*Cos[c + d*x]]*(5*b*B + 11*a*C + 3*b*C*Cos[c + d*x])*Sin[c + d*x])/(30*d)","C",1
831,1,442,296,3.9174516,"\int (a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{4 \tan (c+d x) \sqrt{a+b \cos (c+d x)} \left(3 a^2 B+2 b^2 C \cos (c+d x)\right)+\frac{8 b \left(9 a^2 C+9 a b B+b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 i \left(-3 a^2 B+14 a b C+6 b^2 B\right) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}+\frac{2 \left(12 a^3 C+27 a^2 b B+14 a b^2 C+6 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{12 d}","-\frac{\left(3 a^2 B-14 a b C-6 b^2 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{a^2 (2 a C+5 b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{\left(3 a^3 B+4 a^2 b C+12 a b^2 B+2 b^3 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}-\frac{b (3 a B-2 b C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{a B \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}",1,"((8*b*(9*a*b*B + 9*a^2*C + b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(27*a^2*b*B + 6*b^3*B + 12*a^3*C + 14*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + ((2*I)*(-3*a^2*B + 6*b^2*B + 14*a*b*C)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b*Sqrt[-(a + b)^(-1)]) + 4*Sqrt[a + b*Cos[c + d*x]]*(3*a^2*B + 2*b^2*C*Cos[c + d*x])*Tan[c + d*x])/(12*d)","C",1
832,1,451,315,5.7482846,"\int (a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\frac{8 b \left(a^2 B+12 a b C+4 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 i \left(-4 a^2 C-9 a b B+8 b^2 C\right) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}+\frac{2 \left(8 a^3 B+36 a^2 b C+21 a b^2 B+8 b^3 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+4 a \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)} ((4 a C+9 b B) \cos (c+d x)+2 a B)}{16 d}","-\frac{\left(4 a^2 C+9 a b B-8 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{a \left(4 a^2 B+20 a b C+15 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{\left(4 a^3 C+11 a^2 b B+16 a b^2 C+8 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{a (4 a C+7 b B) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{a B \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}",1,"((8*b*(a^2*B + 4*b^2*B + 12*a*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(8*a^3*B + 21*a*b^2*B + 36*a^2*b*C + 8*b^3*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + ((2*I)*(-9*a*b*B - 4*a^2*C + 8*b^2*C)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b*Sqrt[-(a + b)^(-1)]) + 4*a*Sqrt[a + b*Cos[c + d*x]]*(2*a*B + (9*b*B + 4*a*C)*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(16*d)","C",1
833,1,486,376,6.053123,"\int (a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{\frac{8 b \left(6 a^2 C+13 a b B+24 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+4 \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)} \left(\left(8 a^2 B+27 a b C+\frac{33 b^2 B}{2}\right) \sin (2 (c+d x))+8 a^2 B \tan (c+d x)+2 a (6 a C+13 b B) \sin (c+d x)\right)-\frac{2 i \left(16 a^2 B+54 a b C+33 b^2 B\right) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}+\frac{2 \left(48 a^3 C+104 a^2 b B+126 a b^2 C-3 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{96 d}","\frac{\left(16 a^2 B+54 a b C+33 b^2 B\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d}-\frac{\left(16 a^2 B+54 a b C+33 b^2 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(16 a^3 B+66 a^2 b C+59 a b^2 B+48 b^3 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{a+b \cos (c+d x)}}+\frac{\left(8 a^3 C+20 a^2 b B+30 a b^2 C+5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 d \sqrt{a+b \cos (c+d x)}}+\frac{a (2 a C+3 b B) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{a B \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}",1,"((8*b*(13*a*b*B + 6*a^2*C + 24*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(104*a^2*b*B - 3*b^3*B + 48*a^3*C + 126*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(16*a^2*B + 33*b^2*B + 54*a*b*C)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b*Sqrt[-(a + b)^(-1)]) + 4*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*(2*a*(13*b*B + 6*a*C)*Sin[c + d*x] + (8*a^2*B + (33*b^2*B)/2 + 27*a*b*C)*Sin[2*(c + d*x)] + 8*a^2*B*Tan[c + d*x]))/(96*d)","C",1
834,1,729,465,6.7479426,"\int (a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{\sqrt{a+b \cos (c+d x)} \left(\frac{1}{96} \sec ^2(c+d x) \left(36 a^2 B \sin (c+d x)+104 a b C \sin (c+d x)+59 b^2 B \sin (c+d x)\right)+\frac{1}{24} \sec ^3(c+d x) \left(8 a^2 C \sin (c+d x)+17 a b B \sin (c+d x)\right)+\frac{1}{4} a^2 B \tan (c+d x) \sec ^3(c+d x)+\frac{\sec (c+d x) \left(128 a^3 C \sin (c+d x)+284 a^2 b B \sin (c+d x)+264 a b^2 C \sin (c+d x)+15 b^3 B \sin (c+d x)\right)}{192 a}\right)}{d}+\frac{\frac{2 \left(144 a^3 b B+416 a^2 b^2 C+236 a b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \left(-128 a^3 b C-284 a^2 b^2 B-264 a b^3 C-15 b^4 B\right) \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{b \cos (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left(2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right)}+\frac{2 \left(288 a^4 B+832 a^3 b C+436 a^2 b^2 B-24 a b^3 C-45 b^4 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{768 a d}","\frac{\left(36 a^2 B+104 a b C+59 b^2 B\right) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{96 d}+\frac{\left(128 a^3 C+284 a^2 b B+264 a b^2 C+15 b^3 B\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{192 a d}+\frac{\left(128 a^3 C+356 a^2 b B+472 a b^2 C+133 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{192 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(128 a^3 C+284 a^2 b B+264 a b^2 C+15 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{192 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(48 a^4 B+160 a^3 b C+120 a^2 b^2 B+40 a b^3 C-5 b^4 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{64 a d \sqrt{a+b \cos (c+d x)}}+\frac{a (8 a C+11 b B) \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{a B \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}",1,"((2*(144*a^3*b*B + 236*a*b^3*B + 416*a^2*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(288*a^4*B + 436*a^2*b^2*B - 45*b^4*B + 832*a^3*b*C - 24*a*b^3*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(-284*a^2*b^2*B - 15*b^4*B - 128*a^3*b*C - 264*a*b^3*C)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2)))/(768*a*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]^3*(17*a*b*B*Sin[c + d*x] + 8*a^2*C*Sin[c + d*x]))/24 + (Sec[c + d*x]^2*(36*a^2*B*Sin[c + d*x] + 59*b^2*B*Sin[c + d*x] + 104*a*b*C*Sin[c + d*x]))/96 + (Sec[c + d*x]*(284*a^2*b*B*Sin[c + d*x] + 15*b^3*B*Sin[c + d*x] + 128*a^3*C*Sin[c + d*x] + 264*a*b^2*C*Sin[c + d*x]))/(192*a) + (a^2*B*Sec[c + d*x]^3*Tan[c + d*x])/4))/d","C",0
835,1,180,246,0.9054598,"\int \frac{\cos (c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(\left(8 a^2 C-10 a b B+9 b^2 C\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)+b^2 (2 a C+5 b B) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)+2 b \sin (c+d x) (a+b \cos (c+d x)) (-4 a C+5 b B+3 b C \cos (c+d x))}{15 b^3 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(-8 a^2 C+10 a b B-9 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \left(-8 a^3 C+10 a^2 b B-7 a b^2 C+5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 (5 b B-4 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b d}",1,"(2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(5*b*B + 2*a*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (-10*a*b*B + 8*a^2*C + 9*b^2*C)*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + 2*b*(a + b*Cos[c + d*x])*(5*b*B - 4*a*C + 3*b*C*Cos[c + d*x])*Sin[c + d*x])/(15*b^3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
836,1,154,183,0.70091,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \left(2 a^2 C-3 a b B+b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-2 (a+b) (2 a C-3 b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+2 b C \sin (c+d x) (a+b \cos (c+d x))}{3 b^2 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(-2 a^2 C+3 a b B-b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 (3 b B-2 a C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d}",1,"(-2*(a + b)*(-3*b*B + 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + 2*(-3*a*b*B + 2*a^2*C + b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + 2*b*C*(a + b*Cos[c + d*x])*Sin[c + d*x])/(3*b^2*d*Sqrt[a + b*Cos[c + d*x]])","A",1
837,1,93,130,3.2590869,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left((b B-a C) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+C (a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)}{b d \sqrt{a+b \cos (c+d x)}}","\frac{2 (b B-a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*((a + b)*C*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + (b*B - a*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/(b*d*Sqrt[a + b*Cos[c + d*x]])","A",1
838,1,81,118,0.1922118,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(B \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+C F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)}{d \sqrt{a+b \cos (c+d x)}}","\frac{2 B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}",1,"(2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(C*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + B*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)]))/(d*Sqrt[a + b*Cos[c + d*x]])","A",1
839,1,320,216,6.5184616,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\frac{2 (4 a C-3 b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+4 B \tan (c+d x) \sqrt{a+b \cos (c+d x)}-\frac{2 i B \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}}{4 a d}","-\frac{(b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}+\frac{B \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{a d}+\frac{B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{B \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"((2*(-3*b*B + 4*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*B*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b*Sqrt[-(a + b)^(-1)]) + 4*B*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*a*d)","C",1
840,1,420,299,5.7947792,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\frac{2 \left(8 a^2 B-12 a b C+9 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+4 \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)} ((4 a C-3 b B) \cos (c+d x)+2 a B)+\frac{2 i (3 b B-4 a C) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}+\frac{8 a b B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{16 a^2 d}","\frac{\left(4 a^2 B-4 a b C+3 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{(3 b B-4 a C) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a^2 d}+\frac{(3 b B-4 a C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{(b B-4 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{a+b \cos (c+d x)}}+\frac{B \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 a d}",1,"((8*a*b*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(8*a^2*B + 9*b^2*B - 12*a*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + ((2*I)*(3*b*B - 4*a*C)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b*Sqrt[-(a + b)^(-1)]) + 4*Sqrt[a + b*Cos[c + d*x]]*(2*a*B + (-3*b*B + 4*a*C)*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(16*a^2*d)","C",1
841,1,304,387,1.799173,"\int \frac{\cos ^2(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\frac{30 a^3 b (a C-b B) \sin (c+d x)}{b^2-a^2}+\frac{2 b^2 \left(12 a^3 C-10 a^2 b B+3 a b^2 C-5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{(a-b) (a+b)}+\frac{2 \left(48 a^4 C-40 a^3 b B-24 a^2 b^2 C+25 a b^3 B-9 b^4 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)}{(a-b) (a+b)}+3 b^2 C \sin (2 (c+d x)) (a+b \cos (c+d x))+2 b (5 b B-9 a C) \sin (c+d x) (a+b \cos (c+d x))}{15 b^4 d \sqrt{a+b \cos (c+d x)}}","\frac{2 a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-6 a^2 C+5 a b B+b^2 C\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b^2 d \left(a^2-b^2\right)}+\frac{2 \left(-24 a^3 C+20 a^2 b B+9 a b^2 C-5 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^3 d \left(a^2-b^2\right)}+\frac{2 \left(-48 a^3 C+40 a^2 b B-12 a b^2 C+5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-48 a^4 C+40 a^3 b B+24 a^2 b^2 C-25 a b^3 B+9 b^4 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^4 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"((2*b^2*(-10*a^2*b*B - 5*b^3*B + 12*a^3*C + 3*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/((a - b)*(a + b)) + (2*(-40*a^3*b*B + 25*a*b^3*B + 48*a^4*C - 24*a^2*b^2*C - 9*b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/((a - b)*(a + b)) + (30*a^3*b*(-(b*B) + a*C)*Sin[c + d*x])/(-a^2 + b^2) + 2*b*(5*b*B - 9*a*C)*(a + b*Cos[c + d*x])*Sin[c + d*x] + 3*b^2*C*(a + b*Cos[c + d*x])*Sin[2*(c + d*x)])/(15*b^4*d*Sqrt[a + b*Cos[c + d*x]])","A",1
842,1,189,262,1.456569,"\int \frac{\cos (c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 \left(b \sin (c+d x) \left(\frac{a \left(-4 a^2 C+3 a b B+b^2 C\right)}{b^2-a^2}+b C \cos (c+d x)\right)+\frac{\sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left((a-b) \left(8 a^2 C-6 a b B+b^2 C\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(-8 a^3 C+6 a^2 b B+5 a b^2 C-3 b^3 B\right) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)}{a-b}\right)}{3 b^3 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 a^2 (b B-a C) \sin (c+d x)}{b^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-8 a^2 C+6 a b B-b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-8 a^3 C+6 a^2 b B+5 a b^2 C-3 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^2 d}",1,"(2*((Sqrt[(a + b*Cos[c + d*x])/(a + b)]*((6*a^2*b*B - 3*b^3*B - 8*a^3*C + 5*a*b^2*C)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + (a - b)*(-6*a*b*B + 8*a^2*C + b^2*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/(a - b) + b*((a*(3*a*b*B - 4*a^2*C + b^2*C))/(-a^2 + b^2) + b*C*Cos[c + d*x])*Sin[c + d*x]))/(3*b^3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
843,1,170,204,0.8101939,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 \left(\left(a^2-b^2\right) (2 a C-b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-\left((a+b) \left(2 a^2 C-a b B-b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)+a b (a C-b B) \sin (c+d x)\right)}{b^2 d (a-b) (a+b) \sqrt{a+b \cos (c+d x)}}","\frac{2 a (b B-a C) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-2 a^2 C+a b B+b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b^2 d \sqrt{a+b \cos (c+d x)}}",1,"(-2*(-((a + b)*(-(a*b*B) + 2*a^2*C - b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)]) + (a^2 - b^2)*(-(b*B) + 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + a*b*(-(b*B) + a*C)*Sin[c + d*x]))/((a - b)*b^2*(a + b)*d*Sqrt[a + b*Cos[c + d*x]])","A",1
844,1,151,185,0.5650949,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 \left(C \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+b (a C-b B) \sin (c+d x)-\left((a+b) (a C-b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{b d (a-b) (a+b) \sqrt{a+b \cos (c+d x)}}","-\frac{2 (b B-a C) \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 (b B-a C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{a+b \cos (c+d x)}}",1,"(2*(-((a + b)*(-(b*B) + a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)]) + (a^2 - b^2)*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + b*(-(b*B) + a*C)*Sin[c + d*x]))/((a - b)*b*(a + b)*d*Sqrt[a + b*Cos[c + d*x]])","A",1
845,1,460,190,3.9035419,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\cos (c+d x) (B \sec (c+d x)+C) \left(\frac{4 b (b B-a C) \sin (c+d x)}{\left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{\frac{2 \left(2 a^2 B+a b C-3 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{4 a (a C-b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i (b B-a C) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}}{(b-a) (a+b)}\right)}{2 a d (B+C \cos (c+d x))}","\frac{2 b (b B-a C) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 (b B-a C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}",1,"(Cos[c + d*x]*(C + B*Sec[c + d*x])*(-(((4*a*(-(b*B) + a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(2*a^2*B - 3*b^2*B + a*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(b*B - a*C)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b*Sqrt[-(a + b)^(-1)]))/((-a + b)*(a + b))) + (4*b*(b*B - a*C)*Sin[c + d*x])/((a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]])))/(2*a*d*(B + C*Cos[c + d*x]))","C",1
846,1,482,303,5.6595811,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\frac{4 \tan (c+d x) \left(b \left(a^2 B+2 a b C-3 b^2 B\right) \cos (c+d x)+a B \left(a^2-b^2\right)\right)}{\left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\frac{2 i \left(a^2 B+2 a b C-3 b^2 B\right) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\cos (c+d x)+1)}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}+\frac{2 \left(4 a^3 C-7 a^2 b B-6 a b^2 C+9 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{8 a b (a C-b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{(a-b) (a+b)}}{4 a^2 d}","\frac{b \left(a^2 B+2 a b C-3 b^2 B\right) \sin (c+d x)}{a^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{\left(a^2 B+2 a b C-3 b^2 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{(3 b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{B \tan (c+d x)}{a d \sqrt{a+b \cos (c+d x)}}+\frac{B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}",1,"(((-8*a*b*(-(b*B) + a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(-7*a^2*b*B + 9*b^3*B + 4*a^3*C - 6*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + ((2*I)*(a^2*B - 3*b^2*B + 2*a*b*C)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Cos[c + d*x]))/(a - b))]*Csc[c + d*x]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b*Sqrt[-(a + b)^(-1)]))/((a - b)*(a + b)) + (4*(a*(a^2 - b^2)*B + b*(a^2*B - 3*b^2*B + 2*a*b*C)*Cos[c + d*x])*Tan[c + d*x])/((a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]))/(4*a^2*d)","C",1
847,1,334,413,2.9941102,"\int \frac{\cos ^2(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(\frac{\left(\frac{a+b \cos (c+d x)}{a+b}\right)^{3/2} \left(b^2 \left(-4 a^4 C+2 a^3 b B+7 a^2 b^2 C-6 a b^3 B+b^4 C\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-\left(16 a^5 C-8 a^4 b B-28 a^3 b^2 C+15 a^2 b^3 B+8 a b^4 C-3 b^5 B\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{(a-b)^2 (a+b)}+\frac{b \sin (c+d x) \left(16 a^6 C-8 a^5 b B-25 a^4 b^2 C+16 a^3 b^3 B+C \left(b^3-a^2 b\right)^2 \cos (2 (c+d x))+2 a b \left(10 a^4 C-5 a^3 b B-16 a^2 b^2 C+9 a b^3 B+2 b^4 C\right) \cos (c+d x)+b^6 C\right)}{2 \left(a^2-b^2\right)^2}\right)}{3 b^4 d (a+b \cos (c+d x))^{3/2}}","\frac{2 a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(-2 a^2 C+a b B+b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)}-\frac{2 a^2 \left(-6 a^3 C+3 a^2 b B+10 a b^2 C-7 b^3 B\right) \sin (c+d x)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-16 a^4 C+8 a^3 b B+16 a^2 b^2 C-9 a b^3 B+b^4 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^4 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-16 a^5 C+8 a^4 b B+28 a^3 b^2 C-15 a^2 b^3 B-8 a b^4 C+3 b^5 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^4 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*((((a + b*Cos[c + d*x])/(a + b))^(3/2)*(b^2*(2*a^3*b*B - 6*a*b^3*B - 4*a^4*C + 7*a^2*b^2*C + b^4*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - (-8*a^4*b*B + 15*a^2*b^3*B - 3*b^5*B + 16*a^5*C - 28*a^3*b^2*C + 8*a*b^4*C)*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])))/((a - b)^2*(a + b)) + (b*(-8*a^5*b*B + 16*a^3*b^3*B + 16*a^6*C - 25*a^4*b^2*C + b^6*C + 2*a*b*(-5*a^3*b*B + 9*a*b^3*B + 10*a^4*C - 16*a^2*b^2*C + 2*b^4*C)*Cos[c + d*x] + (-(a^2*b) + b^3)^2*C*Cos[2*(c + d*x)])*Sin[c + d*x])/(2*(a^2 - b^2)^2)))/(3*b^4*d*(a + b*Cos[c + d*x])^(3/2))","A",1
848,1,274,331,2.3644236,"\int \frac{\cos (c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(\frac{\left(\frac{a+b \cos (c+d x)}{a+b}\right)^{3/2} \left(b^2 \left(2 a^3 C+a^2 b B-6 a b^2 C+3 b^3 B\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(8 a^4 C-2 a^3 b B-15 a^2 b^2 C+6 a b^3 B+3 b^4 C\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{(a-b)^2 (a+b)}-\frac{a b \sin (c+d x) \left(b \left(5 a^3 C-2 a^2 b B-9 a b^2 C+6 b^3 B\right) \cos (c+d x)+a \left(4 a^3 C-a^2 b B-8 a b^2 C+5 b^3 B\right)\right)}{\left(a^2-b^2\right)^2}\right)}{3 b^3 d (a+b \cos (c+d x))^{3/2}}","-\frac{2 a^2 (b B-a C) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 a \left(-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-8 a^3 C+2 a^2 b B+9 a b^2 C-3 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-8 a^4 C+2 a^3 b B+15 a^2 b^2 C-6 a b^3 B-3 b^4 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*((((a + b*Cos[c + d*x])/(a + b))^(3/2)*(b^2*(a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C - 15*a^2*b^2*C + 3*b^4*C)*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])))/((a - b)^2*(a + b)) - (a*b*(a*(-(a^2*b*B) + 5*b^3*B + 4*a^3*C - 8*a*b^2*C) + b*(-2*a^2*b*B + 6*b^3*B + 5*a^3*C - 9*a*b^2*C)*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2))/(3*b^3*d*(a + b*Cos[c + d*x])^(3/2))","A",1
849,1,224,307,2.0588114,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(\frac{b \sin (c+d x) \left(b \left(2 a^3 C+a^2 b B-6 a b^2 C+3 b^3 B\right) \cos (c+d x)+a \left(a^3 C+2 a^2 b B-5 a b^2 C+2 b^3 B\right)\right)}{\left(a^2-b^2\right)^2}-\frac{\left(\frac{a+b \cos (c+d x)}{a+b}\right)^{3/2} \left(\left(2 a^3 C+a^2 b B-6 a b^2 C+3 b^3 B\right) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-(a-b) \left(2 a^2 C+a b B-3 b^2 C\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)}{(a-b)^2}\right)}{3 b^2 d (a+b \cos (c+d x))^{3/2}}","\frac{2 a (b B-a C) \sin (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(2 a^2 C+a b B-3 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(2 a^3 C+a^2 b B-6 a b^2 C+3 b^3 B\right) \sin (c+d x)}{3 b d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(2 a^3 C+a^2 b B-6 a b^2 C+3 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*(-((((a + b*Cos[c + d*x])/(a + b))^(3/2)*((a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - (a - b)*(a*b*B + 2*a^2*C - 3*b^2*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/(a - b)^2) + (b*(a*(2*a^2*b*B + 2*b^3*B + a^3*C - 5*a*b^2*C) + b*(a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2))/(3*b^2*d*(a + b*Cos[c + d*x])^(3/2))","A",1
850,1,193,275,1.6782451,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(\frac{\sin (c+d x) \left(2 a^3 C+b \left(a^2 C-4 a b B+3 b^2 C\right) \cos (c+d x)-5 a^2 b B+2 a b^2 C+b^3 B\right)}{\left(a^2-b^2\right)^2}-\frac{\left(\frac{a+b \cos (c+d x)}{a+b}\right)^{3/2} \left(\left(a^2 C-4 a b B+3 b^2 C\right) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-(a-b) (a C-b B) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)}{b (a-b)^2}\right)}{3 d (a+b \cos (c+d x))^{3/2}}","-\frac{2 \left(a^2 (-C)+4 a b B-3 b^2 C\right) \sin (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 (b B-a C) \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 (b B-a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 (-C)+4 a b B-3 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*(-((((a + b*Cos[c + d*x])/(a + b))^(3/2)*((-4*a*b*B + a^2*C + 3*b^2*C)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - (a - b)*(-(b*B) + a*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/((a - b)^2*b)) + ((-5*a^2*b*B + b^3*B + 2*a^3*C + 2*a*b^2*C + b*(-4*a*b*B + a^2*C + 3*b^2*C)*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2))/(3*d*(a + b*Cos[c + d*x])^(3/2))","A",1
851,1,743,349,6.817123,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2),x]","\frac{\cos (c+d x) \sqrt{a+b \cos (c+d x)} (B \sec (c+d x)+C) \left(-\frac{2 \left(a b C \sin (c+d x)-b^2 B \sin (c+d x)\right)}{3 a \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{2 \left(4 a^3 b C \sin (c+d x)-7 a^2 b^2 B \sin (c+d x)+3 b^4 B \sin (c+d x)\right)}{3 a^2 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}\right)}{d (B+C \cos (c+d x))}+\frac{\cos (c+d x) (B \sec (c+d x)+C) \left(-\frac{2 i \left(4 a^3 b C-7 a^2 b^2 B+3 b^4 B\right) \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{b \cos (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left(2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right)}+\frac{2 \left(6 a^4 B+4 a^3 b C-19 a^2 b^2 B+9 b^4 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 \left(6 a^4 C-12 a^3 b B+2 a^2 b^2 C+4 a b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}\right)}{6 a^2 d (a-b)^2 (a+b)^2 (B+C \cos (c+d x))}","\frac{2 b (b B-a C) \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 (b B-a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 B \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 b \left(-4 a^3 C+7 a^2 b B-3 b^3 B\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-4 a^3 C+7 a^2 b B-3 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(Cos[c + d*x]*(C + B*Sec[c + d*x])*((2*(-12*a^3*b*B + 4*a*b^3*B + 6*a^4*C + 2*a^2*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(6*a^4*B - 19*a^2*b^2*B + 9*b^4*B + 4*a^3*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(-7*a^2*b^2*B + 3*b^4*B + 4*a^3*b*C)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2))))/(6*a^2*(a - b)^2*(a + b)^2*d*(B + C*Cos[c + d*x])) + (Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*(C + B*Sec[c + d*x])*((-2*(-(b^2*B*Sin[c + d*x]) + a*b*C*Sin[c + d*x]))/(3*a*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) - (2*(-7*a^2*b^2*B*Sin[c + d*x] + 3*b^4*B*Sin[c + d*x] + 4*a^3*b*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/(d*(B + C*Cos[c + d*x]))","C",0
852,1,750,437,7.125599,"\int \frac{\left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[((B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{a+b \cos (c+d x)} \left(\frac{B \tan (c+d x)}{a^3}+\frac{2 \left(a b^2 C \sin (c+d x)-b^3 B \sin (c+d x)\right)}{3 a^2 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{2 \left(7 a^3 b^2 C \sin (c+d x)-10 a^2 b^3 B \sin (c+d x)-3 a b^4 C \sin (c+d x)+6 b^5 B \sin (c+d x)\right)}{3 a^3 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}\right)}{d}+\frac{\frac{2 \left(-24 a^4 b C+36 a^3 b^2 B+8 a^2 b^3 C-20 a b^4 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \left(-3 a^4 b B-14 a^3 b^2 C+26 a^2 b^3 B+6 a b^4 C-15 b^5 B\right) \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{b \cos (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left(2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right)}+\frac{2 \left(12 a^5 C-33 a^4 b B-38 a^3 b^2 C+86 a^2 b^3 B+18 a b^4 C-45 b^5 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{12 a^3 d (b-a)^2 (a+b)^2}","-\frac{(5 b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^3 d \sqrt{a+b \cos (c+d x)}}+\frac{b \left(3 a^2 B+2 a b C-5 b^2 B\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{\left(3 a^2 B+2 a b C-5 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{b \left(3 a^4 B+14 a^3 b C-26 a^2 b^2 B-6 a b^3 C+15 b^4 B\right) \sin (c+d x)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{\left(3 a^4 B+14 a^3 b C-26 a^2 b^2 B-6 a b^3 C+15 b^4 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{B \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}",1,"((2*(36*a^3*b^2*B - 20*a*b^4*B - 24*a^4*b*C + 8*a^2*b^3*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(-33*a^4*b*B + 86*a^2*b^3*B - 45*b^5*B + 12*a^5*C - 38*a^3*b^2*C + 18*a*b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(-3*a^4*b*B + 26*a^2*b^3*B - 15*b^5*B - 14*a^3*b^2*C + 6*a*b^4*C)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2)))/(12*a^3*(-a + b)^2*(a + b)^2*d) + (Sqrt[a + b*Cos[c + d*x]]*((2*(-(b^3*B*Sin[c + d*x]) + a*b^2*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) + (2*(-10*a^2*b^3*B*Sin[c + d*x] + 6*b^5*B*Sin[c + d*x] + 7*a^3*b^2*C*Sin[c + d*x] - 3*a*b^4*C*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) + (B*Tan[c + d*x])/a^3))/d","C",0
853,1,125,170,1.4295129,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{300 (a C+b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+84 (9 a B+7 b C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} (7 (36 a B+43 b C) \cos (c+d x)+5 (18 (a C+b B) \cos (2 (c+d x))+78 a C+78 b B+7 b C \cos (3 (c+d x))))}{630 d}","\frac{10 (a C+b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 (9 a B+7 b C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (a C+b B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 (9 a B+7 b C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 (a C+b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}",1,"(84*(9*a*B + 7*b*C)*EllipticE[(c + d*x)/2, 2] + 300*(b*B + a*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*(36*a*B + 43*b*C)*Cos[c + d*x] + 5*(78*b*B + 78*a*C + 18*(b*B + a*C)*Cos[2*(c + d*x)] + 7*b*C*Cos[3*(c + d*x)]))*Sin[c + d*x])/(630*d)","A",1
854,1,103,140,0.975843,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{10 (7 a B+5 b C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+126 (a C+b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} (42 (a C+b B) \cos (c+d x)+70 a B+15 b C \cos (2 (c+d x))+65 b C)}{105 d}","\frac{2 (7 a B+5 b C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{6 (a C+b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (a C+b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 (7 a B+5 b C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}",1,"(126*(b*B + a*C)*EllipticE[(c + d*x)/2, 2] + 10*(7*a*B + 5*b*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(70*a*B + 65*b*C + 42*(b*B + a*C)*Cos[c + d*x] + 15*b*C*Cos[2*(c + d*x)])*Sin[c + d*x])/(105*d)","A",1
855,1,86,108,0.4513566,"\int \frac{(a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{2 \left(5 (a C+b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 (5 a B+3 b C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} (5 a C+5 b B+3 b C \cos (c+d x))\right)}{15 d}","\frac{2 (a C+b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (5 a B+3 b C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (a C+b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(2*(3*(5*a*B + 3*b*C)*EllipticE[(c + d*x)/2, 2] + 5*(b*B + a*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(5*b*B + 5*a*C + 3*b*C*Cos[c + d*x])*Sin[c + d*x]))/(15*d)","A",1
856,1,67,75,0.2390496,"\int \frac{(a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{2 \left((3 a B+b C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 (a C+b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b C \sin (c+d x) \sqrt{\cos (c+d x)}\right)}{3 d}","\frac{2 (3 a B+b C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (a C+b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(2*(3*(b*B + a*C)*EllipticE[(c + d*x)/2, 2] + (3*a*B + b*C)*EllipticF[(c + d*x)/2, 2] + b*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x]))/(3*d)","A",1
857,1,64,71,0.3660863,"\int \frac{(a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 \left((a C+b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+(b C-a B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{a B \sin (c+d x)}{\sqrt{\cos (c+d x)}}\right)}{d}","\frac{2 (a C+b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 (a B-b C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(2*((-(a*B) + b*C)*EllipticE[(c + d*x)/2, 2] + (b*B + a*C)*EllipticF[(c + d*x)/2, 2] + (a*B*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/d","A",1
858,1,107,103,0.5065406,"\int \frac{(a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{2 \left((a B+3 b C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-3 (a C+b B) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+a B \tan (c+d x)+3 a C \sin (c+d x)+3 b B \sin (c+d x)\right)}{3 d \sqrt{\cos (c+d x)}}","\frac{2 (a B+3 b C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 (a C+b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 (a C+b B) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 a B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(2*(-3*(b*B + a*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + (a*B + 3*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 3*b*B*Sin[c + d*x] + 3*a*C*Sin[c + d*x] + a*B*Tan[c + d*x]))/(3*d*Sqrt[Cos[c + d*x]])","A",1
859,1,134,140,0.8531343,"\int \frac{(a+b \cos (c+d x)) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{10 (a C+b B) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 (3 a B+5 b C) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+9 a B \sin (2 (c+d x))+6 a B \tan (c+d x)+10 a C \sin (c+d x)+10 b B \sin (c+d x)+15 b C \sin (2 (c+d x))}{15 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 (a C+b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 (3 a B+5 b C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (a C+b B) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (3 a B+5 b C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a B \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-6*(3*a*B + 5*b*C)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 10*(b*B + a*C)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 10*b*B*Sin[c + d*x] + 10*a*C*Sin[c + d*x] + 9*a*B*Sin[2*(c + d*x)] + 15*b*C*Sin[2*(c + d*x)] + 6*a*B*Tan[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",1
860,1,196,264,1.7393591,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{1200 \left(11 a^2 C+22 a b B+9 b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3696 \left(9 a^2 B+14 a b C+7 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(154 \left(36 a^2 B+86 a b C+43 b^2 B\right) \cos (c+d x)+180 \left(11 a^2 C+22 a b B+16 b^2 C\right) \cos (2 (c+d x))+15 \left(572 a^2 C+1144 a b B+21 b^2 C \cos (4 (c+d x))+531 b^2 C\right)+770 b (2 a C+b B) \cos (3 (c+d x))\right)}{27720 d}","\frac{2 \left(9 a^2 B+14 a b C+7 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(9 a^2 B+14 a b C+7 b^2 B\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 \left(11 a (a C+2 b B)+9 b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 \left(11 a (a C+2 b B)+9 b^2 C\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{10 \left(11 a (a C+2 b B)+9 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 b (13 a C+11 b B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d}",1,"(3696*(9*a^2*B + 7*b^2*B + 14*a*b*C)*EllipticE[(c + d*x)/2, 2] + 1200*(22*a*b*B + 11*a^2*C + 9*b^2*C)*EllipticF[(c + d*x)/2, 2] + 2*Sqrt[Cos[c + d*x]]*(154*(36*a^2*B + 43*b^2*B + 86*a*b*C)*Cos[c + d*x] + 180*(22*a*b*B + 11*a^2*C + 16*b^2*C)*Cos[2*(c + d*x)] + 770*b*(b*B + 2*a*C)*Cos[3*(c + d*x)] + 15*(1144*a*b*B + 572*a^2*C + 531*b^2*C + 21*b^2*C*Cos[4*(c + d*x)]))*Sin[c + d*x])/(27720*d)","A",1
861,1,167,223,1.4561088,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{60 \left(7 a^2 B+10 a b C+5 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+84 \left(9 a^2 C+18 a b B+7 b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} \left(7 \left(36 a^2 C+72 a b B+43 b^2 C\right) \cos (c+d x)+5 \left(84 a^2 B+18 b (2 a C+b B) \cos (2 (c+d x))+156 a b C+78 b^2 B+7 b^2 C \cos (3 (c+d x))\right)\right)}{630 d}","\frac{2 \left(7 a^2 B+10 a b C+5 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(7 a^2 B+10 a b C+5 b^2 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 \left(9 a (a C+2 b B)+7 b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(9 a (a C+2 b B)+7 b^2 C\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 b (11 a C+9 b B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d}",1,"(84*(18*a*b*B + 9*a^2*C + 7*b^2*C)*EllipticE[(c + d*x)/2, 2] + 60*(7*a^2*B + 5*b^2*B + 10*a*b*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*(72*a*b*B + 36*a^2*C + 43*b^2*C)*Cos[c + d*x] + 5*(84*a^2*B + 78*b^2*B + 156*a*b*C + 18*b*(b*B + 2*a*C)*Cos[2*(c + d*x)] + 7*b^2*C*Cos[3*(c + d*x)]))*Sin[c + d*x])/(630*d)","A",1
862,1,139,182,1.0716887,"\int \frac{(a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{10 \left(7 a^2 C+14 a b B+5 b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+42 \left(5 a^2 B+6 a b C+3 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} \left(5 \left(14 a^2 C+28 a b B+3 b^2 C \cos (2 (c+d x))+13 b^2 C\right)+42 b (2 a C+b B) \cos (c+d x)\right)}{105 d}","\frac{2 \left(5 a^2 B+6 a b C+3 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(7 a (a C+2 b B)+5 b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(7 a (a C+2 b B)+5 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b (9 a C+7 b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}{7 d}",1,"(42*(5*a^2*B + 3*b^2*B + 6*a*b*C)*EllipticE[(c + d*x)/2, 2] + 10*(14*a*b*B + 7*a^2*C + 5*b^2*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(42*b*(b*B + 2*a*C)*Cos[c + d*x] + 5*(28*a*b*B + 14*a^2*C + 13*b^2*C + 3*b^2*C*Cos[2*(c + d*x)]))*Sin[c + d*x])/(105*d)","A",1
863,1,106,140,0.6087802,"\int \frac{(a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{2 \left(5 \left(3 a^2 B+2 a b C+b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 \left(5 a^2 C+10 a b B+3 b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b \sin (c+d x) \sqrt{\cos (c+d x)} (10 a C+5 b B+3 b C \cos (c+d x))\right)}{15 d}","\frac{2 \left(3 a^2 B+2 a b C+b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 \left(5 a (a C+2 b B)+3 b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b (7 a C+5 b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 b C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}{5 d}",1,"(2*(3*(10*a*b*B + 5*a^2*C + 3*b^2*C)*EllipticE[(c + d*x)/2, 2] + 5*(3*a^2*B + b^2*B + 2*a*b*C)*EllipticF[(c + d*x)/2, 2] + b*Sqrt[Cos[c + d*x]]*(5*b*B + 10*a*C + 3*b*C*Cos[c + d*x])*Sin[c + d*x]))/(15*d)","A",1
864,1,102,121,0.6493153,"\int \frac{(a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 \left(\left(3 a^2 C+6 a b B+b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\left(-3 a^2 B+6 a b C+3 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{\sin (c+d x) \left(3 a^2 B+b^2 C \cos (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)}{3 d}","\frac{2 \left(3 a^2 C+6 a b B+b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(a^2 B-2 a b C-b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 b^2 C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(2*((-3*a^2*B + 3*b^2*B + 6*a*b*C)*EllipticE[(c + d*x)/2, 2] + (6*a*b*B + 3*a^2*C + b^2*C)*EllipticF[(c + d*x)/2, 2] + ((3*a^2*B + b^2*C*Cos[c + d*x])*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(3*d)","A",1
865,1,105,126,1.1640628,"\int \frac{(a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{2 \left(\left(a^2 B+6 a b C+3 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-3 \left(a^2 C+2 a b B-b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{a \sin (c+d x) (3 (a C+2 b B) \cos (c+d x)+a B)}{\cos ^{\frac{3}{2}}(c+d x)}\right)}{3 d}","\frac{2 \left(a^2 B+6 a b C+3 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(a^2 C+2 a b B-b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a (a C+2 b B) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(2*(-3*(2*a*b*B + a^2*C - b^2*C)*EllipticE[(c + d*x)/2, 2] + (a^2*B + 3*b^2*B + 6*a*b*C)*EllipticF[(c + d*x)/2, 2] + (a*(a*B + 3*(2*b*B + a*C)*Cos[c + d*x])*Sin[c + d*x])/Cos[c + d*x]^(3/2)))/(3*d)","A",1
866,1,175,172,1.1258945,"\int \frac{(a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{10 \left(a^2 C+2 a b B+3 b^2 C\right) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 \left(3 a^2 B+10 a b C+5 b^2 B\right) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+9 a^2 B \sin (2 (c+d x))+6 a^2 B \tan (c+d x)+10 a^2 C \sin (c+d x)+20 a b B \sin (c+d x)+30 a b C \sin (2 (c+d x))+15 b^2 B \sin (2 (c+d x))}{15 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 \left(a^2 C+2 a b B+3 b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(3 a^2 B+10 a b C+5 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(3 a^2 B+10 a b C+5 b^2 B\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 B \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a (a C+2 b B) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-6*(3*a^2*B + 5*b^2*B + 10*a*b*C)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 10*(2*a*b*B + a^2*C + 3*b^2*C)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 20*a*b*B*Sin[c + d*x] + 10*a^2*C*Sin[c + d*x] + 9*a^2*B*Sin[2*(c + d*x)] + 15*b^2*B*Sin[2*(c + d*x)] + 30*a*b*C*Sin[2*(c + d*x)] + 6*a^2*B*Tan[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",1
867,1,191,214,4.6124547,"\int \frac{(a+b \cos (c+d x))^2 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{2 \left(5 \left(5 a^2 B+14 a b C+7 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-21 \left(3 a^2 C+6 a b B+5 b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{5 \left(5 a^2 B+14 a b C+7 b^2 B\right) \sin (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)}+\frac{21 \left(3 a^2 C+6 a b B+5 b^2 C\right) \sin (c+d x)}{\sqrt{\cos (c+d x)}}+\frac{15 a^2 B \sin (c+d x)}{\cos ^{\frac{7}{2}}(c+d x)}+\frac{21 a (a C+2 b B) \sin (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)}\right)}{105 d}","\frac{2 \left(5 a^2 B+14 a b C+7 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 \left(3 a^2 C+6 a b B+5 b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(5 a^2 B+14 a b C+7 b^2 B\right) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(3 a^2 C+6 a b B+5 b^2 C\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 B \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a (a C+2 b B) \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(2*(-21*(6*a*b*B + 3*a^2*C + 5*b^2*C)*EllipticE[(c + d*x)/2, 2] + 5*(5*a^2*B + 7*b^2*B + 14*a*b*C)*EllipticF[(c + d*x)/2, 2] + (15*a^2*B*Sin[c + d*x])/Cos[c + d*x]^(7/2) + (21*a*(2*b*B + a*C)*Sin[c + d*x])/Cos[c + d*x]^(5/2) + (5*(5*a^2*B + 7*b^2*B + 14*a*b*C)*Sin[c + d*x])/Cos[c + d*x]^(3/2) + (21*(6*a*b*B + 3*a^2*C + 5*b^2*C)*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(105*d)","A",1
868,1,235,305,2.0242083,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{240 \left(77 a^3 B+165 a^2 b C+165 a b^2 B+45 b^3 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3696 \left(9 a^3 C+27 a^2 b B+21 a b^2 C+7 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(180 b \left(33 a^2 C+33 a b B+16 b^2 C\right) \cos (2 (c+d x))+154 \left(36 a^3 C+108 a^2 b B+129 a b^2 C+43 b^3 B\right) \cos (c+d x)+15 \left(616 a^3 B+1716 a^2 b C+1716 a b^2 B+21 b^3 C \cos (4 (c+d x))+531 b^3 C\right)+770 b^2 (3 a C+b B) \cos (3 (c+d x))\right)}{27720 d}","\frac{2 b \left(26 a^2 C+33 a b B+9 b^2 C\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{2 \left(77 a^3 B+165 a^2 b C+165 a b^2 B+45 b^3 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 \left(9 a^3 C+27 a^2 b B+21 a b^2 C+7 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(9 a^3 C+27 a^2 b B+21 a b^2 C+7 b^3 B\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 \left(77 a^3 B+165 a^2 b C+165 a b^2 B+45 b^3 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 b^2 (15 a C+11 b B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}",1,"(3696*(27*a^2*b*B + 7*b^3*B + 9*a^3*C + 21*a*b^2*C)*EllipticE[(c + d*x)/2, 2] + 240*(77*a^3*B + 165*a*b^2*B + 165*a^2*b*C + 45*b^3*C)*EllipticF[(c + d*x)/2, 2] + 2*Sqrt[Cos[c + d*x]]*(154*(108*a^2*b*B + 43*b^3*B + 36*a^3*C + 129*a*b^2*C)*Cos[c + d*x] + 180*b*(33*a*b*B + 33*a^2*C + 16*b^2*C)*Cos[2*(c + d*x)] + 770*b^2*(b*B + 3*a*C)*Cos[3*(c + d*x)] + 15*(616*a^3*B + 1716*a*b^2*B + 1716*a^2*b*C + 531*b^3*C + 21*b^3*C*Cos[4*(c + d*x)]))*Sin[c + d*x])/(27720*d)","A",1
869,1,197,255,1.2188488,"\int \frac{(a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{60 \left(7 a^3 C+21 a^2 b B+15 a b^2 C+5 b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+84 \left(15 a^3 B+27 a^2 b C+27 a b^2 B+7 b^3 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} \left(7 b \left(108 a^2 C+108 a b B+43 b^2 C\right) \cos (c+d x)+5 \left(84 a^3 C+252 a^2 b B+18 b^2 (3 a C+b B) \cos (2 (c+d x))+234 a b^2 C+78 b^3 B+7 b^3 C \cos (3 (c+d x))\right)\right)}{630 d}","\frac{2 b \left(22 a^2 C+27 a b B+7 b^2 C\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 \left(7 a^3 C+21 a^2 b B+15 a b^2 C+5 b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(15 a^3 B+27 a^2 b C+27 a b^2 B+7 b^3 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(7 a^3 C+21 a^2 b B+15 a b^2 C+5 b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b^2 (13 a C+9 b B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}",1,"(84*(15*a^3*B + 27*a*b^2*B + 27*a^2*b*C + 7*b^3*C)*EllipticE[(c + d*x)/2, 2] + 60*(21*a^2*b*B + 5*b^3*B + 7*a^3*C + 15*a*b^2*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*b*(108*a*b*B + 108*a^2*C + 43*b^2*C)*Cos[c + d*x] + 5*(252*a^2*b*B + 78*b^3*B + 84*a^3*C + 234*a*b^2*C + 18*b^2*(b*B + 3*a*C)*Cos[2*(c + d*x)] + 7*b^3*C*Cos[3*(c + d*x)]))*Sin[c + d*x])/(630*d)","A",1
870,1,158,205,1.3136137,"\int \frac{(a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{b \sin (c+d x) \sqrt{\cos (c+d x)} \left(5 \left(42 a^2 C+42 a b B+3 b^2 C \cos (2 (c+d x))+13 b^2 C\right)+42 b (3 a C+b B) \cos (c+d x)\right)+10 \left(21 a^3 B+21 a^2 b C+21 a b^2 B+5 b^3 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+42 \left(5 a^3 C+15 a^2 b B+9 a b^2 C+3 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{105 d}","\frac{2 b \left(18 a^2 C+21 a b B+5 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 \left(21 a^3 B+21 a^2 b C+21 a b^2 B+5 b^3 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(5 a^3 C+15 a^2 b B+9 a b^2 C+3 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b^2 (11 a C+7 b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 b C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}",1,"(42*(15*a^2*b*B + 3*b^3*B + 5*a^3*C + 9*a*b^2*C)*EllipticE[(c + d*x)/2, 2] + 10*(21*a^3*B + 21*a*b^2*B + 21*a^2*b*C + 5*b^3*C)*EllipticF[(c + d*x)/2, 2] + b*Sqrt[Cos[c + d*x]]*(42*b*(b*B + 3*a*C)*Cos[c + d*x] + 5*(42*a*b*B + 42*a^2*C + 13*b^2*C + 3*b^2*C*Cos[2*(c + d*x)]))*Sin[c + d*x])/(105*d)","A",1
871,1,150,202,1.1775855,"\int \frac{(a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{\frac{\sin (c+d x) \left(3 \left(10 a^3 B+b^3 C \cos (2 (c+d x))+b^3 C\right)+10 b^2 (3 a C+b B) \cos (c+d x)\right)}{\sqrt{\cos (c+d x)}}+10 \left(3 a^3 C+9 a^2 b B+3 a b^2 C+b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\left(-30 a^3 B+90 a^2 b C+90 a b^2 B+18 b^3 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}","-\frac{2 b \left(6 a^2 B-3 a b C-b^2 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 \left(3 a^3 C+9 a^2 b B+3 a b^2 C+b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(5 a^3 B-15 a^2 b C-15 a b^2 B-3 b^3 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{2 b^2 (5 a B-b C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt{\cos (c+d x)}}",1,"((-30*a^3*B + 90*a*b^2*B + 90*a^2*b*C + 18*b^3*C)*EllipticE[(c + d*x)/2, 2] + 10*(9*a^2*b*B + b^3*B + 3*a^3*C + 3*a*b^2*C)*EllipticF[(c + d*x)/2, 2] + ((10*b^2*(b*B + 3*a*C)*Cos[c + d*x] + 3*(10*a^3*B + b^3*C + b^3*C*Cos[2*(c + d*x)]))*Sin[c + d*x])/Sqrt[Cos[c + d*x]])/(15*d)","A",1
872,1,165,192,1.0999983,"\int \frac{(a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{2 a^3 B \tan (c+d x)+6 a^3 C \sin (c+d x)+18 a^2 b B \sin (c+d x)+2 \left(a^3 B+9 a^2 b C+9 a b^2 B+b^3 C\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 \left(a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b^3 C \sin (2 (c+d x))}{3 d \sqrt{\cos (c+d x)}}","\frac{2 a^2 (3 a C+7 b B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 \left(a^3 B+9 a^2 b C+9 a b^2 B+b^3 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(a^3 C+3 a^2 b B-3 a b^2 C-b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 b^2 (a B-b C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-6*(3*a^2*b*B - b^3*B + a^3*C - 3*a*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*(a^3*B + 9*a*b^2*B + 9*a^2*b*C + b^3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 18*a^2*b*B*Sin[c + d*x] + 6*a^3*C*Sin[c + d*x] + b^3*C*Sin[2*(c + d*x)] + 2*a^3*B*Tan[c + d*x])/(3*d*Sqrt[Cos[c + d*x]])","A",1
873,1,176,204,2.1774976,"\int \frac{(a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{6 a^3 B \tan (c+d x)+9 a \left(a^2 B+5 a b C+5 b^2 B\right) \sin (2 (c+d x))+10 a^2 (a C+3 b B) \sin (c+d x)+10 \left(a^3 C+3 a^2 b B+9 a b^2 C+3 b^3 B\right) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 \left(3 a^3 B+15 a^2 b C+15 a b^2 B-5 b^3 C\right) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 a \left(3 a^2 B+15 a b C+14 b^2 B\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 (5 a C+9 b B) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(a^3 C+3 a^2 b B+9 a b^2 C+3 b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(3 a^3 B+15 a^2 b C+15 a b^2 B-5 b^3 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-6*(3*a^3*B + 15*a*b^2*B + 15*a^2*b*C - 5*b^3*C)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 10*(3*a^2*b*B + 3*b^3*B + a^3*C + 9*a*b^2*C)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 10*a^2*(3*b*B + a*C)*Sin[c + d*x] + 9*a*(a^2*B + 5*b^2*B + 5*a*b*C)*Sin[2*(c + d*x)] + 6*a^3*B*Tan[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",1
874,1,221,255,3.6860262,"\int \frac{(a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{2 \left(\frac{15 a^3 B \sin (c+d x)}{\cos ^{\frac{7}{2}}(c+d x)}+\frac{5 a \left(5 a^2 B+21 a b C+21 b^2 B\right) \sin (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)}+\frac{21 a^2 (a C+3 b B) \sin (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)}+5 \left(5 a^3 B+21 a^2 b C+21 a b^2 B+21 b^3 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-21 \left(3 a^3 C+9 a^2 b B+15 a b^2 C+5 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{21 \left(3 a^3 C+9 a^2 b B+15 a b^2 C+5 b^3 B\right) \sin (c+d x)}{\sqrt{\cos (c+d x)}}\right)}{105 d}","\frac{2 a \left(5 a^2 B+21 a b C+18 b^2 B\right) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (7 a C+11 b B) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(5 a^3 B+21 a^2 b C+21 a b^2 B+21 b^3 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 \left(3 a^3 C+9 a^2 b B+15 a b^2 C+5 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(3 a^3 C+9 a^2 b B+15 a b^2 C+5 b^3 B\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(2*(-21*(9*a^2*b*B + 5*b^3*B + 3*a^3*C + 15*a*b^2*C)*EllipticE[(c + d*x)/2, 2] + 5*(5*a^3*B + 21*a*b^2*B + 21*a^2*b*C + 21*b^3*C)*EllipticF[(c + d*x)/2, 2] + (15*a^3*B*Sin[c + d*x])/Cos[c + d*x]^(7/2) + (21*a^2*(3*b*B + a*C)*Sin[c + d*x])/Cos[c + d*x]^(5/2) + (5*a*(5*a^2*B + 21*b^2*B + 21*a*b*C)*Sin[c + d*x])/Cos[c + d*x]^(3/2) + (21*(9*a^2*b*B + 5*b^3*B + 3*a^3*C + 15*a*b^2*C)*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(105*d)","A",1
875,1,266,305,5.0091295,"\int \frac{(a+b \cos (c+d x))^3 \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","\frac{2 \left(\frac{35 a^3 B \sin (c+d x)}{\cos ^{\frac{9}{2}}(c+d x)}+\frac{7 a \left(7 a^2 B+27 a b C+27 b^2 B\right) \sin (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)}+\frac{45 a^2 (a C+3 b B) \sin (c+d x)}{\cos ^{\frac{7}{2}}(c+d x)}+15 \left(5 a^3 C+15 a^2 b B+21 a b^2 C+7 b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-21 \left(7 a^3 B+27 a^2 b C+27 a b^2 B+15 b^3 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{15 \left(5 a^3 C+15 a^2 b B+21 a b^2 C+7 b^3 B\right) \sin (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)}+\frac{21 \left(7 a^3 B+27 a^2 b C+27 a b^2 B+15 b^3 C\right) \sin (c+d x)}{\sqrt{\cos (c+d x)}}\right)}{315 d}","\frac{2 a \left(7 a^2 B+27 a b C+22 b^2 B\right) \sin (c+d x)}{45 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a^2 (9 a C+13 b B) \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 \left(5 a^3 C+15 a^2 b B+21 a b^2 C+7 b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 \left(7 a^3 B+27 a^2 b C+27 a b^2 B+15 b^3 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(5 a^3 C+15 a^2 b B+21 a b^2 C+7 b^3 B\right) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(7 a^3 B+27 a^2 b C+27 a b^2 B+15 b^3 C\right) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(2*(-21*(7*a^3*B + 27*a*b^2*B + 27*a^2*b*C + 15*b^3*C)*EllipticE[(c + d*x)/2, 2] + 15*(15*a^2*b*B + 7*b^3*B + 5*a^3*C + 21*a*b^2*C)*EllipticF[(c + d*x)/2, 2] + (35*a^3*B*Sin[c + d*x])/Cos[c + d*x]^(9/2) + (45*a^2*(3*b*B + a*C)*Sin[c + d*x])/Cos[c + d*x]^(7/2) + (7*a*(7*a^2*B + 27*b^2*B + 27*a*b*C)*Sin[c + d*x])/Cos[c + d*x]^(5/2) + (15*(15*a^2*b*B + 7*b^3*B + 5*a^3*C + 21*a*b^2*C)*Sin[c + d*x])/Cos[c + d*x]^(3/2) + (21*(7*a^3*B + 27*a*b^2*B + 27*a^2*b*C + 15*b^3*C)*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(315*d)","A",1
876,1,305,246,2.8384041,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(70 a^2 C+42 b (b B-a C) \cos (c+d x)-70 a b B+15 b^2 C \cos (2 (c+d x))+65 b^2 C\right)+\frac{4 \left(-28 a^2 C+28 a b B+25 b^2 C\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}-\frac{42 \left(5 a^2+3 b^2\right) (a C-b B) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}+\frac{2 \left(-35 a^3 C+35 a^2 b B-13 a b^2 C+63 b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{210 b^3 d}","\frac{2 a^4 (b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a+b)}+\frac{2 \left(5 a^2+3 b^2\right) (b B-a C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^4 d}-\frac{2 \left(-7 a^2 C+7 a b B-5 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 b^3 d}-\frac{2 \left(-21 a^4 C+21 a^3 b B-7 a^2 b^2 C+7 a b^3 B-5 b^4 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b^5 d}+\frac{2 (b B-a C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b^2 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 b d}",1,"((2*(35*a^2*b*B + 63*b^3*B - 35*a^3*C - 13*a*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (4*(28*a*b*B - 28*a^2*C + 25*b^2*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + 2*Sqrt[Cos[c + d*x]]*(-70*a*b*B + 70*a^2*C + 65*b^2*C + 42*b*(b*B - a*C)*Cos[c + d*x] + 15*b^2*C*Cos[2*(c + d*x)])*Sin[c + d*x] - (42*(5*a^2 + 3*b^2)*(-(b*B) + a*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/(210*b^3*d)","A",1
877,1,260,182,2.3446926,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{\frac{2 b^2 \left(5 a^2 C-5 a b B+9 b^2 C\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{6 \left(5 a^2 C-5 a b B+3 b^2 C\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a \sqrt{\sin ^2(c+d x)}}+4 b^2 \sin (c+d x) \sqrt{\cos (c+d x)} (-5 a C+5 b B+3 b C \cos (c+d x))+2 b^2 (4 a C+5 b B) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{30 b^4 d}","-\frac{2 a^3 (b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a+b)}+\frac{2 \left(3 a^2+b^2\right) (b B-a C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d}-\frac{2 \left(-5 a^2 C+5 a b B-3 b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^3 d}+\frac{2 (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b d}",1,"((2*b^2*(-5*a*b*B + 5*a^2*C + 9*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + 2*b^2*(5*b*B + 4*a*C)*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)) + 4*b^2*Sqrt[Cos[c + d*x]]*(5*b*B - 5*a*C + 3*b*C*Cos[c + d*x])*Sin[c + d*x] + (6*(-5*a*b*B + 5*a^2*C + 3*b^2*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*Sqrt[Sin[c + d*x]^2]))/(30*b^4*d)","A",1
878,1,207,137,1.4394325,"\int \frac{\sqrt{\cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{\frac{3 (b B-a C) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}+\frac{(3 b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+C \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)+2 C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d}","\frac{2 a^2 (b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a+b)}-\frac{2 \left(-3 a^2 C+3 a b B-b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^3 d}+\frac{2 (b B-a C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d}",1,"(((3*b*B - a*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + C*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)) + 2*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x] + (3*(b*B - a*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/(3*b*d)","A",1
879,1,128,89,0.8898432,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])),x]","\frac{b B \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)-\frac{2 C \sin (c+d x) \left(-(a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+a \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{\sqrt{\sin ^2(c+d x)}}}{b^2 d}","\frac{2 (b B-a C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}-\frac{2 a (b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}+\frac{2 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(b*B*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)) - (2*C*(b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] - (a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + a*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/Sqrt[Sin[c + d*x]^2])/(b^2*d)","A",1
880,1,58,61,0.2044824,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])),x]","\frac{2 \left((b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)+C (a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{b d (a+b)}","\frac{2 (b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a+b)}+\frac{2 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(2*((a + b)*C*EllipticF[(c + d*x)/2, 2] + (b*B - a*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(b*(a + b)*d)","A",1
881,1,206,86,2.438876,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])),x]","\frac{-\frac{2 B \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}+\frac{2 (2 a C-3 b B) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}-\frac{2 a B \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{b}+\frac{4 B \sin (c+d x)}{\sqrt{\cos (c+d x)}}}{2 a d}","-\frac{2 (b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a d (a+b)}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{2 B \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}",1,"((2*(-3*b*B + 2*a*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) - (2*a*B*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)))/b + (4*B*Sin[c + d*x])/Sqrt[Cos[c + d*x]] - (2*B*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/(2*a*d)","B",1
882,1,260,150,2.1883859,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])),x]","\frac{\frac{2 a \left(2 a^2 B-9 a b C+9 b^2 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{6 (b B-a C) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{b \sqrt{\sin ^2(c+d x)}}+\frac{a \left(8 a b B-6 a^2 C\right) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{b}+\frac{4 a^2 B \sin (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)}+\frac{12 a (a C-b B) \sin (c+d x)}{\sqrt{\cos (c+d x)}}}{6 a^3 d}","\frac{2 (b B-a C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{2 b (b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a+b)}-\frac{2 (b B-a C) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{2 B \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}",1,"((2*a*(2*a^2*B + 9*b^2*B - 9*a*b*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (a*(8*a*b*B - 6*a^2*C)*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)))/b + (4*a^2*B*Sin[c + d*x])/Cos[c + d*x]^(3/2) + (12*a*(-(b*B) + a*C)*Sin[c + d*x])/Sqrt[Cos[c + d*x]] + (6*(b*B - a*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(b*Sqrt[Sin[c + d*x]^2]))/(6*a^3*d)","A",1
883,1,309,217,4.3223272,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*(a + b*Cos[c + d*x])),x]","\frac{-\frac{2 a \left(9 a^2 B-20 a b C+20 b^2 B\right) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{b}+\frac{2 \left(3 \left(3 a^2 B-5 a b C+5 b^2 B\right) \sin (2 (c+d x))+6 a^2 B \tan (c+d x)+10 a (a C-b B) \sin (c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)}-\frac{6 \left(3 a^2 B-5 a b C+5 b^2 B\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}+\frac{2 \left(10 a^3 C-19 a^2 b B+45 a b^2 C-45 b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{30 a^3 d}","-\frac{2 b^2 (b B-a C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a+b)}-\frac{2 (b B-a C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{2 (b B-a C) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \left(3 a^2 B-5 a b C+5 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^3 d}+\frac{2 \left(3 a^2 B-5 a b C+5 b^2 B\right) \sin (c+d x)}{5 a^3 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x)}{5 a d \cos ^{\frac{5}{2}}(c+d x)}",1,"((2*(-19*a^2*b*B - 45*b^3*B + 10*a^3*C + 45*a*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) - (2*a*(9*a^2*B + 20*b^2*B - 20*a*b*C)*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)))/b - (6*(3*a^2*B + 5*b^2*B - 5*a*b*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]) + (2*(10*a*(-(b*B) + a*C)*Sin[c + d*x] + 3*(3*a^2*B + 5*b^2*B - 5*a*b*C)*Sin[2*(c + d*x)] + 6*a^2*B*Tan[c + d*x]))/Cos[c + d*x]^(3/2))/(30*a^3*d)","A",1
884,1,369,389,4.8415997,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{4 \sqrt{\cos (c+d x)} \left(\frac{15 a^3 (b B-a C) \sin (c+d x)}{\left(a^2-b^2\right) (a+b \cos (c+d x))}+10 (b B-2 a C) \sin (c+d x)+3 b C \sin (2 (c+d x))\right)+\frac{\frac{8 \left(14 a^3 C-10 a^2 b B+a b^2 C-5 b^3 B\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{2 \left(35 a^4 C-25 a^3 b B-32 a^2 b^2 C+40 a b^3 B-18 b^4 C\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{6 \left(35 a^4 C-25 a^3 b B-24 a^2 b^2 C+20 a b^3 B-6 b^4 C\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}}{(a-b) (a+b)}}{60 b^3 d}","\frac{a (b B-a C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\left(-7 a^2 C+5 a b B+2 b^2 C\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b^2 d \left(a^2-b^2\right)}+\frac{\left(-7 a^3 C+5 a^2 b B+4 a b^2 C-2 b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)}-\frac{a^3 \left(-7 a^3 C+5 a^2 b B+9 a b^2 C-7 b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a-b) (a+b)^2}-\frac{\left(-35 a^4 C+25 a^3 b B+24 a^2 b^2 C-20 a b^3 B+6 b^4 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^4 d \left(a^2-b^2\right)}+\frac{\left(-21 a^5 C+15 a^4 b B+20 a^3 b^2 C-16 a^2 b^3 B+4 a b^4 C-2 b^5 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^5 d \left(a^2-b^2\right)}",1,"(((2*(-25*a^3*b*B + 40*a*b^3*B + 35*a^4*C - 32*a^2*b^2*C - 18*b^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(-10*a^2*b*B - 5*b^3*B + 14*a^3*C + a*b^2*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (6*(-25*a^3*b*B + 20*a*b^3*B + 35*a^4*C - 24*a^2*b^2*C - 6*b^4*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b)) + 4*Sqrt[Cos[c + d*x]]*(10*(b*B - 2*a*C)*Sin[c + d*x] + (15*a^3*(b*B - a*C)*Sin[c + d*x])/((a^2 - b^2)*(a + b*Cos[c + d*x])) + 3*b*C*Sin[2*(c + d*x)]))/(60*b^3*d)","A",1
885,1,318,303,3.2411777,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)} \left(\frac{3 a^2 (a C-b B)}{\left(a^2-b^2\right) (a+b \cos (c+d x))}+2 C\right)-\frac{\frac{8 \left(2 a^2 C-3 a b B+b^2 C\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{2 \left(5 a^3 C-3 a^2 b B-8 a b^2 C+6 b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{6 \left(5 a^3 C-3 a^2 b B-4 a b^2 C+2 b^3 B\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}}{(a-b) (a+b)}}{12 b^2 d}","\frac{a (b B-a C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\left(-5 a^2 C+3 a b B+2 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \left(a^2-b^2\right)}+\frac{\left(-5 a^3 C+3 a^2 b B+4 a b^2 C-2 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}+\frac{a^2 \left(-5 a^3 C+3 a^2 b B+7 a b^2 C-5 b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a-b) (a+b)^2}-\frac{\left(-15 a^4 C+9 a^3 b B+16 a^2 b^2 C-12 a b^3 B+2 b^4 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d \left(a^2-b^2\right)}",1,"(4*Sqrt[Cos[c + d*x]]*(2*C + (3*a^2*(-(b*B) + a*C))/((a^2 - b^2)*(a + b*Cos[c + d*x])))*Sin[c + d*x] - ((2*(-3*a^2*b*B + 6*b^3*B + 5*a^3*C - 8*a*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(-3*a*b*B + 2*a^2*C + b^2*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (6*(-3*a^2*b*B + 2*b^3*B + 5*a^3*C - 4*a*b^2*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b)))/(12*b^2*d)","A",1
886,1,280,224,2.713989,"\int \frac{\sqrt{\cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{\frac{\frac{2 \left(a^2 C+a b B-2 b^2 C\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{2 \left(3 a^2 C-a b B-2 b^2 C\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}+\frac{8 (a C-b B) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}}{(a-b) (a+b)}-\frac{4 a (a C-b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{\left(a^2-b^2\right) (a+b \cos (c+d x))}}{4 b d}","-\frac{\left(-3 a^2 C+a b B+2 b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}+\frac{a (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\left(-3 a^3 C+a^2 b B+4 a b^2 C-2 b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}-\frac{a \left(-3 a^3 C+a^2 b B+5 a b^2 C-3 b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}",1,"((-4*a*(-(b*B) + a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*(a + b*Cos[c + d*x])) + ((2*(a*b*B + a^2*C - 2*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(-(b*B) + a*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (2*(-(a*b*B) + 3*a^2*C - 2*b^2*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b)))/(4*b*d)","A",1
887,1,260,198,2.3947656,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2),x]","\frac{\frac{4 (a C-b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{\left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\frac{2 (b B-a C) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}+\frac{2 (a C-b B) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{(4 a B-4 b C) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{b}}{(b-a) (a+b)}}{4 d}","\frac{\left(a^2 C+a b B-2 b^2 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}+\frac{(b B-a C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}-\frac{(b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\left(a^3 C+a^2 b B-3 a b^2 C+b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a-b) (a+b)^2}",1,"((4*(-(b*B) + a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*(a + b*Cos[c + d*x])) - ((2*(-(b*B) + a*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + ((4*a*B - 4*b*C)*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)))/b + (2*(b*B - a*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/((-a + b)*(a + b)))/(4*d)","A",1
888,1,274,200,2.7383074,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2),x]","\frac{\frac{4 b (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{\left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\frac{2 \left(4 a^2 B-a b C-3 b^2 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{2 (a C-b B) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}+\frac{4 a (a C-b B) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{b}}{(a-b) (a+b)}}{4 a d}","-\frac{(b B-a C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}-\frac{(b B-a C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \left(a^2-b^2\right)}+\frac{b (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\left(a^3 (-C)+3 a^2 b B-a b^2 C-b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d (a-b) (a+b)^2}",1,"((4*b*(b*B - a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*(a + b*Cos[c + d*x])) + ((2*(4*a^2*B - 3*b^2*B - a*b*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (4*a*(-(b*B) + a*C)*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)))/b + (2*(-(b*B) + a*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b)))/(4*a*d)","A",1
889,1,316,256,4.1755822,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2),x]","\frac{4 \sqrt{\cos (c+d x)} \left(\frac{b^2 (b B-a C) \sin (c+d x)}{\left(b^2-a^2\right) (a+b \cos (c+d x))}+2 B \tan (c+d x)\right)-\frac{-\frac{8 a \left(a^2 B+a b C-2 b^2 B\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{b (a+b)}-\frac{2 \left(2 a^2 B+a b C-3 b^2 B\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}+\frac{2 \left(4 a^3 C-10 a^2 b B-3 a b^2 C+9 b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{(b-a) (a+b)}}{4 a^2 d}","\frac{(b B-a C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \left(a^2-b^2\right)}-\frac{\left(2 a^2 B+a b C-3 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{\left(2 a^2 B+a b C-3 b^2 B\right) \sin (c+d x)}{a^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}+\frac{b (b B-a C) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}-\frac{\left(-3 a^3 C+5 a^2 b B+a b^2 C-3 b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a-b) (a+b)^2}",1,"(-(((2*(-10*a^2*b*B + 9*b^3*B + 4*a^3*C - 3*a*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) - (8*a*(a^2*B - 2*b^2*B + a*b*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(b*(a + b)) - (2*(2*a^2*B - 3*b^2*B + a*b*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/((-a + b)*(a + b))) + 4*Sqrt[Cos[c + d*x]]*((b^2*(b*B - a*C)*Sin[c + d*x])/((-a^2 + b^2)*(a + b*Cos[c + d*x])) + 2*B*Tan[c + d*x]))/(4*a^2*d)","A",1
890,1,427,345,6.8702133,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])^2),x]","\frac{\sqrt{\cos (c+d x)} \left(\frac{2 \sec (c+d x) (a C \sin (c+d x)-2 b B \sin (c+d x))}{a^3}+\frac{2 B \tan (c+d x) \sec (c+d x)}{3 a^2}+\frac{b^4 B \sin (c+d x)-a b^3 C \sin (c+d x)}{a^3 \left(a^2-b^2\right) (a+b \cos (c+d x))}\right)}{d}+\frac{\frac{2 \left(-6 a^3 b C+12 a^2 b^2 B+9 a b^3 C-15 b^4 B\right) \sin (c+d x) \cos (2 (c+d x)) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{1-\cos ^2(c+d x)} \left(2 \cos ^2(c+d x)-1\right)}+\frac{\left(-12 a^4 C+28 a^3 b B+24 a^2 b^2 C-40 a b^3 B\right) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{b}+\frac{2 \left(4 a^4 B-30 a^3 b C+44 a^2 b^2 B+27 a b^3 C-45 b^4 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{12 a^3 d (a-b) (a+b)}","\frac{\left(2 a^2 B+3 a b C-5 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d \left(a^2-b^2\right)}+\frac{b (b B-a C) \sin (c+d x)}{a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac{\left(2 a^2 B+3 a b C-5 b^2 B\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}+\frac{\left(-2 a^3 C+4 a^2 b B+3 a b^2 C-5 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \left(a^2-b^2\right)}+\frac{b \left(-5 a^3 C+7 a^2 b B+3 a b^2 C-5 b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}-\frac{\left(-2 a^3 C+4 a^2 b B+3 a b^2 C-5 b^3 B\right) \sin (c+d x)}{a^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}",1,"((2*(4*a^4*B + 44*a^2*b^2*B - 45*b^4*B - 30*a^3*b*C + 27*a*b^3*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + ((28*a^3*b*B - 40*a*b^3*B - 12*a^4*C + 24*a^2*b^2*C)*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)))/b + (2*(12*a^2*b^2*B - 15*b^4*B - 6*a^3*b*C + 9*a*b^3*C)*Cos[2*(c + d*x)]*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[1 - Cos[c + d*x]^2]*(-1 + 2*Cos[c + d*x]^2)))/(12*a^3*(a - b)*(a + b)*d) + (Sqrt[Cos[c + d*x]]*((2*Sec[c + d*x]*(-2*b*B*Sin[c + d*x] + a*C*Sin[c + d*x]))/a^3 + (b^4*B*Sin[c + d*x] - a*b^3*C*Sin[c + d*x])/(a^3*(a^2 - b^2)*(a + b*Cos[c + d*x])) + (2*B*Sec[c + d*x]*Tan[c + d*x])/(3*a^2)))/d","A",1
891,1,462,461,4.774341,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)} \left(35 a^6 C-15 a^5 b B-57 a^4 b^2 C+33 a^3 b^3 B+4 C \left(b^3-a^2 b\right)^2 \cos (2 (c+d x))+a b \left(49 a^4 C-21 a^3 b B-83 a^2 b^2 C+39 a b^3 B+16 b^4 C\right) \cos (c+d x)+4 b^6 C\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{\frac{16 \left(7 a^4 C-3 a^3 b B-14 a^2 b^2 C+12 a b^3 B-2 b^4 C\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{2 \left(35 a^5 C-15 a^4 b B-73 a^3 b^2 C+21 a^2 b^3 B+56 a b^4 C-24 b^5 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{6 \left(35 a^5 C-15 a^4 b B-65 a^3 b^2 C+29 a^2 b^3 B+24 a b^4 C-8 b^5 B\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{48 b^3 d}","\frac{a (b B-a C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{a \left(-7 a^3 C+3 a^2 b B+13 a b^2 C-9 b^3 B\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\left(-35 a^4 C+15 a^3 b B+61 a^2 b^2 C-33 a b^3 B-8 b^4 C\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{12 b^3 d \left(a^2-b^2\right)^2}+\frac{\left(-35 a^5 C+15 a^4 b B+65 a^3 b^2 C-29 a^2 b^3 B-24 a b^4 C+8 b^5 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d \left(a^2-b^2\right)^2}+\frac{a^2 \left(-35 a^5 C+15 a^4 b B+86 a^3 b^2 C-38 a^2 b^3 B-63 a b^4 C+35 b^5 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^5 d (a-b)^2 (a+b)^3}-\frac{\left(-105 a^6 C+45 a^5 b B+223 a^4 b^2 C-99 a^3 b^3 B-128 a^2 b^4 C+72 a b^5 B-8 b^6 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 b^5 d \left(a^2-b^2\right)^2}",1,"((4*Sqrt[Cos[c + d*x]]*(-15*a^5*b*B + 33*a^3*b^3*B + 35*a^6*C - 57*a^4*b^2*C + 4*b^6*C + a*b*(-21*a^3*b*B + 39*a*b^3*B + 49*a^4*C - 83*a^2*b^2*C + 16*b^4*C)*Cos[c + d*x] + 4*(-(a^2*b) + b^3)^2*C*Cos[2*(c + d*x)])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) - ((2*(-15*a^4*b*B + 21*a^2*b^3*B - 24*b^5*B + 35*a^5*C - 73*a^3*b^2*C + 56*a*b^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (16*(-3*a^3*b*B + 12*a*b^3*B + 7*a^4*C - 14*a^2*b^2*C - 2*b^4*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (6*(-15*a^4*b*B + 29*a^2*b^3*B - 8*b^5*B + 35*a^5*C - 65*a^3*b^2*C + 24*a*b^4*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(48*b^3*d)","A",1
892,1,390,367,4.8233907,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{\frac{8 \left(a^3 C+a^2 b B-4 a b^2 C+2 b^3 B\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{\left(5 a^4 C-a^3 b B-7 a^2 b^2 C-5 a b^3 B+8 b^4 C\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{\left(15 a^4 C-3 a^3 b B-29 a^2 b^2 C+9 a b^3 B+8 b^4 C\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}-\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)} \left(b \left(7 a^3 C-3 a^2 b B-13 a b^2 C+9 b^3 B\right) \cos (c+d x)+a \left(5 a^3 C-a^2 b B-11 a b^2 C+7 b^3 B\right)\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}}{8 b^2 d}","\frac{a (b B-a C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{a \left(-5 a^3 C+a^2 b B+11 a b^2 C-7 b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\left(-15 a^4 C+3 a^3 b B+29 a^2 b^2 C-9 a b^3 B-8 b^4 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d \left(a^2-b^2\right)^2}+\frac{\left(-15 a^5 C+3 a^4 b B+33 a^3 b^2 C-5 a^2 b^3 B-24 a b^4 C+8 b^5 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d \left(a^2-b^2\right)^2}-\frac{a \left(-15 a^5 C+3 a^4 b B+38 a^3 b^2 C-6 a^2 b^3 B-35 a b^4 C+15 b^5 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d (a-b)^2 (a+b)^3}",1,"((-2*a*Sqrt[Cos[c + d*x]]*(a*(-(a^2*b*B) + 7*b^3*B + 5*a^3*C - 11*a*b^2*C) + b*(-3*a^2*b*B + 9*b^3*B + 7*a^3*C - 13*a*b^2*C)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) + (((-(a^3*b*B) - 5*a*b^3*B + 5*a^4*C - 7*a^2*b^2*C + 8*b^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(a^2*b*B + 2*b^3*B + a^3*C - 4*a*b^2*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + ((-3*a^3*b*B + 9*a*b^3*B + 15*a^4*C - 29*a^2*b^2*C + 8*b^4*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(8*b^2*d)","A",1
893,1,360,344,3.6494374,"\int \frac{\sqrt{\cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(b \left(3 a^3 C+a^2 b B-9 a b^2 C+5 b^3 B\right) \cos (c+d x)+a \left(a^3 C+3 a^2 b B-7 a b^2 C+3 b^3 B\right)\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{-\frac{8 \left(a^2 C-3 a b B+2 b^2 C\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{\left(a^3 C-5 a^2 b B+5 a b^2 C-b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{\left(3 a^3 C+a^2 b B-9 a b^2 C+5 b^3 B\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{8 b d}","\frac{a (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{\left(3 a^3 C+a^2 b B-9 a b^2 C+5 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d \left(a^2-b^2\right)^2}+\frac{\left(3 a^3 C+a^2 b B-9 a b^2 C+5 b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(3 a^4 C+a^3 b B-5 a^2 b^2 C-7 a b^3 B+8 b^4 C\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d \left(a^2-b^2\right)^2}-\frac{\left(3 a^5 C+a^4 b B-6 a^3 b^2 C-10 a^2 b^3 B+15 a b^4 C-3 b^5 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d (a-b)^2 (a+b)^3}",1,"((2*Sqrt[Cos[c + d*x]]*(a*(3*a^2*b*B + 3*b^3*B + a^3*C - 7*a*b^2*C) + b*(a^2*b*B + 5*b^3*B + 3*a^3*C - 9*a*b^2*C)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) - (((-5*a^2*b*B - b^3*B + a^3*C + 5*a*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) - (8*(-3*a*b*B + a^2*C + 2*b^2*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + ((a^2*b*B + 5*b^3*B + 3*a^3*C - 9*a*b^2*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(8*b*d)","A",1
894,1,365,337,4.5992754,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3),x]","\frac{\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)} \left(b \left(a^3 C-5 a^2 b B+5 a b^2 C-b^3 B\right) \cos (c+d x)+a \left(3 a^3 C-7 a^2 b B+3 a b^2 C+b^3 B\right)\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\frac{16 a \left(2 a^2 B-3 a b C+b^2 B\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{b (a+b)}+\frac{2 \left(5 a^3 C-9 a^2 b B+a b^2 C+3 b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}-\frac{2 \left(a^3 C-5 a^2 b B+5 a b^2 C-b^3 B\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{16 a d}","-\frac{(b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\left(a^3 C+3 a^2 b B-7 a b^2 C+3 b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d \left(a^2-b^2\right)^2}+\frac{\left(a^3 (-C)+5 a^2 b B-5 a b^2 C+b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b d \left(a^2-b^2\right)^2}-\frac{\left(a^3 (-C)+5 a^2 b B-5 a b^2 C+b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 a d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\left(a^5 C+3 a^4 b B-10 a^3 b^2 C+10 a^2 b^3 B-3 a b^4 C-b^5 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^2 d (a-b)^2 (a+b)^3}",1,"((4*Sqrt[Cos[c + d*x]]*(a*(-7*a^2*b*B + b^3*B + 3*a^3*C + 3*a*b^2*C) + b*(-5*a^2*b*B - b^3*B + a^3*C + 5*a*b^2*C)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) + ((2*(-9*a^2*b*B + 3*b^3*B + 5*a^3*C + a*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (16*a*(2*a^2*B + b^2*B - 3*a*b*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(b*(a + b)) - (2*(-5*a^2*b*B - b^3*B + a^3*C + 5*a*b^2*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(16*a*d)","A",1
895,1,383,345,4.8636108,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3),x]","\frac{\frac{\frac{8 a \left(2 a^3 C-4 a^2 b B+a b^2 C+b^3 B\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{b (a+b)}+\frac{\left(5 a^3 C-9 a^2 b B+a b^2 C+3 b^3 B\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}+\frac{\left(16 a^4 B-9 a^3 b C-19 a^2 b^2 B+3 a b^3 C+9 b^4 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{(a-b)^2 (a+b)^2}-\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} \left(b \left(5 a^3 C-9 a^2 b B+a b^2 C+3 b^3 B\right) \cos (c+d x)+a \left(7 a^3 C-11 a^2 b B-a b^2 C+5 b^3 B\right)\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}}{8 a^2 d}","\frac{b (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{\left(-3 a^3 C+7 a^2 b B-3 a b^2 C-b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b d \left(a^2-b^2\right)^2}-\frac{\left(-5 a^3 C+9 a^2 b B-a b^2 C-3 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d \left(a^2-b^2\right)^2}+\frac{b \left(-5 a^3 C+9 a^2 b B-a b^2 C-3 b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(-3 a^5 C+15 a^4 b B-10 a^3 b^2 C-6 a^2 b^3 B+a b^4 C+3 b^5 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b d (a-b)^2 (a+b)^3}",1,"((-2*b*Sqrt[Cos[c + d*x]]*(a*(-11*a^2*b*B + 5*b^3*B + 7*a^3*C - a*b^2*C) + b*(-9*a^2*b*B + 3*b^3*B + 5*a^3*C + a*b^2*C)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) + (((16*a^4*B - 19*a^2*b^2*B + 9*b^4*B - 9*a^3*b*C + 3*a*b^3*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*a*(-4*a^2*b*B + b^3*B + 2*a^3*C + a*b^2*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(b*(a + b)) + ((-9*a^2*b*B + 3*b^3*B + 5*a^3*C + a*b^2*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(8*a^2*d)","A",1
896,1,458,420,5.5488361,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^3),x]","\frac{\frac{\sqrt{\cos (c+d x)} \left(16 B \left(a^3-a b^2\right)^2 \tan (c+d x)+b^2 \left(8 a^4 B+9 a^3 b C-29 a^2 b^2 B-3 a b^3 C+15 b^4 B\right) \sin (2 (c+d x))+2 a b \left(16 a^4 B+11 a^3 b C-47 a^2 b^2 B-5 a b^3 C+25 b^4 B\right) \sin (c+d x)\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{\frac{8 a \left(2 a^4 B+4 a^3 b C-10 a^2 b^2 B-a b^3 C+5 b^4 B\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{b (a+b)}+\frac{\left(8 a^4 B+9 a^3 b C-29 a^2 b^2 B-3 a b^3 C+15 b^4 B\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}+\frac{\left(-16 a^5 C+56 a^4 b B+19 a^3 b^2 C-95 a^2 b^3 B-9 a b^4 C+45 b^5 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{(a-b)^2 (a+b)^2}}{8 a^3 d}","\frac{b (b B-a C) \sin (c+d x)}{2 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}+\frac{\left(-7 a^3 C+11 a^2 b B+a b^2 C-5 b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d \left(a^2-b^2\right)^2}+\frac{b \left(-7 a^3 C+11 a^2 b B+a b^2 C-5 b^3 B\right) \sin (c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}-\frac{\left(8 a^4 B+9 a^3 b C-29 a^2 b^2 B-3 a b^3 C+15 b^4 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}+\frac{\left(8 a^4 B+9 a^3 b C-29 a^2 b^2 B-3 a b^3 C+15 b^4 B\right) \sin (c+d x)}{4 a^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}-\frac{\left(-15 a^5 C+35 a^4 b B+6 a^3 b^2 C-38 a^2 b^3 B-3 a b^4 C+15 b^5 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d (a-b)^2 (a+b)^3}",1,"(-((((56*a^4*b*B - 95*a^2*b^3*B + 45*b^5*B - 16*a^5*C + 19*a^3*b^2*C - 9*a*b^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*a*(2*a^4*B - 10*a^2*b^2*B + 5*b^4*B + 4*a^3*b*C - a*b^3*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(b*(a + b)) + ((8*a^4*B - 29*a^2*b^2*B + 15*b^4*B + 9*a^3*b*C - 3*a*b^3*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2)) + (Sqrt[Cos[c + d*x]]*(2*a*b*(16*a^4*B - 47*a^2*b^2*B + 25*b^4*B + 11*a^3*b*C - 5*a*b^3*C)*Sin[c + d*x] + b^2*(8*a^4*B - 29*a^2*b^2*B + 15*b^4*B + 9*a^3*b*C - 3*a*b^3*C)*Sin[2*(c + d*x)] + 16*(a^3 - a*b^2)^2*B*Tan[c + d*x]))/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2))/(8*a^3*d)","A",1
897,1,1224,560,6.3255312,"\int \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{(6 b B+a C) \sin (c+d x)}{12 b}+\frac{1}{6} C \sin (2 (c+d x))\right)}{d}-\frac{-\frac{4 a \left(C a^2-18 b B a-16 b^2 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-24 B b^2-28 a C b\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(3 C a^2-6 b B a-16 b^2 C\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{48 b d}","\frac{\left(-3 a^2 C+6 a b B+16 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 b^2 d \sqrt{\cos (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(-3 a^2 C+6 a b B+16 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^2 d}+\frac{\sqrt{a+b} \left(a^3 (-C)+2 a^2 b B-4 a b^2 C-8 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^3 d}+\frac{\sqrt{a+b} (a+2 b) (-3 a C+6 b B+8 b C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^2 d}+\frac{(2 b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 b d}",1,"-1/48*((-4*a*(-18*a*b*B + a^2*C - 16*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-24*b^2*B - 28*a*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-6*a*b*B + 3*a^2*C - 16*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(b*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(((6*b*B + a*C)*Sin[c + d*x])/(12*b) + (C*Sin[2*(c + d*x)])/6))/d","C",0
898,1,1175,473,21.1046256,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{C \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{2 d}+\frac{-\frac{4 a (4 b B+3 a C) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a (8 a B+4 b C) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 (4 b B+a C) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{8 d}","-\frac{\sqrt{a+b} \left(a^2 (-C)+4 a b B+4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}+\frac{(a C+4 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} (a C+2 b (2 B+C)) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}-\frac{(a-b) \sqrt{a+b} (a C+4 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}",1,"(C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + ((-4*a*(4*b*B + 3*a*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(8*a*B + 4*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(4*b*B + a*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(8*d)","C",1
899,1,408,385,11.4678021,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{\sqrt{\cos (c+d x)} \left(-4 (a (C-B)+b B) \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+8 b B \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+2 C (a+b) \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+4 a C \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+2 a C \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \tan \left(\frac{1}{2} (c+d x)\right)-b C \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \tan \left(\frac{1}{2} (c+d x)\right)+b C \sin \left(\frac{3}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sec \left(\frac{1}{2} (c+d x)\right)\right)}{2 d \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{a+b \cos (c+d x)}}","\frac{\sqrt{a+b} (2 B+C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}-\frac{\sqrt{a+b} (a C+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{C (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}",1,"(Sqrt[Cos[c + d*x]]*(2*(a + b)*C*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 4*(b*B + a*(-B + C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 8*b*B*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 4*a*C*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + b*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sin[(3*(c + d*x))/2] + 2*a*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Tan[(c + d*x)/2] - b*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Tan[(c + d*x)/2]))/(2*d*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[a + b*Cos[c + d*x]])","A",1
900,1,273,351,12.5105151,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 (a (B+C)+b (B-C)) \sqrt{\cos (c+d x)+1} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+\frac{2 B \tan \left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))}{\sqrt{\cos (c+d x)}}-2 B (a+b) \sqrt{\cos (c+d x)+1} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+4 b C \sqrt{\cos (c+d x)+1} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}","\frac{2 \sqrt{a+b} (b B-a (B-C)) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}+\frac{2 B (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}",1,"(-2*(a + b)*B*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*(b*(B - C) + a*(B + C))*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 4*b*C*Sqrt[1 + Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (2*B*(a + b*Cos[c + d*x])*Tan[(c + d*x)/2])/Sqrt[Cos[c + d*x]])/(d*Sqrt[a + b*Cos[c + d*x]])","A",1
901,1,407,284,13.3548807,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \sec (c+d x) (3 a C \sin (c+d x)+b B \sin (c+d x))}{3 a}+\frac{2}{3} B \tan (c+d x) \sec (c+d x)\right)}{d}+\frac{4 \left(\frac{\cos (c+d x)}{\cos (c+d x)+1}\right)^{3/2} \sqrt{\cos (c+d x)+1} \cos ^2\left(\frac{1}{2} (c+d x)\right)^{5/2} \sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} \left(-\left((3 a C+b B) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))\right)+2 a (a+b) (B+3 C) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-2 (a+b) (3 a C+b B) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{3 a d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}","\frac{2 (a-b) \sqrt{a+b} (3 a C+b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d}+\frac{2 (a-b) \sqrt{a+b} (B-3 C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}+\frac{2 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(4*(Cos[(c + d*x)/2]^2)^(5/2)*(Cos[c + d*x]/(1 + Cos[c + d*x]))^(3/2)*Sqrt[1 + Cos[c + d*x]]*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*(-2*(a + b)*(b*B + 3*a*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(B + 3*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (b*B + 3*a*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*a*d*Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]*(b*B*Sin[c + d*x] + 3*a*C*Sin[c + d*x]))/(3*a) + (2*B*Sec[c + d*x]*Tan[c + d*x])/3))/d","A",0
902,1,1315,350,6.3759883,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 (b B \sin (c+d x)+5 a C \sin (c+d x)) \sec ^2(c+d x)}{15 a}+\frac{2}{5} B \tan (c+d x) \sec ^2(c+d x)+\frac{2 \left(9 B \sin (c+d x) a^2+5 b C \sin (c+d x) a-2 b^2 B \sin (c+d x)\right) \sec (c+d x)}{15 a^2}\right)}{d}-\frac{-\frac{4 a \left(-5 C a^3+2 b B a^2+5 b^2 C a-2 b^3 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(9 B a^3+5 b C a^2-2 b^2 B a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-2 B b^3+5 a C b^2+9 a^2 B b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{15 a^2 d}","-\frac{2 (a-b) \sqrt{a+b} (9 a B-5 a C+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \left(9 a^2 B+5 a b C-2 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d}+\frac{2 (5 a C+b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"-1/15*((-4*a*(2*a^2*b*B - 2*b^3*B - 5*a^3*C + 5*a*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(9*a^3*B - 2*a*b^2*B + 5*a^2*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(9*a^2*b*B - 2*b^3*B + 5*a*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a^2*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^2*(b*B*Sin[c + d*x] + 5*a*C*Sin[c + d*x]))/(15*a) + (2*Sec[c + d*x]*(9*a^2*B*Sin[c + d*x] - 2*b^2*B*Sin[c + d*x] + 5*a*b*C*Sin[c + d*x]))/(15*a^2) + (2*B*Sec[c + d*x]^2*Tan[c + d*x])/5))/d","C",1
903,1,1408,433,6.4779035,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{-\frac{4 a \left(25 B a^4-14 b C a^3-17 b^2 B a^2+14 b^3 C a-8 b^4 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-63 C a^4-19 b B a^3+14 b^2 C a^2-8 b^3 B a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-8 B b^4+14 a C b^3-19 a^2 B b^2-63 a^3 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{105 a^3 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 (b B \sin (c+d x)+7 a C \sin (c+d x)) \sec ^3(c+d x)}{35 a}+\frac{2}{7} B \tan (c+d x) \sec ^3(c+d x)+\frac{2 \left(25 B \sin (c+d x) a^2+7 b C \sin (c+d x) a-4 b^2 B \sin (c+d x)\right) \sec ^2(c+d x)}{105 a^2}+\frac{2 \left(63 C \sin (c+d x) a^3+19 b B \sin (c+d x) a^2-14 b^2 C \sin (c+d x) a+8 b^3 B \sin (c+d x)\right) \sec (c+d x)}{105 a^3}\right)}{d}","\frac{2 \left(25 a^2 B+7 a b C-4 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(a^2 (25 B-63 C)+2 a b (3 B-7 C)+8 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(63 a^3 C+19 a^2 b B-14 a b^2 C+8 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d}+\frac{2 (7 a C+b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"((-4*a*(25*a^4*B - 17*a^2*b^2*B - 8*b^4*B - 14*a^3*b*C + 14*a*b^3*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-19*a^3*b*B - 8*a*b^3*B - 63*a^4*C + 14*a^2*b^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-19*a^2*b^2*B - 8*b^4*B - 63*a^3*b*C + 14*a*b^3*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(105*a^3*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^3*(b*B*Sin[c + d*x] + 7*a*C*Sin[c + d*x]))/(35*a) + (2*Sec[c + d*x]^2*(25*a^2*B*Sin[c + d*x] - 4*b^2*B*Sin[c + d*x] + 7*a*b*C*Sin[c + d*x]))/(105*a^2) + (2*Sec[c + d*x]*(19*a^2*b*B*Sin[c + d*x] + 8*b^3*B*Sin[c + d*x] + 63*a^3*C*Sin[c + d*x] - 14*a*b^2*C*Sin[c + d*x]))/(105*a^3) + (2*B*Sec[c + d*x]^3*Tan[c + d*x])/7))/d","C",0
904,1,1284,670,6.4139931,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{\left(3 C a^2+56 b B a+42 b^2 C\right) \sin (c+d x)}{96 b}+\frac{1}{48} (8 b B+9 a C) \sin (2 (c+d x))+\frac{1}{16} b C \sin (3 (c+d x))\right)}{d}-\frac{-\frac{4 a \left(3 C a^3-136 b B a^2-228 b^2 C a-128 b^3 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-144 C b^3-416 a B b^2-228 a^2 C b\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(9 C a^3-24 b B a^2-156 b^2 C a-128 b^3 B\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{384 b d}","\frac{\left(-3 a^2 C+8 a b B+12 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{32 b d}+\frac{\left(-9 a^3 C+24 a^2 b B+156 a b^2 C+128 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{192 b^2 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(9 a^3 C-6 a^2 b (4 B+C)-4 a b^2 (28 B+39 C)-8 b^3 (16 B+9 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b^2 d}-\frac{(a-b) \sqrt{a+b} \left(-9 a^3 C+24 a^2 b B+156 a b^2 C+128 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 a b^2 d}+\frac{\sqrt{a+b} \left(-3 a^4 C+8 a^3 b B-24 a^2 b^2 C-96 a b^3 B-48 b^4 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^3 d}+\frac{(8 b B-3 a C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{24 b d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}",1,"-1/384*((-4*a*(-136*a^2*b*B - 128*b^3*B + 3*a^3*C - 228*a*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-416*a*b^2*B - 228*a^2*b*C - 144*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-24*a^2*b*B - 128*b^3*B + 9*a^3*C - 156*a*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(b*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(((56*a*b*B + 3*a^2*C + 42*b^2*C)*Sin[c + d*x])/(96*b) + ((8*b*B + 9*a*C)*Sin[2*(c + d*x)])/48 + (b*C*Sin[3*(c + d*x)])/16))/d","C",0
905,1,1227,566,6.3173979,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{-\frac{4 a \left(17 C a^2+42 b B a+16 b^2 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(48 B a^2+52 b C a+24 b^2 B\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(3 C a^2+30 b B a+16 b^2 C\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{48 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{1}{12} (6 b B+7 a C) \sin (c+d x)+\frac{1}{6} b C \sin (2 (c+d x))\right)}{d}","\frac{\left(3 a^2 C+30 a b B+16 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(3 a^2 C+30 a b B+14 a b C+12 b^2 B+16 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b d}-\frac{(a-b) \sqrt{a+b} \left(3 a^2 C+30 a b B+16 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b d}-\frac{\sqrt{a+b} \left(a^3 (-C)+6 a^2 b B+12 a b^2 C+8 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^2 d}+\frac{(7 a C+6 b B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{12 d}+\frac{b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}",1,"((-4*a*(42*a*b*B + 17*a^2*C + 16*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(48*a^2*B + 24*b^2*B + 52*a*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(30*a*b*B + 3*a^2*C + 16*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(48*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(((6*b*B + 7*a*C)*Sin[c + d*x])/12 + (b*C*Sin[2*(c + d*x)])/6))/d","C",0
906,1,1198,472,6.356854,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{b C \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{2 d}+\frac{-\frac{4 a \left(8 B a^2+7 b C a+4 b^2 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(8 C a^2+16 b B a+4 b^2 C\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(4 B b^2+5 a C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{8 d}","-\frac{\sqrt{a+b} \left(3 a^2 C+12 a b B+4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}+\frac{(5 a C+4 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} (8 a B+5 a C+4 b B+2 b C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}-\frac{(a-b) \sqrt{a+b} (5 a C+4 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a d}+\frac{b C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}",1,"(b*C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + ((-4*a*(8*a^2*B + 4*b^2*B + 7*a*b*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(16*a*b*B + 8*a^2*C + 4*b^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(4*b^2*B + 5*a*b*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(8*d)","C",1
907,1,1196,449,6.3452555,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 a B \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{\frac{4 a \left(-2 C a^2-2 b B a-b^2 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+4 a \left(2 B a^2-4 b C a-2 b^2 B\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)-2 \left(2 a b B-b^2 C\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{2 d}","-\frac{(2 a B-b C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} (2 a (B-C)-b (4 B+C)) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}+\frac{(a-b) \sqrt{a+b} (2 a B-b C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}-\frac{\sqrt{a+b} (3 a C+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}+\frac{2 a B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(2*a*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + ((4*a*(-2*a*b*B - 2*a^2*C - b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) + 4*a*(2*a^2*B - 2*b^2*B - 4*a*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) - 2*(2*a*b*B - b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(2*d)","C",1
908,1,1236,418,6.3760471,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{-\frac{4 a \left(B a^2+3 b C a-b^2 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-3 C a^2-4 b B a+3 b^2 C\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-4 B b^2-3 a C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{3 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{3} \sec (c+d x) (4 b B \sin (c+d x)+3 a C \sin (c+d x))+\frac{2}{3} a B \sec (c+d x) \tan (c+d x)\right)}{d}","\frac{2 \sqrt{a+b} \left(a^2 (B-3 C)-a b (4 B-6 C)+3 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}+\frac{2 (a-b) \sqrt{a+b} (3 a C+4 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}+\frac{2 a B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}",1,"((-4*a*(a^2*B - b^2*B + 3*a*b*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-4*a*b*B - 3*a^2*C + 3*b^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-4*b^2*B - 3*a*b*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(3*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]*(4*b*B*Sin[c + d*x] + 3*a*C*Sin[c + d*x]))/3 + (2*a*B*Sec[c + d*x]*Tan[c + d*x])/3))/d","C",1
909,1,1314,353,6.4629646,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{15} (6 b B \sin (c+d x)+5 a C \sin (c+d x)) \sec ^2(c+d x)+\frac{2}{5} a B \tan (c+d x) \sec ^2(c+d x)+\frac{2 \left(9 B \sin (c+d x) a^2+20 b C \sin (c+d x) a+3 b^2 B \sin (c+d x)\right) \sec (c+d x)}{15 a}\right)}{d}-\frac{-\frac{4 a \left(-5 C a^3-3 b B a^2+5 b^2 C a+3 b^3 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(9 B a^3+20 b C a^2+3 b^2 B a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(3 B b^3+20 a C b^2+9 a^2 B b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{15 a d}","\frac{2 (a-b) \sqrt{a+b} \left(9 a^2 B+20 a b C+3 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^2 d}+\frac{2 (5 a C+6 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 (a-b) \sqrt{a+b} (9 a B-5 a C-3 b B+15 b C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d}+\frac{2 a B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"-1/15*((-4*a*(-3*a^2*b*B + 3*b^3*B - 5*a^3*C + 5*a*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(9*a^3*B + 3*a*b^2*B + 20*a^2*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(9*a^2*b*B + 3*b^3*B + 20*a*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^2*(6*b*B*Sin[c + d*x] + 5*a*C*Sin[c + d*x]))/15 + (2*Sec[c + d*x]*(9*a^2*B*Sin[c + d*x] + 3*b^2*B*Sin[c + d*x] + 20*a*b*C*Sin[c + d*x]))/(15*a) + (2*a*B*Sec[c + d*x]^2*Tan[c + d*x])/5))/d","C",1
910,1,1407,433,6.5462987,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{-\frac{4 a \left(25 B a^4+21 b C a^3-31 b^2 B a^2-21 b^3 C a+6 b^4 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-63 C a^4-82 b B a^3-21 b^2 C a^2+6 b^3 B a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(6 B b^4-21 a C b^3-82 a^2 B b^2-63 a^3 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{105 a^2 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{35} (8 b B \sin (c+d x)+7 a C \sin (c+d x)) \sec ^3(c+d x)+\frac{2}{7} a B \tan (c+d x) \sec ^3(c+d x)+\frac{2 \left(25 B \sin (c+d x) a^2+42 b C \sin (c+d x) a+3 b^2 B \sin (c+d x)\right) \sec ^2(c+d x)}{105 a}+\frac{2 \left(63 C \sin (c+d x) a^3+82 b B \sin (c+d x) a^2+21 b^2 C \sin (c+d x) a-6 b^3 B \sin (c+d x)\right) \sec (c+d x)}{105 a^2}\right)}{d}","\frac{2 \left(25 a^2 B+42 a b C+3 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 (a-b) \sqrt{a+b} \left(-\left(a^2 (25 B-63 C)\right)+3 a b (19 B-7 C)+6 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \left(63 a^3 C+82 a^2 b B+21 a b^2 C-6 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d}+\frac{2 (7 a C+8 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"((-4*a*(25*a^4*B - 31*a^2*b^2*B + 6*b^4*B + 21*a^3*b*C - 21*a*b^3*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-82*a^3*b*B + 6*a*b^3*B - 63*a^4*C - 21*a^2*b^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-82*a^2*b^2*B + 6*b^4*B - 63*a^3*b*C - 21*a*b^3*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(105*a^2*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^3*(8*b*B*Sin[c + d*x] + 7*a*C*Sin[c + d*x]))/35 + (2*Sec[c + d*x]^2*(25*a^2*B*Sin[c + d*x] + 3*b^2*B*Sin[c + d*x] + 42*a*b*C*Sin[c + d*x]))/(105*a) + (2*Sec[c + d*x]*(82*a^2*b*B*Sin[c + d*x] - 6*b^3*B*Sin[c + d*x] + 63*a^3*C*Sin[c + d*x] + 21*a*b^2*C*Sin[c + d*x]))/(105*a^2) + (2*a*B*Sec[c + d*x]^3*Tan[c + d*x])/7))/d","C",0
911,1,1353,779,6.5430123,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{1}{40} C \sin (4 (c+d x)) b^2+\frac{1}{160} (10 b B+21 a C) \sin (3 (c+d x)) b+\frac{1}{480} \left(93 C a^2+170 b B a+88 b^2 C\right) \sin (2 (c+d x))+\frac{\left(15 C a^3+590 b B a^2+898 b^2 C a+420 b^3 B\right) \sin (c+d x)}{960 b}\right)}{d}-\frac{-\frac{4 a \left(15 C a^4-1330 b B a^3-3236 b^2 C a^2-3560 b^3 B a-1024 b^4 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-1440 B b^4-4624 a C b^3-6440 a^2 B b^2-2292 a^3 C b\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(45 C a^4-150 b B a^3-1692 b^2 C a^2-2840 b^3 B a-1024 b^4 C\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{3840 b d}","\frac{\left(-15 a^2 C+50 a b B+64 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{240 b d}+\frac{\left(-15 a^3 C+50 a^2 b B+172 a b^2 C+120 b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{320 b d}+\frac{\left(-45 a^4 C+150 a^3 b B+1692 a^2 b^2 C+2840 a b^3 B+1024 b^4 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(45 a^4 C-30 a^3 b (5 B+C)-4 a^2 b^2 (295 B+423 C)-8 a b^3 (355 B+193 C)-16 b^4 (45 B+64 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{1920 b^2 d}-\frac{(a-b) \sqrt{a+b} \left(-45 a^4 C+150 a^3 b B+1692 a^2 b^2 C+2840 a b^3 B+1024 b^4 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{1920 a b^2 d}+\frac{\sqrt{a+b} \left(-3 a^5 C+10 a^4 b B-40 a^3 b^2 C-240 a^2 b^3 B-240 a b^4 C-96 b^5 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{128 b^3 d}+\frac{(10 b B-3 a C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{40 b d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{7/2}}{5 b d}",1,"-1/3840*((-4*a*(-1330*a^3*b*B - 3560*a*b^3*B + 15*a^4*C - 3236*a^2*b^2*C - 1024*b^4*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-6440*a^2*b^2*B - 1440*b^4*B - 2292*a^3*b*C - 4624*a*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-150*a^3*b*B - 2840*a*b^3*B + 45*a^4*C - 1692*a^2*b^2*C - 1024*b^4*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(b*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(((590*a^2*b*B + 420*b^3*B + 15*a^3*C + 898*a*b^2*C)*Sin[c + d*x])/(960*b) + ((170*a*b*B + 93*a^2*C + 88*b^2*C)*Sin[2*(c + d*x)])/480 + (b*(10*b*B + 21*a*C)*Sin[3*(c + d*x)])/160 + (b^2*C*Sin[4*(c + d*x)])/40))/d","C",0
912,1,1287,664,6.4247749,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{-\frac{4 a \left(133 C a^3+472 b B a^2+356 b^2 C a+128 b^3 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(384 B a^3+644 b C a^2+608 b^2 B a+144 b^3 C\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(15 C a^3+264 b B a^2+284 b^2 C a+128 b^3 B\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{384 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{1}{16} C \sin (3 (c+d x)) b^2+\frac{1}{48} (8 b B+17 a C) \sin (2 (c+d x)) b+\frac{1}{96} \left(59 C a^2+104 b B a+42 b^2 C\right) \sin (c+d x)\right)}{d}","\frac{\left(5 a^2 C+24 a b B+12 b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{32 d}+\frac{\left(15 a^3 C+264 a^2 b B+284 a b^2 C+128 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{192 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(15 a^3 C+2 a^2 b (132 B+59 C)+4 a b^2 (52 B+71 C)+8 b^3 (16 B+9 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b d}-\frac{(a-b) \sqrt{a+b} \left(15 a^3 C+264 a^2 b B+284 a b^2 C+128 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 a b d}-\frac{\sqrt{a+b} \left(-5 a^4 C+40 a^3 b B+120 a^2 b^2 C+160 a b^3 B+48 b^4 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d}+\frac{(11 a C+8 b B) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{24 d}+\frac{b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}",1,"((-4*a*(472*a^2*b*B + 128*b^3*B + 133*a^3*C + 356*a*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(384*a^3*B + 608*a*b^2*B + 644*a^2*b*C + 144*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(264*a^2*b*B + 128*b^3*B + 15*a^3*C + 284*a*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(384*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(((104*a*b*B + 59*a^2*C + 42*b^2*C)*Sin[c + d*x])/96 + (b*(8*b*B + 17*a*C)*Sin[2*(c + d*x)])/48 + (b^2*C*Sin[3*(c + d*x)])/16))/d","C",0
913,1,1251,563,6.5079595,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{-\frac{4 a \left(48 B a^3+59 b C a^2+66 b^2 B a+16 b^3 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(48 C a^3+144 b B a^2+76 b^2 C a+24 b^3 B\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(16 C b^3+54 a B b^2+33 a^2 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{48 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{1}{6} C \sin (2 (c+d x)) b^2+\frac{1}{12} (6 b B+13 a C) \sin (c+d x) b\right)}{d}","\frac{\left(33 a^2 C+54 a b B+16 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(a^2 (48 B+33 C)+a b (54 B+26 C)+4 b^2 (3 B+4 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 d}-\frac{(a-b) \sqrt{a+b} \left(33 a^2 C+54 a b B+16 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a d}-\frac{\sqrt{a+b} \left(5 a^3 C+30 a^2 b B+20 a b^2 C+8 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b d}+\frac{b (3 a C+2 b B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{b C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}",1,"((-4*a*(48*a^3*B + 66*a*b^2*B + 59*a^2*b*C + 16*b^3*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(144*a^2*b*B + 24*b^3*B + 48*a^3*C + 76*a*b^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(54*a*b^2*B + 33*a^2*b*C + 16*b^3*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(48*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((b*(6*b*B + 13*a*C)*Sin[c + d*x])/12 + (b^2*C*Sin[2*(c + d*x)])/6))/d","C",0
914,1,1241,547,6.5460191,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{\frac{4 a \left(-8 C a^3-16 b B a^2-11 b^2 C a-4 b^3 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+4 a \left(8 B a^3-24 b C a^2-24 b^2 B a-4 b^3 C\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)-2 \left(-4 B b^3-9 a C b^2+8 a^2 B b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{8 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(2 B \tan (c+d x) a^2+\frac{1}{2} b^2 C \sin (c+d x)\right)}{d}","-\frac{\left(8 a^2 B-9 a b C-4 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(8 a^2 (B-C)-3 a b (8 B+3 C)-2 b^2 (2 B+C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}+\frac{(a-b) \sqrt{a+b} \left(8 a^2 B-9 a b C-4 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a d}-\frac{\sqrt{a+b} \left(15 a^2 C+20 a b B+4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}-\frac{b (4 a B-b C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt{\cos (c+d x)}}",1,"((4*a*(-16*a^2*b*B - 4*b^3*B - 8*a^3*C - 11*a*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) + 4*a*(8*a^3*B - 24*a*b^2*B - 24*a^2*b*C - 4*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) - 2*(8*a^2*b*B - 4*b^3*B - 9*a*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(8*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((b^2*C*Sin[c + d*x])/2 + 2*a^2*B*Tan[c + d*x]))/d","C",0
915,1,1269,536,6.5211919,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{-\frac{4 a \left(2 B a^3+12 b C a^2+4 b^2 B a+3 b^3 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-6 C a^3-14 b B a^2+18 b^2 C a+6 b^3 B\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(3 C b^3-14 a B b^2-6 a^2 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{6 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{3} B \sec (c+d x) \tan (c+d x) a^2+\frac{2}{3} \sec (c+d x) \left(3 C \sin (c+d x) a^2+7 b B \sin (c+d x) a\right)\right)}{d}","-\frac{\left(6 a^2 C+14 a b B-3 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(-2 a^2 (B-3 C)+2 a b (7 B-9 C)-3 b^2 (6 B+C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 d}+\frac{(a-b) \sqrt{a+b} \left(6 a^2 C+14 a b B-3 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}+\frac{2 a (a C+2 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{b \sqrt{a+b} (5 a C+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"((-4*a*(2*a^3*B + 4*a*b^2*B + 12*a^2*b*C + 3*b^3*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-14*a^2*b*B + 6*b^3*B - 6*a^3*C + 18*a*b^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-14*a*b^2*B - 6*a^2*b*C + 3*b^3*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(6*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]*(7*a*b*B*Sin[c + d*x] + 3*a^2*C*Sin[c + d*x]))/3 + (2*a^2*B*Sec[c + d*x]*Tan[c + d*x])/3))/d","C",1
916,1,1319,493,6.5564362,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{\frac{4 a \left(-5 C a^3-8 b B a^2-10 b^2 C a+8 b^3 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+4 a \left(9 B a^3+35 b C a^2+23 b^2 B a-15 b^3 C\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)-2 \left(23 B b^3+35 a C b^2+9 a^2 B b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{15 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{15} \left(5 C \sin (c+d x) a^2+11 b B \sin (c+d x) a\right) \sec ^2(c+d x)+\frac{2}{5} a^2 B \tan (c+d x) \sec ^2(c+d x)+\frac{2}{15} \left(9 B \sin (c+d x) a^2+35 b C \sin (c+d x) a+23 b^2 B \sin (c+d x)\right) \sec (c+d x)\right)}{d}","\frac{2 (a-b) \sqrt{a+b} \left(9 a^2 B+35 a b C+23 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d}+\frac{2 \sqrt{a+b} \left(-\left(a^3 (9 B-5 C)\right)+a^2 b (17 B-35 C)-a b^2 (23 B-45 C)+15 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d}-\frac{2 b^2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}+\frac{2 a (5 a C+8 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"((4*a*(-8*a^2*b*B + 8*b^3*B - 5*a^3*C - 10*a*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) + 4*a*(9*a^3*B + 23*a*b^2*B + 35*a^2*b*C - 15*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) - 2*(9*a^2*b*B + 23*b^3*B + 35*a*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(15*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^2*(11*a*b*B*Sin[c + d*x] + 5*a^2*C*Sin[c + d*x]))/15 + (2*Sec[c + d*x]*(9*a^2*B*Sin[c + d*x] + 23*b^2*B*Sin[c + d*x] + 35*a*b*C*Sin[c + d*x]))/15 + (2*a^2*B*Sec[c + d*x]^2*Tan[c + d*x])/5))/d","C",0
917,1,1409,434,6.6472383,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{-\frac{4 a \left(25 B a^4+56 b C a^3-10 b^2 B a^2-56 b^3 C a-15 b^4 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-63 C a^4-145 b B a^3-161 b^2 C a^2-15 b^3 B a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-15 B b^4-161 a C b^3-145 a^2 B b^2-63 a^3 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{105 a d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{35} \left(7 C \sin (c+d x) a^2+15 b B \sin (c+d x) a\right) \sec ^3(c+d x)+\frac{2}{7} a^2 B \tan (c+d x) \sec ^3(c+d x)+\frac{2}{105} \left(25 B \sin (c+d x) a^2+77 b C \sin (c+d x) a+45 b^2 B \sin (c+d x)\right) \sec ^2(c+d x)+\frac{2 \left(63 C \sin (c+d x) a^3+145 b B \sin (c+d x) a^2+161 b^2 C \sin (c+d x) a+15 b^3 B \sin (c+d x)\right) \sec (c+d x)}{105 a}\right)}{d}","\frac{2 \left(25 a^2 B+77 a b C+45 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(a^2 (25 B-63 C)-8 a b (15 B-7 C)+15 b^2 (B-7 C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a d}+\frac{2 (a-b) \sqrt{a+b} \left(63 a^3 C+145 a^2 b B+161 a b^2 C+15 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d}+\frac{2 a (7 a C+10 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"((-4*a*(25*a^4*B - 10*a^2*b^2*B - 15*b^4*B + 56*a^3*b*C - 56*a*b^3*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-145*a^3*b*B - 15*a*b^3*B - 63*a^4*C - 161*a^2*b^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-145*a^2*b^2*B - 15*b^4*B - 63*a^3*b*C - 161*a*b^3*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(105*a*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^3*(15*a*b*B*Sin[c + d*x] + 7*a^2*C*Sin[c + d*x]))/35 + (2*Sec[c + d*x]^2*(25*a^2*B*Sin[c + d*x] + 45*b^2*B*Sin[c + d*x] + 77*a*b*C*Sin[c + d*x]))/105 + (2*Sec[c + d*x]*(145*a^2*b*B*Sin[c + d*x] + 15*b^3*B*Sin[c + d*x] + 63*a^3*C*Sin[c + d*x] + 161*a*b^2*C*Sin[c + d*x]))/(105*a) + (2*a^2*B*Sec[c + d*x]^3*Tan[c + d*x])/7))/d","C",0
918,1,1517,522,6.7567308,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{63} \left(9 C \sin (c+d x) a^2+19 b B \sin (c+d x) a\right) \sec ^4(c+d x)+\frac{2}{9} a^2 B \tan (c+d x) \sec ^4(c+d x)+\frac{2}{315} \left(49 B \sin (c+d x) a^2+135 b C \sin (c+d x) a+75 b^2 B \sin (c+d x)\right) \sec ^3(c+d x)+\frac{2 \left(75 C \sin (c+d x) a^3+163 b B \sin (c+d x) a^2+135 b^2 C \sin (c+d x) a+5 b^3 B \sin (c+d x)\right) \sec ^2(c+d x)}{315 a}+\frac{2 \left(147 B \sin (c+d x) a^4+435 b C \sin (c+d x) a^3+279 b^2 B \sin (c+d x) a^2+45 b^3 C \sin (c+d x) a-10 b^4 B \sin (c+d x)\right) \sec (c+d x)}{315 a^2}\right)}{d}-\frac{-\frac{4 a \left(-75 C a^5-114 b B a^4+30 b^2 C a^3+124 b^3 B a^2+45 b^4 C a-10 b^5 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(147 B a^5+435 b C a^4+279 b^2 B a^3+45 b^3 C a^2-10 b^4 B a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-10 B b^5+45 a C b^4+279 a^2 B b^3+435 a^3 C b^2+147 a^4 B b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{315 a^2 d}","\frac{2 \left(49 a^2 B+135 a b C+75 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(75 a^3 C+163 a^2 b B+135 a b^2 C+5 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 (a-b) \sqrt{a+b} \left(3 a^3 (49 B-25 C)-6 a^2 b (19 B-60 C)+15 a b^2 (11 B-3 C)+10 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \left(147 a^4 B+435 a^3 b C+279 a^2 b^2 B+45 a b^3 C-10 b^4 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d}+\frac{2 a (3 a C+4 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"-1/315*((-4*a*(-114*a^4*b*B + 124*a^2*b^3*B - 10*b^5*B - 75*a^5*C + 30*a^3*b^2*C + 45*a*b^4*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(147*a^5*B + 279*a^3*b^2*B - 10*a*b^4*B + 435*a^4*b*C + 45*a^2*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(147*a^4*b*B + 279*a^2*b^3*B - 10*b^5*B + 435*a^3*b^2*C + 45*a*b^4*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a^2*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^4*(19*a*b*B*Sin[c + d*x] + 9*a^2*C*Sin[c + d*x]))/63 + (2*Sec[c + d*x]^3*(49*a^2*B*Sin[c + d*x] + 75*b^2*B*Sin[c + d*x] + 135*a*b*C*Sin[c + d*x]))/315 + (2*Sec[c + d*x]^2*(163*a^2*b*B*Sin[c + d*x] + 5*b^3*B*Sin[c + d*x] + 75*a^3*C*Sin[c + d*x] + 135*a*b^2*C*Sin[c + d*x]))/(315*a) + (2*Sec[c + d*x]*(147*a^4*B*Sin[c + d*x] + 279*a^2*b^2*B*Sin[c + d*x] - 10*b^4*B*Sin[c + d*x] + 435*a^3*b*C*Sin[c + d*x] + 45*a*b^3*C*Sin[c + d*x]))/(315*a^2) + (2*a^2*B*Sec[c + d*x]^4*Tan[c + d*x])/9))/d","C",0
919,1,1640,622,6.8827634,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{15}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(15/2),x]","\frac{-\frac{4 a \left(675 B a^6+1254 b C a^5-390 b^2 B a^4-1364 b^3 C a^3-245 b^4 B a^2+110 b^5 C a-40 b^6 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-1617 C a^6-3705 b B a^5-3069 b^2 C a^4-255 b^3 B a^3+110 b^4 C a^2-40 b^5 B a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-40 B b^6+110 a C b^5-255 a^2 B b^4-3069 a^3 C b^3-3705 a^4 B b^2-1617 a^5 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{3465 a^3 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{99} \left(11 C \sin (c+d x) a^2+23 b B \sin (c+d x) a\right) \sec ^5(c+d x)+\frac{2}{11} a^2 B \tan (c+d x) \sec ^5(c+d x)+\frac{2}{693} \left(81 B \sin (c+d x) a^2+209 b C \sin (c+d x) a+113 b^2 B \sin (c+d x)\right) \sec ^4(c+d x)+\frac{2 \left(539 C \sin (c+d x) a^3+1145 b B \sin (c+d x) a^2+825 b^2 C \sin (c+d x) a+15 b^3 B \sin (c+d x)\right) \sec ^3(c+d x)}{3465 a}+\frac{2 \left(675 B \sin (c+d x) a^4+1793 b C \sin (c+d x) a^3+1025 b^2 B \sin (c+d x) a^2+55 b^3 C \sin (c+d x) a-20 b^4 B \sin (c+d x)\right) \sec ^2(c+d x)}{3465 a^2}+\frac{2 \left(1617 C \sin (c+d x) a^5+3705 b B \sin (c+d x) a^4+3069 b^2 C \sin (c+d x) a^3+255 b^3 B \sin (c+d x) a^2-110 b^4 C \sin (c+d x) a+40 b^5 B \sin (c+d x)\right) \sec (c+d x)}{3465 a^3}\right)}{d}","\frac{2 \left(81 a^2 B+209 a b C+113 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{693 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 \left(539 a^3 C+1145 a^2 b B+825 a b^2 C+15 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3465 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(675 a^4 B+1793 a^3 b C+1025 a^2 b^2 B+55 a b^3 C-20 b^4 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3465 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(3 a^4 (225 B-539 C)-6 a^3 b (505 B-209 C)+15 a^2 b^2 (19 B-121 C)+10 a b^3 (3 B-11 C)+40 b^4 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3465 a^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(1617 a^5 C+3705 a^4 b B+3069 a^3 b^2 C+255 a^2 b^3 B-110 a b^4 C+40 b^5 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3465 a^4 d}+\frac{2 a (11 a C+14 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 a B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac{11}{2}}(c+d x)}",1,"((-4*a*(675*a^6*B - 390*a^4*b^2*B - 245*a^2*b^4*B - 40*b^6*B + 1254*a^5*b*C - 1364*a^3*b^3*C + 110*a*b^5*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-3705*a^5*b*B - 255*a^3*b^3*B - 40*a*b^5*B - 1617*a^6*C - 3069*a^4*b^2*C + 110*a^2*b^4*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-3705*a^4*b^2*B - 255*a^2*b^4*B - 40*b^6*B - 1617*a^5*b*C - 3069*a^3*b^3*C + 110*a*b^5*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(3465*a^3*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^5*(23*a*b*B*Sin[c + d*x] + 11*a^2*C*Sin[c + d*x]))/99 + (2*Sec[c + d*x]^4*(81*a^2*B*Sin[c + d*x] + 113*b^2*B*Sin[c + d*x] + 209*a*b*C*Sin[c + d*x]))/693 + (2*Sec[c + d*x]^3*(1145*a^2*b*B*Sin[c + d*x] + 15*b^3*B*Sin[c + d*x] + 539*a^3*C*Sin[c + d*x] + 825*a*b^2*C*Sin[c + d*x]))/(3465*a) + (2*Sec[c + d*x]^2*(675*a^4*B*Sin[c + d*x] + 1025*a^2*b^2*B*Sin[c + d*x] - 20*b^4*B*Sin[c + d*x] + 1793*a^3*b*C*Sin[c + d*x] + 55*a*b^3*C*Sin[c + d*x]))/(3465*a^2) + (2*Sec[c + d*x]*(3705*a^4*b*B*Sin[c + d*x] + 255*a^2*b^3*B*Sin[c + d*x] + 40*b^5*B*Sin[c + d*x] + 1617*a^5*C*Sin[c + d*x] + 3069*a^3*b^2*C*Sin[c + d*x] - 110*a*b^4*C*Sin[c + d*x]))/(3465*a^3) + (2*a^2*B*Sec[c + d*x]^5*Tan[c + d*x])/11))/d","C",0
920,1,1229,571,6.4175622,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{-\frac{4 a \left(5 C a^2-6 b B a+16 b^2 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(24 B b^2+4 a C b\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(15 C a^2-18 b B a+16 b^2 C\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{48 b^2 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{(6 b B-5 a C) \sin (c+d x)}{12 b^2}+\frac{C \sin (2 (c+d x))}{6 b}\right)}{d}","-\frac{\left(-15 a^2 C+18 a b B-16 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 b^3 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(-15 a^2 C+18 a b B+10 a b C-12 b^2 B-16 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^3 d}+\frac{(a-b) \sqrt{a+b} \left(-15 a^2 C+18 a b B-16 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^3 d}-\frac{\sqrt{a+b} \left(-5 a^3 C+6 a^2 b B-4 a b^2 C+8 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^4 d}+\frac{(6 b B-5 a C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{12 b^2 d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d}",1,"((-4*a*(-6*a*b*B + 5*a^2*C + 16*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(24*b^2*B + 4*a*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-18*a*b*B + 15*a^2*C + 16*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(48*b^2*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(((6*b*B - 5*a*C)*Sin[c + d*x])/(12*b^2) + (C*Sin[2*(c + d*x)])/(6*b)))/d","C",0
921,1,1175,479,12.7848802,"\int \frac{\sqrt{\cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{C \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{2 b d}+\frac{-\frac{4 a (4 b B-a C) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-16 a b C \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 (4 b B-3 a C) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{8 b d}","\frac{\sqrt{a+b} \left(-3 a^2 C+4 a b B-4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d}+\frac{(4 b B-3 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b^2 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} (3 a C-2 b (2 B+C)) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}-\frac{(a-b) \sqrt{a+b} (4 b B-3 a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b^2 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 b d}",1,"(C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b*d) + ((-4*a*(4*b*B - a*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 16*a*b*C*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(4*b*B - 3*a*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(8*b*d)","C",0
922,1,4017,391,17.6064346,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]),x]","\text{Result too large to show}","-\frac{\sqrt{a+b} (2 b B-a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}+\frac{C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d}-\frac{C (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d}",1,"((1 + Cos[c + d*x])^(3/2)*((B*Sqrt[Cos[c + d*x]])/Sqrt[a + b*Cos[c + d*x]] + (C*Cos[c + d*x]^(3/2))/Sqrt[a + b*Cos[c + d*x]])*Sec[(c + d*x)/2]^2*((2*I)*(a - b)*C*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))] + (4*I)*(b*B - a*C)*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))] - (8*I)*b*B*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))] + (4*I)*a*C*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))] + b*Sqrt[(a - b)/(a + b)]*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sin[(3*(c + d*x))/2] + 2*a*Sqrt[(a - b)/(a + b)]*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Tan[(c + d*x)/2] - b*Sqrt[(a - b)/(a + b)]*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Tan[(c + d*x)/2]))/(4*b*Sqrt[(a - b)/(a + b)]*d*Sqrt[a + b*Cos[c + d*x]]*(((1 + Cos[c + d*x])^(3/2)*Sec[(c + d*x)/2]^2*Sin[c + d*x]*((2*I)*(a - b)*C*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))] + (4*I)*(b*B - a*C)*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))] - (8*I)*b*B*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))] + (4*I)*a*C*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))] + b*Sqrt[(a - b)/(a + b)]*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sin[(3*(c + d*x))/2] + 2*a*Sqrt[(a - b)/(a + b)]*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Tan[(c + d*x)/2] - b*Sqrt[(a - b)/(a + b)]*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Tan[(c + d*x)/2]))/(8*Sqrt[(a - b)/(a + b)]*(a + b*Cos[c + d*x])^(3/2)) - (3*Sqrt[1 + Cos[c + d*x]]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*((2*I)*(a - b)*C*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))] + (4*I)*(b*B - a*C)*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))] - (8*I)*b*B*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))] + (4*I)*a*C*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))] + b*Sqrt[(a - b)/(a + b)]*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sin[(3*(c + d*x))/2] + 2*a*Sqrt[(a - b)/(a + b)]*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Tan[(c + d*x)/2] - b*Sqrt[(a - b)/(a + b)]*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Tan[(c + d*x)/2]))/(8*b*Sqrt[(a - b)/(a + b)]*Sqrt[a + b*Cos[c + d*x]]) + ((1 + Cos[c + d*x])^(3/2)*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]*((2*I)*(a - b)*C*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))] + (4*I)*(b*B - a*C)*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))] - (8*I)*b*B*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))] + (4*I)*a*C*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))] + b*Sqrt[(a - b)/(a + b)]*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sin[(3*(c + d*x))/2] + 2*a*Sqrt[(a - b)/(a + b)]*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Tan[(c + d*x)/2] - b*Sqrt[(a - b)/(a + b)]*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Tan[(c + d*x)/2]))/(4*b*Sqrt[(a - b)/(a + b)]*Sqrt[a + b*Cos[c + d*x]]) + ((1 + Cos[c + d*x])^(3/2)*Sec[(c + d*x)/2]^2*((3*b*Sqrt[(a - b)/(a + b)]*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Cos[(3*(c + d*x))/2]*Sec[(c + d*x)/2])/2 + a*Sqrt[(a - b)/(a + b)]*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]^2 - (b*Sqrt[(a - b)/(a + b)]*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]^2)/2 + (I*(a - b)*C*EllipticE[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + ((2*I)*(b*B - a*C)*EllipticF[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - ((4*I)*b*B*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + ((2*I)*a*C*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*Sqrt[(a - b)/(a + b)]*C*Sec[(c + d*x)/2]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x]))*Sin[(3*(c + d*x))/2])/(2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]) + (a*Sqrt[(a - b)/(a + b)]*C*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x]))*Tan[(c + d*x)/2])/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (b*Sqrt[(a - b)/(a + b)]*C*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x]))*Tan[(c + d*x)/2])/(2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]) + (b*Sqrt[(a - b)/(a + b)]*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sin[(3*(c + d*x))/2]*Tan[(c + d*x)/2])/2 - (2*Sqrt[(a - b)/(a + b)]*(b*B - a*C)*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 + ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a - b)*Sqrt[(a - b)/(a + b)]*C*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2])/Sqrt[1 + ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + (4*b*Sqrt[(a - b)/(a + b)]*B*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[1 + ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) - (2*a*Sqrt[(a - b)/(a + b)]*C*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[1 + ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])))/(4*b*Sqrt[(a - b)/(a + b)]*Sqrt[a + b*Cos[c + d*x]])))","C",0
923,1,144,228,1.5310344,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{2 \sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} \left((B-C) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{d \sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \cos (c+d x)}}","\frac{2 B \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d}",1,"(2*Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*((B - C)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]))/(d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2])","A",1
924,1,299,230,13.1934017,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{2 \left(B \sin (c+d x) (a+b \cos (c+d x))-\frac{2 \sqrt{2} \cos ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} \left(-2 a (B+C) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+B \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))+2 B (a+b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{(\cos (c+d x)+1)^{3/2}}\right)}{a d \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}","\frac{2 B (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d}-\frac{2 \sqrt{a+b} (B-C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}",1,"(2*(B*(a + b*Cos[c + d*x])*Sin[c + d*x] - (2*Sqrt[2]*(Cos[(c + d*x)/2]^2)^(3/2)*(2*(a + b)*B*Cos[(c + d*x)/2]^2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 2*a*(B + C)*Cos[(c + d*x)/2]^2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + B*Cos[c + d*x]*(a + b*Cos[c + d*x])*Tan[(c + d*x)/2]))/(1 + Cos[c + d*x])^(3/2)))/(a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",1
925,1,416,290,15.6822511,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \sec (c+d x) (3 a C \sin (c+d x)-2 b B \sin (c+d x))}{3 a^2}+\frac{2 B \tan (c+d x) \sec (c+d x)}{3 a}\right)}{d}+\frac{8 \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \cos ^2\left(\frac{1}{2} (c+d x)\right)^{7/2} \sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} \left((2 b B-3 a C) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))+2 a (a (B+3 C)-2 b B) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-2 (a+b) (3 a C-2 b B) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x) (\cos (c+d x)+1)^{3/2} \sqrt{a+b \cos (c+d x)}}","-\frac{2 (a-b) \sqrt{a+b} (2 b B-3 a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d}+\frac{2 \sqrt{a+b} (a (B-3 C)+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d}+\frac{2 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 a d \cos ^{\frac{3}{2}}(c+d x)}",1,"(8*(Cos[(c + d*x)/2]^2)^(7/2)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*(-2*(a + b)*(-2*b*B + 3*a*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(-2*b*B + a*(B + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (2*b*B - 3*a*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*a^2*d*Cos[c + d*x]^(3/2)*(1 + Cos[c + d*x])^(3/2)*Sqrt[a + b*Cos[c + d*x]]) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]*(-2*b*B*Sin[c + d*x] + 3*a*C*Sin[c + d*x]))/(3*a^2) + (2*B*Sec[c + d*x]*Tan[c + d*x])/(3*a)))/d","A",0
926,1,1319,363,6.4162789,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{9}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 (5 a C \sin (c+d x)-4 b B \sin (c+d x)) \sec ^2(c+d x)}{15 a^2}+\frac{2 B \tan (c+d x) \sec ^2(c+d x)}{5 a}+\frac{2 \left(9 B \sin (c+d x) a^2-10 b C \sin (c+d x) a+8 b^2 B \sin (c+d x)\right) \sec (c+d x)}{15 a^3}\right)}{d}-\frac{-\frac{4 a \left(-5 C a^3+7 b B a^2-10 b^2 C a+8 b^3 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(9 B a^3-10 b C a^2+8 b^2 B a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(8 B b^3-10 a C b^2+9 a^2 B b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{15 a^3 d}","-\frac{2 (4 b B-5 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(9 a^2 B-10 a b C+8 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^4 d}-\frac{2 \sqrt{a+b} \left(a^2 (9 B-5 C)-2 a b (B+5 C)+8 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d}+\frac{2 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 a d \cos ^{\frac{5}{2}}(c+d x)}",1,"-1/15*((-4*a*(7*a^2*b*B + 8*b^3*B - 5*a^3*C - 10*a*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(9*a^3*B + 8*a*b^2*B - 10*a^2*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(9*a^2*b*B + 8*b^3*B - 10*a*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a^3*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^2*(-4*b*B*Sin[c + d*x] + 5*a*C*Sin[c + d*x]))/(15*a^2) + (2*Sec[c + d*x]*(9*a^2*B*Sin[c + d*x] + 8*b^2*B*Sin[c + d*x] - 10*a*b*C*Sin[c + d*x]))/(15*a^3) + (2*B*Sec[c + d*x]^2*Tan[c + d*x])/(5*a)))/d","C",1
927,1,1297,620,6.5718764,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{C \sin (c+d x)}{2 b^2}-\frac{2 \left(a^3 C \sin (c+d x)-a^2 b B \sin (c+d x)\right)}{b^2 \left(b^2-a^2\right) (a+b \cos (c+d x))}\right)}{d}-\frac{-\frac{4 a \left(5 C a^3-4 b B a^2-5 b^2 C a+4 b^3 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(4 C b^3-8 a B b^2+4 a^2 C b\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(15 C a^3-12 b B a^2-7 b^2 C a+4 b^3 B\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{8 (a-b) b^2 (a+b) d}","\frac{2 a (b B-a C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{\left(-5 a^2 C+4 a b B+b^2 C\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 b^2 d \left(a^2-b^2\right)}+\frac{\sqrt{a+b} \left(-15 a^2 C+12 a b B-4 b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^4 d}+\frac{\left(-15 a^2 C+a b (12 B-5 C)+2 b^2 (2 B+C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{a+b}}+\frac{\left(-15 a^3 C+12 a^2 b B+7 a b^2 C-4 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{\left(-15 a^3 C+12 a^2 b B+7 a b^2 C-4 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b^3 d \sqrt{a+b}}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((C*Sin[c + d*x])/(2*b^2) - (2*(-(a^2*b*B*Sin[c + d*x]) + a^3*C*Sin[c + d*x]))/(b^2*(-a^2 + b^2)*(a + b*Cos[c + d*x]))))/d - ((-4*a*(-4*a^2*b*B + 4*b^3*B + 5*a^3*C - 5*a*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-8*a*b^2*B + 4*a^2*b*C + 4*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-12*a^2*b*B + 4*b^3*B + 15*a^3*C - 7*a*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(8*(a - b)*b^2*(a + b)*d)","C",0
928,1,1234,500,6.3900827,"\int \frac{\sqrt{\cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 \sqrt{\cos (c+d x)} \left(a^2 C \sin (c+d x)-a b B \sin (c+d x)\right)}{b \left(b^2-a^2\right) d \sqrt{a+b \cos (c+d x)}}+\frac{-\frac{4 a \left(a^2 C-b^2 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(2 a b C-2 b^2 B\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(3 C a^2-2 b B a-b^2 C\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{2 (a-b) b (a+b) d}","-\frac{\left(-3 a^2 C+2 a b B+b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{b^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}+\frac{2 a (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\left(-3 a^2 C+2 a b B+b^2 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b}}-\frac{\sqrt{a+b} (2 b B-3 a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d}-\frac{(-3 a C+2 b B-b C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d \sqrt{a+b}}",1,"(2*Sqrt[Cos[c + d*x]]*(-(a*b*B*Sin[c + d*x]) + a^2*C*Sin[c + d*x]))/(b*(-a^2 + b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((-4*a*(a^2*C - b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-2*b^2*B + 2*a*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-2*a*b*B + 3*a^2*C - b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(2*(a - b)*b*(a + b)*d)","C",0
929,1,1012,416,18.0083491,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{2 \sqrt{\cos (c+d x)} (a C \sin (c+d x)-b B \sin (c+d x))}{\left(a^2-b^2\right) d \sqrt{a+b \cos (c+d x)}}-\frac{2 (b B-a C) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)-4 a (a B-b C) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{(b-a) (a+b) d}","\frac{2 a (b B-a C) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d}+\frac{2 (b B-a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{a+b}}-\frac{2 (b B-a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{a+b}}",1,"(2*Sqrt[Cos[c + d*x]]*(-(b*B*Sin[c + d*x]) + a*C*Sin[c + d*x]))/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (-4*a*(a*B - b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(b*B - a*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/((-a + b)*(a + b)*d)","C",1
930,1,1223,284,6.3617392,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{-\frac{4 a \left(a^2 B-b^2 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(a^2 C-a b B\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(a b C-b^2 B\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{a (a-b) (a+b) d}-\frac{2 \sqrt{\cos (c+d x)} \left(a b C \sin (c+d x)-b^2 B \sin (c+d x)\right)}{a \left(a^2-b^2\right) d \sqrt{a+b \cos (c+d x)}}","-\frac{2 (b B-a C) \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 (b B-a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d \sqrt{a+b}}+\frac{2 (B+C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{a+b}}",1,"(-2*Sqrt[Cos[c + d*x]]*(-(b^2*B*Sin[c + d*x]) + a*b*C*Sin[c + d*x]))/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((-4*a*(a^2*B - b^2*B)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-(a*b*B) + a^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-(b^2*B) + a*b*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a*(a - b)*(a + b)*d)","C",1
931,1,1281,305,6.4967653,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{-\frac{4 a \left(-C a^3+2 b B a^2+b^2 C a-2 b^3 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(B a^3+b C a^2-2 b^2 B a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-2 B b^3+a C b^2+a^2 B b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{a^2 (b-a) (a+b) d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \left(a b^2 C \sin (c+d x)-b^3 B \sin (c+d x)\right)}{a^2 \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{2 B \tan (c+d x)}{a^2}\right)}{d}","\frac{2 b (b B-a C) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{2 (a (B-C)+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d \sqrt{a+b}}+\frac{2 \left(a^2 B+a b C-2 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^3 d \sqrt{a+b}}",1,"((-4*a*(2*a^2*b*B - 2*b^3*B - a^3*C + a*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(a^3*B - 2*a*b^2*B + a^2*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(a^2*b*B - 2*b^3*B + a*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a^2*(-a + b)*(a + b)*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*(-(b^3*B*Sin[c + d*x]) + a*b^2*C*Sin[c + d*x]))/(a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])) + (2*B*Tan[c + d*x])/a^2))/d","C",1
932,1,1357,393,6.704843,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{-\frac{4 a \left(B a^4-6 b C a^3+7 b^2 B a^2+6 b^3 C a-8 b^4 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-3 C a^4+5 b B a^3+6 b^2 C a^2-8 b^3 B a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-8 B b^4+6 a C b^3+5 a^2 B b^2-3 a^3 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{3 a^3 (a-b) (a+b) d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \sec (c+d x) (3 a C \sin (c+d x)-5 b B \sin (c+d x))}{3 a^3}-\frac{2 \left(a b^3 C \sin (c+d x)-b^4 B \sin (c+d x)\right)}{a^3 \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{2 B \sec (c+d x) \tan (c+d x)}{3 a^2}\right)}{d}","\frac{2 (a+2 b) (a (B-3 C)+4 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b}}+\frac{2 \left(a^2 B+3 a b C-4 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 a^2 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b (b B-a C) \sin (c+d x)}{a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-3 a^3 C+5 a^2 b B+6 a b^2 C-8 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b}}",1,"((-4*a*(a^4*B + 7*a^2*b^2*B - 8*b^4*B - 6*a^3*b*C + 6*a*b^3*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(5*a^3*b*B - 8*a*b^3*B - 3*a^4*C + 6*a^2*b^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(5*a^2*b^2*B - 8*b^4*B - 3*a^3*b*C + 6*a*b^3*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(3*a^3*(a - b)*(a + b)*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]*(-5*b*B*Sin[c + d*x] + 3*a*C*Sin[c + d*x]))/(3*a^3) - (2*(-(b^4*B*Sin[c + d*x]) + a*b^3*C*Sin[c + d*x]))/(a^3*(a^2 - b^2)*(a + b*Cos[c + d*x])) + (2*B*Sec[c + d*x]*Tan[c + d*x])/(3*a^2)))/d","C",0
933,1,1396,674,6.7148423,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(-\frac{2 \left(a^3 C \sin (c+d x)-a^2 b B \sin (c+d x)\right)}{3 b^2 \left(b^2-a^2\right) (a+b \cos (c+d x))^2}-\frac{2 \left(6 C \sin (c+d x) a^4-3 b B \sin (c+d x) a^3-10 b^2 C \sin (c+d x) a^2+7 b^3 B \sin (c+d x) a\right)}{3 b^2 \left(b^2-a^2\right)^2 (a+b \cos (c+d x))}\right)}{d}+\frac{-\frac{4 a \left(5 C a^4-2 b B a^3-8 b^2 C a^2+2 b^3 B a+3 b^4 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(6 B b^4-12 a C b^3+2 a^2 B b^2+4 a^3 C b\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(15 C a^4-6 b B a^3-26 b^2 C a^2+14 b^3 B a+3 b^4 C\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{6 (a-b)^2 b^2 (a+b)^2 d}","\frac{2 a (b B-a C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 a \left(-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{\left(-15 a^3 C+a^2 b (6 B-5 C)+a b^2 (2 B+21 C)-3 b^3 (4 B-C)\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 b^3 d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{\left(-15 a^4 C+6 a^3 b B+26 a^2 b^2 C-14 a b^3 B-3 b^4 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}+\frac{\left(-15 a^4 C+6 a^3 b B+26 a^2 b^2 C-14 a b^3 B-3 b^4 C\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^3 d (a-b) (a+b)^{3/2}}-\frac{\sqrt{a+b} (2 b B-5 a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^4 d}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((-2*(-(a^2*b*B*Sin[c + d*x]) + a^3*C*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) - (2*(-3*a^3*b*B*Sin[c + d*x] + 7*a*b^3*B*Sin[c + d*x] + 6*a^4*C*Sin[c + d*x] - 10*a^2*b^2*C*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)^2*(a + b*Cos[c + d*x]))))/d + ((-4*a*(-2*a^3*b*B + 2*a*b^3*B + 5*a^4*C - 8*a^2*b^2*C + 3*b^4*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(2*a^2*b^2*B + 6*b^4*B + 4*a^3*b*C - 12*a*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-6*a^3*b*B + 14*a*b^3*B + 15*a^4*C - 26*a^2*b^2*C + 3*b^4*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(6*(a - b)^2*b^2*(a + b)^2*d)","C",0
934,1,1342,545,6.5249168,"\int \frac{\sqrt{\cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \left(a^2 C \sin (c+d x)-a b B \sin (c+d x)\right)}{3 b \left(b^2-a^2\right) (a+b \cos (c+d x))^2}+\frac{2 \left(3 C \sin (c+d x) a^3-7 b^2 C \sin (c+d x) a+4 b^3 B \sin (c+d x)\right)}{3 b \left(b^2-a^2\right)^2 (a+b \cos (c+d x))}\right)}{d}-\frac{-\frac{4 a \left(C a^3-b B a^2-b^2 C a+b^3 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-3 C b^3+4 a B b^2-a^2 C b\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(3 C a^3-7 b^2 C a+4 b^3 B\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{3 (a-b)^2 b (a+b)^2 d}","\frac{2 \left(3 a^3 C-7 a b^2 C+4 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^2 d (a-b) (a+b)^{3/2}}+\frac{2 a (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 a \left(3 a^3 C-7 a b^2 C+4 b^3 B\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-3 a^3 C-a^2 b C+a b^2 B+6 a b^2 C-3 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^2 d (a-b) (a+b)^{3/2}}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*(-(a*b*B*Sin[c + d*x]) + a^2*C*Sin[c + d*x]))/(3*b*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (2*(4*b^3*B*Sin[c + d*x] + 3*a^3*C*Sin[c + d*x] - 7*a*b^2*C*Sin[c + d*x]))/(3*b*(-a^2 + b^2)^2*(a + b*Cos[c + d*x]))))/d - ((-4*a*(-(a^2*b*B) + b^3*B + a^3*C - a*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(4*a*b^2*B - a^2*b*C - 3*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(4*b^3*B + 3*a^3*C - 7*a*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(3*(a - b)^2*b*(a + b)^2*d)","C",0
935,1,1335,391,6.4420842,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 (a C \sin (c+d x)-b B \sin (c+d x))}{3 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{2 \left(B \sin (c+d x) b^3-4 a C \sin (c+d x) b^2+3 a^2 B \sin (c+d x) b\right)}{3 a \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}\right)}{d}+\frac{-\frac{4 a \left(C a^3-b B a^2-b^2 C a+b^3 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(3 B a^3-4 b C a^2+b^2 B a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(B b^3-4 a C b^2+3 a^2 B b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{3 a (a-b)^2 (a+b)^2 d}","\frac{2 \left(3 a^2 B-4 a b C+b^2 B\right) \sin (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{2 (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(3 a^2 B-4 a b C+b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d (a-b) (a+b)^{3/2}}+\frac{2 (3 a B+a C-b B-3 b C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d (a-b) (a+b)^{3/2}}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*(-(b*B*Sin[c + d*x]) + a*C*Sin[c + d*x]))/(3*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) - (2*(3*a^2*b*B*Sin[c + d*x] + b^3*B*Sin[c + d*x] - 4*a*b^2*C*Sin[c + d*x]))/(3*a*(a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/d + ((-4*a*(-(a^2*b*B) + b^3*B + a^3*C - a*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(3*a^3*B + a*b^2*B - 4*a^2*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(3*a^2*b*B + b^3*B - 4*a*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(3*a*(a - b)^2*(a + b)^2*d)","C",1
936,1,1384,429,6.589695,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(5/2)),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(-\frac{2 \left(a b C \sin (c+d x)-b^2 B \sin (c+d x)\right)}{3 a \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{2 \left(2 B \sin (c+d x) b^4+a C \sin (c+d x) b^3-6 a^2 B \sin (c+d x) b^2+3 a^3 C \sin (c+d x) b\right)}{3 a^2 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}\right)}{d}+\frac{-\frac{4 a \left(3 B a^4-b C a^3-5 b^2 B a^2+b^3 C a+2 b^4 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(3 C a^4-6 b B a^3+b^2 C a^2+2 b^3 B a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(2 B b^4+a C b^3-6 a^2 B b^2+3 a^3 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{3 a^2 (a-b)^2 (a+b)^2 d}","\frac{2 b (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(-3 a^2 (B+C)+a b (3 B+C)+2 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{2 \left(-3 a^3 C+6 a^2 b B-a b^2 C-2 b^3 B\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-3 a^3 C+6 a^2 b B-a b^2 C-2 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d (a-b) (a+b)^{3/2}}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((-2*(-(b^2*B*Sin[c + d*x]) + a*b*C*Sin[c + d*x]))/(3*a*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) - (2*(-6*a^2*b^2*B*Sin[c + d*x] + 2*b^4*B*Sin[c + d*x] + 3*a^3*b*C*Sin[c + d*x] + a*b^3*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/d + ((-4*a*(3*a^4*B - 5*a^2*b^2*B + 2*b^4*B - a^3*b*C + a*b^3*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-6*a^3*b*B + 2*a*b^3*B + 3*a^4*C + a^2*b^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-6*a^2*b^2*B + 2*b^4*B + 3*a^3*b*C + a*b^3*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(3*a^2*(a - b)^2*(a + b)^2*d)","C",0
937,1,1431,456,6.7138707,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(5/2)),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \left(a b^2 C \sin (c+d x)-b^3 B \sin (c+d x)\right)}{3 a^2 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{2 \left(5 B \sin (c+d x) b^5-2 a C \sin (c+d x) b^4-9 a^2 B \sin (c+d x) b^3+6 a^3 C \sin (c+d x) b^2\right)}{3 a^3 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{2 B \tan (c+d x)}{a^3}\right)}{d}-\frac{-\frac{4 a \left(-3 C a^5+9 b B a^4+5 b^2 C a^3-17 b^3 B a^2-2 b^4 C a+8 b^5 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(3 B a^5+6 b C a^4-15 b^2 B a^3-2 b^3 C a^2+8 b^4 B a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(8 B b^5-2 a C b^4-15 a^2 B b^3+6 a^3 C b^2+3 a^4 B b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{3 a^3 (a-b)^2 (a+b)^2 d}","\frac{2 b (b B-a C) \sin (c+d x)}{3 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}+\frac{2 b \left(-5 a^3 C+8 a^2 b B+a b^2 C-4 b^3 B\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-3 a^3 (B-C)-3 a^2 b (3 B+C)+2 a b^2 (3 B-C)+8 b^3 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{2 \left(3 a^4 B+6 a^3 b C-15 a^2 b^2 B-2 a b^3 C+8 b^4 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d (a-b) (a+b)^{3/2}}",1,"-1/3*((-4*a*(9*a^4*b*B - 17*a^2*b^3*B + 8*b^5*B - 3*a^5*C + 5*a^3*b^2*C - 2*a*b^4*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B + 6*a^4*b*C - 2*a^2*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(3*a^4*b*B - 15*a^2*b^3*B + 8*b^5*B + 6*a^3*b^2*C - 2*a*b^4*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a^3*(a - b)^2*(a + b)^2*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*(-(b^3*B*Sin[c + d*x]) + a*b^2*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) + (2*(-9*a^2*b^3*B*Sin[c + d*x] + 5*b^5*B*Sin[c + d*x] + 6*a^3*b^2*C*Sin[c + d*x] - 2*a*b^4*C*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) + (2*B*Tan[c + d*x])/a^3))/d","C",0
938,1,117,156,0.5872978,"\int \cos ^2(c+d x) (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{-160 \sin ^3(c+d x) (a B+A b+2 b C)+480 \sin (c+d x) (a B+A b+b C)+15 (4 (c+d x) (4 a A+3 a C+3 b B)+8 \sin (2 (c+d x)) (a (A+C)+b B)+(a C+b B) \sin (4 (c+d x)))+96 b C \sin ^5(c+d x)}{480 d}","-\frac{\sin ^3(c+d x) (5 a B+5 A b+4 b C)}{15 d}+\frac{\sin (c+d x) (5 a B+5 A b+4 b C)}{5 d}+\frac{\sin (c+d x) \cos (c+d x) (4 a A+3 a C+3 b B)}{8 d}+\frac{1}{8} x (4 a A+3 a C+3 b B)+\frac{(a C+b B) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{b C \sin (c+d x) \cos ^4(c+d x)}{5 d}",1,"(480*(A*b + a*B + b*C)*Sin[c + d*x] - 160*(A*b + a*B + 2*b*C)*Sin[c + d*x]^3 + 96*b*C*Sin[c + d*x]^5 + 15*(4*(4*a*A + 3*b*B + 3*a*C)*(c + d*x) + 8*(b*B + a*(A + C))*Sin[2*(c + d*x)] + (b*B + a*C)*Sin[4*(c + d*x)]))/(480*d)","A",1
939,1,118,128,0.3314521,"\int \cos (c+d x) (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{24 \sin (c+d x) (4 a A+3 a C+3 b B)+24 \sin (2 (c+d x)) (a B+A b+b C)+48 a B c+48 a B d x+8 a C \sin (3 (c+d x))+48 A b c+48 A b d x+8 b B \sin (3 (c+d x))+3 b C \sin (4 (c+d x))+36 b c C+36 b C d x}{96 d}","\frac{\sin (c+d x) (3 a A+2 a C+2 b B)}{3 d}+\frac{\sin (c+d x) \cos (c+d x) (4 a B+4 A b+3 b C)}{8 d}+\frac{1}{8} x (4 a B+4 A b+3 b C)+\frac{(a C+b B) \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac{b C \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"(48*A*b*c + 48*a*B*c + 36*b*c*C + 48*A*b*d*x + 48*a*B*d*x + 36*b*C*d*x + 24*(4*a*A + 3*b*B + 3*a*C)*Sin[c + d*x] + 24*(A*b + a*B + b*C)*Sin[2*(c + d*x)] + 8*b*B*Sin[3*(c + d*x)] + 8*a*C*Sin[3*(c + d*x)] + 3*b*C*Sin[4*(c + d*x)])/(96*d)","A",1
940,1,85,80,0.2022808,"\int (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{3 \sin (c+d x) (4 a B+4 A b+3 b C)+12 a A d x+3 (a C+b B) \sin (2 (c+d x))+6 a c C+6 a C d x+6 b B c+6 b B d x+b C \sin (3 (c+d x))}{12 d}","\frac{\sin (c+d x) (a B+A b+b C)}{d}+\frac{1}{2} x (a (2 A+C)+b B)+\frac{(a C+b B) \sin (c+d x) \cos (c+d x)}{2 d}-\frac{b C \sin ^3(c+d x)}{3 d}",1,"(6*b*B*c + 6*a*c*C + 12*a*A*d*x + 6*b*B*d*x + 6*a*C*d*x + 3*(4*A*b + 4*a*B + 3*b*C)*Sin[c + d*x] + 3*(b*B + a*C)*Sin[2*(c + d*x)] + b*C*Sin[3*(c + d*x)])/(12*d)","A",1
941,1,68,69,0.1293169,"\int (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{4 a A \tanh ^{-1}(\sin (c+d x))+4 (a C+b B) \sin (c+d x)+4 a B d x+4 A b d x+b C \sin (2 (c+d x))+2 b c C+2 b C d x}{4 d}","\frac{1}{2} x (2 a B+2 A b+b C)+\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(a C+b B) \sin (c+d x)}{d}+\frac{b C \sin (c+d x) \cos (c+d x)}{2 d}",1,"(2*b*c*C + 4*A*b*d*x + 4*a*B*d*x + 2*b*C*d*x + 4*a*A*ArcTanh[Sin[c + d*x]] + 4*(b*B + a*C)*Sin[c + d*x] + b*C*Sin[2*(c + d*x)])/(4*d)","A",1
942,1,71,52,0.0249281,"\int (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{a A \tan (c+d x)}{d}+\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+a C x+\frac{A b \tanh ^{-1}(\sin (c+d x))}{d}+b B x+\frac{b C \sin (c) \cos (d x)}{d}+\frac{b C \cos (c) \sin (d x)}{d}","\frac{(a B+A b) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A \tan (c+d x)}{d}+x (a C+b B)+\frac{b C \sin (c+d x)}{d}",1,"b*B*x + a*C*x + (A*b*ArcTanh[Sin[c + d*x]])/d + (a*B*ArcTanh[Sin[c + d*x]])/d + (b*C*Cos[d*x]*Sin[c])/d + (b*C*Cos[c]*Sin[d*x])/d + (a*A*Tan[c + d*x])/d","A",1
943,1,92,69,0.0442162,"\int (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{a A \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a B \tan (c+d x)}{d}+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A b \tan (c+d x)}{d}+\frac{b B \tanh ^{-1}(\sin (c+d x))}{d}+b C x","\frac{(a (A+2 C)+2 b B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(a B+A b) \tan (c+d x)}{d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}+b C x",1,"b*C*x + (a*A*ArcTanh[Sin[c + d*x]])/(2*d) + (b*B*ArcTanh[Sin[c + d*x]])/d + (a*C*ArcTanh[Sin[c + d*x]])/d + (A*b*Tan[c + d*x])/d + (a*B*Tan[c + d*x])/d + (a*A*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
944,1,73,101,0.5746515,"\int (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{3 (a B+A b+2 b C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(3 (a B+A b) \sec (c+d x)+2 a A \tan ^2(c+d x)+6 a (A+C)+6 b B\right)}{6 d}","\frac{\tan (c+d x) (2 a A+3 a C+3 b B)}{3 d}+\frac{(a B+A b+2 b C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(a B+A b) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"(3*(A*b + a*B + 2*b*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(6*b*B + 6*a*(A + C) + 3*(A*b + a*B)*Sec[c + d*x] + 2*a*A*Tan[c + d*x]^2))/(6*d)","A",1
945,1,100,137,0.7311056,"\int (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{3 (3 a A+4 a C+4 b B) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(3 \sec (c+d x) (3 a A+4 a C+4 b B)+8 (a B+A b) \tan ^2(c+d x)+24 (a B+A b+b C)+6 a A \sec ^3(c+d x)\right)}{24 d}","\frac{\tan (c+d x) (2 a B+2 A b+3 b C)}{3 d}+\frac{(3 a A+4 a C+4 b B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec (c+d x) (3 a A+4 a C+4 b B)}{8 d}+\frac{(a B+A b) \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"(3*(3*a*A + 4*b*B + 4*a*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(24*(A*b + a*B + b*C) + 3*(3*a*A + 4*b*B + 4*a*C)*Sec[c + d*x] + 6*a*A*Sec[c + d*x]^3 + 8*(A*b + a*B)*Tan[c + d*x]^2))/(24*d)","A",1
946,1,123,165,1.3698007,"\int (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{15 (3 a B+3 A b+4 b C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(8 \left(5 \tan ^2(c+d x) (a (2 A+C)+b B)+15 (a (A+C)+b B)+3 a A \tan ^4(c+d x)\right)+15 \sec (c+d x) (3 a B+3 A b+4 b C)+30 (a B+A b) \sec ^3(c+d x)\right)}{120 d}","\frac{\tan ^3(c+d x) (4 a A+5 a C+5 b B)}{15 d}+\frac{\tan (c+d x) (4 a A+5 a C+5 b B)}{5 d}+\frac{(3 a B+3 A b+4 b C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec (c+d x) (3 a B+3 A b+4 b C)}{8 d}+\frac{(a B+A b) \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{a A \tan (c+d x) \sec ^4(c+d x)}{5 d}",1,"(15*(3*A*b + 3*a*B + 4*b*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*(3*A*b + 3*a*B + 4*b*C)*Sec[c + d*x] + 30*(A*b + a*B)*Sec[c + d*x]^3 + 8*(15*(b*B + a*(A + C)) + 5*(b*B + a*(2*A + C))*Tan[c + d*x]^2 + 3*a*A*Tan[c + d*x]^4)))/(120*d)","A",1
947,1,169,224,0.8388545,"\int \cos (c+d x) (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{60 (c+d x) \left(4 a^2 B+8 a A b+6 a b C+3 b^2 B\right)+60 \sin (c+d x) \left(a^2 (8 A+6 C)+12 a b B+b^2 (6 A+5 C)\right)+120 \sin (2 (c+d x)) \left(a^2 B+2 a b (A+C)+b^2 B\right)+10 \sin (3 (c+d x)) \left(4 a^2 C+8 a b B+4 A b^2+5 b^2 C\right)+15 b (2 a C+b B) \sin (4 (c+d x))+6 b^2 C \sin (5 (c+d x))}{480 d}","\frac{\sin (c+d x) \left(5 a^2 (3 A+2 C)+20 a b B+2 b^2 (5 A+4 C)\right)}{15 d}+\frac{\sin (c+d x) \cos ^2(c+d x) \left(2 a^2 C+10 a b B+5 A b^2+4 b^2 C\right)}{15 d}+\frac{\sin (c+d x) \cos (c+d x) \left(4 a^2 B+8 a A b+6 a b C+3 b^2 B\right)}{8 d}+\frac{1}{8} x \left(4 a^2 B+8 a A b+6 a b C+3 b^2 B\right)+\frac{b (2 a C+5 b B) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{5 d}",1,"(60*(8*a*A*b + 4*a^2*B + 3*b^2*B + 6*a*b*C)*(c + d*x) + 60*(12*a*b*B + b^2*(6*A + 5*C) + a^2*(8*A + 6*C))*Sin[c + d*x] + 120*(a^2*B + b^2*B + 2*a*b*(A + C))*Sin[2*(c + d*x)] + 10*(4*A*b^2 + 8*a*b*B + 4*a^2*C + 5*b^2*C)*Sin[3*(c + d*x)] + 15*b*(b*B + 2*a*C)*Sin[4*(c + d*x)] + 6*b^2*C*Sin[5*(c + d*x)])/(480*d)","A",1
948,1,137,191,0.6200189,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{12 (c+d x) \left(4 a^2 (2 A+C)+8 a b B+b^2 (4 A+3 C)\right)+24 \sin (c+d x) \left(4 a^2 B+8 a A b+6 a b C+3 b^2 B\right)+24 \sin (2 (c+d x)) \left(a^2 C+2 a b B+A b^2+b^2 C\right)+8 b (2 a C+b B) \sin (3 (c+d x))+3 b^2 C \sin (4 (c+d x))}{96 d}","\frac{\sin (c+d x) \cos (c+d x) \left(-2 a^2 C+8 a b B+12 A b^2+9 b^2 C\right)}{24 d}+\frac{1}{8} x \left(4 a^2 (2 A+C)+8 a b B+b^2 (4 A+3 C)\right)+\frac{\sin (c+d x) \left(a^3 (-C)+4 a^2 b B+4 a b^2 (3 A+2 C)+4 b^3 B\right)}{6 b d}+\frac{(4 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^2}{12 b d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d}",1,"(12*(8*a*b*B + 4*a^2*(2*A + C) + b^2*(4*A + 3*C))*(c + d*x) + 24*(8*a*A*b + 4*a^2*B + 3*b^2*B + 6*a*b*C)*Sin[c + d*x] + 24*(A*b^2 + 2*a*b*B + a^2*C + b^2*C)*Sin[2*(c + d*x)] + 8*b*(b*B + 2*a*C)*Sin[3*(c + d*x)] + 3*b^2*C*Sin[4*(c + d*x)])/(96*d)","A",1
949,1,158,134,0.5401243,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{6 (c+d x) \left(2 a^2 B+2 a b (2 A+C)+b^2 B\right)+3 \sin (c+d x) \left(4 a^2 C+8 a b B+4 A b^2+3 b^2 C\right)-12 a^2 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 a^2 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+3 b (2 a C+b B) \sin (2 (c+d x))+b^2 C \sin (3 (c+d x))}{12 d}","\frac{\sin (c+d x) \left(2 a^2 C+6 a b B+3 A b^2+2 b^2 C\right)}{3 d}+\frac{1}{2} x \left(2 a^2 B+2 a b (2 A+C)+b^2 B\right)+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b (2 a C+3 b B) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}",1,"(6*(2*a^2*B + b^2*B + 2*a*b*(2*A + C))*(c + d*x) - 12*a^2*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*a^2*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 3*(4*A*b^2 + 8*a*b*B + 4*a^2*C + 3*b^2*C)*Sin[c + d*x] + 3*b*(b*B + 2*a*C)*Sin[2*(c + d*x)] + b^2*C*Sin[3*(c + d*x)])/(12*d)","A",1
950,1,155,126,1.0668594,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{2 (c+d x) \left(2 a^2 C+4 a b B+2 A b^2+b^2 C\right)+\tan (c+d x) \left(4 a^2 A+4 b (2 a C+b B) \cos (c+d x)+b^2 C \cos (2 (c+d x))+b^2 C\right)-4 a (a B+2 A b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 a (a B+2 A b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}","\frac{1}{2} x \left(2 a^2 C+4 a b B+2 A b^2+b^2 C\right)-\frac{b \sin (c+d x) (2 a A-2 a C-b B)}{d}+\frac{a (a B+2 A b) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^2}{d}-\frac{b^2 (2 A-C) \sin (c+d x) \cos (c+d x)}{2 d}",1,"(2*(2*A*b^2 + 4*a*b*B + 2*a^2*C + b^2*C)*(c + d*x) - 4*a*(2*A*b + a*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*a*(2*A*b + a*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (4*a^2*A + b^2*C + 4*b*(b*B + 2*a*C)*Cos[c + d*x] + b^2*C*Cos[2*(c + d*x)])*Tan[c + d*x])/(4*d)","A",1
951,1,277,118,1.7690783,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{-2 \left(a^2 (A+2 C)+4 a b B+2 A b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \left(a^2 (A+2 C)+4 a b B+2 A b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{a^2 A}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{a^2 A}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{4 a (a B+2 A b) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 a (a B+2 A b) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+4 b (c+d x) (2 a C+b B)+4 b^2 C \sin (c+d x)}{4 d}","\frac{\left(a^2 (A+2 C)+4 a b B+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (a B+A b) \tan (c+d x)}{d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}+b x (2 a C+b B)-\frac{b^2 (A-2 C) \sin (c+d x)}{2 d}",1,"(4*b*(b*B + 2*a*C)*(c + d*x) - 2*(2*A*b^2 + 4*a*b*B + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(2*A*b^2 + 4*a*b*B + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a^2*A)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (4*a*(2*A*b + a*B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (a^2*A)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*a*(2*A*b + a*B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 4*b^2*C*Sin[c + d*x])/(4*d)","B",1
952,1,104,141,0.6504743,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{3 \left(a^2 B+2 a b (A+2 C)+2 b^2 B\right) \tanh ^{-1}(\sin (c+d x))+3 \tan (c+d x) \left(2 a^2 (A+C)+a (a B+2 A b) \sec (c+d x)+4 a b B+2 A b^2\right)+2 a^2 A \tan ^3(c+d x)+6 b^2 C d x}{6 d}","\frac{\tan (c+d x) \left(a^2 (2 A+3 C)+6 a b B+2 A b^2\right)}{3 d}+\frac{\left(a^2 B+2 a b (A+2 C)+2 b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (3 a B+2 A b) \tan (c+d x) \sec (c+d x)}{6 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}+b^2 C x",1,"(6*b^2*C*d*x + 3*(a^2*B + 2*b^2*B + 2*a*b*(A + 2*C))*ArcTanh[Sin[c + d*x]] + 3*(2*A*b^2 + 4*a*b*B + 2*a^2*(A + C) + a*(2*A*b + a*B)*Sec[c + d*x])*Tan[c + d*x] + 2*a^2*A*Tan[c + d*x]^3)/(6*d)","A",1
953,1,137,184,1.1532963,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{3 \left(a^2 (3 A+4 C)+8 a b B+4 b^2 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(8 \left(3 a^2 B+a (a B+2 A b) \tan ^2(c+d x)+6 a b (A+C)+3 b^2 B\right)+3 \sec (c+d x) \left(a^2 (3 A+4 C)+8 a b B+4 A b^2\right)+6 a^2 A \sec ^3(c+d x)\right)}{24 d}","\frac{\tan (c+d x) \left(2 a^2 B+4 a A b+6 a b C+3 b^2 B\right)}{3 d}+\frac{\left(a^2 (3 A+4 C)+8 a b B+4 b^2 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec (c+d x) \left(a^2 (3 A+4 C)+8 a b B+2 A b^2\right)}{8 d}+\frac{a (2 a B+A b) \tan (c+d x) \sec ^2(c+d x)}{6 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{4 d}",1,"(3*(8*a*b*B + 4*b^2*(A + 2*C) + a^2*(3*A + 4*C))*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(3*(4*A*b^2 + 8*a*b*B + a^2*(3*A + 4*C))*Sec[c + d*x] + 6*a^2*A*Sec[c + d*x]^3 + 8*(3*a^2*B + 3*b^2*B + 6*a*b*(A + C) + a*(2*A*b + a*B)*Tan[c + d*x]^2)))/(24*d)","A",1
954,1,167,232,2.5082075,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{15 \left(3 a^2 B+6 a A b+8 a b C+4 b^2 B\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(8 \left(5 \tan ^2(c+d x) \left(a^2 (2 A+C)+2 a b B+A b^2\right)+15 \left(a^2 (A+C)+2 a b B+b^2 (A+C)\right)+3 a^2 A \tan ^4(c+d x)\right)+15 \sec (c+d x) \left(3 a^2 B+6 a A b+8 a b C+4 b^2 B\right)+30 a (a B+2 A b) \sec ^3(c+d x)\right)}{120 d}","\frac{\tan (c+d x) \left(2 a^2 (4 A+5 C)+20 a b B+5 b^2 (2 A+3 C)\right)}{15 d}+\frac{\left(3 a^2 B+6 a A b+8 a b C+4 b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec ^2(c+d x) \left(a^2 (4 A+5 C)+10 a b B+2 A b^2\right)}{15 d}+\frac{\tan (c+d x) \sec (c+d x) \left(3 a^2 B+6 a A b+8 a b C+4 b^2 B\right)}{8 d}+\frac{a (5 a B+2 A b) \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}",1,"(15*(6*a*A*b + 3*a^2*B + 4*b^2*B + 8*a*b*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*(6*a*A*b + 3*a^2*B + 4*b^2*B + 8*a*b*C)*Sec[c + d*x] + 30*a*(2*A*b + a*B)*Sec[c + d*x]^3 + 8*(15*(2*a*b*B + a^2*(A + C) + b^2*(A + C)) + 5*(A*b^2 + 2*a*b*B + a^2*(2*A + C))*Tan[c + d*x]^2 + 3*a^2*A*Tan[c + d*x]^4)))/(120*d)","A",1
955,1,368,327,1.2209672,"\int \cos (c+d x) (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{480 a^3 B c+480 a^3 B d x+80 a^3 C \sin (3 (c+d x))+1440 a^2 A b c+1440 a^2 A b d x+240 a^2 b B \sin (3 (c+d x))+90 a^2 b C \sin (4 (c+d x))+1080 a^2 b c C+1080 a^2 b C d x+120 \sin (c+d x) \left(a^3 (8 A+6 C)+18 a^2 b B+3 a b^2 (6 A+5 C)+5 b^3 B\right)+15 \sin (2 (c+d x)) \left(16 a^3 B+48 a^2 b (A+C)+48 a b^2 B+b^3 (16 A+15 C)\right)+240 a A b^2 \sin (3 (c+d x))+90 a b^2 B \sin (4 (c+d x))+1080 a b^2 B c+1080 a b^2 B d x+300 a b^2 C \sin (3 (c+d x))+36 a b^2 C \sin (5 (c+d x))+30 A b^3 \sin (4 (c+d x))+360 A b^3 c+360 A b^3 d x+100 b^3 B \sin (3 (c+d x))+12 b^3 B \sin (5 (c+d x))+45 b^3 C \sin (4 (c+d x))+5 b^3 C \sin (6 (c+d x))+300 b^3 c C+300 b^3 C d x}{960 d}","\frac{b \sin (c+d x) \cos ^3(c+d x) \left(6 a^2 C+42 a b B+30 A b^2+25 b^2 C\right)}{120 d}+\frac{\sin (c+d x) \left(5 a^3 (3 A+2 C)+30 a^2 b B+6 a b^2 (5 A+4 C)+8 b^3 B\right)}{15 d}+\frac{\sin (c+d x) \cos ^2(c+d x) \left(a^3 C+12 a^2 b B+3 a b^2 (5 A+4 C)+4 b^3 B\right)}{15 d}+\frac{\sin (c+d x) \cos (c+d x) \left(8 a^3 B+6 a^2 b (4 A+3 C)+18 a b^2 B+b^3 (6 A+5 C)\right)}{16 d}+\frac{1}{16} x \left(8 a^3 B+6 a^2 b (4 A+3 C)+18 a b^2 B+b^3 (6 A+5 C)\right)+\frac{(a C+2 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{10 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}",1,"(1440*a^2*A*b*c + 360*A*b^3*c + 480*a^3*B*c + 1080*a*b^2*B*c + 1080*a^2*b*c*C + 300*b^3*c*C + 1440*a^2*A*b*d*x + 360*A*b^3*d*x + 480*a^3*B*d*x + 1080*a*b^2*B*d*x + 1080*a^2*b*C*d*x + 300*b^3*C*d*x + 120*(18*a^2*b*B + 5*b^3*B + 3*a*b^2*(6*A + 5*C) + a^3*(8*A + 6*C))*Sin[c + d*x] + 15*(16*a^3*B + 48*a*b^2*B + 48*a^2*b*(A + C) + b^3*(16*A + 15*C))*Sin[2*(c + d*x)] + 240*a*A*b^2*Sin[3*(c + d*x)] + 240*a^2*b*B*Sin[3*(c + d*x)] + 100*b^3*B*Sin[3*(c + d*x)] + 80*a^3*C*Sin[3*(c + d*x)] + 300*a*b^2*C*Sin[3*(c + d*x)] + 30*A*b^3*Sin[4*(c + d*x)] + 90*a*b^2*B*Sin[4*(c + d*x)] + 90*a^2*b*C*Sin[4*(c + d*x)] + 45*b^3*C*Sin[4*(c + d*x)] + 12*b^3*B*Sin[5*(c + d*x)] + 36*a*b^2*C*Sin[5*(c + d*x)] + 5*b^3*C*Sin[6*(c + d*x)])/(960*d)","A",1
956,1,288,277,0.9694097,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{480 a^3 A c+480 a^3 A d x+240 a^3 c C+240 a^3 C d x+720 a^2 b B c+720 a^2 b B d x+120 a^2 b C \sin (3 (c+d x))+60 \sin (c+d x) \left(8 a^3 B+6 a^2 b (4 A+3 C)+18 a b^2 B+b^3 (6 A+5 C)\right)+120 \sin (2 (c+d x)) \left(a^3 C+3 a^2 b B+3 a b^2 (A+C)+b^3 B\right)+720 a A b^2 c+720 a A b^2 d x+120 a b^2 B \sin (3 (c+d x))+45 a b^2 C \sin (4 (c+d x))+540 a b^2 c C+540 a b^2 C d x+40 A b^3 \sin (3 (c+d x))+15 b^3 B \sin (4 (c+d x))+180 b^3 B c+180 b^3 B d x+50 b^3 C \sin (3 (c+d x))+6 b^3 C \sin (5 (c+d x))}{480 d}","\frac{\sin (c+d x) \cos (c+d x) \left(-6 a^3 C+30 a^2 b B+a b^2 (100 A+71 C)+45 b^3 B\right)}{120 d}+\frac{1}{8} x \left(4 a^3 (2 A+C)+12 a^2 b B+3 a b^2 (4 A+3 C)+3 b^3 B\right)+\frac{\sin (c+d x) \left(-3 a^4 C+15 a^3 b B+4 a^2 b^2 (20 A+13 C)+60 a b^3 B+4 b^4 (5 A+4 C)\right)}{30 b d}+\frac{\sin (c+d x) \left(3 a (5 b B-a C)+4 b^2 (5 A+4 C)\right) (a+b \cos (c+d x))^2}{60 b d}+\frac{(5 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^3}{20 b d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^4}{5 b d}",1,"(480*a^3*A*c + 720*a*A*b^2*c + 720*a^2*b*B*c + 180*b^3*B*c + 240*a^3*c*C + 540*a*b^2*c*C + 480*a^3*A*d*x + 720*a*A*b^2*d*x + 720*a^2*b*B*d*x + 180*b^3*B*d*x + 240*a^3*C*d*x + 540*a*b^2*C*d*x + 60*(8*a^3*B + 18*a*b^2*B + 6*a^2*b*(4*A + 3*C) + b^3*(6*A + 5*C))*Sin[c + d*x] + 120*(3*a^2*b*B + b^3*B + a^3*C + 3*a*b^2*(A + C))*Sin[2*(c + d*x)] + 40*A*b^3*Sin[3*(c + d*x)] + 120*a*b^2*B*Sin[3*(c + d*x)] + 120*a^2*b*C*Sin[3*(c + d*x)] + 50*b^3*C*Sin[3*(c + d*x)] + 15*b^3*B*Sin[4*(c + d*x)] + 45*a*b^2*C*Sin[4*(c + d*x)] + 6*b^3*C*Sin[5*(c + d*x)])/(480*d)","A",1
957,1,218,207,0.9788949,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{-96 a^3 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+96 a^3 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+24 b \sin (2 (c+d x)) \left(3 a^2 C+3 a b B+A b^2+b^2 C\right)+12 (c+d x) \left(8 a^3 B+12 a^2 b (2 A+C)+12 a b^2 B+b^3 (4 A+3 C)\right)+24 \sin (c+d x) \left(4 a^3 C+12 a^2 b B+3 a b^2 (4 A+3 C)+3 b^3 B\right)+8 b^2 (3 a C+b B) \sin (3 (c+d x))+3 b^3 C \sin (4 (c+d x))}{96 d}","\frac{a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b \sin (c+d x) \cos (c+d x) \left(6 a^2 C+20 a b B+12 A b^2+9 b^2 C\right)}{24 d}+\frac{\sin (c+d x) \left(3 a^3 C+16 a^2 b B+6 a b^2 (3 A+2 C)+4 b^3 B\right)}{6 d}+\frac{1}{8} x \left(8 a^3 B+12 a^2 b (2 A+C)+12 a b^2 B+b^3 (4 A+3 C)\right)+\frac{(3 a C+4 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}",1,"(12*(8*a^3*B + 12*a*b^2*B + 12*a^2*b*(2*A + C) + b^3*(4*A + 3*C))*(c + d*x) - 96*a^3*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 96*a^3*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 24*(12*a^2*b*B + 3*b^3*B + 4*a^3*C + 3*a*b^2*(4*A + 3*C))*Sin[c + d*x] + 24*b*(A*b^2 + 3*a*b*B + 3*a^2*C + b^2*C)*Sin[2*(c + d*x)] + 8*b^2*(b*B + 3*a*C)*Sin[3*(c + d*x)] + 3*b^3*C*Sin[4*(c + d*x)])/(96*d)","A",1
958,1,266,192,1.2988836,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{\frac{12 a^3 A \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{12 a^3 A \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+3 b \sin (c+d x) \left(3 \left(4 a^2 C+4 a b B+b^2 C\right)+4 A b^2\right)-12 a^2 (a B+3 A b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 a^2 (a B+3 A b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+6 (c+d x) \left(2 a^3 C+6 a^2 b B+3 a b^2 (2 A+C)+b^3 B\right)+3 b^2 (3 a C+b B) \sin (2 (c+d x))+b^3 C \sin (3 (c+d x))}{12 d}","\frac{b \sin (c+d x) \left(-\left(a^2 (6 A-8 C)\right)+9 a b B+b^2 (3 A+2 C)\right)}{3 d}+\frac{a^2 (a B+3 A b) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} x \left(2 a^3 C+6 a^2 b B+3 a b^2 (2 A+C)+b^3 B\right)-\frac{b^2 \sin (c+d x) \cos (c+d x) (6 a A-5 a C-3 b B)}{6 d}-\frac{b (3 A-C) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^3}{d}",1,"(6*(6*a^2*b*B + b^3*B + 2*a^3*C + 3*a*b^2*(2*A + C))*(c + d*x) - 12*a^2*(3*A*b + a*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*a^2*(3*A*b + a*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (12*a^3*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (12*a^3*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 3*b*(4*A*b^2 + 3*(4*a*b*B + 4*a^2*C + b^2*C))*Sin[c + d*x] + 3*b^2*(b*B + 3*a*C)*Sin[2*(c + d*x)] + b^3*C*Sin[3*(c + d*x)])/(12*d)","A",1
959,1,318,204,3.2065622,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{\frac{a^3 A}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{a^3 A}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+2 b (c+d x) \left(6 a^2 C+6 a b B+2 A b^2+b^2 C\right)-2 a \left(a^2 (A+2 C)+6 a b B+6 A b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 a \left(a^2 (A+2 C)+6 a b B+6 A b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{4 a^2 (a B+3 A b) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 a^2 (a B+3 A b) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+4 b^2 (3 a C+b B) \sin (c+d x)+b^3 C \sin (2 (c+d x))}{4 d}","-\frac{b \sin (c+d x) \left(4 a^2 B+9 a A b-6 a b C-2 b^2 B\right)}{2 d}+\frac{a \left(a^2 (A+2 C)+6 a b B+6 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{1}{2} b x \left(6 a^2 C+6 a b B+2 A b^2+b^2 C\right)-\frac{b^2 \sin (c+d x) \cos (c+d x) (2 a B+4 A b-b C)}{2 d}+\frac{(2 a B+3 A b) \tan (c+d x) (a+b \cos (c+d x))^2}{2 d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^3}{2 d}",1,"(2*b*(2*A*b^2 + 6*a*b*B + 6*a^2*C + b^2*C)*(c + d*x) - 2*a*(6*A*b^2 + 6*a*b*B + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*a*(6*A*b^2 + 6*a*b*B + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a^3*A)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (4*a^2*(3*A*b + a*B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (a^3*A)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*a^2*(3*A*b + a*B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 4*b^2*(b*B + 3*a*C)*Sin[c + d*x] + b^3*C*Sin[2*(c + d*x)])/(4*d)","A",1
960,1,429,196,5.6939024,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\frac{2 a^3 A \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2 a^3 A \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{4 a \sin \left(\frac{1}{2} (c+d x)\right) \left(a^2 (2 A+3 C)+9 a b B+9 A b^2\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 a \sin \left(\frac{1}{2} (c+d x)\right) \left(a^2 (2 A+3 C)+9 a b B+9 A b^2\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+\frac{a^2 (a (A+3 B)+9 A b)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{a^2 (a (A+3 B)+9 A b)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-6 \left(a^3 B+3 a^2 b (A+2 C)+6 a b^2 B+2 A b^3\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 \left(a^3 B+3 a^2 b (A+2 C)+6 a b^2 B+2 A b^3\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+12 b^2 (c+d x) (3 a C+b B)+12 b^3 C \sin (c+d x)}{12 d}","\frac{a \tan (c+d x) \left(a^2 (2 A+3 C)+6 a b B+3 A b^2\right)}{3 d}+\frac{\left(a^3 B+3 a^2 b (A+2 C)+6 a b^2 B+2 A b^3\right) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{b^2 \sin (c+d x) (3 a B+5 A b-6 b C)}{6 d}+\frac{(a B+A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}+b^2 x (3 a C+b B)",1,"(12*b^2*(b*B + 3*a*C)*(c + d*x) - 6*(2*A*b^3 + a^3*B + 6*a*b^2*B + 3*a^2*b*(A + 2*C))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*(2*A*b^3 + a^3*B + 6*a*b^2*B + 3*a^2*b*(A + 2*C))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a^2*(9*A*b + a*(A + 3*B)))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (2*a^3*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (4*a*(9*A*b^2 + 9*a*b*B + a^2*(2*A + 3*C))*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (2*a^3*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - (a^2*(9*A*b + a*(A + 3*B)))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*a*(9*A*b^2 + 9*a*b*B + a^2*(2*A + 3*C))*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 12*b^3*C*Sin[c + d*x])/(12*d)","B",1
961,1,165,223,1.3691189,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{8 a^2 (a B+3 A b) \tan ^3(c+d x)+3 \left(a^3 (3 A+4 C)+12 a^2 b B+12 a b^2 (A+2 C)+8 b^3 B\right) \tanh ^{-1}(\sin (c+d x))+3 \tan (c+d x) \left(2 a^3 A \sec ^3(c+d x)+a \sec (c+d x) \left(a^2 (3 A+4 C)+12 a b B+12 A b^2\right)+8 \left(a^3 B+3 a^2 b (A+C)+3 a b^2 B+A b^3\right)\right)+24 b^3 C d x}{24 d}","\frac{a \tan (c+d x) \sec (c+d x) \left(3 a^2 (3 A+4 C)+20 a b B+6 A b^2\right)}{24 d}+\frac{\tan (c+d x) \left(4 a^3 B+6 a^2 b (2 A+3 C)+16 a b^2 B+3 A b^3\right)}{6 d}+\frac{\left(a^3 (3 A+4 C)+12 a^2 b B+12 a b^2 (A+2 C)+8 b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(4 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}+b^3 C x",1,"(24*b^3*C*d*x + 3*(12*a^2*b*B + 8*b^3*B + 12*a*b^2*(A + 2*C) + a^3*(3*A + 4*C))*ArcTanh[Sin[c + d*x]] + 3*(8*(A*b^3 + a^3*B + 3*a*b^2*B + 3*a^2*b*(A + C)) + a*(12*A*b^2 + 12*a*b*B + a^2*(3*A + 4*C))*Sec[c + d*x] + 2*a^3*A*Sec[c + d*x]^3)*Tan[c + d*x] + 8*a^2*(3*A*b + a*B)*Tan[c + d*x]^3)/(24*d)","A",1
962,1,204,278,4.8649533,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{15 \left(3 a^3 B+3 a^2 b (3 A+4 C)+12 a b^2 B+4 b^3 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(30 a^2 (a B+3 A b) \sec ^3(c+d x)+8 \left(3 a^3 A \tan ^4(c+d x)+5 a \tan ^2(c+d x) \left(a^2 (2 A+C)+3 a b B+3 A b^2\right)+15 \left(a^3 (A+C)+3 a^2 b B+3 a b^2 (A+C)+b^3 B\right)\right)+15 \sec (c+d x) \left(3 a^3 B+3 a^2 b (3 A+4 C)+12 a b^2 B+4 A b^3\right)\right)}{120 d}","\frac{a \tan (c+d x) \sec ^2(c+d x) \left(2 a^2 (4 A+5 C)+15 a b B+3 A b^2\right)}{30 d}+\frac{\tan (c+d x) \left(2 a^3 (4 A+5 C)+30 a^2 b B+15 a b^2 (2 A+3 C)+15 b^3 B\right)}{15 d}+\frac{\left(3 a^3 B+3 a^2 b (3 A+4 C)+12 a b^2 B+4 b^3 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec (c+d x) \left(15 a^3 B+15 a^2 b (3 A+4 C)+50 a b^2 B+6 A b^3\right)}{40 d}+\frac{(5 a B+3 A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{20 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{5 d}",1,"(15*(3*a^3*B + 12*a*b^2*B + 4*b^3*(A + 2*C) + 3*a^2*b*(3*A + 4*C))*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*(4*A*b^3 + 3*a^3*B + 12*a*b^2*B + 3*a^2*b*(3*A + 4*C))*Sec[c + d*x] + 30*a^2*(3*A*b + a*B)*Sec[c + d*x]^3 + 8*(15*(3*a^2*b*B + b^3*B + a^3*(A + C) + 3*a*b^2*(A + C)) + 5*a*(3*A*b^2 + 3*a*b*B + a^2*(2*A + C))*Tan[c + d*x]^2 + 3*a^3*A*Tan[c + d*x]^4)))/(120*d)","A",1
963,1,252,336,2.9140941,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","\frac{15 \left(a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(40 a^3 A \sec ^5(c+d x)+10 a \sec ^3(c+d x) \left(a^2 (5 A+6 C)+18 a b B+18 A b^2\right)+16 \left(3 a^2 (a B+3 A b) \tan ^4(c+d x)+5 \tan ^2(c+d x) \left(2 a^3 B+3 a^2 b (2 A+C)+3 a b^2 B+A b^3\right)+15 \left(a^3 B+3 a^2 b (A+C)+3 a b^2 B+b^3 (A+C)\right)\right)+15 \sec (c+d x) \left(a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right)\right)}{240 d}","\frac{a \tan (c+d x) \sec ^3(c+d x) \left(5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right)}{120 d}+\frac{\tan (c+d x) \left(8 a^3 B+6 a^2 b (4 A+5 C)+30 a b^2 B+5 b^3 (2 A+3 C)\right)}{15 d}+\frac{\left(a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\tan (c+d x) \sec ^2(c+d x) \left(4 a^3 B+3 a^2 b (4 A+5 C)+12 a b^2 B+A b^3\right)}{15 d}+\frac{\tan (c+d x) \sec (c+d x) \left(a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right)}{16 d}+\frac{(2 a B+A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{10 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}",1,"(15*(18*a^2*b*B + 8*b^3*B + 6*a*b^2*(3*A + 4*C) + a^3*(5*A + 6*C))*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*(18*a^2*b*B + 8*b^3*B + 6*a*b^2*(3*A + 4*C) + a^3*(5*A + 6*C))*Sec[c + d*x] + 10*a*(18*A*b^2 + 18*a*b*B + a^2*(5*A + 6*C))*Sec[c + d*x]^3 + 40*a^3*A*Sec[c + d*x]^5 + 16*(15*(a^3*B + 3*a*b^2*B + 3*a^2*b*(A + C) + b^3*(A + C)) + 5*(A*b^3 + 2*a^3*B + 3*a*b^2*B + 3*a^2*b*(2*A + C))*Tan[c + d*x]^2 + 3*a^2*(3*A*b + a*B)*Tan[c + d*x]^4)))/(240*d)","A",1
964,1,528,445,1.4570106,"\int \cos (c+d x) (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{3360 a^4 B c+3360 a^4 B d x+560 a^4 C \sin (3 (c+d x))+13440 a^3 A b c+13440 a^3 A b d x+2240 a^3 b B \sin (3 (c+d x))+840 a^3 b C \sin (4 (c+d x))+10080 a^3 b c C+10080 a^3 b C d x+3360 a^2 A b^2 \sin (3 (c+d x))+1260 a^2 b^2 B \sin (4 (c+d x))+15120 a^2 b^2 B c+15120 a^2 b^2 B d x+4200 a^2 b^2 C \sin (3 (c+d x))+504 a^2 b^2 C \sin (5 (c+d x))+105 \sin (c+d x) \left(16 a^4 (4 A+3 C)+192 a^3 b B+48 a^2 b^2 (6 A+5 C)+160 a b^3 B+5 b^4 (8 A+7 C)\right)+105 \sin (2 (c+d x)) \left(16 a^4 B+64 a^3 b (A+C)+96 a^2 b^2 B+4 a b^3 (16 A+15 C)+15 b^4 B\right)+840 a A b^3 \sin (4 (c+d x))+10080 a A b^3 c+10080 a A b^3 d x+2800 a b^3 B \sin (3 (c+d x))+336 a b^3 B \sin (5 (c+d x))+1260 a b^3 C \sin (4 (c+d x))+140 a b^3 C \sin (6 (c+d x))+8400 a b^3 c C+8400 a b^3 C d x+700 A b^4 \sin (3 (c+d x))+84 A b^4 \sin (5 (c+d x))+315 b^4 B \sin (4 (c+d x))+35 b^4 B \sin (6 (c+d x))+2100 b^4 B c+2100 b^4 B d x+735 b^4 C \sin (3 (c+d x))+147 b^4 C \sin (5 (c+d x))+15 b^4 C \sin (7 (c+d x))}{6720 d}","\frac{\sin (c+d x) \cos ^2(c+d x) \left(4 a^2 C+21 a b B+14 A b^2+12 b^2 C\right) (a+b \cos (c+d x))^2}{70 d}+\frac{b \sin (c+d x) \cos ^3(c+d x) \left(24 a^3 C+336 a^2 b B+4 a b^2 (126 A+103 C)+175 b^3 B\right)}{840 d}+\frac{\sin (c+d x) \left(35 a^4 (3 A+2 C)+280 a^3 b B+84 a^2 b^2 (5 A+4 C)+224 a b^3 B+8 b^4 (7 A+6 C)\right)}{105 d}+\frac{\sin (c+d x) \cos ^2(c+d x) \left(4 a^4 C+91 a^3 b B+3 a^2 b^2 (63 A+50 C)+112 a b^3 B+4 b^4 (7 A+6 C)\right)}{105 d}+\frac{\sin (c+d x) \cos (c+d x) \left(8 a^4 B+8 a^3 b (4 A+3 C)+36 a^2 b^2 B+4 a b^3 (6 A+5 C)+5 b^4 B\right)}{16 d}+\frac{1}{16} x \left(8 a^4 B+8 a^3 b (4 A+3 C)+36 a^2 b^2 B+4 a b^3 (6 A+5 C)+5 b^4 B\right)+\frac{(4 a C+7 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{42 d}+\frac{C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d}",1,"(13440*a^3*A*b*c + 10080*a*A*b^3*c + 3360*a^4*B*c + 15120*a^2*b^2*B*c + 2100*b^4*B*c + 10080*a^3*b*c*C + 8400*a*b^3*c*C + 13440*a^3*A*b*d*x + 10080*a*A*b^3*d*x + 3360*a^4*B*d*x + 15120*a^2*b^2*B*d*x + 2100*b^4*B*d*x + 10080*a^3*b*C*d*x + 8400*a*b^3*C*d*x + 105*(192*a^3*b*B + 160*a*b^3*B + 16*a^4*(4*A + 3*C) + 48*a^2*b^2*(6*A + 5*C) + 5*b^4*(8*A + 7*C))*Sin[c + d*x] + 105*(16*a^4*B + 96*a^2*b^2*B + 15*b^4*B + 64*a^3*b*(A + C) + 4*a*b^3*(16*A + 15*C))*Sin[2*(c + d*x)] + 3360*a^2*A*b^2*Sin[3*(c + d*x)] + 700*A*b^4*Sin[3*(c + d*x)] + 2240*a^3*b*B*Sin[3*(c + d*x)] + 2800*a*b^3*B*Sin[3*(c + d*x)] + 560*a^4*C*Sin[3*(c + d*x)] + 4200*a^2*b^2*C*Sin[3*(c + d*x)] + 735*b^4*C*Sin[3*(c + d*x)] + 840*a*A*b^3*Sin[4*(c + d*x)] + 1260*a^2*b^2*B*Sin[4*(c + d*x)] + 315*b^4*B*Sin[4*(c + d*x)] + 840*a^3*b*C*Sin[4*(c + d*x)] + 1260*a*b^3*C*Sin[4*(c + d*x)] + 84*A*b^4*Sin[5*(c + d*x)] + 336*a*b^3*B*Sin[5*(c + d*x)] + 504*a^2*b^2*C*Sin[5*(c + d*x)] + 147*b^4*C*Sin[5*(c + d*x)] + 35*b^4*B*Sin[6*(c + d*x)] + 140*a*b^3*C*Sin[6*(c + d*x)] + 15*b^4*C*Sin[7*(c + d*x)])/(6720*d)","A",1
965,1,432,375,1.3926301,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{960 a^4 A c+960 a^4 A d x+480 a^4 c C+480 a^4 C d x+1920 a^3 b B c+1920 a^3 b B d x+320 a^3 b C \sin (3 (c+d x))+2880 a^2 A b^2 c+2880 a^2 A b^2 d x+480 a^2 b^2 B \sin (3 (c+d x))+180 a^2 b^2 C \sin (4 (c+d x))+2160 a^2 b^2 c C+2160 a^2 b^2 C d x+120 \sin (c+d x) \left(8 a^4 B+8 a^3 b (4 A+3 C)+36 a^2 b^2 B+4 a b^3 (6 A+5 C)+5 b^4 B\right)+15 \sin (2 (c+d x)) \left(16 a^4 C+64 a^3 b B+96 a^2 b^2 (A+C)+64 a b^3 B+b^4 (16 A+15 C)\right)+320 a A b^3 \sin (3 (c+d x))+120 a b^3 B \sin (4 (c+d x))+1440 a b^3 B c+1440 a b^3 B d x+400 a b^3 C \sin (3 (c+d x))+48 a b^3 C \sin (5 (c+d x))+30 A b^4 \sin (4 (c+d x))+360 A b^4 c+360 A b^4 d x+100 b^4 B \sin (3 (c+d x))+12 b^4 B \sin (5 (c+d x))+45 b^4 C \sin (4 (c+d x))+5 b^4 C \sin (6 (c+d x))+300 b^4 c C+300 b^4 C d x}{960 d}","\frac{\sin (c+d x) \left(-4 a^3 C+24 a^2 b B+a b^2 (70 A+53 C)+32 b^3 B\right) (a+b \cos (c+d x))^2}{120 b d}+\frac{\sin (c+d x) \cos (c+d x) \left(-8 a^4 C+48 a^3 b B+2 a^2 b^2 (130 A+89 C)+232 a b^3 B+15 b^4 (6 A+5 C)\right)}{240 d}+\frac{1}{16} x \left(8 a^4 (2 A+C)+32 a^3 b B+12 a^2 b^2 (4 A+3 C)+24 a b^3 B+b^4 (6 A+5 C)\right)+\frac{\sin (c+d x) \left(-4 a^5 C+24 a^4 b B+a^3 b^2 (190 A+121 C)+224 a^2 b^3 B+32 a b^4 (5 A+4 C)+32 b^5 B\right)}{60 b d}+\frac{\sin (c+d x) \left(4 a (6 b B-a C)+5 b^2 (6 A+5 C)\right) (a+b \cos (c+d x))^3}{120 b d}+\frac{(6 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^4}{30 b d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}",1,"(960*a^4*A*c + 2880*a^2*A*b^2*c + 360*A*b^4*c + 1920*a^3*b*B*c + 1440*a*b^3*B*c + 480*a^4*c*C + 2160*a^2*b^2*c*C + 300*b^4*c*C + 960*a^4*A*d*x + 2880*a^2*A*b^2*d*x + 360*A*b^4*d*x + 1920*a^3*b*B*d*x + 1440*a*b^3*B*d*x + 480*a^4*C*d*x + 2160*a^2*b^2*C*d*x + 300*b^4*C*d*x + 120*(8*a^4*B + 36*a^2*b^2*B + 5*b^4*B + 8*a^3*b*(4*A + 3*C) + 4*a*b^3*(6*A + 5*C))*Sin[c + d*x] + 15*(64*a^3*b*B + 64*a*b^3*B + 16*a^4*C + 96*a^2*b^2*(A + C) + b^4*(16*A + 15*C))*Sin[2*(c + d*x)] + 320*a*A*b^3*Sin[3*(c + d*x)] + 480*a^2*b^2*B*Sin[3*(c + d*x)] + 100*b^4*B*Sin[3*(c + d*x)] + 320*a^3*b*C*Sin[3*(c + d*x)] + 400*a*b^3*C*Sin[3*(c + d*x)] + 30*A*b^4*Sin[4*(c + d*x)] + 120*a*b^3*B*Sin[4*(c + d*x)] + 180*a^2*b^2*C*Sin[4*(c + d*x)] + 45*b^4*C*Sin[4*(c + d*x)] + 12*b^4*B*Sin[5*(c + d*x)] + 48*a*b^3*C*Sin[5*(c + d*x)] + 5*b^4*C*Sin[6*(c + d*x)])/(960*d)","A",1
966,1,382,290,1.2602394,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{-480 a^4 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+480 a^4 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+480 a^4 B c+480 a^4 B d x+1920 a^3 A b c+1920 a^3 A b d x+960 a^3 b c C+960 a^3 b C d x+1440 a^2 b^2 B c+1440 a^2 b^2 B d x+240 a^2 b^2 C \sin (3 (c+d x))+120 b \sin (2 (c+d x)) \left(4 a^3 C+6 a^2 b B+4 a b^2 (A+C)+b^3 B\right)+60 \sin (c+d x) \left(8 a^4 C+32 a^3 b B+12 a^2 b^2 (4 A+3 C)+24 a b^3 B+b^4 (6 A+5 C)\right)+960 a A b^3 c+960 a A b^3 d x+160 a b^3 B \sin (3 (c+d x))+60 a b^3 C \sin (4 (c+d x))+720 a b^3 c C+720 a b^3 C d x+40 A b^4 \sin (3 (c+d x))+15 b^4 B \sin (4 (c+d x))+180 b^4 B c+180 b^4 B d x+50 b^4 C \sin (3 (c+d x))+6 b^4 C \sin (5 (c+d x))}{480 d}","\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{\sin (c+d x) \left(12 a^2 C+35 a b B+20 A b^2+16 b^2 C\right) (a+b \cos (c+d x))^2}{60 d}+\frac{b \sin (c+d x) \cos (c+d x) \left(24 a^3 C+130 a^2 b B+4 a b^2 (40 A+29 C)+45 b^3 B\right)}{120 d}+\frac{\sin (c+d x) \left(12 a^4 C+95 a^3 b B+2 a^2 b^2 (85 A+56 C)+80 a b^3 B+4 b^4 (5 A+4 C)\right)}{30 d}+\frac{1}{8} x \left(8 a^4 B+16 a^3 b (2 A+C)+24 a^2 b^2 B+4 a b^3 (4 A+3 C)+3 b^4 B\right)+\frac{(4 a C+5 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{20 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}",1,"(1920*a^3*A*b*c + 960*a*A*b^3*c + 480*a^4*B*c + 1440*a^2*b^2*B*c + 180*b^4*B*c + 960*a^3*b*c*C + 720*a*b^3*c*C + 1920*a^3*A*b*d*x + 960*a*A*b^3*d*x + 480*a^4*B*d*x + 1440*a^2*b^2*B*d*x + 180*b^4*B*d*x + 960*a^3*b*C*d*x + 720*a*b^3*C*d*x - 480*a^4*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 480*a^4*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 60*(32*a^3*b*B + 24*a*b^3*B + 8*a^4*C + 12*a^2*b^2*(4*A + 3*C) + b^4*(6*A + 5*C))*Sin[c + d*x] + 120*b*(6*a^2*b*B + b^3*B + 4*a^3*C + 4*a*b^2*(A + C))*Sin[2*(c + d*x)] + 40*A*b^4*Sin[3*(c + d*x)] + 160*a*b^3*B*Sin[3*(c + d*x)] + 240*a^2*b^2*C*Sin[3*(c + d*x)] + 50*b^4*C*Sin[3*(c + d*x)] + 15*b^4*B*Sin[4*(c + d*x)] + 60*a*b^3*C*Sin[4*(c + d*x)] + 6*b^4*C*Sin[5*(c + d*x)])/(480*d)","A",1
967,1,383,273,3.0468831,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{b^2 \sec (c+d x) \left(3 \sin (3 (c+d x)) \left(48 a^2 C+32 a b B+8 A b^2+9 b^2 C\right)+b (8 (4 a C+b B) \sin (4 (c+d x))+3 b C \sin (5 (c+d x)))\right)+32 b \sin (c+d x) \left(24 a^3 C+36 a^2 b B+4 a b^2 (6 A+5 C)+5 b^3 B\right)+24 \left(8 a^4 B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+8 a^4 c C+8 a^4 C d x-8 a^3 (a B+4 A b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+32 a^3 A b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+32 a^3 b B c+32 a^3 b B d x+48 a^2 A b^2 c+48 a^2 A b^2 d x+24 a^2 b^2 c C+24 a^2 b^2 C d x+\tan (c+d x) \left(8 a^4 A+6 a^2 b^2 C+4 a b^3 B+b^4 (A+C)\right)+16 a b^3 B c+16 a b^3 B d x+4 A b^4 c+4 A b^4 d x+3 b^4 c C+3 b^4 C d x\right)}{192 d}","\frac{a^3 (a B+4 A b) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \sin (c+d x) \cos (c+d x) \left(-\left(a^2 (24 A-26 C)\right)+32 a b B+3 b^2 (4 A+3 C)\right)}{24 d}+\frac{b \sin (c+d x) \left(-\left(a^3 (12 A-19 C)\right)+34 a^2 b B+8 a b^2 (3 A+2 C)+4 b^3 B\right)}{6 d}+\frac{1}{8} x \left(8 a^4 C+32 a^3 b B+24 a^2 b^2 (2 A+C)+16 a b^3 B+b^4 (4 A+3 C)\right)-\frac{b \sin (c+d x) (12 a A-7 a C-4 b B) (a+b \cos (c+d x))^2}{12 d}-\frac{b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^4}{d}",1,"(32*b*(36*a^2*b*B + 5*b^3*B + 24*a^3*C + 4*a*b^2*(6*A + 5*C))*Sin[c + d*x] + b^2*Sec[c + d*x]*(3*(8*A*b^2 + 32*a*b*B + 48*a^2*C + 9*b^2*C)*Sin[3*(c + d*x)] + b*(8*(b*B + 4*a*C)*Sin[4*(c + d*x)] + 3*b*C*Sin[5*(c + d*x)])) + 24*(48*a^2*A*b^2*c + 4*A*b^4*c + 32*a^3*b*B*c + 16*a*b^3*B*c + 8*a^4*c*C + 24*a^2*b^2*c*C + 3*b^4*c*C + 48*a^2*A*b^2*d*x + 4*A*b^4*d*x + 32*a^3*b*B*d*x + 16*a*b^3*B*d*x + 8*a^4*C*d*x + 24*a^2*b^2*C*d*x + 3*b^4*C*d*x - 8*a^3*(4*A*b + a*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 32*a^3*A*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 8*a^4*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (8*a^4*A + 4*a*b^3*B + 6*a^2*b^2*C + b^4*(A + C))*Tan[c + d*x]))/(192*d)","A",1
968,1,367,274,4.5837766,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{\frac{3 a^4 A}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{3 a^4 A}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{12 a^3 (a B+4 A b) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{12 a^3 (a B+4 A b) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+3 b^2 \sin (c+d x) \left(24 a^2 C+16 a b B+4 A b^2+3 b^2 C\right)-6 a^2 \left(a^2 (A+2 C)+8 a b B+12 A b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 a^2 \left(a^2 (A+2 C)+8 a b B+12 A b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+6 b (c+d x) \left(8 a^3 C+12 a^2 b B+4 a b^2 (2 A+C)+b^3 B\right)+3 b^3 (4 a C+b B) \sin (2 (c+d x))+b^4 C \sin (3 (c+d x))}{12 d}","\frac{a^2 \left(a^2 (A+2 C)+8 a b B+12 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{b^2 \sin (c+d x) \cos (c+d x) \left(6 a^2 B+2 a b (9 A-4 C)-3 b^2 B\right)}{6 d}-\frac{b \sin (c+d x) \left(12 a^3 B+a^2 b (39 A-34 C)-24 a b^2 B-2 b^3 (3 A+2 C)\right)}{6 d}+\frac{1}{2} b x \left(8 a^3 C+12 a^2 b B+4 a b^2 (2 A+C)+b^3 B\right)-\frac{b \sin (c+d x) (6 a B+15 A b-2 b C) (a+b \cos (c+d x))^2}{6 d}+\frac{(a B+2 A b) \tan (c+d x) (a+b \cos (c+d x))^3}{d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^4}{2 d}",1,"(6*b*(12*a^2*b*B + b^3*B + 8*a^3*C + 4*a*b^2*(2*A + C))*(c + d*x) - 6*a^2*(12*A*b^2 + 8*a*b*B + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*a^2*(12*A*b^2 + 8*a*b*B + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (3*a^4*A)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (12*a^3*(4*A*b + a*B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (3*a^4*A)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (12*a^3*(4*A*b + a*B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 3*b^2*(4*A*b^2 + 16*a*b*B + 24*a^2*C + 3*b^2*C)*Sin[c + d*x] + 3*b^3*(b*B + 4*a*C)*Sin[2*(c + d*x)] + b^4*C*Sin[3*(c + d*x)])/(12*d)","A",1
969,1,351,303,2.2696464,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\sec ^3(c+d x) \left(36 b^2 (c+d x) \cos (c+d x) \left(12 a^2 C+8 a b B+2 A b^2+b^2 C\right)+12 b^2 (c+d x) \cos (3 (c+d x)) \left(12 a^2 C+8 a b B+2 A b^2+b^2 C\right)-48 a \cos ^3(c+d x) \left(a^3 B+4 a^2 b (A+2 C)+12 a b^2 B+8 A b^3\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+2 \sin (c+d x) \left(32 a^4 A+24 a^4 C+96 a^3 b B+144 a^2 A b^2+12 \cos (c+d x) \left(2 a^4 B+8 a^3 A b+12 a b^3 C+3 b^4 B\right)+4 \cos (2 (c+d x)) \left(a^4 (4 A+6 C)+24 a^3 b B+36 a^2 A b^2+3 b^4 C\right)+48 a b^3 C \cos (3 (c+d x))+12 b^4 B \cos (3 (c+d x))+3 b^4 C \cos (4 (c+d x))+9 b^4 C\right)\right)}{96 d}","-\frac{b^2 \sin (c+d x) \cos (c+d x) \left(a^2 (4 A+6 C)+18 a b B+3 b^2 (6 A-C)\right)}{6 d}+\frac{\tan (c+d x) \left(a^2 (4 A+6 C)+15 a b B+12 A b^2\right) (a+b \cos (c+d x))^2}{6 d}+\frac{1}{2} b^2 x \left(12 a^2 C+8 a b B+2 A b^2+b^2 C\right)-\frac{b \sin (c+d x) \left(4 a^3 (2 A+3 C)+39 a^2 b B+4 a b^2 (11 A-6 C)-6 b^3 B\right)}{6 d}+\frac{a \left(a^3 B+4 a^2 b (A+2 C)+12 a b^2 B+8 A b^3\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(3 a B+4 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^3}{6 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^4}{3 d}",1,"(Sec[c + d*x]^3*(36*b^2*(2*A*b^2 + 8*a*b*B + 12*a^2*C + b^2*C)*(c + d*x)*Cos[c + d*x] + 12*b^2*(2*A*b^2 + 8*a*b*B + 12*a^2*C + b^2*C)*(c + d*x)*Cos[3*(c + d*x)] - 48*a*(8*A*b^3 + a^3*B + 12*a*b^2*B + 4*a^2*b*(A + 2*C))*Cos[c + d*x]^3*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 2*(32*a^4*A + 144*a^2*A*b^2 + 96*a^3*b*B + 24*a^4*C + 9*b^4*C + 12*(8*a^3*A*b + 2*a^4*B + 3*b^4*B + 12*a*b^3*C)*Cos[c + d*x] + 4*(36*a^2*A*b^2 + 24*a^3*b*B + 3*b^4*C + a^4*(4*A + 6*C))*Cos[2*(c + d*x)] + 12*b^4*B*Cos[3*(c + d*x)] + 48*a*b^3*C*Cos[3*(c + d*x)] + 3*b^4*C*Cos[4*(c + d*x)])*Sin[c + d*x]))/(96*d)","A",1
970,1,462,293,2.5034174,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{32 a \tan (c+d x) \sec ^2(c+d x) \left(2 a^3 B+a^2 (8 A b+6 b C)+9 a b^2 B+6 A b^3\right)+3 \tan (c+d x) \sec ^3(c+d x) \left(a^4 (11 A+4 C)+16 a^3 b B+24 a^2 A b^2+4 b^4 C\right)-12 \left(a^4 (3 A+4 C)+16 a^3 b B+24 a^2 b^2 (A+2 C)+32 a b^3 B+8 A b^4\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+\sec ^4(c+d x) \left(9 a^4 A \sin (3 (c+d x))+8 a^4 B \sin (4 (c+d x))+12 a^4 C \sin (3 (c+d x))+32 a^3 A b \sin (4 (c+d x))+48 a^3 b B \sin (3 (c+d x))+48 a^3 b C \sin (4 (c+d x))+72 a^2 A b^2 \sin (3 (c+d x))+72 a^2 b^2 B \sin (4 (c+d x))+48 a A b^3 \sin (4 (c+d x))+48 b^3 (c+d x) (4 a C+b B) \cos (2 (c+d x))+12 b^3 (c+d x) (4 a C+b B) \cos (4 (c+d x))+144 a b^3 c C+144 a b^3 C d x+36 b^4 B c+36 b^4 B d x+18 b^4 C \sin (3 (c+d x))+6 b^4 C \sin (5 (c+d x))\right)}{96 d}","-\frac{b^2 \sin (c+d x) \left(3 a^2 (3 A+4 C)+32 a b B+2 b^2 (13 A-12 C)\right)}{24 d}+\frac{\tan (c+d x) \sec (c+d x) \left(a^2 (3 A+4 C)+8 a b B+4 A b^2\right) (a+b \cos (c+d x))^2}{8 d}+\frac{a \tan (c+d x) \left(8 a^3 B+a^2 b (23 A+36 C)+36 a b^2 B+12 A b^3\right)}{12 d}+\frac{\left(a^4 (3 A+4 C)+16 a^3 b B+24 a^2 b^2 (A+2 C)+32 a b^3 B+8 A b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(a B+A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}+b^3 x (4 a C+b B)",1,"(-12*(8*A*b^4 + 16*a^3*b*B + 32*a*b^3*B + 24*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c + d*x]^4*(36*b^4*B*c + 144*a*b^3*c*C + 36*b^4*B*d*x + 144*a*b^3*C*d*x + 48*b^3*(b*B + 4*a*C)*(c + d*x)*Cos[2*(c + d*x)] + 12*b^3*(b*B + 4*a*C)*(c + d*x)*Cos[4*(c + d*x)] + 9*a^4*A*Sin[3*(c + d*x)] + 72*a^2*A*b^2*Sin[3*(c + d*x)] + 48*a^3*b*B*Sin[3*(c + d*x)] + 12*a^4*C*Sin[3*(c + d*x)] + 18*b^4*C*Sin[3*(c + d*x)] + 32*a^3*A*b*Sin[4*(c + d*x)] + 48*a*A*b^3*Sin[4*(c + d*x)] + 8*a^4*B*Sin[4*(c + d*x)] + 72*a^2*b^2*B*Sin[4*(c + d*x)] + 48*a^3*b*C*Sin[4*(c + d*x)] + 6*b^4*C*Sin[5*(c + d*x)]) + 32*a*(6*A*b^3 + 2*a^3*B + 9*a*b^2*B + a^2*(8*A*b + 6*b*C))*Sec[c + d*x]^2*Tan[c + d*x] + 3*(24*a^2*A*b^2 + 16*a^3*b*B + 4*b^4*C + a^4*(11*A + 4*C))*Sec[c + d*x]^3*Tan[c + d*x])/(96*d)","A",1
971,1,230,314,1.8913316,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{24 a^4 A \tan ^5(c+d x)+40 a^2 \tan ^3(c+d x) \left(a^2 (2 A+C)+4 a b B+6 A b^2\right)+15 \left(3 a^4 B+4 a^3 b (3 A+4 C)+24 a^2 b^2 B+16 a b^3 (A+2 C)+8 b^4 B\right) \tanh ^{-1}(\sin (c+d x))+15 \tan (c+d x) \left(2 a^3 (a B+4 A b) \sec ^3(c+d x)+a \sec (c+d x) \left(3 a^3 B+4 a^2 b (3 A+4 C)+24 a b^2 B+16 A b^3\right)+8 \left(a^4 (A+C)+4 a^3 b B+6 a^2 b^2 (A+C)+4 a b^3 B+A b^4\right)\right)+120 b^4 C d x}{120 d}","\frac{\tan (c+d x) \sec ^2(c+d x) \left(4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right) (a+b \cos (c+d x))^2}{60 d}+\frac{a \tan (c+d x) \sec (c+d x) \left(45 a^3 B+4 a^2 b (29 A+40 C)+130 a b^2 B+24 A b^3\right)}{120 d}+\frac{\tan (c+d x) \left(4 a^4 (4 A+5 C)+80 a^3 b B+2 a^2 b^2 (56 A+85 C)+95 a b^3 B+12 A b^4\right)}{30 d}+\frac{\left(3 a^4 B+4 a^3 b (3 A+4 C)+24 a^2 b^2 B+16 a b^3 (A+2 C)+8 b^4 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(5 a B+4 A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{20 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^4}{5 d}+b^4 C x",1,"(120*b^4*C*d*x + 15*(3*a^4*B + 24*a^2*b^2*B + 8*b^4*B + 16*a*b^3*(A + 2*C) + 4*a^3*b*(3*A + 4*C))*ArcTanh[Sin[c + d*x]] + 15*(8*(A*b^4 + 4*a^3*b*B + 4*a*b^3*B + a^4*(A + C) + 6*a^2*b^2*(A + C)) + a*(16*A*b^3 + 3*a^3*B + 24*a*b^2*B + 4*a^2*b*(3*A + 4*C))*Sec[c + d*x] + 2*a^3*(4*A*b + a*B)*Sec[c + d*x]^3)*Tan[c + d*x] + 40*a^2*(6*A*b^2 + 4*a*b*B + a^2*(2*A + C))*Tan[c + d*x]^3 + 24*a^4*A*Tan[c + d*x]^5)/(120*d)","A",1
972,1,366,381,6.3088825,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","\frac{a^4 A \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac{5 a^4 A \left(2 \tan (c+d x) \sec ^3(c+d x)+3 \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)\right)}{48 d}+\frac{a^3 (a B+4 A b) \left(3 \tan ^5(c+d x)+10 \tan ^3(c+d x)+15 \tan (c+d x)\right)}{15 d}+\frac{b^2 \left(6 a^2 C+4 a b B+A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 \tan (c+d x) \sec ^3(c+d x) \left(a (a C+4 b B)+6 A b^2\right)}{4 d}+\frac{b^2 \tan (c+d x) \sec (c+d x) \left(6 a^2 C+4 a b B+A b^2\right)}{2 d}+\frac{3 a^2 \left(a (a C+4 b B)+6 A b^2\right) \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)}{8 d}+\frac{2 a b \left(\tan ^3(c+d x)+3 \tan (c+d x)\right) \left(a (2 a C+3 b B)+2 A b^2\right)}{3 d}+\frac{b^3 (4 a C+b B) \tan (c+d x)}{d}+\frac{b^4 C \tanh ^{-1}(\sin (c+d x))}{d}","\frac{\tan (c+d x) \sec ^3(c+d x) \left(5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right) (a+b \cos (c+d x))^2}{120 d}+\frac{a \tan (c+d x) \sec ^2(c+d x) \left(16 a^3 B+a^2 b (39 A+50 C)+36 a b^2 B+4 A b^3\right)}{60 d}+\frac{\tan (c+d x) \left(8 a^4 B+8 a^3 b (4 A+5 C)+60 a^2 b^2 B+20 a b^3 (2 A+3 C)+15 b^4 B\right)}{15 d}+\frac{\left(a^4 (5 A+6 C)+24 a^3 b B+12 a^2 b^2 (3 A+4 C)+32 a b^3 B+8 b^4 (A+2 C)\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\tan (c+d x) \sec (c+d x) \left(15 a^4 (5 A+6 C)+360 a^3 b B+10 a^2 b^2 (49 A+66 C)+336 a b^3 B+24 A b^4\right)}{240 d}+\frac{(3 a B+2 A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{15 d}+\frac{A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^4}{6 d}",1,"(b^4*C*ArcTanh[Sin[c + d*x]])/d + (b^2*(A*b^2 + 4*a*b*B + 6*a^2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (b^3*(b*B + 4*a*C)*Tan[c + d*x])/d + (b^2*(A*b^2 + 4*a*b*B + 6*a^2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a^2*(6*A*b^2 + a*(4*b*B + a*C))*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a^4*A*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) + (3*a^2*(6*A*b^2 + a*(4*b*B + a*C))*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x]))/(8*d) + (2*a*b*(2*A*b^2 + a*(3*b*B + 2*a*C))*(3*Tan[c + d*x] + Tan[c + d*x]^3))/(3*d) + (a^3*(4*A*b + a*B)*(15*Tan[c + d*x] + 10*Tan[c + d*x]^3 + 3*Tan[c + d*x]^5))/(15*d) + (5*a^4*A*(2*Sec[c + d*x]^3*Tan[c + d*x] + 3*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x])))/(48*d)","A",1
973,1,341,454,3.5956169,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^8(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^8,x]","\frac{105 \left(5 a^4 B+4 a^3 b (5 A+6 C)+36 a^2 b^2 B+8 a b^3 (3 A+4 C)+8 b^4 B\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(280 a^3 (a B+4 A b) \sec ^5(c+d x)+70 a \sec ^3(c+d x) \left(5 a^3 B+4 a^2 b (5 A+6 C)+36 a b^2 B+24 A b^3\right)+16 \left(15 a^4 A \tan ^6(c+d x)+21 a^2 \tan ^4(c+d x) \left(a^2 (3 A+C)+4 a b B+6 A b^2\right)+35 \tan ^2(c+d x) \left(a^4 (3 A+2 C)+8 a^3 b B+6 a^2 b^2 (2 A+C)+4 a b^3 B+A b^4\right)+105 \left(a^4 (A+C)+4 a^3 b B+6 a^2 b^2 (A+C)+4 a b^3 B+b^4 (A+C)\right)\right)+105 \sec (c+d x) \left(5 a^4 B+4 a^3 b (5 A+6 C)+36 a^2 b^2 B+8 a b^3 (3 A+4 C)+8 b^4 B\right)\right)}{1680 d}","\frac{\tan (c+d x) \sec ^4(c+d x) \left(2 a^2 (6 A+7 C)+21 a b B+4 A b^2\right) (a+b \cos (c+d x))^2}{70 d}+\frac{a \tan (c+d x) \sec ^3(c+d x) \left(175 a^3 B+a^2 (412 A b+504 b C)+336 a b^2 B+24 A b^3\right)}{840 d}+\frac{\tan (c+d x) \left(8 a^4 (6 A+7 C)+224 a^3 b B+84 a^2 b^2 (4 A+5 C)+280 a b^3 B+35 b^4 (2 A+3 C)\right)}{105 d}+\frac{\left(5 a^4 B+4 a^3 b (5 A+6 C)+36 a^2 b^2 B+8 a b^3 (3 A+4 C)+8 b^4 B\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\tan (c+d x) \sec ^2(c+d x) \left(4 a^4 (6 A+7 C)+112 a^3 b B+3 a^2 b^2 (50 A+63 C)+91 a b^3 B+4 A b^4\right)}{105 d}+\frac{\tan (c+d x) \sec (c+d x) \left(5 a^4 B+4 a^3 b (5 A+6 C)+36 a^2 b^2 B+8 a b^3 (3 A+4 C)+8 b^4 B\right)}{16 d}+\frac{(7 a B+4 A b) \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{42 d}+\frac{A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}",1,"(105*(5*a^4*B + 36*a^2*b^2*B + 8*b^4*B + 8*a*b^3*(3*A + 4*C) + 4*a^3*b*(5*A + 6*C))*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(105*(5*a^4*B + 36*a^2*b^2*B + 8*b^4*B + 8*a*b^3*(3*A + 4*C) + 4*a^3*b*(5*A + 6*C))*Sec[c + d*x] + 70*a*(24*A*b^3 + 5*a^3*B + 36*a*b^2*B + 4*a^2*b*(5*A + 6*C))*Sec[c + d*x]^3 + 280*a^3*(4*A*b + a*B)*Sec[c + d*x]^5 + 16*(105*(4*a^3*b*B + 4*a*b^3*B + a^4*(A + C) + 6*a^2*b^2*(A + C) + b^4*(A + C)) + 35*(A*b^4 + 8*a^3*b*B + 4*a*b^3*B + 6*a^2*b^2*(2*A + C) + a^4*(3*A + 2*C))*Tan[c + d*x]^2 + 21*a^2*(6*A*b^2 + 4*a*b*B + a^2*(3*A + C))*Tan[c + d*x]^4 + 15*a^4*A*Tan[c + d*x]^6)))/(1680*d)","A",1
974,1,287,256,1.1317937,"\int (a+b \cos (c+d x))^3 \left(a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2),x]","\frac{-480 a^5 c C-480 a^5 C d x+480 a^4 b B c+480 a^4 b B d x-480 a^3 b^2 c C-480 a^3 b^2 C d x+1440 a^2 b^3 B c+1440 a^2 b^3 B d x+80 a^2 b^3 C \sin (3 (c+d x))+120 b^2 \left(-2 a^3 C+6 a^2 b B+3 a b^2 C+b^3 B\right) \sin (2 (c+d x))+60 b \left(-24 a^4 C+32 a^3 b B+12 a^2 b^2 C+24 a b^3 B+5 b^4 C\right) \sin (c+d x)+160 a b^4 B \sin (3 (c+d x))+45 a b^4 C \sin (4 (c+d x))+540 a b^4 c C+540 a b^4 C d x+15 b^5 B \sin (4 (c+d x))+180 b^5 B c+180 b^5 B d x+50 b^5 C \sin (3 (c+d x))+6 b^5 C \sin (5 (c+d x))}{480 d}","\frac{b \left(-23 a^2 C+35 a b B+16 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^2}{60 d}+\frac{b^2 \left(-106 a^3 C+130 a^2 b B+71 a b^2 C+45 b^3 B\right) \sin (c+d x) \cos (c+d x)}{120 d}+\frac{b \left(-83 a^4 C+95 a^3 b B+32 a^2 b^2 C+80 a b^3 B+16 b^4 C\right) \sin (c+d x)}{30 d}+\frac{1}{8} x \left(-8 a^5 C+8 a^4 b B-8 a^3 b^2 C+24 a^2 b^3 B+9 a b^4 C+3 b^5 B\right)+\frac{b (5 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^3}{20 d}+\frac{b C \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}",1,"(480*a^4*b*B*c + 1440*a^2*b^3*B*c + 180*b^5*B*c - 480*a^5*c*C - 480*a^3*b^2*c*C + 540*a*b^4*c*C + 480*a^4*b*B*d*x + 1440*a^2*b^3*B*d*x + 180*b^5*B*d*x - 480*a^5*C*d*x - 480*a^3*b^2*C*d*x + 540*a*b^4*C*d*x + 60*b*(32*a^3*b*B + 24*a*b^3*B - 24*a^4*C + 12*a^2*b^2*C + 5*b^4*C)*Sin[c + d*x] + 120*b^2*(6*a^2*b*B + b^3*B - 2*a^3*C + 3*a*b^2*C)*Sin[2*(c + d*x)] + 160*a*b^4*B*Sin[3*(c + d*x)] + 80*a^2*b^3*C*Sin[3*(c + d*x)] + 50*b^5*C*Sin[3*(c + d*x)] + 15*b^5*B*Sin[4*(c + d*x)] + 45*a*b^4*C*Sin[4*(c + d*x)] + 6*b^5*C*Sin[5*(c + d*x)])/(480*d)","A",1
975,1,134,176,0.6185368,"\int (a+b \cos (c+d x))^2 \left(a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2),x]","\frac{-12 (c+d x) \left(8 a^4 C-8 a^3 b B-12 a b^3 B-3 b^4 C\right)+24 b \left(-8 a^3 C+12 a^2 b B+6 a b^2 C+3 b^3 B\right) \sin (c+d x)+24 b^3 (3 a B+b C) \sin (2 (c+d x))+8 b^3 (2 a C+b B) \sin (3 (c+d x))+3 b^4 C \sin (4 (c+d x))}{96 d}","\frac{b^2 \left(-14 a^2 C+20 a b B+9 b^2 C\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(-8 a^4 C+8 a^3 b B+12 a b^3 B+3 b^4 C\right)+\frac{b \left(-13 a^3 C+16 a^2 b B+8 a b^2 C+4 b^3 B\right) \sin (c+d x)}{6 d}+\frac{b (4 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{b C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}",1,"(-12*(-8*a^3*b*B - 12*a*b^3*B + 8*a^4*C - 3*b^4*C)*(c + d*x) + 24*b*(12*a^2*b*B + 3*b^3*B - 8*a^3*C + 6*a*b^2*C)*Sin[c + d*x] + 24*b^3*(3*a*B + b*C)*Sin[2*(c + d*x)] + 8*b^3*(b*B + 2*a*C)*Sin[3*(c + d*x)] + 3*b^4*C*Sin[4*(c + d*x)])/(96*d)","A",1
976,1,102,120,0.4394997,"\int (a+b \cos (c+d x)) \left(a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])*(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2),x]","\frac{3 b \left(-4 a^2 C+8 a b B+3 b^2 C\right) \sin (c+d x)-6 (c+d x) \left(2 a^3 C-2 a^2 b B-a b^2 C-b^3 B\right)+3 b^2 (a C+b B) \sin (2 (c+d x))+b^3 C \sin (3 (c+d x))}{12 d}","\frac{2 b \left(-2 a^2 C+3 a b B+b^2 C\right) \sin (c+d x)}{3 d}+\frac{1}{2} x \left(-2 a^3 C+2 a^2 b B+a b^2 C+b^3 B\right)+\frac{b^2 (3 b B-a C) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{b C \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}",1,"(-6*(-2*a^2*b*B - b^3*B + 2*a^3*C - a*b^2*C)*(c + d*x) + 3*b*(8*a*b*B - 4*a^2*C + 3*b^2*C)*Sin[c + d*x] + 3*b^2*(b*B + a*C)*Sin[2*(c + d*x)] + b^3*C*Sin[3*(c + d*x)])/(12*d)","A",1
977,1,238,279,0.9032117,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{24 b^2 \sin (2 (c+d x)) \left(a^2 C-a b B+A b^2+b^2 C\right)+\frac{192 a^3 \left(a (a C-b B)+A b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+24 b \sin (c+d x) \left(-4 a^3 C+4 a^2 b B-a b^2 (4 A+3 C)+3 b^3 B\right)+12 (c+d x) \left(8 a^4 C-8 a^3 b B+4 a^2 b^2 (2 A+C)-4 a b^3 B+b^4 (4 A+3 C)\right)+8 b^3 (b B-a C) \sin (3 (c+d x))+3 b^4 C \sin (4 (c+d x))}{96 b^5 d}","-\frac{2 a^3 \left(A b^2-a (b B-a C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d \sqrt{a-b} \sqrt{a+b}}+\frac{\sin (c+d x) \cos (c+d x) \left(4 a^2 C-4 a b B+4 A b^2+3 b^2 C\right)}{8 b^3 d}+\frac{\sin (c+d x) \left(-3 a^3 C+3 a^2 b B-a b^2 (3 A+2 C)+2 b^3 B\right)}{3 b^4 d}-\frac{x \left(-8 a^4 C+8 a^3 b B-4 a^2 b^2 (2 A+C)+4 a b^3 B-b^4 (4 A+3 C)\right)}{8 b^5}+\frac{(b B-a C) \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d}+\frac{C \sin (c+d x) \cos ^3(c+d x)}{4 b d}",1,"(12*(-8*a^3*b*B - 4*a*b^3*B + 8*a^4*C + 4*a^2*b^2*(2*A + C) + b^4*(4*A + 3*C))*(c + d*x) + (192*a^3*(A*b^2 + a*(-(b*B) + a*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + 24*b*(4*a^2*b*B + 3*b^3*B - 4*a^3*C - a*b^2*(4*A + 3*C))*Sin[c + d*x] + 24*b^2*(A*b^2 - a*b*B + a^2*C + b^2*C)*Sin[2*(c + d*x)] + 8*b^3*(b*B - a*C)*Sin[3*(c + d*x)] + 3*b^4*C*Sin[4*(c + d*x)])/(96*b^5*d)","A",1
978,1,179,206,0.6826724,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{3 b \sin (c+d x) \left(4 a^2 C-4 a b B+4 A b^2+3 b^2 C\right)-\frac{24 a^2 \left(a (a C-b B)+A b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}-6 (c+d x) \left(2 a^3 C-2 a^2 b B+a b^2 (2 A+C)-b^3 B\right)+3 b^2 (b B-a C) \sin (2 (c+d x))+b^3 C \sin (3 (c+d x))}{12 b^4 d}","\frac{2 a^2 \left(A b^2-a (b B-a C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d \sqrt{a-b} \sqrt{a+b}}+\frac{\sin (c+d x) \left(3 a^2 C-3 a b B+3 A b^2+2 b^2 C\right)}{3 b^3 d}+\frac{x \left(-2 a^3 C+2 a^2 b B-a b^2 (2 A+C)+b^3 B\right)}{2 b^4}+\frac{(b B-a C) \sin (c+d x) \cos (c+d x)}{2 b^2 d}+\frac{C \sin (c+d x) \cos ^2(c+d x)}{3 b d}",1,"(-6*(-2*a^2*b*B - b^3*B + 2*a^3*C + a*b^2*(2*A + C))*(c + d*x) - (24*a^2*(A*b^2 + a*(-(b*B) + a*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + 3*b*(4*A*b^2 - 4*a*b*B + 4*a^2*C + 3*b^2*C)*Sin[c + d*x] + 3*b^2*(b*B - a*C)*Sin[2*(c + d*x)] + b^3*C*Sin[3*(c + d*x)])/(12*b^4*d)","A",1
979,1,133,144,0.410532,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{2 (c+d x) \left(2 a^2 C-2 a b B+2 A b^2+b^2 C\right)+\frac{8 a \left(a (a C-b B)+A b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+4 b (b B-a C) \sin (c+d x)+b^2 C \sin (2 (c+d x))}{4 b^3 d}","-\frac{2 a \left(A b^2-a (b B-a C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d \sqrt{a-b} \sqrt{a+b}}+\frac{x \left(b^2 (2 A+C)-2 a (b B-a C)\right)}{2 b^3}+\frac{(b B-a C) \sin (c+d x)}{b^2 d}+\frac{C \sin (c+d x) \cos (c+d x)}{2 b d}",1,"(2*(2*A*b^2 - 2*a*b*B + 2*a^2*C + b^2*C)*(c + d*x) + (8*a*(A*b^2 + a*(-(b*B) + a*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + 4*b*(b*B - a*C)*Sin[c + d*x] + b^2*C*Sin[2*(c + d*x)])/(4*b^3*d)","A",1
980,1,92,97,0.2433418,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]),x]","\frac{-\frac{2 \left(a (a C-b B)+A b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+(c+d x) (b B-a C)+b C \sin (c+d x)}{b^2 d}","\frac{2 \left(A b^2-a (b B-a C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d \sqrt{a-b} \sqrt{a+b}}+\frac{x (b B-a C)}{b^2}+\frac{C \sin (c+d x)}{b d}",1,"((b*B - a*C)*(c + d*x) - (2*(A*b^2 + a*(-(b*B) + a*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + b*C*Sin[c + d*x])/(b^2*d)","A",1
981,1,256,94,0.7234333,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x]),x]","\frac{2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \left(2 (\sin (c)+i \cos (c)) \left(a (a C-b B)+A b^2\right) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (b \cos (c)-a)+b \sin (c)\right)}{\sqrt{-\left(\left(a^2-b^2\right) (\cos (c)-i \sin (c))^2\right)}}\right)+\sqrt{-\left(\left(a^2-b^2\right) (\cos (c)-i \sin (c))^2\right)} \left(a C d x-A b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+A b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{a b d \sqrt{\left(b^2-a^2\right) (\cos (2 c)-i \sin (2 c))} (2 A+2 B \cos (c+d x)+C \cos (2 (c+d x))+C)}","-\frac{2 \left(A b^2-a (b B-a C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a b d \sqrt{a-b} \sqrt{a+b}}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a d}+\frac{C x}{b}",1,"(2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*((a*C*d*x - A*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + A*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*Sqrt[-((a^2 - b^2)*(Cos[c] - I*Sin[c])^2)] + 2*(A*b^2 + a*(-(b*B) + a*C))*ArcTan[((I*Cos[c] + Sin[c])*(b*Sin[c] + (-a + b*Cos[c])*Tan[(d*x)/2]))/Sqrt[-((a^2 - b^2)*(Cos[c] - I*Sin[c])^2)]]*(I*Cos[c] + Sin[c])))/(a*b*d*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*(c + d*x)])*Sqrt[(-a^2 + b^2)*(Cos[2*c] - I*Sin[2*c])])","C",1
982,1,339,107,2.7703137,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x]),x]","\frac{2 \cos ^2(c+d x) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(-\frac{2 i (\cos (c)-i \sin (c)) \left(a (a C-b B)+A b^2\right) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (b \cos (c)-a)+b \sin (c)\right)}{\sqrt{-\left(\left(a^2-b^2\right) (\cos (c)-i \sin (c))^2\right)}}\right)}{\sqrt{\left(b^2-a^2\right) (\cos (c)-i \sin (c))^2}}+(A b-a B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+(a B-A b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{a A \sin \left(\frac{d x}{2}\right)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{a A \sin \left(\frac{d x}{2}\right)}{\left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}\right)}{a^2 d (2 A+2 B \cos (c+d x)+C \cos (2 (c+d x))+C)}","\frac{2 \left(A b^2-a (b B-a C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{(A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{A \tan (c+d x)}{a d}",1,"(2*Cos[c + d*x]^2*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*((A*b - a*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + (-(A*b) + a*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - ((2*I)*(A*b^2 + a*(-(b*B) + a*C))*ArcTan[((I*Cos[c] + Sin[c])*(b*Sin[c] + (-a + b*Cos[c])*Tan[(d*x)/2]))/Sqrt[-((a^2 - b^2)*(Cos[c] - I*Sin[c])^2)]]*(Cos[c] - I*Sin[c]))/Sqrt[(-a^2 + b^2)*(Cos[c] - I*Sin[c])^2] + (a*A*Sin[(d*x)/2])/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (a*A*Sin[(d*x)/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(a^2*d*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*(c + d*x)]))","C",0
983,1,314,154,2.220074,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x]),x]","\frac{\frac{8 b \left(a (a C-b B)+A b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}-2 \left(a^2 (A+2 C)-2 a b B+2 A b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \left(a^2 (A+2 C)-2 a b B+2 A b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{a^2 A}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{a^2 A}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{4 a (a B-A b) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 a (a B-A b) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}}{4 a^3 d}","-\frac{2 b \left(A b^2-a (b B-a C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d \sqrt{a-b} \sqrt{a+b}}-\frac{(A b-a B) \tan (c+d x)}{a^2 d}+\frac{\left(a^2 (A+2 C)-2 a b B+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{A \tan (c+d x) \sec (c+d x)}{2 a d}",1,"((8*b*(A*b^2 + a*(-(b*B) + a*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] - 2*(2*A*b^2 - 2*a*b*B + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(2*A*b^2 - 2*a*b*B + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a^2*A)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (4*a*(-(A*b) + a*B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (a^2*A)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*a*(-(A*b) + a*B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(4*a^3*d)","B",1
984,1,466,214,2.9143972,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + b*Cos[c + d*x]),x]","\frac{\frac{2 a^3 A \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2 a^3 A \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{4 a \sin \left(\frac{1}{2} (c+d x)\right) \left(a^2 (2 A+3 C)-3 a b B+3 A b^2\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 a \sin \left(\frac{1}{2} (c+d x)\right) \left(a^2 (2 A+3 C)-3 a b B+3 A b^2\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}-\frac{24 b^2 \left(a (a C-b B)+A b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+\frac{a^2 (a (A+3 B)-3 A b)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{a^2 (a (A+3 B)-3 A b)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+6 \left(a^3 (-B)+a^2 b (A+2 C)-2 a b^2 B+2 A b^3\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 \left(a^3 B-a^2 b (A+2 C)+2 a b^2 B-2 A b^3\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{12 a^4 d}","\frac{2 b^2 \left(A b^2-a (b B-a C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d \sqrt{a-b} \sqrt{a+b}}-\frac{(A b-a B) \tan (c+d x) \sec (c+d x)}{2 a^2 d}+\frac{\tan (c+d x) \left(a^2 (2 A+3 C)-3 a b B+3 A b^2\right)}{3 a^3 d}-\frac{\left(a^3 (-B)+a^2 b (A+2 C)-2 a b^2 B+2 A b^3\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{A \tan (c+d x) \sec ^2(c+d x)}{3 a d}",1,"((-24*b^2*(A*b^2 + a*(-(b*B) + a*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + 6*(2*A*b^3 - a^3*B - 2*a*b^2*B + a^2*b*(A + 2*C))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*(-2*A*b^3 + a^3*B + 2*a*b^2*B - a^2*b*(A + 2*C))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a^2*(-3*A*b + a*(A + 3*B)))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (2*a^3*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (4*a*(3*A*b^2 - 3*a*b*B + a^2*(2*A + 3*C))*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (2*a^3*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - (a^2*(-3*A*b + a*(A + 3*B)))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*a*(3*A*b^2 - 3*a*b*B + a^2*(2*A + 3*C))*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(12*a^4*d)","B",1
985,1,406,285,1.4973703,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5)/(a + b*Cos[c + d*x]),x]","\frac{\frac{96 b^3 \left(a (a C-b B)+A b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+a \tan (c+d x) \sec ^3(c+d x) \left(21 a^3 A+8 a^3 B \cos (3 (c+d x))+12 a^3 C+3 a \cos (2 (c+d x)) \left(a^2 (3 A+4 C)-4 a b B+4 A b^2\right)-8 a^2 A b \cos (3 (c+d x))-12 a^2 b B-12 a^2 b C \cos (3 (c+d x))+4 \cos (c+d x) \left(10 a^3 B-a^2 b (10 A+9 C)+9 a b^2 B-9 A b^3\right)+12 a A b^2+12 a b^2 B \cos (3 (c+d x))-12 A b^3 \cos (3 (c+d x))\right)-6 \left(a^4 (3 A+4 C)-4 a^3 b B+4 a^2 b^2 (A+2 C)-8 a b^3 B+8 A b^4\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 \left(a^4 (3 A+4 C)-4 a^3 b B+4 a^2 b^2 (A+2 C)-8 a b^3 B+8 A b^4\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{48 a^5 d}","-\frac{2 b^3 \left(A b^2-a (b B-a C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d \sqrt{a-b} \sqrt{a+b}}-\frac{(A b-a B) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d}+\frac{\tan (c+d x) \sec (c+d x) \left(a^2 (3 A+4 C)-4 a b B+4 A b^2\right)}{8 a^3 d}-\frac{\tan (c+d x) \left(-2 a^3 B+a^2 b (2 A+3 C)-3 a b^2 B+3 A b^3\right)}{3 a^4 d}+\frac{\left(a^4 (3 A+4 C)-4 a^3 b B+4 a^2 b^2 (A+2 C)-8 a b^3 B+8 A b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 a^5 d}+\frac{A \tan (c+d x) \sec ^3(c+d x)}{4 a d}",1,"((96*b^3*(A*b^2 + a*(-(b*B) + a*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] - 6*(8*A*b^4 - 4*a^3*b*B - 8*a*b^3*B + 4*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*(8*A*b^4 - 4*a^3*b*B - 8*a*b^3*B + 4*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + a*(21*a^3*A + 12*a*A*b^2 - 12*a^2*b*B + 12*a^3*C + 4*(-9*A*b^3 + 10*a^3*B + 9*a*b^2*B - a^2*b*(10*A + 9*C))*Cos[c + d*x] + 3*a*(4*A*b^2 - 4*a*b*B + a^2*(3*A + 4*C))*Cos[2*(c + d*x)] - 8*a^2*A*b*Cos[3*(c + d*x)] - 12*A*b^3*Cos[3*(c + d*x)] + 8*a^3*B*Cos[3*(c + d*x)] + 12*a*b^2*B*Cos[3*(c + d*x)] - 12*a^2*b*C*Cos[3*(c + d*x)])*Sec[c + d*x]^3*Tan[c + d*x])/(48*a^5*d)","A",1
986,1,256,398,1.8485914,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{\frac{12 a^3 b \sin (c+d x) \left(a (a C-b B)+A b^2\right)}{(a-b) (a+b) (a+b \cos (c+d x))}+3 b \sin (c+d x) \left(12 a^2 C-8 a b B+4 A b^2+3 b^2 C\right)+6 (c+d x) \left(-8 a^3 C+6 a^2 b B-2 a b^2 (2 A+C)+b^3 B\right)+\frac{24 a^2 \left(4 a^4 C-3 a^3 b B+a^2 b^2 (2 A-5 C)+4 a b^3 B-3 A b^4\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}+3 b^2 (b B-2 a C) \sin (2 (c+d x))+b^3 C \sin (3 (c+d x))}{12 b^5 d}","-\frac{\sin (c+d x) \cos ^3(c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \cos ^2(c+d x) \left(4 a^2 C-3 a b B+3 A b^2-b^2 C\right)}{3 b^2 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \cos (c+d x) \left(-4 a^3 C+3 a^2 b B-2 a b^2 (A-C)-b^3 B\right)}{2 b^3 d \left(a^2-b^2\right)}+\frac{x \left(-8 a^3 C+6 a^2 b B-2 a b^2 (2 A+C)+b^3 B\right)}{2 b^5}-\frac{\sin (c+d x) \left(-12 a^4 C+9 a^3 b B-a^2 b^2 (6 A-7 C)-6 a b^3 B+b^4 (3 A+2 C)\right)}{3 b^4 d \left(a^2-b^2\right)}+\frac{2 a^2 \left(4 a^4 C-3 a^3 b B+2 a^2 A b^2-5 a^2 b^2 C+4 a b^3 B-3 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{3/2} (a+b)^{3/2}}",1,"(6*(6*a^2*b*B + b^3*B - 8*a^3*C - 2*a*b^2*(2*A + C))*(c + d*x) + (24*a^2*(-3*A*b^4 - 3*a^3*b*B + 4*a*b^3*B + a^2*b^2*(2*A - 5*C) + 4*a^4*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + 3*b*(4*A*b^2 - 8*a*b*B + 12*a^2*C + 3*b^2*C)*Sin[c + d*x] + (12*a^3*b*(A*b^2 + a*(-(b*B) + a*C))*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])) + 3*b^2*(b*B - 2*a*C)*Sin[2*(c + d*x)] + b^3*C*Sin[3*(c + d*x)])/(12*b^5*d)","A",1
987,1,208,303,1.4896942,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{2 (c+d x) \left(6 a^2 C-4 a b B+2 A b^2+b^2 C\right)-\frac{4 a^2 b \sin (c+d x) \left(a (a C-b B)+A b^2\right)}{(a-b) (a+b) (a+b \cos (c+d x))}-\frac{8 a \left(3 a^4 C-2 a^3 b B+a^2 b^2 (A-4 C)+3 a b^3 B-2 A b^4\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}+4 b (b B-2 a C) \sin (c+d x)+b^2 C \sin (2 (c+d x))}{4 b^4 d}","-\frac{\sin (c+d x) \cos ^2(c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \cos (c+d x) \left(3 a^2 C-2 a b B+2 A b^2-b^2 C\right)}{2 b^2 d \left(a^2-b^2\right)}+\frac{x \left(6 a^2 C-4 a b B+2 A b^2+b^2 C\right)}{2 b^4}+\frac{\sin (c+d x) \left(-3 a^3 C+2 a^2 b B-a b^2 (A-2 C)-b^3 B\right)}{b^3 d \left(a^2-b^2\right)}-\frac{2 a \left(3 a^4 C-2 a^3 b B+a^2 A b^2-4 a^2 b^2 C+3 a b^3 B-2 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{3/2} (a+b)^{3/2}}",1,"(2*(2*A*b^2 - 4*a*b*B + 6*a^2*C + b^2*C)*(c + d*x) - (8*a*(-2*A*b^4 - 2*a^3*b*B + 3*a*b^3*B + a^2*b^2*(A - 4*C) + 3*a^4*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + 4*b*(b*B - 2*a*C)*Sin[c + d*x] - (4*a^2*b*(A*b^2 + a*(-(b*B) + a*C))*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])) + b^2*C*Sin[2*(c + d*x)])/(4*b^4*d)","A",1
988,1,159,168,1.0267449,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{-\frac{2 \left(a \left(-2 a^3 C+a^2 b B+3 a b^2 C-2 b^3 B\right)+A b^4\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}+\frac{a b \sin (c+d x) \left(a (a C-b B)+A b^2\right)}{(a-b) (a+b) (a+b \cos (c+d x))}+(c+d x) (b B-2 a C)+b C \sin (c+d x)}{b^3 d}","\frac{a \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{2 \left(-2 a^4 C+a^3 b B+3 a^2 b^2 C-2 a b^3 B+A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{x (b B-2 a C)}{b^3}+\frac{C \sin (c+d x)}{b^2 d}",1,"((b*B - 2*a*C)*(c + d*x) - (2*(A*b^4 + a*(a^2*b*B - 2*b^3*B - 2*a^3*C + 3*a*b^2*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + b*C*Sin[c + d*x] + (a*b*(A*b^2 + a*(-(b*B) + a*C))*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])))/(b^3*d)","A",1
989,1,131,139,0.693567,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2,x]","\frac{-\frac{2 \left(a^3 C-a b^2 (A+2 C)+b^3 B\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}-\frac{b \sin (c+d x) \left(a (a C-b B)+A b^2\right)}{(a-b) (a+b) (a+b \cos (c+d x))}+C (c+d x)}{b^2 d}","\frac{2 \left(a^3 (-C)+a A b^2+2 a b^2 C-b^3 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{C x}{b^2}",1,"(C*(c + d*x) - (2*(b^3*B + a^3*C - a*b^2*(A + 2*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) - (b*(A*b^2 + a*(-(b*B) + a*C))*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])))/(b^2*d)","A",1
990,1,319,147,3.0643012,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^2,x]","\frac{2 \cos (c+d x) (A \sec (c+d x)+B+C \cos (c+d x)) \left(\frac{2 i (\cos (c)-i \sin (c))^3 \left(a^3 B-a^2 b (2 A+C)+A b^3\right) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (b \cos (c)-a)+b \sin (c)\right)}{\sqrt{-\left(\left(a^2-b^2\right) (\cos (c)-i \sin (c))^2\right)}}\right)}{\left(\left(b^2-a^2\right) (\cos (c)-i \sin (c))^2\right)^{3/2}}+\frac{a \left(a (a C-b B)+A b^2\right) (b \sin (d x)-a \sin (c))}{b (a-b) (a+b) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) (a+b \cos (c+d x))}-A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{a^2 d (2 A+2 B \cos (c+d x)+C \cos (2 (c+d x))+C)}","\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{A \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{2 \left(a^3 (-B)+2 a^2 A b+a^2 b C-A b^3\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}",1,"(2*Cos[c + d*x]*(B + C*Cos[c + d*x] + A*Sec[c + d*x])*(-(A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]) + A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + ((2*I)*(A*b^3 + a^3*B - a^2*b*(2*A + C))*ArcTan[((I*Cos[c] + Sin[c])*(b*Sin[c] + (-a + b*Cos[c])*Tan[(d*x)/2]))/Sqrt[-((a^2 - b^2)*(Cos[c] - I*Sin[c])^2)]]*(Cos[c] - I*Sin[c])^3)/((-a^2 + b^2)*(Cos[c] - I*Sin[c])^2)^(3/2) + (a*(A*b^2 + a*(-(b*B) + a*C))*(-(a*Sin[c]) + b*Sin[d*x]))/((a - b)*b*(a + b)*(a + b*Cos[c + d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2]))))/(a^2*d*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*(c + d*x)]))","C",0
991,1,331,211,1.8445906,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^2,x]","\frac{2 \cos ^2(c+d x) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(\frac{2 \left(a^4 C-2 a^3 b B+3 a^2 A b^2+a b^3 B-2 A b^4\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}-\frac{a b \sin (c+d x) \left(a (a C-b B)+A b^2\right)}{(a-b) (a+b) (a+b \cos (c+d x))}+(2 A b-a B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+(a B-2 A b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{a A \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{a A \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}\right)}{a^3 d (2 A+2 B \cos (c+d x)+C \cos (2 (c+d x))+C)}","-\frac{(2 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{\tan (c+d x) \left(-\left(a^2 (A-C)\right)-a b B+2 A b^2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{\tan (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{2 \left(a^4 C-2 a^3 b B+3 a^2 A b^2+a b^3 B-2 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{3/2} (a+b)^{3/2}}",1,"(2*Cos[c + d*x]^2*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*((2*(3*a^2*A*b^2 - 2*A*b^4 - 2*a^3*b*B + a*b^3*B + a^4*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + (2*A*b - a*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + (-2*A*b + a*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (a*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - (a*b*(A*b^2 + a*(-(b*B) + a*C))*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x]))))/(a^3*d*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*(c + d*x)]))","A",0
992,1,389,307,5.7664295,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^2,x]","\frac{-2 \left(a^2 (A+2 C)-4 a b B+6 A b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \left(a^2 (A+2 C)-4 a b B+6 A b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{a^2 A}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{a^2 A}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{8 b \left(-2 a^4 C+3 a^3 b B+a^2 b^2 (C-4 A)-2 a b^3 B+3 A b^4\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}+\frac{4 a b^2 \sin (c+d x) \left(a (a C-b B)+A b^2\right)}{(a-b) (a+b) (a+b \cos (c+d x))}+\frac{4 a (a B-2 A b) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 a (a B-2 A b) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}}{4 a^4 d}","-\frac{\tan (c+d x) \sec (c+d x) \left(-\left(a^2 (A-2 C)\right)-2 a b B+3 A b^2\right)}{2 a^2 d \left(a^2-b^2\right)}+\frac{\tan (c+d x) \sec (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\left(a^2 (A+2 C)-4 a b B+6 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{\tan (c+d x) \left(a^3 B-a^2 b (2 A-C)-2 a b^2 B+3 A b^3\right)}{a^3 d \left(a^2-b^2\right)}-\frac{2 b \left(2 a^4 C-3 a^3 b B+4 a^2 A b^2-a^2 b^2 C+2 a b^3 B-3 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{3/2} (a+b)^{3/2}}",1,"((8*b*(3*A*b^4 + 3*a^3*b*B - 2*a*b^3*B - 2*a^4*C + a^2*b^2*(-4*A + C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) - 2*(6*A*b^2 - 4*a*b*B + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(6*A*b^2 - 4*a*b*B + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a^2*A)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (4*a*(-2*A*b + a*B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (a^2*A)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*a*(-2*A*b + a*B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + (4*a*b^2*(A*b^2 + a*(-(b*B) + a*C))*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])))/(4*a^4*d)","A",1
993,1,519,405,3.3224525,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + b*Cos[c + d*x])^2,x]","\frac{6 \left(a^3 (-B)+2 a^2 b (A+2 C)-6 a b^2 B+8 A b^3\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 \left(a^3 B-2 a^2 b (A+2 C)+6 a b^2 B-8 A b^3\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-\frac{24 b^2 \left(-3 a^4 C+4 a^3 b B+a^2 b^2 (2 C-5 A)-3 a b^3 B+4 A b^4\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}+\frac{a \tan (c+d x) \sec ^2(c+d x) \left(8 a^5 A+6 a^5 C+2 a^4 A b \cos (3 (c+d x))-9 a^4 b B+3 a^4 b C \cos (3 (c+d x))+4 a^3 A b^2-6 a^3 b^2 B \cos (3 (c+d x))-6 a^3 b^2 C+7 a^2 A b^3 \cos (3 (c+d x))+a \left(a^2-b^2\right) \cos (2 (c+d x)) \left(a^2 (4 A+6 C)-9 a b B+12 A b^2\right)+9 a^2 b^3 B-6 a^2 b^3 C \cos (3 (c+d x))+\cos (c+d x) \left(6 a^5 B+a^4 (9 b C-2 A b)-24 a^3 b^2 B+a^2 b^3 (29 A-18 C)+27 a b^4 B-36 A b^5\right)-12 a A b^4+9 a b^4 B \cos (3 (c+d x))-12 A b^5 \cos (3 (c+d x))\right)}{\left(a^2-b^2\right) (a+b \cos (c+d x))}}{12 a^5 d}","-\frac{\tan (c+d x) \sec ^2(c+d x) \left(-\left(a^2 (A-3 C)\right)-3 a b B+4 A b^2\right)}{3 a^2 d \left(a^2-b^2\right)}+\frac{\tan (c+d x) \sec ^2(c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\tan (c+d x) \sec (c+d x) \left(a^3 B-2 a^2 b (A-C)-3 a b^2 B+4 A b^3\right)}{2 a^3 d \left(a^2-b^2\right)}-\frac{\left(a^3 (-B)+2 a^2 b (A+2 C)-6 a b^2 B+8 A b^3\right) \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}-\frac{\tan (c+d x) \left(-\left(a^4 (2 A+3 C)\right)+6 a^3 b B-a^2 b^2 (7 A-6 C)-9 a b^3 B+12 A b^4\right)}{3 a^4 d \left(a^2-b^2\right)}+\frac{2 b^2 \left(3 a^4 C-4 a^3 b B+5 a^2 A b^2-2 a^2 b^2 C+3 a b^3 B-4 A b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{3/2} (a+b)^{3/2}}",1,"((-24*b^2*(4*A*b^4 + 4*a^3*b*B - 3*a*b^3*B - 3*a^4*C + a^2*b^2*(-5*A + 2*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + 6*(8*A*b^3 - a^3*B - 6*a*b^2*B + 2*a^2*b*(A + 2*C))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*(-8*A*b^3 + a^3*B + 6*a*b^2*B - 2*a^2*b*(A + 2*C))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a*(8*a^5*A + 4*a^3*A*b^2 - 12*a*A*b^4 - 9*a^4*b*B + 9*a^2*b^3*B + 6*a^5*C - 6*a^3*b^2*C + (-36*A*b^5 + 6*a^5*B - 24*a^3*b^2*B + 27*a*b^4*B + a^2*b^3*(29*A - 18*C) + a^4*(-2*A*b + 9*b*C))*Cos[c + d*x] + a*(a^2 - b^2)*(12*A*b^2 - 9*a*b*B + a^2*(4*A + 6*C))*Cos[2*(c + d*x)] + 2*a^4*A*b*Cos[3*(c + d*x)] + 7*a^2*A*b^3*Cos[3*(c + d*x)] - 12*A*b^5*Cos[3*(c + d*x)] - 6*a^3*b^2*B*Cos[3*(c + d*x)] + 9*a*b^4*B*Cos[3*(c + d*x)] + 3*a^4*b*C*Cos[3*(c + d*x)] - 6*a^2*b^3*C*Cos[3*(c + d*x)])*Sec[c + d*x]^2*Tan[c + d*x])/((a^2 - b^2)*(a + b*Cos[c + d*x])))/(12*a^5*d)","A",1
994,1,883,456,5.1394693,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{16 a \left(12 C a^6-6 b B a^5+b^2 (2 A-29 C) a^4+15 b^3 B a^3-5 b^4 (A-4 C) a^2-12 b^5 B a+6 A b^6\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}+\frac{96 c C a^8+96 C d x a^8-48 b B c a^7-48 b B d x a^7-96 b C \sin (c+d x) a^7+16 A b^2 c a^6-136 b^2 c C a^6+16 A b^2 d x a^6-136 b^2 C d x a^6+48 b^2 B \sin (c+d x) a^6-72 b^2 C \sin (2 (c+d x)) a^6+72 b^3 B c a^5+72 b^3 B d x a^5-16 A b^3 \sin (c+d x) a^5+160 b^3 C \sin (c+d x) a^5+36 b^3 B \sin (2 (c+d x)) a^5-8 b^3 C \sin (3 (c+d x)) a^5-24 A b^4 c a^4-12 b^4 c C a^4-24 A b^4 d x a^4-12 b^4 C d x a^4-84 b^4 B \sin (c+d x) a^4-12 A b^4 \sin (2 (c+d x)) a^4+130 b^4 C \sin (2 (c+d x)) a^4+4 b^4 B \sin (3 (c+d x)) a^4+b^4 C \sin (4 (c+d x)) a^4+40 A b^5 \sin (c+d x) a^3-32 b^5 C \sin (c+d x) a^3-64 b^5 B \sin (2 (c+d x)) a^3+16 b^5 C \sin (3 (c+d x)) a^3+48 b^6 c C a^2+48 b^6 C d x a^2+8 b^6 B \sin (c+d x) a^2+24 A b^6 \sin (2 (c+d x)) a^2-48 b^6 C \sin (2 (c+d x)) a^2-8 b^6 B \sin (3 (c+d x)) a^2-2 b^6 C \sin (4 (c+d x)) a^2-24 b^7 B c a-24 b^7 B d x a+16 b \left(a^2-b^2\right)^2 \left(12 C a^2-6 b B a+2 A b^2+b^2 C\right) (c+d x) \cos (c+d x) a-8 b^7 C \sin (c+d x) a+16 b^7 B \sin (2 (c+d x)) a-8 b^7 C \sin (3 (c+d x)) a+8 A b^8 c+4 b^8 c C+8 A b^8 d x+4 b^8 C d x+4 \left(b^3-a^2 b\right)^2 \left(12 C a^2-6 b B a+2 A b^2+b^2 C\right) (c+d x) \cos (2 (c+d x))+4 b^8 B \sin (c+d x)+2 b^8 C \sin (2 (c+d x))+4 b^8 B \sin (3 (c+d x))+b^8 C \sin (4 (c+d x))}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}}{16 b^5 d}","-\frac{\sin (c+d x) \cos ^3(c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{x \left(12 a^2 C-6 a b B+2 A b^2+b^2 C\right)}{2 b^5}+\frac{\sin (c+d x) \cos ^2(c+d x) \left(a \left(-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right)+3 A b^4\right)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\sin (c+d x) \cos (c+d x) \left(-6 a^4 C+3 a^3 b B-a^2 b^2 (A-10 C)-6 a b^3 B+b^4 (4 A-C)\right)}{2 b^3 d \left(a^2-b^2\right)^2}+\frac{\sin (c+d x) \left(-12 a^5 C+6 a^4 b B-a^3 b^2 (2 A-21 C)-11 a^2 b^3 B+a b^4 (5 A-6 C)+2 b^5 B\right)}{2 b^4 d \left(a^2-b^2\right)^2}-\frac{a \left(12 a^6 C-6 a^5 b B+a^4 b^2 (2 A-29 C)+15 a^3 b^3 B-5 a^2 b^4 (A-4 C)-12 a b^5 B+6 A b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{5/2} (a+b)^{5/2}}",1,"((16*a*(6*A*b^6 - 6*a^5*b*B + 15*a^3*b^3*B - 12*a*b^5*B + a^4*b^2*(2*A - 29*C) - 5*a^2*b^4*(A - 4*C) + 12*a^6*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (16*a^6*A*b^2*c - 24*a^4*A*b^4*c + 8*A*b^8*c - 48*a^7*b*B*c + 72*a^5*b^3*B*c - 24*a*b^7*B*c + 96*a^8*c*C - 136*a^6*b^2*c*C - 12*a^4*b^4*c*C + 48*a^2*b^6*c*C + 4*b^8*c*C + 16*a^6*A*b^2*d*x - 24*a^4*A*b^4*d*x + 8*A*b^8*d*x - 48*a^7*b*B*d*x + 72*a^5*b^3*B*d*x - 24*a*b^7*B*d*x + 96*a^8*C*d*x - 136*a^6*b^2*C*d*x - 12*a^4*b^4*C*d*x + 48*a^2*b^6*C*d*x + 4*b^8*C*d*x + 16*a*b*(a^2 - b^2)^2*(2*A*b^2 - 6*a*b*B + 12*a^2*C + b^2*C)*(c + d*x)*Cos[c + d*x] + 4*(-(a^2*b) + b^3)^2*(2*A*b^2 - 6*a*b*B + 12*a^2*C + b^2*C)*(c + d*x)*Cos[2*(c + d*x)] - 16*a^5*A*b^3*Sin[c + d*x] + 40*a^3*A*b^5*Sin[c + d*x] + 48*a^6*b^2*B*Sin[c + d*x] - 84*a^4*b^4*B*Sin[c + d*x] + 8*a^2*b^6*B*Sin[c + d*x] + 4*b^8*B*Sin[c + d*x] - 96*a^7*b*C*Sin[c + d*x] + 160*a^5*b^3*C*Sin[c + d*x] - 32*a^3*b^5*C*Sin[c + d*x] - 8*a*b^7*C*Sin[c + d*x] - 12*a^4*A*b^4*Sin[2*(c + d*x)] + 24*a^2*A*b^6*Sin[2*(c + d*x)] + 36*a^5*b^3*B*Sin[2*(c + d*x)] - 64*a^3*b^5*B*Sin[2*(c + d*x)] + 16*a*b^7*B*Sin[2*(c + d*x)] - 72*a^6*b^2*C*Sin[2*(c + d*x)] + 130*a^4*b^4*C*Sin[2*(c + d*x)] - 48*a^2*b^6*C*Sin[2*(c + d*x)] + 2*b^8*C*Sin[2*(c + d*x)] + 4*a^4*b^4*B*Sin[3*(c + d*x)] - 8*a^2*b^6*B*Sin[3*(c + d*x)] + 4*b^8*B*Sin[3*(c + d*x)] - 8*a^5*b^3*C*Sin[3*(c + d*x)] + 16*a^3*b^5*C*Sin[3*(c + d*x)] - 8*a*b^7*C*Sin[3*(c + d*x)] + a^4*b^4*C*Sin[4*(c + d*x)] - 2*a^2*b^6*C*Sin[4*(c + d*x)] + b^8*C*Sin[4*(c + d*x)])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2))/(16*b^5*d)","A",1
995,1,573,314,2.9505356,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{-12 a^7 c C-12 a^7 C d x+4 a^6 b B c+4 a^6 b B d x+12 a^6 b C \sin (c+d x)-4 a^5 b^2 B \sin (c+d x)+9 a^5 b^2 C \sin (2 (c+d x))+18 a^5 b^2 c C+18 a^5 b^2 C d x-3 a^4 b^3 B \sin (2 (c+d x))-6 a^4 b^3 B c-6 a^4 b^3 B d x-21 a^4 b^3 C \sin (c+d x)+a^4 b^3 C \sin (3 (c+d x))+a^3 A b^4 \sin (2 (c+d x))+10 a^3 b^4 B \sin (c+d x)-16 a^3 b^4 C \sin (2 (c+d x))-6 a^2 A b^5 \sin (c+d x)+6 a^2 b^5 B \sin (2 (c+d x))+2 a^2 b^5 C \sin (c+d x)-2 a^2 b^5 C \sin (3 (c+d x))+2 \left(b^3-a^2 b\right)^2 (c+d x) (b B-3 a C) \cos (2 (c+d x))-8 a b \left(a^2-b^2\right)^2 (c+d x) (3 a C-b B) \cos (c+d x)-4 a A b^6 \sin (2 (c+d x))+4 a b^6 C \sin (2 (c+d x))-6 a b^6 c C-6 a b^6 C d x+2 b^7 B c+2 b^7 B d x+b^7 C \sin (c+d x)+b^7 C \sin (3 (c+d x))}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{4 \left(6 a^6 C-2 a^5 b B-15 a^4 b^2 C+5 a^3 b^3 B+a^2 b^4 (A+12 C)-6 a b^5 B+2 A b^6\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}}{4 b^4 d}","-\frac{\sin (c+d x) \cos ^2(c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\sin (c+d x) \left(3 a^2 C-a b B+A b^2-2 b^2 C\right)}{2 b^3 d \left(a^2-b^2\right)}-\frac{a \sin (c+d x) \left(-3 a^4 C+a^3 b B+a^2 b^2 (A+6 C)-4 a b^3 B+2 A b^4\right)}{2 b^3 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(6 a^6 C-2 a^5 b B-15 a^4 b^2 C+5 a^3 b^3 B+a^2 b^4 (A+12 C)-6 a b^5 B+2 A b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{x (b B-3 a C)}{b^4}",1,"((-4*(2*A*b^6 - 2*a^5*b*B + 5*a^3*b^3*B - 6*a*b^5*B + 6*a^6*C - 15*a^4*b^2*C + a^2*b^4*(A + 12*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (4*a^6*b*B*c - 6*a^4*b^3*B*c + 2*b^7*B*c - 12*a^7*c*C + 18*a^5*b^2*c*C - 6*a*b^6*c*C + 4*a^6*b*B*d*x - 6*a^4*b^3*B*d*x + 2*b^7*B*d*x - 12*a^7*C*d*x + 18*a^5*b^2*C*d*x - 6*a*b^6*C*d*x - 8*a*b*(a^2 - b^2)^2*(-(b*B) + 3*a*C)*(c + d*x)*Cos[c + d*x] + 2*(-(a^2*b) + b^3)^2*(b*B - 3*a*C)*(c + d*x)*Cos[2*(c + d*x)] - 6*a^2*A*b^5*Sin[c + d*x] - 4*a^5*b^2*B*Sin[c + d*x] + 10*a^3*b^4*B*Sin[c + d*x] + 12*a^6*b*C*Sin[c + d*x] - 21*a^4*b^3*C*Sin[c + d*x] + 2*a^2*b^5*C*Sin[c + d*x] + b^7*C*Sin[c + d*x] + a^3*A*b^4*Sin[2*(c + d*x)] - 4*a*A*b^6*Sin[2*(c + d*x)] - 3*a^4*b^3*B*Sin[2*(c + d*x)] + 6*a^2*b^5*B*Sin[2*(c + d*x)] + 9*a^5*b^2*C*Sin[2*(c + d*x)] - 16*a^3*b^4*C*Sin[2*(c + d*x)] + 4*a*b^6*C*Sin[2*(c + d*x)] + a^4*b^3*C*Sin[3*(c + d*x)] - 2*a^2*b^5*C*Sin[3*(c + d*x)] + b^7*C*Sin[3*(c + d*x)])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2))/(4*b^4*d)","A",1
996,1,225,233,1.8207277,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{2 \left(2 a^5 C-5 a^3 b^2 C-a^2 b^3 B+3 a b^4 (A+2 C)-2 b^5 B\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}+\frac{b \sin (c+d x) \left(-3 a^4 C+a^3 b B+a^2 b^2 (A+6 C)-4 a b^3 B+2 A b^4\right)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}+\frac{a b \sin (c+d x) \left(a (a C-b B)+A b^2\right)}{(a-b) (a+b) (a+b \cos (c+d x))^2}+2 C (c+d x)}{2 b^3 d}","\frac{a \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\left(-2 a^5 C+5 a^3 b^2 C+a^2 b^3 B-3 a b^4 (A+2 C)+2 b^5 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{\sin (c+d x) \left(-3 a^4 C+a^3 b B+a^2 b^2 (A+6 C)-4 a b^3 B+2 A b^4\right)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{C x}{b^3}",1,"(2*C*(c + d*x) + (2*(-(a^2*b^3*B) - 2*b^5*B + 2*a^5*C - 5*a^3*b^2*C + 3*a*b^4*(A + 2*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (a*b*(A*b^2 + a*(-(b*B) + a*C))*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^2) + (b*(2*A*b^4 + a^3*b*B - 4*a*b^3*B - 3*a^4*C + a^2*b^2*(A + 6*C))*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])))/(2*b^3*d)","A",1
997,1,192,202,1.124153,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3,x]","\frac{-\frac{2 \left(a^2 (2 A+C)-3 a b B+b^2 (A+2 C)\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}+\frac{\sin (c+d x) \left(a^3 C+a^2 b B-a b^2 (3 A+4 C)+2 b^3 B\right)}{b (a-b)^2 (a+b)^2 (a+b \cos (c+d x))}+\frac{\sin (c+d x) \left(a (a C-b B)+A b^2\right)}{b (b-a) (a+b) (a+b \cos (c+d x))^2}}{2 d}","-\frac{\left(-\left(a^2 (2 A+C)\right)+3 a b B-b^2 (A+2 C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\sin (c+d x) \left(a^3 C+a^2 b B-a b^2 (3 A+4 C)+2 b^3 B\right)}{2 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}",1,"((-2*(-3*a*b*B + a^2*(2*A + C) + b^2*(A + 2*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + ((A*b^2 + a*(-(b*B) + a*C))*Sin[c + d*x])/(b*(-a + b)*(a + b)*(a + b*Cos[c + d*x])^2) + ((a^2*b*B + 2*b^3*B + a^3*C - a*b^2*(3*A + 4*C))*Sin[c + d*x])/((a - b)^2*b*(a + b)^2*(a + b*Cos[c + d*x])))/(2*d)","A",1
998,1,473,238,4.7061071,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^3,x]","\frac{\cos (c+d x) (A \sec (c+d x)+B+C \cos (c+d x)) \left(\frac{a \left(b \sec (c) \left(b \left(a \sin (2 c+d x) \left(2 a^3 B-a^2 b (4 A+3 C)+a b^2 B+A b^3\right)+\sin (c+2 d x) \left(a^4 C-3 a^3 b B+a^2 b^2 (5 A+2 C)-2 A b^4\right)\right)+a \sin (d x) \left(4 a^4 C-10 a^3 b B+a^2 b^2 (16 A+5 C)+a b^3 B-7 A b^4\right)\right)-\left(2 a^2+b^2\right) \tan (c) \left(a^4 C-3 a^3 b B+a^2 b^2 (5 A+2 C)-2 A b^4\right)\right)}{b \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{4 i (\cos (c)-i \sin (c)) \left(2 a^5 B-3 a^4 b (2 A+C)+a^3 b^2 B+5 a^2 A b^3-2 A b^5\right) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (b \cos (c)-a)+b \sin (c)\right)}{\sqrt{-\left(\left(a^2-b^2\right) (\cos (c)-i \sin (c))^2\right)}}\right)}{\left(a^2-b^2\right)^2 \sqrt{-\left(\left(a^2-b^2\right) (\cos (c)-i \sin (c))^2\right)}}-4 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{2 a^3 d (2 A+2 B \cos (c+d x)+C \cos (2 (c+d x))+C)}","\frac{A \tanh ^{-1}(\sin (c+d x))}{a^3 d}+\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{\sin (c+d x) \left(a^4 (-C)+3 a^3 b B-a^2 b^2 (5 A+2 C)+2 A b^4\right)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(2 a^5 B-3 a^4 b (2 A+C)+a^3 b^2 B+5 a^2 A b^3-2 A b^5\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}",1,"(Cos[c + d*x]*(B + C*Cos[c + d*x] + A*Sec[c + d*x])*(-4*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - ((4*I)*(5*a^2*A*b^3 - 2*A*b^5 + 2*a^5*B + a^3*b^2*B - 3*a^4*b*(2*A + C))*ArcTan[((I*Cos[c] + Sin[c])*(b*Sin[c] + (-a + b*Cos[c])*Tan[(d*x)/2]))/Sqrt[-((a^2 - b^2)*(Cos[c] - I*Sin[c])^2)]]*(Cos[c] - I*Sin[c]))/((a^2 - b^2)^2*Sqrt[-((a^2 - b^2)*(Cos[c] - I*Sin[c])^2)]) + (a*(b*Sec[c]*(a*(-7*A*b^4 - 10*a^3*b*B + a*b^3*B + 4*a^4*C + a^2*b^2*(16*A + 5*C))*Sin[d*x] + b*(a*(A*b^3 + 2*a^3*B + a*b^2*B - a^2*b*(4*A + 3*C))*Sin[2*c + d*x] + (-2*A*b^4 - 3*a^3*b*B + a^4*C + a^2*b^2*(5*A + 2*C))*Sin[c + 2*d*x])) - (2*a^2 + b^2)*(-2*A*b^4 - 3*a^3*b*B + a^4*C + a^2*b^2*(5*A + 2*C))*Tan[c]))/(b*(a^2 - b^2)^2*(a + b*Cos[c + d*x])^2)))/(2*a^3*d*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*(c + d*x)]))","C",1
999,1,444,339,2.5537723,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^3,x]","\frac{\cos (c+d x) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(\frac{2 a \sin (c+d x) \left(4 a^6 A-6 a^4 A b^2-3 a^4 b^2 C+5 a^3 b^3 B-7 a^2 A b^4+2 a b \cos (c+d x) \left(4 a^4 (A-C)+6 a^3 b B+a^2 b^2 (C-16 A)-3 a b^3 B+9 A b^4\right)+b^2 \cos (2 (c+d x)) \left(a^4 (2 A-3 C)+5 a^3 b B-11 a^2 A b^2-2 a b^3 B+6 A b^4\right)-2 a b^5 B+6 A b^6\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{8 \cos (c+d x) \left(2 a^6 C-6 a^5 b B+a^4 b^2 (12 A+C)+5 a^3 b^3 B-15 a^2 A b^4-2 a b^5 B+6 A b^6\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}+8 (3 A b-a B) \cos (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+8 (a B-3 A b) \cos (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{4 a^4 d (2 A+2 B \cos (c+d x)+C \cos (2 (c+d x))+C)}","-\frac{(3 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac{\tan (c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{\tan (c+d x) \left(-\left(a^4 (2 A-3 C)\right)-5 a^3 b B+11 a^2 A b^2+2 a b^3 B-6 A b^4\right)}{2 a^3 d \left(a^2-b^2\right)^2}-\frac{\tan (c+d x) \left(-2 a^4 C+4 a^3 b B-a^2 b^2 (6 A+C)-a b^3 B+3 A b^4\right)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\left(-2 a^6 C+6 a^5 b B-a^4 b^2 (12 A+C)-5 a^3 b^3 B+15 a^2 A b^4+2 a b^5 B-6 A b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{5/2} (a+b)^{5/2}}",1,"(Cos[c + d*x]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*((-8*(-15*a^2*A*b^4 + 6*A*b^6 - 6*a^5*b*B + 5*a^3*b^3*B - 2*a*b^5*B + 2*a^6*C + a^4*b^2*(12*A + C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]]*Cos[c + d*x])/(-a^2 + b^2)^(5/2) + 8*(3*A*b - a*B)*Cos[c + d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 8*(-3*A*b + a*B)*Cos[c + d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*a*(4*a^6*A - 6*a^4*A*b^2 - 7*a^2*A*b^4 + 6*A*b^6 + 5*a^3*b^3*B - 2*a*b^5*B - 3*a^4*b^2*C + 2*a*b*(9*A*b^4 + 6*a^3*b*B - 3*a*b^3*B + 4*a^4*(A - C) + a^2*b^2*(-16*A + C))*Cos[c + d*x] + b^2*(-11*a^2*A*b^2 + 6*A*b^4 + 5*a^3*b*B - 2*a*b^3*B + a^4*(2*A - 3*C))*Cos[2*(c + d*x)])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2)))/(4*a^4*d*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*(c + d*x)]))","A",0
1000,1,606,462,3.8610973,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^3,x]","\frac{-8 \left(a^2 (A+2 C)-6 a b B+12 A b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+8 \left(a^2 (A+2 C)-6 a b B+12 A b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{16 b \left(6 a^6 C-12 a^5 b B+5 a^4 b^2 (4 A-C)+15 a^3 b^3 B+a^2 b^4 (2 C-29 A)-6 a b^5 B+12 A b^6\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}+\frac{2 a \tan (c+d x) \sec (c+d x) \left(4 a^7 A+8 a^6 b B-30 a^5 A b^2+2 a^5 b^2 B \cos (3 (c+d x))+12 a^5 b^2 C-6 a^4 A b^3 \cos (3 (c+d x))-32 a^4 b^3 B+5 a^4 b^3 C \cos (3 (c+d x))+68 a^3 A b^4-11 a^3 b^4 B \cos (3 (c+d x))-6 a^3 b^4 C+21 a^2 A b^5 \cos (3 (c+d x))+18 a^2 b^5 B-2 a^2 b^5 C \cos (3 (c+d x))+2 a b \cos (2 (c+d x)) \left(4 a^5 B+a^4 (6 b C-11 A b)-16 a^3 b^2 B+a^2 b^3 (32 A-3 C)+9 a b^4 B-18 A b^5\right)+\cos (c+d x) \left(8 a^7 B-16 a^6 A b-10 a^5 b^2 B+a^4 b^3 (14 A+15 C)-25 a^3 b^4 B+a^2 b^5 (47 A-6 C)+18 a b^6 B-36 A b^7\right)-36 a A b^6+6 a b^6 B \cos (3 (c+d x))-12 A b^7 \cos (3 (c+d x))\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}}{16 a^5 d}","\frac{\tan (c+d x) \sec (c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\left(a^2 (A+2 C)-6 a b B+12 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}+\frac{\tan (c+d x) \sec (c+d x) \left(a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 a b^3 B+6 A b^4\right)}{2 a^3 d \left(a^2-b^2\right)^2}+\frac{\tan (c+d x) \sec (c+d x) \left(3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\tan (c+d x) \left(-2 a^5 B+a^4 b (6 A-5 C)+11 a^3 b^2 B-a^2 b^3 (21 A-2 C)-6 a b^4 B+12 A b^5\right)}{2 a^4 d \left(a^2-b^2\right)^2}-\frac{b \left(6 a^6 C-12 a^5 b B+5 a^4 b^2 (4 A-C)+15 a^3 b^3 B-a^2 b^4 (29 A-2 C)-6 a b^5 B+12 A b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{5/2} (a+b)^{5/2}}",1,"((16*b*(12*A*b^6 - 12*a^5*b*B + 15*a^3*b^3*B - 6*a*b^5*B + 5*a^4*b^2*(4*A - C) + 6*a^6*C + a^2*b^4*(-29*A + 2*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) - 8*(12*A*b^2 - 6*a*b*B + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 8*(12*A*b^2 - 6*a*b*B + a^2*(A + 2*C))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*a*(4*a^7*A - 30*a^5*A*b^2 + 68*a^3*A*b^4 - 36*a*A*b^6 + 8*a^6*b*B - 32*a^4*b^3*B + 18*a^2*b^5*B + 12*a^5*b^2*C - 6*a^3*b^4*C + (-16*a^6*A*b - 36*A*b^7 + 8*a^7*B - 10*a^5*b^2*B - 25*a^3*b^4*B + 18*a*b^6*B + a^2*b^5*(47*A - 6*C) + a^4*b^3*(14*A + 15*C))*Cos[c + d*x] + 2*a*b*(-18*A*b^5 + 4*a^5*B - 16*a^3*b^2*B + 9*a*b^4*B + a^2*b^3*(32*A - 3*C) + a^4*(-11*A*b + 6*b*C))*Cos[2*(c + d*x)] - 6*a^4*A*b^3*Cos[3*(c + d*x)] + 21*a^2*A*b^5*Cos[3*(c + d*x)] - 12*A*b^7*Cos[3*(c + d*x)] + 2*a^5*b^2*B*Cos[3*(c + d*x)] - 11*a^3*b^4*B*Cos[3*(c + d*x)] + 6*a*b^6*B*Cos[3*(c + d*x)] + 5*a^4*b^3*C*Cos[3*(c + d*x)] - 2*a^2*b^5*C*Cos[3*(c + d*x)])*Sec[c + d*x]*Tan[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2))/(16*a^5*d)","A",1
1001,1,658,649,6.9174462,"\int \frac{\cos ^4(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^4,x]","\frac{(c+d x) \left(20 a^2 C-8 a b B+2 A b^2+b^2 C\right)}{2 b^6 d}+\frac{a^6 C \sin (c+d x)-a^5 b B \sin (c+d x)+a^4 A b^2 \sin (c+d x)}{3 b^5 d \left(b^2-a^2\right) (a+b \cos (c+d x))^3}+\frac{13 a^7 C \sin (c+d x)-10 a^6 b B \sin (c+d x)+7 a^5 A b^2 \sin (c+d x)-18 a^5 b^2 C \sin (c+d x)+15 a^4 b^3 B \sin (c+d x)-12 a^3 A b^4 \sin (c+d x)}{6 b^5 d \left(b^2-a^2\right)^2 (a+b \cos (c+d x))^2}+\frac{47 a^8 C \sin (c+d x)-26 a^7 b B \sin (c+d x)+11 a^6 A b^2 \sin (c+d x)-122 a^6 b^2 C \sin (c+d x)+71 a^5 b^3 B \sin (c+d x)-32 a^4 A b^4 \sin (c+d x)+90 a^4 b^4 C \sin (c+d x)-60 a^3 b^5 B \sin (c+d x)+36 a^2 A b^6 \sin (c+d x)}{6 b^5 d \left(b^2-a^2\right)^3 (a+b \cos (c+d x))}+\frac{a \left(20 a^8 C-8 a^7 b B+2 a^6 A b^2-69 a^6 b^2 C+28 a^5 b^3 B-7 a^4 A b^4+84 a^4 b^4 C-35 a^3 b^5 B+8 a^2 A b^6-40 a^2 b^6 C+20 a b^7 B-8 A b^8\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{b^6 d \left(a^2-b^2\right)^3 \sqrt{b^2-a^2}}+\frac{(4 a C-b B) \left(-\frac{\sin (c+d x)}{2 b^5}-\frac{i \cos (c+d x)}{2 b^5}\right)}{d}+\frac{(4 a C-b B) \left(-\frac{\sin (c+d x)}{2 b^5}+\frac{i \cos (c+d x)}{2 b^5}\right)}{d}+\frac{C \sin (2 (c+d x))}{4 b^4 d}","-\frac{\sin (c+d x) \cos ^4(c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{x \left(20 a^2 C-8 a b B+2 A b^2+b^2 C\right)}{2 b^6}+\frac{\sin (c+d x) \cos ^3(c+d x) \left(-5 a^4 C+2 a^3 b B+a^2 b^2 (A+10 C)-7 a b^3 B+4 A b^4\right)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{\sin (c+d x) \cos ^2(c+d x) \left(20 a^6 C-8 a^5 b B+a^4 b^2 (2 A-53 C)+20 a^3 b^3 B+a^2 b^4 (A+48 C)-27 a b^5 B+12 A b^6\right)}{6 b^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{\sin (c+d x) \cos (c+d x) \left(-10 a^6 C+4 a^5 b B-a^4 b^2 (A-27 C)-11 a^3 b^3 B+a^2 b^4 (2 A-23 C)+12 a b^5 B-b^6 (6 A-C)\right)}{2 b^4 d \left(a^2-b^2\right)^3}+\frac{\sin (c+d x) \left(-60 a^7 C+24 a^6 b B-a^5 b^2 (6 A-167 C)-68 a^4 b^3 B+a^3 b^4 (17 A-146 C)+65 a^2 b^5 B-2 a b^6 (13 A-12 C)-6 b^7 B\right)}{6 b^5 d \left(a^2-b^2\right)^3}+\frac{a \left(-20 a^8 C+8 a^7 b B-a^6 b^2 (2 A-69 C)-28 a^5 b^3 B+7 a^4 b^4 (A-12 C)+35 a^3 b^5 B-8 a^2 b^6 (A-5 C)-20 a b^7 B+8 A b^8\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^6 d \sqrt{a-b} \sqrt{a+b} \left(a^2-b^2\right)^3}",1,"((2*A*b^2 - 8*a*b*B + 20*a^2*C + b^2*C)*(c + d*x))/(2*b^6*d) + (a*(2*a^6*A*b^2 - 7*a^4*A*b^4 + 8*a^2*A*b^6 - 8*A*b^8 - 8*a^7*b*B + 28*a^5*b^3*B - 35*a^3*b^5*B + 20*a*b^7*B + 20*a^8*C - 69*a^6*b^2*C + 84*a^4*b^4*C - 40*a^2*b^6*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(b^6*(a^2 - b^2)^3*Sqrt[-a^2 + b^2]*d) + ((-(b*B) + 4*a*C)*(((-1/2*I)*Cos[c + d*x])/b^5 - Sin[c + d*x]/(2*b^5)))/d + ((-(b*B) + 4*a*C)*(((I/2)*Cos[c + d*x])/b^5 - Sin[c + d*x]/(2*b^5)))/d + (a^4*A*b^2*Sin[c + d*x] - a^5*b*B*Sin[c + d*x] + a^6*C*Sin[c + d*x])/(3*b^5*(-a^2 + b^2)*d*(a + b*Cos[c + d*x])^3) + (7*a^5*A*b^2*Sin[c + d*x] - 12*a^3*A*b^4*Sin[c + d*x] - 10*a^6*b*B*Sin[c + d*x] + 15*a^4*b^3*B*Sin[c + d*x] + 13*a^7*C*Sin[c + d*x] - 18*a^5*b^2*C*Sin[c + d*x])/(6*b^5*(-a^2 + b^2)^2*d*(a + b*Cos[c + d*x])^2) + (11*a^6*A*b^2*Sin[c + d*x] - 32*a^4*A*b^4*Sin[c + d*x] + 36*a^2*A*b^6*Sin[c + d*x] - 26*a^7*b*B*Sin[c + d*x] + 71*a^5*b^3*B*Sin[c + d*x] - 60*a^3*b^5*B*Sin[c + d*x] + 47*a^8*C*Sin[c + d*x] - 122*a^6*b^2*C*Sin[c + d*x] + 90*a^4*b^4*C*Sin[c + d*x])/(6*b^5*(-a^2 + b^2)^3*d*(a + b*Cos[c + d*x])) + (C*Sin[2*(c + d*x)])/(4*b^4*d)","C",1
1002,1,532,461,6.7890739,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^4,x]","\frac{a^5 (-C) \sin (c+d x)+a^4 b B \sin (c+d x)-a^3 A b^2 \sin (c+d x)}{3 b^4 d \left(b^2-a^2\right) (a+b \cos (c+d x))^3}+\frac{-10 a^6 C \sin (c+d x)+7 a^5 b B \sin (c+d x)-4 a^4 A b^2 \sin (c+d x)+15 a^4 b^2 C \sin (c+d x)-12 a^3 b^3 B \sin (c+d x)+9 a^2 A b^4 \sin (c+d x)}{6 b^4 d \left(b^2-a^2\right)^2 (a+b \cos (c+d x))^2}+\frac{-26 a^7 C \sin (c+d x)+11 a^6 b B \sin (c+d x)-2 a^5 A b^2 \sin (c+d x)+71 a^5 b^2 C \sin (c+d x)-32 a^4 b^3 B \sin (c+d x)+5 a^3 A b^4 \sin (c+d x)-60 a^3 b^4 C \sin (c+d x)+36 a^2 b^5 B \sin (c+d x)-18 a A b^6 \sin (c+d x)}{6 b^4 d \left(b^2-a^2\right)^3 (a+b \cos (c+d x))}-\frac{\left(8 a^8 C-2 a^7 b B-28 a^6 b^2 C+7 a^5 b^3 B+35 a^4 b^4 C-8 a^3 b^5 B-3 a^2 A b^6-20 a^2 b^6 C+8 a b^7 B-2 A b^8\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{b^5 d \left(a^2-b^2\right)^3 \sqrt{b^2-a^2}}+\frac{(c+d x) (b B-4 a C)}{b^5 d}+\frac{C \sin (c+d x)}{b^4 d}","-\frac{\sin (c+d x) \cos ^3(c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}-\frac{\sin (c+d x) \left(-12 a^4 C+3 a^3 b B+23 a^2 b^2 C-8 a b^3 B+5 A b^4-6 b^4 C\right)}{6 b^4 d \left(a^2-b^2\right)^2}+\frac{\sin (c+d x) \cos ^2(c+d x) \left(-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{a \sin (c+d x) \left(4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right)}{2 b^4 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{\left(-8 a^8 C+2 a^7 b B+28 a^6 b^2 C-7 a^5 b^3 B-35 a^4 b^4 C+8 a^3 b^5 B+a^2 b^6 (3 A+20 C)-8 a b^7 B+2 A b^8\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{x (b B-4 a C)}{b^5}",1,"((b*B - 4*a*C)*(c + d*x))/(b^5*d) - ((-3*a^2*A*b^6 - 2*A*b^8 - 2*a^7*b*B + 7*a^5*b^3*B - 8*a^3*b^5*B + 8*a*b^7*B + 8*a^8*C - 28*a^6*b^2*C + 35*a^4*b^4*C - 20*a^2*b^6*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(b^5*(a^2 - b^2)^3*Sqrt[-a^2 + b^2]*d) + (C*Sin[c + d*x])/(b^4*d) + (-(a^3*A*b^2*Sin[c + d*x]) + a^4*b*B*Sin[c + d*x] - a^5*C*Sin[c + d*x])/(3*b^4*(-a^2 + b^2)*d*(a + b*Cos[c + d*x])^3) + (-4*a^4*A*b^2*Sin[c + d*x] + 9*a^2*A*b^4*Sin[c + d*x] + 7*a^5*b*B*Sin[c + d*x] - 12*a^3*b^3*B*Sin[c + d*x] - 10*a^6*C*Sin[c + d*x] + 15*a^4*b^2*C*Sin[c + d*x])/(6*b^4*(-a^2 + b^2)^2*d*(a + b*Cos[c + d*x])^2) + (-2*a^5*A*b^2*Sin[c + d*x] + 5*a^3*A*b^4*Sin[c + d*x] - 18*a*A*b^6*Sin[c + d*x] + 11*a^6*b*B*Sin[c + d*x] - 32*a^4*b^3*B*Sin[c + d*x] + 36*a^2*b^5*B*Sin[c + d*x] - 26*a^7*C*Sin[c + d*x] + 71*a^5*b^2*C*Sin[c + d*x] - 60*a^3*b^4*C*Sin[c + d*x])/(6*b^4*(-a^2 + b^2)^3*d*(a + b*Cos[c + d*x]))","A",1
1003,1,863,349,4.4459196,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^4,x]","\frac{\frac{24 c C a^9+24 C d x a^9-24 b C \sin (c+d x) a^8-36 b^2 c C a^7-36 b^2 C d x a^7-30 b^2 C \sin (2 (c+d x)) a^7+6 b^3 c C \cos (3 (c+d x)) a^6+6 b^3 C d x \cos (3 (c+d x)) a^6+57 b^3 C \sin (c+d x) a^6-11 b^3 C \sin (3 (c+d x)) a^6-36 b^4 c C a^5-36 b^4 C d x a^5+18 b^4 B \sin (c+d x) a^5+6 A b^4 \sin (2 (c+d x)) a^5+90 b^4 C \sin (2 (c+d x)) a^5+2 b^4 B \sin (3 (c+d x)) a^5-18 b^5 c C \cos (3 (c+d x)) a^4-18 b^5 C d x \cos (3 (c+d x)) a^4-51 A b^5 \sin (c+d x) a^4-72 b^5 C \sin (c+d x) a^4+6 b^5 B \sin (2 (c+d x)) a^4+A b^5 \sin (3 (c+d x)) a^4+32 b^5 C \sin (3 (c+d x)) a^4+84 b^6 c C a^3+84 b^6 C d x a^3+39 b^6 B \sin (c+d x) a^3-54 A b^6 \sin (2 (c+d x)) a^3-120 b^6 C \sin (2 (c+d x)) a^3-5 b^6 B \sin (3 (c+d x)) a^3+18 b^7 c C \cos (3 (c+d x)) a^2+18 b^7 C d x \cos (3 (c+d x)) a^2-18 A b^7 \sin (c+d x) a^2-36 b^7 C \sin (c+d x) a^2+54 b^7 B \sin (2 (c+d x)) a^2-10 A b^7 \sin (3 (c+d x)) a^2-36 b^7 C \sin (3 (c+d x)) a^2-36 b^8 c C a-36 b^8 C d x a+36 b^2 \left(a^2-b^2\right)^3 C (c+d x) \cos (2 (c+d x)) a+18 b^8 B \sin (c+d x) a-12 A b^8 \sin (2 (c+d x)) a+18 b^8 B \sin (3 (c+d x)) a+18 b \left(a^2-b^2\right)^3 \left(4 a^2+b^2\right) C (c+d x) \cos (c+d x)-6 b^9 c C \cos (3 (c+d x))-6 b^9 C d x \cos (3 (c+d x))-6 A b^9 \sin (c+d x)-6 A b^9 \sin (3 (c+d x))}{\left(a^2-b^2\right)^3 (a+b \cos (c+d x))^3}-\frac{24 \left(2 C a^7-7 b^2 C a^5-b^4 (A-8 C) a^3+3 b^5 B a^2-4 b^6 (A+2 C) a+2 b^7 B\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{7/2}}}{24 b^4 d}","-\frac{\sin (c+d x) \cos ^2(c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}-\frac{a \sin (c+d x) \left(-3 a^4 C+a^2 b^2 (3 A+8 C)-5 a b^3 B+2 A b^4\right)}{6 b^3 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{\left(2 a^7 C-7 a^5 b^2 C-a^3 b^4 (A-8 C)+3 a^2 b^5 B-4 a b^6 (A+2 C)+2 b^7 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\sin (c+d x) \left(9 a^6 C-a^4 b^2 (3 A+28 C)+a^3 b^3 B+2 a^2 b^4 (7 A+17 C)-16 a b^5 B+4 A b^6\right)}{6 b^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{C x}{b^4}",1,"((-24*(3*a^2*b^5*B + 2*b^7*B - a^3*b^4*(A - 8*C) + 2*a^7*C - 7*a^5*b^2*C - 4*a*b^6*(A + 2*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) + (24*a^9*c*C - 36*a^7*b^2*c*C - 36*a^5*b^4*c*C + 84*a^3*b^6*c*C - 36*a*b^8*c*C + 24*a^9*C*d*x - 36*a^7*b^2*C*d*x - 36*a^5*b^4*C*d*x + 84*a^3*b^6*C*d*x - 36*a*b^8*C*d*x + 18*b*(a^2 - b^2)^3*(4*a^2 + b^2)*C*(c + d*x)*Cos[c + d*x] + 36*a*b^2*(a^2 - b^2)^3*C*(c + d*x)*Cos[2*(c + d*x)] + 6*a^6*b^3*c*C*Cos[3*(c + d*x)] - 18*a^4*b^5*c*C*Cos[3*(c + d*x)] + 18*a^2*b^7*c*C*Cos[3*(c + d*x)] - 6*b^9*c*C*Cos[3*(c + d*x)] + 6*a^6*b^3*C*d*x*Cos[3*(c + d*x)] - 18*a^4*b^5*C*d*x*Cos[3*(c + d*x)] + 18*a^2*b^7*C*d*x*Cos[3*(c + d*x)] - 6*b^9*C*d*x*Cos[3*(c + d*x)] - 51*a^4*A*b^5*Sin[c + d*x] - 18*a^2*A*b^7*Sin[c + d*x] - 6*A*b^9*Sin[c + d*x] + 18*a^5*b^4*B*Sin[c + d*x] + 39*a^3*b^6*B*Sin[c + d*x] + 18*a*b^8*B*Sin[c + d*x] - 24*a^8*b*C*Sin[c + d*x] + 57*a^6*b^3*C*Sin[c + d*x] - 72*a^4*b^5*C*Sin[c + d*x] - 36*a^2*b^7*C*Sin[c + d*x] + 6*a^5*A*b^4*Sin[2*(c + d*x)] - 54*a^3*A*b^6*Sin[2*(c + d*x)] - 12*a*A*b^8*Sin[2*(c + d*x)] + 6*a^4*b^5*B*Sin[2*(c + d*x)] + 54*a^2*b^7*B*Sin[2*(c + d*x)] - 30*a^7*b^2*C*Sin[2*(c + d*x)] + 90*a^5*b^4*C*Sin[2*(c + d*x)] - 120*a^3*b^6*C*Sin[2*(c + d*x)] + a^4*A*b^5*Sin[3*(c + d*x)] - 10*a^2*A*b^7*Sin[3*(c + d*x)] - 6*A*b^9*Sin[3*(c + d*x)] + 2*a^5*b^4*B*Sin[3*(c + d*x)] - 5*a^3*b^6*B*Sin[3*(c + d*x)] + 18*a*b^8*B*Sin[3*(c + d*x)] - 11*a^6*b^3*C*Sin[3*(c + d*x)] + 32*a^4*b^5*C*Sin[3*(c + d*x)] - 36*a^2*b^7*C*Sin[3*(c + d*x)])/((a^2 - b^2)^3*(a + b*Cos[c + d*x])^3))/(24*b^4*d)","B",1
1004,1,307,314,1.725222,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^4,x]","\frac{\frac{2 \sin (c+d x) \left(12 a^5 A+10 a^5 C-25 a^4 b B+22 a^3 A b^2+17 a^3 b^2 C-14 a^2 b^3 B+6 \cos (c+d x) \left(a^5 B+a^4 b (2 A+C)-9 a^3 b^2 B+9 a^2 b^3 (A+C)-2 a b^4 B-A b^5\right)+\cos (2 (c+d x)) \left(2 a^5 C+a^4 b B+a^3 b^2 (2 A-5 C)-10 a^2 b^3 B+a b^4 (13 A+18 C)-6 b^5 B\right)+11 a A b^4+18 a b^4 C-6 b^5 B\right)}{(a+b \cos (c+d x))^3}-\frac{24 \left(a^3 B-a^2 b (4 A+3 C)+4 a b^2 B-b^3 (A+2 C)\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}}{24 d \left(a^2-b^2\right)^3}","\frac{a \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{\left(a^3 B-a^2 b (4 A+3 C)+4 a b^2 B-b^3 (A+2 C)\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac{\sin (c+d x) \left(-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\sin (c+d x) \left(2 a^5 C+a^4 b B+a^3 b^2 (2 A-5 C)-10 a^2 b^3 B+a b^4 (13 A+18 C)-6 b^5 B\right)}{6 b^2 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}",1,"((-24*(a^3*B + 4*a*b^2*B - b^3*(A + 2*C) - a^2*b*(4*A + 3*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (2*(12*a^5*A + 22*a^3*A*b^2 + 11*a*A*b^4 - 25*a^4*b*B - 14*a^2*b^3*B - 6*b^5*B + 10*a^5*C + 17*a^3*b^2*C + 18*a*b^4*C + 6*(-(A*b^5) + a^5*B - 9*a^3*b^2*B - 2*a*b^4*B + 9*a^2*b^3*(A + C) + a^4*b*(2*A + C))*Cos[c + d*x] + (a^4*b*B - 10*a^2*b^3*B - 6*b^5*B + a^3*b^2*(2*A - 5*C) + 2*a^5*C + a*b^4*(13*A + 18*C))*Cos[2*(c + d*x)])*Sin[c + d*x])/(a + b*Cos[c + d*x])^3)/(24*(a^2 - b^2)^3*d)","A",1
1005,1,301,299,1.5883578,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4,x]","\frac{\frac{2 \sin (c+d x) \left(12 a^5 B-36 a^4 A b-25 a^4 b C+22 a^3 b^2 B-a^2 A b^3-14 a^2 b^3 C+b \cos (2 (c+d x)) \left(a^4 C+2 a^3 b B-a^2 b^2 (11 A+10 C)+13 a b^3 B-2 b^4 (2 A+3 C)\right)+6 \cos (c+d x) \left(a^5 C+2 a^4 b B-9 a^3 b^2 (A+C)+9 a^2 b^3 B-a b^4 (A+2 C)-b^5 B\right)+11 a b^4 B-8 A b^5-6 b^5 C\right)}{(a+b \cos (c+d x))^3}-\frac{24 \left(a^3 (2 A+C)-4 a^2 b B+a b^2 (3 A+4 C)-b^3 B\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}}{24 d \left(a^2-b^2\right)^3}","-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}-\frac{\left(-\left(a^3 (2 A+C)\right)+4 a^2 b B-a b^2 (3 A+4 C)+b^3 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac{\sin (c+d x) \left(a^3 C+2 a^2 b B-a b^2 (5 A+6 C)+3 b^3 B\right)}{6 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\sin (c+d x) \left(a^4 C+2 a^3 b B-a^2 b^2 (11 A+10 C)+13 a b^3 B-2 b^4 (2 A+3 C)\right)}{6 b d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}",1,"((-24*(-4*a^2*b*B - b^3*B + a^3*(2*A + C) + a*b^2*(3*A + 4*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (2*(-36*a^4*A*b - a^2*A*b^3 - 8*A*b^5 + 12*a^5*B + 22*a^3*b^2*B + 11*a*b^4*B - 25*a^4*b*C - 14*a^2*b^3*C - 6*b^5*C + 6*(2*a^4*b*B + 9*a^2*b^3*B - b^5*B + a^5*C - 9*a^3*b^2*(A + C) - a*b^4*(A + 2*C))*Cos[c + d*x] + b*(2*a^3*b*B + 13*a*b^3*B + a^4*C - 2*b^4*(2*A + 3*C) - a^2*b^2*(11*A + 10*C))*Cos[2*(c + d*x)])*Sin[c + d*x])/(a + b*Cos[c + d*x])^3)/(24*(a^2 - b^2)^3*d)","A",1
1006,1,587,345,6.7953345,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^4} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^4,x]","\frac{\cos (c+d x) (A \sec (c+d x)+B+C \cos (c+d x)) \left(-\frac{6 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a^4}+\frac{6 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^4}+\frac{2 \sec (c) \left(a (a C-b B)+A b^2\right) (b \sin (d x)-a \sin (c))}{a b \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{a \tan (c) \left(3 a^3 B-a^2 b (6 A+5 C)+2 a b^2 B+A b^3\right)+\sec (c) \sin (d x) \left(2 a^4 C-5 a^3 b B+a^2 b^2 (8 A+3 C)-3 A b^4\right)}{a^2 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{6 i (\cos (c)-i \sin (c)) \left(2 a^7 B-4 a^6 b (2 A+C)+3 a^5 b^2 B+a^4 b^3 (8 A-C)-7 a^2 A b^5+2 A b^7\right) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (b \cos (c)-a)+b \sin (c)\right)}{\sqrt{-\left(\left(a^2-b^2\right) (\cos (c)-i \sin (c))^2\right)}}\right)}{a^4 \left(a^2-b^2\right)^3 \sqrt{-\left(\left(a^2-b^2\right) (\cos (c)-i \sin (c))^2\right)}}+\frac{3 a \tan (c) \left(2 a^5 B-2 a^4 b (3 A+2 C)+3 a^3 b^2 B+a^2 b^3 (2 A-C)-A b^5\right)+\sec (c) \sin (d x) \left(2 a^6 C-11 a^5 b B+13 a^4 b^2 (2 A+C)-4 a^3 b^3 B-17 a^2 A b^4+6 A b^6\right)}{\left(a^3-a b^2\right)^3 (a+b \cos (c+d x))}\right)}{3 d (2 A+2 B \cos (c+d x)+C \cos (2 (c+d x))+C)}","\frac{A \tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}-\frac{\sin (c+d x) \left(-2 a^4 C+5 a^3 b B-a^2 b^2 (8 A+3 C)+3 A b^4\right)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{\left(-2 a^7 B+4 a^6 b (2 A+C)-3 a^5 b^2 B-a^4 b^3 (8 A-C)+7 a^2 A b^5-2 A b^7\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\sin (c+d x) \left(-2 a^6 C+11 a^5 b B-13 a^4 b^2 (2 A+C)+4 a^3 b^3 B+17 a^2 A b^4-6 A b^6\right)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}",1,"(Cos[c + d*x]*(B + C*Cos[c + d*x] + A*Sec[c + d*x])*((-6*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/a^4 + (6*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/a^4 - ((6*I)*(-7*a^2*A*b^5 + 2*A*b^7 + 2*a^7*B + 3*a^5*b^2*B + a^4*b^3*(8*A - C) - 4*a^6*b*(2*A + C))*ArcTan[((I*Cos[c] + Sin[c])*(b*Sin[c] + (-a + b*Cos[c])*Tan[(d*x)/2]))/Sqrt[-((a^2 - b^2)*(Cos[c] - I*Sin[c])^2)]]*(Cos[c] - I*Sin[c]))/(a^4*(a^2 - b^2)^3*Sqrt[-((a^2 - b^2)*(Cos[c] - I*Sin[c])^2)]) + (2*(A*b^2 + a*(-(b*B) + a*C))*Sec[c]*(-(a*Sin[c]) + b*Sin[d*x]))/(a*b*(a^2 - b^2)*(a + b*Cos[c + d*x])^3) + ((-17*a^2*A*b^4 + 6*A*b^6 - 11*a^5*b*B - 4*a^3*b^3*B + 2*a^6*C + 13*a^4*b^2*(2*A + C))*Sec[c]*Sin[d*x] + 3*a*(-(A*b^5) + 2*a^5*B + 3*a^3*b^2*B + a^2*b^3*(2*A - C) - 2*a^4*b*(3*A + 2*C))*Tan[c])/((a^3 - a*b^2)^3*(a + b*Cos[c + d*x])) + ((-3*A*b^4 - 5*a^3*b*B + 2*a^4*C + a^2*b^2*(8*A + 3*C))*Sec[c]*Sin[d*x] + a*(A*b^3 + 3*a^3*B + 2*a*b^2*B - a^2*b*(6*A + 5*C))*Tan[c])/(a^2*(a^2 - b^2)^2*(a + b*Cos[c + d*x])^2)))/(3*d*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*(c + d*x)]))","C",1
1007,1,709,480,4.9906031,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^4,x]","\frac{\cos (c+d x) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(\frac{48 \cos (c+d x) \left(2 a^8 C-8 a^7 b B+a^6 b^2 (20 A+3 C)+8 a^5 b^3 B-35 a^4 A b^4-7 a^3 b^5 B+28 a^2 A b^6+2 a b^7 B-8 A b^8\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{7/2}}+\frac{2 a \sin (c+d x) \left(24 a^9 A-36 a^7 A b^2-54 a^7 b^2 C+6 a^6 A b^3 \cos (3 (c+d x))+120 a^6 b^3 B-11 a^6 b^3 C \cos (3 (c+d x))-246 a^5 A b^4+26 a^5 b^4 B \cos (3 (c+d x))-6 a^5 b^4 C-65 a^4 A b^5 \cos (3 (c+d x))-90 a^4 b^5 B-4 a^4 b^5 C \cos (3 (c+d x))+318 a^3 A b^6-17 a^3 b^6 B \cos (3 (c+d x))+68 a^2 A b^7 \cos (3 (c+d x))+30 a^2 b^7 B+6 a b^2 \cos (2 (c+d x)) \left(a^6 (6 A-9 C)+20 a^5 b B-a^4 b^2 (53 A+C)-15 a^3 b^3 B+57 a^2 A b^4+5 a b^5 B-20 A b^6\right)-b \cos (c+d x) \left(-72 a^8 (A-C)-144 a^7 b B+a^6 b^2 (438 A+13 C)+50 a^5 b^3 B-5 a^4 b^4 (61 A-4 C)+7 a^3 b^5 B-28 a^2 A b^6-18 a b^7 B+72 A b^8\right)-120 a A b^8+6 a b^8 B \cos (3 (c+d x))-24 A b^9 \cos (3 (c+d x))\right)}{\left(a^2-b^2\right)^3 (a+b \cos (c+d x))^3}+48 (4 A b-a B) \cos (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+48 (a B-4 A b) \cos (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{24 a^5 d (2 A+2 B \cos (c+d x)+C \cos (2 (c+d x))+C)}","-\frac{(4 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^5 d}+\frac{\tan (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}-\frac{\tan (c+d x) \left(-3 a^4 C+6 a^3 b B-a^2 b^2 (9 A+2 C)-a b^3 B+4 A b^4\right)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\tan (c+d x) \left(a^6 (6 A-11 C)+26 a^5 b B-a^4 b^2 (65 A+4 C)-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right)}{6 a^4 d \left(a^2-b^2\right)^3}-\frac{\tan (c+d x) \left(-2 a^6 C+6 a^5 b B-3 a^4 b^2 (4 A+C)-2 a^3 b^3 B+11 a^2 A b^4+a b^5 B-4 A b^6\right)}{2 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{\left(-2 a^8 C+8 a^7 b B-a^6 b^2 (20 A+3 C)-8 a^5 b^3 B+35 a^4 A b^4+7 a^3 b^5 B-28 a^2 A b^6-2 a b^7 B+8 A b^8\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{7/2} (a+b)^{7/2}}",1,"(Cos[c + d*x]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*((48*(-35*a^4*A*b^4 + 28*a^2*A*b^6 - 8*A*b^8 - 8*a^7*b*B + 8*a^5*b^3*B - 7*a^3*b^5*B + 2*a*b^7*B + 2*a^8*C + a^6*b^2*(20*A + 3*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]]*Cos[c + d*x])/(-a^2 + b^2)^(7/2) + 48*(4*A*b - a*B)*Cos[c + d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 48*(-4*A*b + a*B)*Cos[c + d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*a*(24*a^9*A - 36*a^7*A*b^2 - 246*a^5*A*b^4 + 318*a^3*A*b^6 - 120*a*A*b^8 + 120*a^6*b^3*B - 90*a^4*b^5*B + 30*a^2*b^7*B - 54*a^7*b^2*C - 6*a^5*b^4*C - b*(-28*a^2*A*b^6 + 72*A*b^8 - 144*a^7*b*B + 50*a^5*b^3*B + 7*a^3*b^5*B - 18*a*b^7*B - 5*a^4*b^4*(61*A - 4*C) - 72*a^8*(A - C) + a^6*b^2*(438*A + 13*C))*Cos[c + d*x] + 6*a*b^2*(57*a^2*A*b^4 - 20*A*b^6 + 20*a^5*b*B - 15*a^3*b^3*B + 5*a*b^5*B + a^6*(6*A - 9*C) - a^4*b^2*(53*A + C))*Cos[2*(c + d*x)] + 6*a^6*A*b^3*Cos[3*(c + d*x)] - 65*a^4*A*b^5*Cos[3*(c + d*x)] + 68*a^2*A*b^7*Cos[3*(c + d*x)] - 24*A*b^9*Cos[3*(c + d*x)] + 26*a^5*b^4*B*Cos[3*(c + d*x)] - 17*a^3*b^6*B*Cos[3*(c + d*x)] + 6*a*b^8*B*Cos[3*(c + d*x)] - 11*a^6*b^3*C*Cos[3*(c + d*x)] - 4*a^4*b^5*C*Cos[3*(c + d*x)])*Sin[c + d*x])/((a^2 - b^2)^3*(a + b*Cos[c + d*x])^3)))/(24*a^5*d*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*(c + d*x)]))","A",0
1008,1,686,657,6.5035734,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^4,x]","\frac{\sec (c+d x) (a B \sin (c+d x)-4 A b \sin (c+d x))}{a^5 d}+\frac{A \tan (c+d x) \sec (c+d x)}{2 a^4 d}+\frac{\left(a^2 (-A)-2 a^2 C+8 a b B-20 A b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^6 d}+\frac{\left(a^2 A+2 a^2 C-8 a b B+20 A b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^6 d}+\frac{a^2 b^2 C \sin (c+d x)-a b^3 B \sin (c+d x)+A b^4 \sin (c+d x)}{3 a^3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{8 a^4 b^2 C \sin (c+d x)-11 a^3 b^3 B \sin (c+d x)+14 a^2 A b^4 \sin (c+d x)-3 a^2 b^4 C \sin (c+d x)+6 a b^5 B \sin (c+d x)-9 A b^6 \sin (c+d x)}{6 a^4 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{26 a^6 b^2 C \sin (c+d x)-47 a^5 b^3 B \sin (c+d x)+74 a^4 A b^4 \sin (c+d x)-17 a^4 b^4 C \sin (c+d x)+50 a^3 b^5 B \sin (c+d x)-95 a^2 A b^6 \sin (c+d x)+6 a^2 b^6 C \sin (c+d x)-18 a b^7 B \sin (c+d x)+36 A b^8 \sin (c+d x)}{6 a^5 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{b \left(8 a^8 C-20 a^7 b B+40 a^6 A b^2-8 a^6 b^2 C+35 a^5 b^3 B-84 a^4 A b^4+7 a^4 b^4 C-28 a^3 b^5 B+69 a^2 A b^6-2 a^2 b^6 C+8 a b^7 B-20 A b^8\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{a^6 d \left(a^2-b^2\right)^3 \sqrt{b^2-a^2}}","\frac{\tan (c+d x) \sec (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{\left(a^2 (A+2 C)-8 a b B+20 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}-\frac{\tan (c+d x) \sec (c+d x) \left(-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{\tan (c+d x) \sec (c+d x) \left(-\left(a^6 (A-6 C)\right)-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right)}{2 a^4 d \left(a^2-b^2\right)^3}+\frac{\tan (c+d x) \sec (c+d x) \left(12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{\tan (c+d x) \left(6 a^7 B-a^6 (24 A b-26 b C)-65 a^5 b^2 B+a^4 b^3 (146 A-17 C)+68 a^3 b^4 B-a^2 b^5 (167 A-6 C)-24 a b^6 B+60 A b^7\right)}{6 a^5 d \left(a^2-b^2\right)^3}+\frac{b \left(-8 a^8 C+20 a^7 b B-8 a^6 b^2 (5 A-C)-35 a^5 b^3 B+7 a^4 b^4 (12 A-C)+28 a^3 b^5 B-a^2 b^6 (69 A-2 C)-8 a b^7 B+20 A b^8\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^6 d \sqrt{a-b} \sqrt{a+b} \left(a^2-b^2\right)^3}",1,"(b*(40*a^6*A*b^2 - 84*a^4*A*b^4 + 69*a^2*A*b^6 - 20*A*b^8 - 20*a^7*b*B + 35*a^5*b^3*B - 28*a^3*b^5*B + 8*a*b^7*B + 8*a^8*C - 8*a^6*b^2*C + 7*a^4*b^4*C - 2*a^2*b^6*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(a^6*(a^2 - b^2)^3*Sqrt[-a^2 + b^2]*d) + ((-(a^2*A) - 20*A*b^2 + 8*a*b*B - 2*a^2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(2*a^6*d) + ((a^2*A + 20*A*b^2 - 8*a*b*B + 2*a^2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(2*a^6*d) + (Sec[c + d*x]*(-4*A*b*Sin[c + d*x] + a*B*Sin[c + d*x]))/(a^5*d) + (A*b^4*Sin[c + d*x] - a*b^3*B*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x])/(3*a^3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (14*a^2*A*b^4*Sin[c + d*x] - 9*A*b^6*Sin[c + d*x] - 11*a^3*b^3*B*Sin[c + d*x] + 6*a*b^5*B*Sin[c + d*x] + 8*a^4*b^2*C*Sin[c + d*x] - 3*a^2*b^4*C*Sin[c + d*x])/(6*a^4*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (74*a^4*A*b^4*Sin[c + d*x] - 95*a^2*A*b^6*Sin[c + d*x] + 36*A*b^8*Sin[c + d*x] - 47*a^5*b^3*B*Sin[c + d*x] + 50*a^3*b^5*B*Sin[c + d*x] - 18*a*b^7*B*Sin[c + d*x] + 26*a^6*b^2*C*Sin[c + d*x] - 17*a^4*b^4*C*Sin[c + d*x] + 6*a^2*b^6*C*Sin[c + d*x])/(6*a^5*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x])) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d)","A",1
1009,1,34,23,0.0319667,"\int \frac{a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x]),x]","-a C x+b B x+\frac{b C \sin (c) \cos (d x)}{d}+\frac{b C \cos (c) \sin (d x)}{d}","x (b B-a C)+\frac{b C \sin (c+d x)}{d}",1,"b*B*x - a*C*x + (b*C*Cos[d*x]*Sin[c])/d + (b*C*Cos[c]*Sin[d*x])/d","A",1
1010,1,68,61,0.1301292,"\int \frac{a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^2,x]","\frac{2 (2 a C-b B) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{d \sqrt{b^2-a^2}}+\frac{C (c+d x)}{d}","\frac{2 (b B-2 a C) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d \sqrt{a-b} \sqrt{a+b}}+C x",1,"(C*(c + d*x))/d + (2*(-(b*B) + 2*a*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(Sqrt[-a^2 + b^2]*d)","A",1
1011,1,107,110,0.3781266,"\int \frac{a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{b (2 a C-b B) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}-\frac{2 \left(a^2 C-a b B+b^2 C\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}}{d}","\frac{2 \left(a^2 (-C)+a b B-b^2 C\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{3/2} (a+b)^{3/2}}-\frac{b (b B-2 a C) \sin (c+d x)}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"((-2*(-(a*b*B) + a^2*C + b^2*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + (b*(-(b*B) + 2*a*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])))/d","A",1
1012,1,171,175,0.7716108,"\int \frac{a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^4,x]","\frac{\frac{b \left(4 a^2 C-3 a b B+2 b^2 C\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}+\frac{2 \left(2 a^3 C-2 a^2 b B+4 a b^2 C-b^3 B\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}+\frac{b (2 a C-b B) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}}{2 d}","-\frac{b \left(-4 a^2 C+3 a b B-2 b^2 C\right) \sin (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{b (b B-2 a C) \sin (c+d x)}{2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\left(-2 a^3 C+2 a^2 b B-4 a b^2 C+b^3 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}",1,"((2*(-2*a^2*b*B - b^3*B + 2*a^3*C + 4*a*b^2*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (b*(-(b*B) + 2*a*C)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^2) + (b*(-3*a*b*B + 4*a^2*C + 2*b^2*C)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])))/(2*d)","A",1
1013,1,246,249,1.0780028,"\int \frac{a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)}{(a+b \cos (c+d x))^5} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^5,x]","\frac{\frac{24 \left(2 a^4 C-2 a^3 b B+7 a^2 b^2 C-3 a b^3 B+b^4 C\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}-\frac{2 b \sin (c+d x) \left(-48 a^5 C+36 a^4 b B-23 a^3 b^2 C+a^2 b^3 B+b^2 \left(-13 a^3 C+11 a^2 b B-17 a b^2 C+4 b^3 B\right) \cos (2 (c+d x))+6 b \left(-11 a^4 C+9 a^3 b B-10 a^2 b^2 C+a b^3 B+b^4 C\right) \cos (c+d x)-19 a b^4 C+8 b^5 B\right)}{(a+b \cos (c+d x))^3}}{24 d \left(a^2-b^2\right)^3}","-\frac{b \left(-7 a^2 C+5 a b B-3 b^2 C\right) \sin (c+d x)}{6 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{b (b B-2 a C) \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}-\frac{b \left(-13 a^3 C+11 a^2 b B-17 a b^2 C+4 b^3 B\right) \sin (c+d x)}{6 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{\left(-2 a^4 C+2 a^3 b B-7 a^2 b^2 C+3 a b^3 B-b^4 C\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}",1,"((24*(-2*a^3*b*B - 3*a*b^3*B + 2*a^4*C + 7*a^2*b^2*C + b^4*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] - (2*b*(36*a^4*b*B + a^2*b^3*B + 8*b^5*B - 48*a^5*C - 23*a^3*b^2*C - 19*a*b^4*C + 6*b*(9*a^3*b*B + a*b^3*B - 11*a^4*C - 10*a^2*b^2*C + b^4*C)*Cos[c + d*x] + b^2*(11*a^2*b*B + 4*b^3*B - 13*a^3*C - 17*a*b^2*C)*Cos[2*(c + d*x)])*Sin[c + d*x])/(a + b*Cos[c + d*x])^3)/(24*(a^2 - b^2)^3*d)","A",1
1014,1,321,416,1.7055949,"\int \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{b (a+b \cos (c+d x)) \left(b \left(\sin (2 (c+d x)) \left(-24 a^2 C+36 a b B+252 A b^2+266 b^2 C\right)+5 b (2 (a C+9 b B) \sin (3 (c+d x))+7 b C \sin (4 (c+d x)))\right)+2 \sin (c+d x) \left(32 a^3 C-48 a^2 b B+3 a b^2 (28 A+19 C)+345 b^3 B\right)\right)+8 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b^2 \left(-4 a^3 C+6 a^2 b B+3 a b^2 (49 A+37 C)+75 b^3 B\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-\left(16 a^4 C-24 a^3 b B+6 a^2 b^2 (7 A+4 C)-57 a b^3 B-21 b^4 (9 A+7 C)\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{1260 b^4 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \sin (c+d x) \left(24 a^2 C-36 a b B+63 A b^2+49 b^2 C\right) (a+b \cos (c+d x))^{3/2}}{315 b^3 d}+\frac{2 \sin (c+d x) \left(-16 a^3 C+24 a^2 b B-6 a b^2 (7 A+6 C)+75 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{315 b^3 d}-\frac{2 \left(a^2-b^2\right) \left(-16 a^3 C+24 a^2 b B-6 a b^2 (7 A+6 C)+75 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-16 a^4 C+24 a^3 b B-6 a^2 b^2 (7 A+4 C)+57 a b^3 B+21 b^4 (9 A+7 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (3 b B-2 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{21 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{9 b d}",1,"(8*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(6*a^2*b*B + 75*b^3*B - 4*a^3*C + 3*a*b^2*(49*A + 37*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - (-24*a^3*b*B - 57*a*b^3*B + 16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x])*(2*(-48*a^2*b*B + 345*b^3*B + 32*a^3*C + 3*a*b^2*(28*A + 19*C))*Sin[c + d*x] + b*((252*A*b^2 + 36*a*b*B - 24*a^2*C + 266*b^2*C)*Sin[2*(c + d*x)] + 5*b*(2*(9*b*B + a*C)*Sin[3*(c + d*x)] + 7*b*C*Sin[4*(c + d*x)]))))/(1260*b^4*d*Sqrt[a + b*Cos[c + d*x]])","A",1
1015,1,249,321,1.2644631,"\int \cos (c+d x) \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{b (a+b \cos (c+d x)) \left(\sin (c+d x) \left(-16 a^2 C+28 a b B+140 A b^2+115 b^2 C\right)+3 b (2 (a C+7 b B) \sin (2 (c+d x))+5 b C \sin (3 (c+d x)))\right)+4 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b^2 \left(2 a^2 C+49 a b B+35 A b^2+25 b^2 C\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(8 a^3 C-14 a^2 b B+a b^2 (35 A+19 C)+63 b^3 B\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{210 b^3 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \sin (c+d x) \left(8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{105 b^2 d}-\frac{2 \left(a^2-b^2\right) \left(8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-8 a^3 C+14 a^2 b B-a b^2 (35 A+19 C)-63 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (7 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}",1,"(4*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(35*A*b^2 + 49*a*b*B + 2*a^2*C + 25*b^2*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (-14*a^2*b*B + 63*b^3*B + 8*a^3*C + a*b^2*(35*A + 19*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x])*((140*A*b^2 + 28*a*b*B - 16*a^2*C + 115*b^2*C)*Sin[c + d*x] + 3*b*(2*(7*b*B + a*C)*Sin[2*(c + d*x)] + 5*b*C*Sin[3*(c + d*x)])))/(210*b^3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
1016,1,189,237,0.8232514,"\int \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(\left(-2 a^2 C+5 a b B+15 A b^2+9 b^2 C\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)+b^2 (15 a A+7 a C+5 b B) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)+2 b \sin (c+d x) (a+b \cos (c+d x)) (a C+5 b B+3 b C \cos (c+d x))}{15 b^2 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(a^2-b^2\right) (5 b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a (5 b B-2 a C)+3 b^2 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (5 b B-2 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}",1,"(2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(15*a*A + 5*b*B + 7*a*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (15*A*b^2 + 5*a*b*B - 2*a^2*C + 9*b^2*C)*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + 2*b*(a + b*Cos[c + d*x])*(5*b*B + a*C + 3*b*C*Cos[c + d*x])*Sin[c + d*x])/(15*b^2*d*Sqrt[a + b*Cos[c + d*x]])","A",1
1017,1,393,240,3.5496331,"\int \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{\frac{4 (3 a B+3 A b+b C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 (a (6 A+C)+3 b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 i (a C+3 b B) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b^2 \sqrt{-\frac{1}{a+b}}}+4 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{6 d}","\frac{2 \left(3 A b^2-C \left(a^2-b^2\right)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 a A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 (a C+3 b B) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}",1,"((4*(3*A*b + 3*a*B + b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(3*b*B + a*(6*A + C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + ((2*I)*(3*b*B + a*C)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b^2*Sqrt[-(a + b)^(-1)]) + 4*C*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(6*d)","C",1
1018,1,385,217,2.6461075,"\int \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{\frac{2 (4 a B+A b+2 b C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i (A-2 C) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}+4 A \tan (c+d x) \sqrt{a+b \cos (c+d x)}+\frac{8 (a C+b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{4 d}","\frac{(a A+2 b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{(2 a B+A b) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{(A-2 C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{A \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{d}",1,"((8*(b*B + a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(A*b + 4*a*B + 2*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(A - 2*C)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b*Sqrt[-(a + b)^(-1)]) + 4*A*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*d)","C",1
1019,1,428,299,5.4262485,"\int \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{\frac{2 \left(8 a^2 (A+2 C)+4 a b B-3 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a \sqrt{a+b \cos (c+d x)}}-\frac{2 i (4 a B+A b) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a^2 b \sqrt{-\frac{1}{a+b}}}+\frac{4 \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)} ((4 a B+A b) \cos (c+d x)+2 a A)}{a}+\frac{8 b (A+4 C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{16 d}","-\frac{\left(-4 a^2 (A+2 C)-4 a b B+A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{a+b \cos (c+d x)}}+\frac{(4 a B+3 A b+8 b C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{(4 a B+A b) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a d}-\frac{(4 a B+A b) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{A \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d}",1,"((8*b*(A + 4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(-3*A*b^2 + 4*a*b*B + 8*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(a*Sqrt[a + b*Cos[c + d*x]]) - ((2*I)*(A*b + 4*a*B)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a^2*b*Sqrt[-(a + b)^(-1)]) + (4*Sqrt[a + b*Cos[c + d*x]]*(2*a*A + (A*b + 4*a*B)*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/a)/(16*d)","C",1
1020,1,661,399,6.6880025,"\int \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\sqrt{a+b \cos (c+d x)} \left(\frac{\sec (c+d x) \left(16 a^2 A \sin (c+d x)+24 a^2 C \sin (c+d x)+6 a b B \sin (c+d x)-3 A b^2 \sin (c+d x)\right)}{24 a^2}+\frac{\sec ^2(c+d x) (6 a B \sin (c+d x)+A b \sin (c+d x))}{12 a}+\frac{1}{3} A \tan (c+d x) \sec ^2(c+d x)\right)}{d}+\frac{\frac{2 \left(24 a^2 b B+4 a A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \sin (c+d x) \cos (2 (c+d x)) \left(-16 a^2 A b-24 a^2 b C-6 a b^2 B+3 A b^3\right) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{b \cos (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left(2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right)}+\frac{2 \left(48 a^3 B+8 a^2 A b+24 a^2 b C-18 a b^2 B+9 A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{96 a^2 d}","-\frac{\tan (c+d x) \left(-8 a^2 (2 A+3 C)-6 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)}}{24 a^2 d}-\frac{\left(-8 a^2 (2 A+3 C)-18 a b B+A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a d \sqrt{a+b \cos (c+d x)}}+\frac{\left(-8 a^2 (2 A+3 C)-6 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(8 a^3 B+4 a^2 b (A+2 C)-2 a b^2 B+A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{(6 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{12 a d}+\frac{A \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}",1,"((2*(4*a*A*b^2 + 24*a^2*b*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(8*a^2*A*b + 9*A*b^3 + 48*a^3*B - 18*a*b^2*B + 24*a^2*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(-16*a^2*A*b + 3*A*b^3 - 6*a*b^2*B - 24*a^2*b*C)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2)))/(96*a^2*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]^2*(A*b*Sin[c + d*x] + 6*a*B*Sin[c + d*x]))/(12*a) + (Sec[c + d*x]*(16*a^2*A*Sin[c + d*x] - 3*A*b^2*Sin[c + d*x] + 6*a*b*B*Sin[c + d*x] + 24*a^2*C*Sin[c + d*x]))/(24*a^2) + (A*Sec[c + d*x]^2*Tan[c + d*x])/3))/d","C",1
1021,1,407,518,2.7246756,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{b (a+b \cos (c+d x)) \left(b \left(5 b \left(\sin (3 (c+d x)) \left(12 a^2 C+440 a b B+396 A b^2+513 b^2 C\right)+7 b ((24 a C+22 b B) \sin (4 (c+d x))+9 b C \sin (5 (c+d x)))\right)+4 \sin (2 (c+d x)) \left(-36 a^3 C+66 a^2 b B+48 a b^2 (33 A+34 C)+1463 b^3 B\right)\right)+2 \sin (c+d x) \left(192 a^4 C-352 a^3 b B+18 a^2 b^2 (44 A+27 C)+8844 a b^3 B+15 b^4 (506 A+435 C)\right)\right)+16 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b^2 \left(-12 a^4 C+22 a^3 b B+9 a^2 b^2 (187 A+141 C)+2046 a b^3 B+75 b^4 (11 A+9 C)\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-\left(48 a^5 C-88 a^4 b B+18 a^3 b^2 (11 A+6 C)-363 a^2 b^3 B-6 a b^4 (451 A+348 C)-1617 b^5 B\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{27720 b^4 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \sin (c+d x) \left(24 a^2 C-44 a b B+99 A b^2+81 b^2 C\right) (a+b \cos (c+d x))^{5/2}}{693 b^3 d}+\frac{2 \sin (c+d x) \left(-48 a^3 C+88 a^2 b B-6 a b^2 (33 A+34 C)+539 b^3 B\right) (a+b \cos (c+d x))^{3/2}}{3465 b^3 d}+\frac{2 \sin (c+d x) \left(-48 a^4 C+88 a^3 b B-18 a^2 b^2 (11 A+8 C)+429 a b^3 B+75 b^4 (11 A+9 C)\right) \sqrt{a+b \cos (c+d x)}}{3465 b^3 d}-\frac{2 \left(a^2-b^2\right) \left(-48 a^4 C+88 a^3 b B-18 a^2 b^2 (11 A+8 C)+429 a b^3 B+75 b^4 (11 A+9 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3465 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-48 a^5 C+88 a^4 b B-18 a^3 b^2 (11 A+6 C)+363 a^2 b^3 B+6 a b^4 (451 A+348 C)+1617 b^5 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3465 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (11 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{99 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{11 b d}",1,"(16*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(22*a^3*b*B + 2046*a*b^3*B - 12*a^4*C + 75*b^4*(11*A + 9*C) + 9*a^2*b^2*(187*A + 141*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - (-88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C + 18*a^3*b^2*(11*A + 6*C) - 6*a*b^4*(451*A + 348*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x])*(2*(-352*a^3*b*B + 8844*a*b^3*B + 192*a^4*C + 18*a^2*b^2*(44*A + 27*C) + 15*b^4*(506*A + 435*C))*Sin[c + d*x] + b*(4*(66*a^2*b*B + 1463*b^3*B - 36*a^3*C + 48*a*b^2*(33*A + 34*C))*Sin[2*(c + d*x)] + 5*b*((396*A*b^2 + 440*a*b*B + 12*a^2*C + 513*b^2*C)*Sin[3*(c + d*x)] + 7*b*((22*b*B + 24*a*C)*Sin[4*(c + d*x)] + 9*b*C*Sin[5*(c + d*x)])))))/(27720*b^4*d*Sqrt[a + b*Cos[c + d*x]])","A",1
1022,1,321,408,1.7631868,"\int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{b (a+b \cos (c+d x)) \left(b \left(2 \sin (2 (c+d x)) \left(6 a^2 C+144 a b B+126 A b^2+133 b^2 C\right)+5 b (2 (10 a C+9 b B) \sin (3 (c+d x))+7 b C \sin (4 (c+d x)))\right)+\sin (c+d x) \left(-32 a^3 C+72 a^2 b B+12 a b^2 (84 A+67 C)+690 b^3 B\right)\right)+8 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b^2 \left(2 a^3 C+153 a^2 b B+6 a b^2 (42 A+31 C)+75 b^3 B\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(8 a^4 C-18 a^3 b B+3 a^2 b^2 (21 A+11 C)+246 a b^3 B+21 b^4 (9 A+7 C)\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{1260 b^3 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \sin (c+d x) \left(8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}-\frac{2 \sin (c+d x) \left(-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{315 b^2 d}+\frac{2 \left(a^2-b^2\right) \left(-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-8 a^4 C+18 a^3 b B-3 a^2 b^2 (21 A+11 C)-246 a b^3 B-21 b^4 (9 A+7 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (9 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}",1,"(8*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(153*a^2*b*B + 75*b^3*B + 2*a^3*C + 6*a*b^2*(42*A + 31*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x])*((72*a^2*b*B + 690*b^3*B - 32*a^3*C + 12*a*b^2*(84*A + 67*C))*Sin[c + d*x] + b*(2*(126*A*b^2 + 144*a*b*B + 6*a^2*C + 133*b^2*C)*Sin[2*(c + d*x)] + 5*b*(2*(9*b*B + 10*a*C)*Sin[3*(c + d*x)] + 7*b*C*Sin[4*(c + d*x)]))))/(1260*b^3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
1023,1,257,315,1.3084984,"\int (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{b (a+b \cos (c+d x)) \left(\sin (c+d x) \left(12 a^2 C+168 a b B+140 A b^2+115 b^2 C\right)+3 b (2 (8 a C+7 b B) \sin (2 (c+d x))+5 b C \sin (3 (c+d x)))\right)+4 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b^2 \left(3 a^2 (35 A+17 C)+84 a b B+5 b^2 (7 A+5 C)\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(-6 a^3 C+21 a^2 b B+2 a b^2 (70 A+41 C)+63 b^3 B\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{210 b^2 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \sin (c+d x) \left(-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{105 b d}-\frac{2 \left(a^2-b^2\right) \left(-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-6 a^3 C+21 a^2 b B+2 a b^2 (70 A+41 C)+63 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}",1,"(4*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(84*a*b*B + 5*b^2*(7*A + 5*C) + 3*a^2*(35*A + 17*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (21*a^2*b*B + 63*b^3*B - 6*a^3*C + 2*a*b^2*(70*A + 41*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x])*((140*A*b^2 + 168*a*b*B + 12*a^2*C + 115*b^2*C)*Sin[c + d*x] + 3*b*(2*(7*b*B + 8*a*C)*Sin[2*(c + d*x)] + 5*b*C*Sin[3*(c + d*x)])))/(210*b^2*d*Sqrt[a + b*Cos[c + d*x]])","A",1
1024,1,455,306,3.4686741,"\int (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{\frac{4 \left(15 a^2 B+6 a b (5 A+2 C)+5 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 \left(3 a^2 (10 A+C)+20 a b B+3 b^2 (5 A+3 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 i \csc (c+d x) \left(3 a^2 C+20 a b B+15 A b^2+9 b^2 C\right) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b^2 \sqrt{-\frac{1}{a+b}}}+4 \sin (c+d x) \sqrt{a+b \cos (c+d x)} (6 a C+5 b B+3 b C \cos (c+d x))}{30 d}","\frac{2 \left(3 a^2 C+20 a b B+15 A b^2+9 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 a^2 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(3 a^3 C+5 a^2 b B-3 a b^2 (5 A+C)-5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 (3 a C+5 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}",1,"((4*(15*a^2*B + 5*b^2*B + 6*a*b*(5*A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(20*a*b*B + 3*a^2*(10*A + C) + 3*b^2*(5*A + 3*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + ((2*I)*(15*A*b^2 + 20*a*b*B + 3*a^2*C + 9*b^2*C)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b^2*Sqrt[-(a + b)^(-1)]) + 4*Sqrt[a + b*Cos[c + d*x]]*(5*b*B + 6*a*C + 3*b*C*Cos[c + d*x])*Sin[c + d*x])/(30*d)","C",1
1025,1,434,286,4.1314751,"\int (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{\frac{8 \left(3 a^2 C+6 a b B+3 A b^2+b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 \left(12 a^2 B+a b (15 A+8 C)+6 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 i \csc (c+d x) (-3 a A+8 a C+6 b B) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}+4 \tan (c+d x) \sqrt{a+b \cos (c+d x)} (3 a A+2 b C \cos (c+d x))}{12 d}","\frac{\left(a^2 (3 A-2 C)+6 a b B+2 b^2 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}-\frac{(3 a A-8 a C-6 b B) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{a (2 a B+3 A b) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}-\frac{b (3 A-2 C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}",1,"((8*(3*A*b^2 + 6*a*b*B + 3*a^2*C + b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(12*a^2*B + 6*b^2*B + a*b*(15*A + 8*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + ((2*I)*(-3*a*A + 6*b*B + 8*a*C)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b*Sqrt[-(a + b)^(-1)]) + 4*Sqrt[a + b*Cos[c + d*x]]*(3*a*A + 2*b*C*Cos[c + d*x])*Tan[c + d*x])/(12*d)","C",1
1026,1,438,307,6.2717285,"\int (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{\frac{2 \left(8 a^2 (A+2 C)+20 a b B+b^2 (A+8 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{8 b (a (A+8 C)+4 b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \csc (c+d x) (4 a B+5 A b-8 b C) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}+4 \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)} ((4 a B+5 A b) \cos (c+d x)+2 a A)}{16 d}","\frac{\left(4 a^2 B+a b (7 A+8 C)+8 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{\left(4 a^2 (A+2 C)+12 a b B+3 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}-\frac{(4 a B+5 A b-8 b C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{(4 a B+3 A b) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}",1,"((8*b*(4*b*B + a*(A + 8*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(20*a*b*B + 8*a^2*(A + 2*C) + b^2*(A + 8*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(5*A*b + 4*a*B - 8*b*C)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b*Sqrt[-(a + b)^(-1)]) + 4*Sqrt[a + b*Cos[c + d*x]]*(2*a*A + (5*A*b + 4*a*B)*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(16*d)","C",1
1027,1,667,399,6.9121061,"\int (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\sqrt{a+b \cos (c+d x)} \left(\frac{\sec (c+d x) \left(16 a^2 A \sin (c+d x)+24 a^2 C \sin (c+d x)+30 a b B \sin (c+d x)+3 A b^2 \sin (c+d x)\right)}{24 a}+\frac{1}{12} \sec ^2(c+d x) (6 a B \sin (c+d x)+7 A b \sin (c+d x))+\frac{1}{3} a A \tan (c+d x) \sec ^2(c+d x)\right)}{d}+\frac{\frac{2 \left(24 a^2 b B+28 a A b^2+96 a b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \sin (c+d x) \cos (2 (c+d x)) \left(-16 a^2 A b-24 a^2 b C-30 a b^2 B-3 A b^3\right) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{b \cos (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left(2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right)}+\frac{2 \left(48 a^3 B+56 a^2 A b+120 a^2 b C+6 a b^2 B-9 A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{96 a d}","\frac{\tan (c+d x) \left(8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)}}{24 a d}+\frac{\left(8 a^2 (2 A+3 C)+42 a b B+b^2 (17 A+48 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{\left(-8 a^3 B-12 a^2 b (A+2 C)-6 a b^2 B+A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 a d \sqrt{a+b \cos (c+d x)}}+\frac{(2 a B+A b) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}",1,"((2*(28*a*A*b^2 + 24*a^2*b*B + 96*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(56*a^2*A*b - 9*A*b^3 + 48*a^3*B + 6*a*b^2*B + 120*a^2*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(-16*a^2*A*b - 3*A*b^3 - 30*a*b^2*B - 24*a^2*b*C)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2)))/(96*a*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]^2*(7*A*b*Sin[c + d*x] + 6*a*B*Sin[c + d*x]))/12 + (Sec[c + d*x]*(16*a^2*A*Sin[c + d*x] + 3*A*b^2*Sin[c + d*x] + 30*a*b*B*Sin[c + d*x] + 24*a^2*C*Sin[c + d*x]))/(24*a) + (a*A*Sec[c + d*x]^2*Tan[c + d*x])/3))/d","C",1
1028,1,783,503,7.0302639,"\int (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{\sqrt{a+b \cos (c+d x)} \left(\frac{\sec ^2(c+d x) \left(36 a^2 A \sin (c+d x)+48 a^2 C \sin (c+d x)+56 a b B \sin (c+d x)+3 A b^2 \sin (c+d x)\right)}{96 a}+\frac{\sec (c+d x) \left(128 a^3 B \sin (c+d x)+156 a^2 A b \sin (c+d x)+240 a^2 b C \sin (c+d x)+24 a b^2 B \sin (c+d x)-9 A b^3 \sin (c+d x)\right)}{192 a^2}+\frac{1}{24} \sec ^3(c+d x) (8 a B \sin (c+d x)+9 A b \sin (c+d x))+\frac{1}{4} a A \tan (c+d x) \sec ^3(c+d x)\right)}{d}+\frac{\frac{2 \left(144 a^3 A b+192 a^3 b C+224 a^2 b^2 B+12 a A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \sin (c+d x) \cos (2 (c+d x)) \left(-128 a^3 b B-156 a^2 A b^2-240 a^2 b^2 C-24 a b^3 B+9 A b^4\right) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{b \cos (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left(2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right)}+\frac{2 \left(288 a^4 A+384 a^4 C+448 a^3 b B-12 a^2 A b^2+48 a^2 b^2 C-72 a b^3 B+27 A b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{768 a^2 d}","\frac{\tan (c+d x) \sec (c+d x) \left(12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)}}{96 a d}-\frac{\tan (c+d x) \left(-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right) \sqrt{a+b \cos (c+d x)}}{192 a^2 d}-\frac{\left(-128 a^3 B-12 a^2 b (19 A+28 C)-136 a b^2 B+3 A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{192 a d \sqrt{a+b \cos (c+d x)}}+\frac{\left(-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{192 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(16 a^4 (3 A+4 C)+96 a^3 b B+24 a^2 b^2 (A+2 C)-8 a b^3 B+3 A b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{64 a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{(8 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}",1,"((2*(144*a^3*A*b + 12*a*A*b^3 + 224*a^2*b^2*B + 192*a^3*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(288*a^4*A - 12*a^2*A*b^2 + 27*A*b^4 + 448*a^3*b*B - 72*a*b^3*B + 384*a^4*C + 48*a^2*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(-156*a^2*A*b^2 + 9*A*b^4 - 128*a^3*b*B - 24*a*b^3*B - 240*a^2*b^2*C)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2)))/(768*a^2*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]^3*(9*A*b*Sin[c + d*x] + 8*a*B*Sin[c + d*x]))/24 + (Sec[c + d*x]^2*(36*a^2*A*Sin[c + d*x] + 3*A*b^2*Sin[c + d*x] + 56*a*b*B*Sin[c + d*x] + 48*a^2*C*Sin[c + d*x]))/(96*a) + (Sec[c + d*x]*(156*a^2*A*b*Sin[c + d*x] - 9*A*b^3*Sin[c + d*x] + 128*a^3*B*Sin[c + d*x] + 24*a*b^2*B*Sin[c + d*x] + 240*a^2*b*C*Sin[c + d*x]))/(192*a^2) + (a*A*Sec[c + d*x]^3*Tan[c + d*x])/4))/d","C",0
1029,1,501,629,4.0399459,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{b (a+b \cos (c+d x)) \left(b \left(5 b \left(7 b \left(4 \sin (4 (c+d x)) \left(159 a^2 C+299 a b B+143 A b^2+220 b^2 C\right)+9 b ((54 a C+26 b B) \sin (5 (c+d x))+11 b C \sin (6 (c+d x)))\right)+2 \sin (3 (c+d x)) \left(60 a^3 C+5876 a^2 b B+a b^2 (10868 A+13939 C)+6669 b^3 B\right)\right)+\sin (2 (c+d x)) \left(-1440 a^4 C+3120 a^3 b B+120 a^2 b^2 (1430 A+1457 C)+321880 a b^3 B+77 b^4 (1976 A+1897 C)\right)\right)+4 \sin (c+d x) \left(960 a^5 C-2080 a^4 b B+10 a^3 b^2 (572 A+331 C)+121290 a^2 b^3 B+3 a b^4 (71214 A+60793 C)+84825 b^5 B\right)\right)+32 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b^2 \left(-60 a^5 C+130 a^4 b B+5 a^3 b^2 (4433 A+3337 C)+43095 a^2 b^3 B+3 a b^4 (12441 A+10277 C)+8775 b^5 B\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-\left(240 a^6 C-520 a^5 b B+10 a^4 b^2 (143 A+76 C)-3315 a^3 b^3 B-3 a^2 b^4 (13299 A+10223 C)-48165 a b^5 B-1617 b^6 (13 A+11 C)\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{720720 b^4 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \sin (c+d x) \left(24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right) (a+b \cos (c+d x))^{7/2}}{1287 b^3 d}+\frac{2 \sin (c+d x) \left(-48 a^3 C+104 a^2 b B-2 a b^2 (143 A+166 C)+1053 b^3 B\right) (a+b \cos (c+d x))^{5/2}}{9009 b^3 d}+\frac{2 \sin (c+d x) \left(-240 a^4 C+520 a^3 b B-10 a^2 b^2 (143 A+124 C)+4355 a b^3 B+539 b^4 (13 A+11 C)\right) (a+b \cos (c+d x))^{3/2}}{45045 b^3 d}+\frac{2 \sin (c+d x) \left(-240 a^5 C+520 a^4 b B-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right) \sqrt{a+b \cos (c+d x)}}{45045 b^3 d}-\frac{2 \left(a^2-b^2\right) \left(-240 a^5 C+520 a^4 b B-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{45045 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-240 a^6 C+520 a^5 b B-10 a^4 b^2 (143 A+76 C)+3315 a^3 b^3 B+3 a^2 b^4 (13299 A+10223 C)+48165 a b^5 B+1617 b^6 (13 A+11 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{45045 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{143 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}",1,"(32*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(130*a^4*b*B + 43095*a^2*b^3*B + 8775*b^5*B - 60*a^5*C + 5*a^3*b^2*(4433*A + 3337*C) + 3*a*b^4*(12441*A + 10277*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - (-520*a^5*b*B - 3315*a^3*b^3*B - 48165*a*b^5*B + 240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x])*(4*(-2080*a^4*b*B + 121290*a^2*b^3*B + 84825*b^5*B + 960*a^5*C + 10*a^3*b^2*(572*A + 331*C) + 3*a*b^4*(71214*A + 60793*C))*Sin[c + d*x] + b*((3120*a^3*b*B + 321880*a*b^3*B - 1440*a^4*C + 120*a^2*b^2*(1430*A + 1457*C) + 77*b^4*(1976*A + 1897*C))*Sin[2*(c + d*x)] + 5*b*(2*(5876*a^2*b*B + 6669*b^3*B + 60*a^3*C + a*b^2*(10868*A + 13939*C))*Sin[3*(c + d*x)] + 7*b*(4*(143*A*b^2 + 299*a*b*B + 159*a^2*C + 220*b^2*C)*Sin[4*(c + d*x)] + 9*b*((26*b*B + 54*a*C)*Sin[5*(c + d*x)] + 11*b*C*Sin[6*(c + d*x)]))))))/(720720*b^4*d*Sqrt[a + b*Cos[c + d*x]])","A",1
1030,1,405,510,2.721854,"\int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{b (a+b \cos (c+d x)) \left(b \left(5 b \left(\sin (3 (c+d x)) \left(452 a^2 C+836 a b B+396 A b^2+513 b^2 C\right)+7 b ((46 a C+22 b B) \sin (4 (c+d x))+9 b C \sin (5 (c+d x)))\right)+4 \sin (2 (c+d x)) \left(30 a^3 C+1650 a^2 b B+5 a b^2 (594 A+619 C)+1463 b^3 B\right)\right)+\sin (c+d x) \left(-320 a^4 C+880 a^3 b B+60 a^2 b^2 (396 A+311 C)+32868 a b^3 B+30 b^4 (506 A+435 C)\right)\right)+16 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b^2 \left(10 a^4 C+1705 a^3 b B+15 a^2 b^2 (297 A+221 C)+2871 a b^3 B+75 b^4 (11 A+9 C)\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(40 a^5 C-110 a^4 b B+15 a^3 b^2 (33 A+17 C)+3069 a^2 b^3 B+15 a b^4 (319 A+247 C)+1617 b^5 B\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{27720 b^3 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \sin (c+d x) \left(8 a^2 C-22 a b B+99 A b^2+81 b^2 C\right) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}-\frac{2 \sin (c+d x) \left(-40 a^3 C+110 a^2 b B-5 a b^2 (99 A+67 C)-539 b^3 B\right) (a+b \cos (c+d x))^{3/2}}{3465 b^2 d}-\frac{2 \sin (c+d x) \left(-40 a^4 C+110 a^3 b B-15 a^2 b^2 (33 A+19 C)-1254 a b^3 B-75 b^4 (11 A+9 C)\right) \sqrt{a+b \cos (c+d x)}}{3465 b^2 d}+\frac{2 \left(a^2-b^2\right) \left(-40 a^4 C+110 a^3 b B-15 a^2 b^2 (33 A+19 C)-1254 a b^3 B-75 b^4 (11 A+9 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3465 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-40 a^5 C+110 a^4 b B-15 a^3 b^2 (33 A+17 C)-3069 a^2 b^3 B-15 a b^4 (319 A+247 C)-1617 b^5 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3465 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (11 b B-4 a C) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}",1,"(16*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(1705*a^3*b*B + 2871*a*b^3*B + 10*a^4*C + 75*b^4*(11*A + 9*C) + 15*a^2*b^2*(297*A + 221*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (-110*a^4*b*B + 3069*a^2*b^3*B + 1617*b^5*B + 40*a^5*C + 15*a^3*b^2*(33*A + 17*C) + 15*a*b^4*(319*A + 247*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x])*((880*a^3*b*B + 32868*a*b^3*B - 320*a^4*C + 60*a^2*b^2*(396*A + 311*C) + 30*b^4*(506*A + 435*C))*Sin[c + d*x] + b*(4*(1650*a^2*b*B + 1463*b^3*B + 30*a^3*C + 5*a*b^2*(594*A + 619*C))*Sin[2*(c + d*x)] + 5*b*((396*A*b^2 + 836*a*b*B + 452*a^2*C + 513*b^2*C)*Sin[3*(c + d*x)] + 7*b*((22*b*B + 46*a*C)*Sin[4*(c + d*x)] + 9*b*C*Sin[5*(c + d*x)])))))/(27720*b^3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
1031,1,327,402,1.8787214,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{b (a+b \cos (c+d x)) \left(b \left(\sin (2 (c+d x)) \left(300 a^2 C+540 a b B+252 A b^2+266 b^2 C\right)+5 b (2 (19 a C+9 b B) \sin (3 (c+d x))+7 b C \sin (4 (c+d x)))\right)+2 \sin (c+d x) \left(20 a^3 C+540 a^2 b B+3 a b^2 (308 A+249 C)+345 b^3 B\right)\right)+8 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b^2 \left(5 a^3 (63 A+31 C)+405 a^2 b B+3 a b^2 (119 A+87 C)+75 b^3 B\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(-10 a^4 C+45 a^3 b B+3 a^2 b^2 (161 A+93 C)+435 a b^3 B+21 b^4 (9 A+7 C)\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{1260 b^2 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \sin (c+d x) \left(-10 a^2 C+45 a b B+63 A b^2+49 b^2 C\right) (a+b \cos (c+d x))^{3/2}}{315 b d}+\frac{2 \sin (c+d x) \left(-10 a^3 C+45 a^2 b B+6 a b^2 (28 A+19 C)+75 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{315 b d}-\frac{2 \left(a^2-b^2\right) \left(-10 a^3 C+45 a^2 b B+6 a b^2 (28 A+19 C)+75 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-10 a^4 C+45 a^3 b B+3 a^2 b^2 (161 A+93 C)+435 a b^3 B+21 b^4 (9 A+7 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (9 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}",1,"(8*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(405*a^2*b*B + 75*b^3*B + 5*a^3*(63*A + 31*C) + 3*a*b^2*(119*A + 87*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (45*a^3*b*B + 435*a*b^3*B - 10*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(161*A + 93*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x])*(2*(540*a^2*b*B + 345*b^3*B + 20*a^3*C + 3*a*b^2*(308*A + 249*C))*Sin[c + d*x] + b*((252*A*b^2 + 540*a*b*B + 300*a^2*C + 266*b^2*C)*Sin[2*(c + d*x)] + 5*b*(2*(9*b*B + 19*a*C)*Sin[3*(c + d*x)] + 7*b*C*Sin[4*(c + d*x)]))))/(1260*b^2*d*Sqrt[a + b*Cos[c + d*x]])","A",1
1032,1,526,383,4.1863202,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{2 \sin (c+d x) \sqrt{a+b \cos (c+d x)} \left(90 a^2 C+6 b (15 a C+7 b B) \cos (c+d x)+154 a b B+70 A b^2+15 b^2 C \cos (2 (c+d x))+65 b^2 C\right)+\frac{4 \left(105 a^3 B+45 a^2 b (7 A+3 C)+119 a b^2 B+5 b^3 (7 A+5 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 \left(15 a^3 (14 A+C)+161 a^2 b B+5 a b^2 (49 A+29 C)+63 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 i \csc (c+d x) \left(15 a^3 C+161 a^2 b B+5 a b^2 (49 A+29 C)+63 b^3 B\right) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\cos (c+d x)+1)}{a-b}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b^2 \sqrt{-\frac{1}{a+b}}}}{210 d}","\frac{2 a^3 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \left(15 a^2 C+56 a b B+35 A b^2+25 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{105 d}+\frac{2 \left(15 a^3 C+161 a^2 b B+5 a b^2 (49 A+29 C)+63 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 \left(15 a^4 C+56 a^3 b B-10 a^2 b^2 (7 A-C)-56 a b^3 B-5 b^4 (7 A+5 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 (5 a C+7 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}",1,"((4*(105*a^3*B + 119*a*b^2*B + 45*a^2*b*(7*A + 3*C) + 5*b^3*(7*A + 5*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(161*a^2*b*B + 63*b^3*B + 15*a^3*(14*A + C) + 5*a*b^2*(49*A + 29*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + ((2*I)*(161*a^2*b*B + 63*b^3*B + 15*a^3*C + 5*a*b^2*(49*A + 29*C))*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Cos[c + d*x]))/(a - b))]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b^2*Sqrt[-(a + b)^(-1)]) + 2*Sqrt[a + b*Cos[c + d*x]]*(70*A*b^2 + 154*a*b*B + 90*a^2*C + 65*b^2*C + 6*b*(7*b*B + 15*a*C)*Cos[c + d*x] + 15*b^2*C*Cos[2*(c + d*x)])*Sin[c + d*x])/(210*d)","C",1
1033,1,502,357,4.2533939,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{60 \sqrt{a+b \cos (c+d x)} \left(a^2 A \tan (c+d x)+\frac{2}{15} b (11 a C+5 b B) \sin (c+d x)+\frac{1}{5} b^2 C \sin (2 (c+d x))\right)+\frac{2 i \csc (c+d x) \left(a^2 (46 C-15 A)+70 a b B+6 b^2 (5 A+3 C)\right) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\cos (c+d x)+1)}{a-b}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}+\frac{8 \left(15 a^3 C+45 a^2 b B+a b^2 (45 A+17 C)+5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 \left(60 a^3 B+a^2 b (135 A+46 C)+70 a b^2 B+6 b^3 (5 A+3 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{60 d}","\frac{\left(-\left(a^2 (15 A-46 C)\right)+70 a b B+6 b^2 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{a^2 (2 a B+5 A b) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{\left(a^3 (15 A-16 C)+20 a^2 b B+4 a b^2 (15 A+4 C)+10 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{a+b \cos (c+d x)}}-\frac{b \sin (c+d x) (15 a A-16 a C-10 b B) \sqrt{a+b \cos (c+d x)}}{15 d}-\frac{b (5 A-2 C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}+\frac{A \tan (c+d x) (a+b \cos (c+d x))^{5/2}}{d}",1,"((8*(45*a^2*b*B + 5*b^3*B + 15*a^3*C + a*b^2*(45*A + 17*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(60*a^3*B + 70*a*b^2*B + 6*b^3*(5*A + 3*C) + a^2*b*(135*A + 46*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + ((2*I)*(70*a*b*B + 6*b^2*(5*A + 3*C) + a^2*(-15*A + 46*C))*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Cos[c + d*x]))/(a - b))]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b*Sqrt[-(a + b)^(-1)]) + 60*Sqrt[a + b*Cos[c + d*x]]*((2*b*(5*b*B + 11*a*C)*Sin[c + d*x])/15 + (b^2*C*Sin[2*(c + d*x)])/5 + a^2*A*Tan[c + d*x]))/(60*d)","C",1
1034,1,492,372,5.2072902,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{\frac{8 b \left(3 a^2 (A+12 C)+36 a b B+4 b^2 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+4 \sec (c+d x) \sqrt{a+b \cos (c+d x)} \left(6 a^2 A \tan (c+d x)+3 a (4 a B+9 A b) \sin (c+d x)+4 b^2 C \sin (2 (c+d x))\right)+\frac{2 i \csc (c+d x) \left(-12 a^2 B-27 a A b+56 a b C+24 b^2 B\right) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}+\frac{2 \left(24 a^3 (A+2 C)+108 a^2 b B+7 a b^2 (9 A+8 C)+24 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{48 d}","-\frac{\left(12 a^2 B+a b (27 A-56 C)-24 b^2 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{12 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{a \left(4 a^2 (A+2 C)+20 a b B+15 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{\left(12 a^3 B+a^2 b (33 A+16 C)+48 a b^2 B+8 b^3 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{12 d \sqrt{a+b \cos (c+d x)}}-\frac{b \sin (c+d x) (12 a B+21 A b-8 b C) \sqrt{a+b \cos (c+d x)}}{12 d}+\frac{(4 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{4 d}+\frac{A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{5/2}}{2 d}",1,"((8*b*(36*a*b*B + 4*b^2*(3*A + C) + 3*a^2*(A + 12*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(108*a^2*b*B + 24*b^3*B + 24*a^3*(A + 2*C) + 7*a*b^2*(9*A + 8*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + ((2*I)*(-27*a*A*b - 12*a^2*B + 24*b^2*B + 56*a*b*C)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b*Sqrt[-(a + b)^(-1)]) + 4*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*(3*a*(9*A*b + 4*a*B)*Sin[c + d*x] + 4*b^2*C*Sin[2*(c + d*x)] + 6*a^2*A*Tan[c + d*x]))/(48*d)","C",1
1035,1,519,407,6.4015465,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{\frac{8 b \left(6 a^2 B+a b (13 A+72 C)+24 b^2 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+4 \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)} \left(\sin (2 (c+d x)) \left(4 a^2 (2 A+3 C)+27 a b B+\frac{33 A b^2}{2}\right)+8 a^2 A \tan (c+d x)+2 a (6 a B+13 A b) \sin (c+d x)\right)-\frac{2 i \csc (c+d x) \left(8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}+\frac{2 \left(48 a^3 B+8 a^2 b (13 A+27 C)+126 a b^2 B-3 b^3 (A-16 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{96 d}","\frac{\tan (c+d x) \left(8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right) \sqrt{a+b \cos (c+d x)}}{24 d}-\frac{\left(8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(8 a^3 (2 A+3 C)+66 a^2 b B+a b^2 (59 A+96 C)+48 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 d \sqrt{a+b \cos (c+d x)}}+\frac{\left(8 a^3 B+20 a^2 b (A+2 C)+30 a b^2 B+5 A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 d \sqrt{a+b \cos (c+d x)}}+\frac{(6 a B+5 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{12 d}+\frac{A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}",1,"((8*b*(6*a^2*B + 24*b^2*B + a*b*(13*A + 72*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(48*a^3*B + 126*a*b^2*B - 3*b^3*(A - 16*C) + 8*a^2*b*(13*A + 27*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(54*a*b*B + 3*b^2*(11*A - 16*C) + 8*a^2*(2*A + 3*C))*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b*Sqrt[-(a + b)^(-1)]) + 4*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*(2*a*(13*A*b + 6*a*B)*Sin[c + d*x] + ((33*A*b^2)/2 + 27*a*b*B + 4*a^2*(2*A + 3*C))*Sin[2*(c + d*x)] + 8*a^2*A*Tan[c + d*x]))/(96*d)","C",1
1036,1,792,502,7.1847,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{\sqrt{a+b \cos (c+d x)} \left(\frac{1}{96} \sec ^2(c+d x) \left(36 a^2 A \sin (c+d x)+48 a^2 C \sin (c+d x)+104 a b B \sin (c+d x)+59 A b^2 \sin (c+d x)\right)+\frac{1}{24} \sec ^3(c+d x) \left(8 a^2 B \sin (c+d x)+17 a A b \sin (c+d x)\right)+\frac{1}{4} a^2 A \tan (c+d x) \sec ^3(c+d x)+\frac{\sec (c+d x) \left(128 a^3 B \sin (c+d x)+284 a^2 A b \sin (c+d x)+432 a^2 b C \sin (c+d x)+264 a b^2 B \sin (c+d x)+15 A b^3 \sin (c+d x)\right)}{192 a}\right)}{d}+\frac{\frac{2 \left(144 a^3 A b+192 a^3 b C+416 a^2 b^2 B+236 a A b^3+768 a b^3 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \sin (c+d x) \cos (2 (c+d x)) \left(-128 a^3 b B-284 a^2 A b^2-432 a^2 b^2 C-264 a b^3 B-15 A b^4\right) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{b \cos (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left(2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right)}+\frac{2 \left(288 a^4 A+384 a^4 C+832 a^3 b B+436 a^2 A b^2+1008 a^2 b^2 C-24 a b^3 B-45 A b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{768 a d}","\frac{\tan (c+d x) \sec (c+d x) \left(4 a^2 (3 A+4 C)+24 a b B+5 A b^2\right) \sqrt{a+b \cos (c+d x)}}{32 d}+\frac{\tan (c+d x) \left(128 a^3 B+4 a^2 b (71 A+108 C)+264 a b^2 B+15 A b^3\right) \sqrt{a+b \cos (c+d x)}}{192 a d}+\frac{\left(128 a^3 B+4 a^2 b (89 A+132 C)+472 a b^2 B+b^3 (133 A+384 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{192 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(128 a^3 B+4 a^2 b (71 A+108 C)+264 a b^2 B+15 A b^3\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{192 a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{\left(-16 a^4 (3 A+4 C)-160 a^3 b B-120 a^2 b^2 (A+2 C)-40 a b^3 B+5 A b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{64 a d \sqrt{a+b \cos (c+d x)}}+\frac{(8 a B+5 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{24 d}+\frac{A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{5/2}}{4 d}",1,"((2*(144*a^3*A*b + 236*a*A*b^3 + 416*a^2*b^2*B + 192*a^3*b*C + 768*a*b^3*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(288*a^4*A + 436*a^2*A*b^2 - 45*A*b^4 + 832*a^3*b*B - 24*a*b^3*B + 384*a^4*C + 1008*a^2*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(-284*a^2*A*b^2 - 15*A*b^4 - 128*a^3*b*B - 264*a*b^3*B - 432*a^2*b^2*C)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2)))/(768*a*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]^3*(17*a*A*b*Sin[c + d*x] + 8*a^2*B*Sin[c + d*x]))/24 + (Sec[c + d*x]^2*(36*a^2*A*Sin[c + d*x] + 59*A*b^2*Sin[c + d*x] + 104*a*b*B*Sin[c + d*x] + 48*a^2*C*Sin[c + d*x]))/96 + (Sec[c + d*x]*(284*a^2*A*b*Sin[c + d*x] + 15*A*b^3*Sin[c + d*x] + 128*a^3*B*Sin[c + d*x] + 264*a*b^2*B*Sin[c + d*x] + 432*a^2*b*C*Sin[c + d*x]))/(192*a) + (a^2*A*Sec[c + d*x]^3*Tan[c + d*x])/4))/d","C",0
1037,1,930,624,7.3432636,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{\frac{2 \left(1440 b B a^4+3088 A b^2 a^3+4160 b^2 C a^3+2360 b^3 B a^2+60 A b^4 a\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 \left(2880 B a^5+6176 A b a^4+8320 b C a^4+4360 b^2 B a^3-492 A b^3 a^2-240 b^3 C a^2-450 b^4 B a+135 A b^5\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \left(45 A b^5-150 a B b^4-1692 a^2 A b^3-2640 a^2 C b^3-2840 a^3 B b^2-1024 a^4 A b-1280 a^4 C b\right) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{\cos (c+d x) b+b}{a-b}} \cos (2 (c+d x)) \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right) \sin (c+d x)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 (a+b \cos (c+d x)) a-b^2+(a+b \cos (c+d x))^2}{b^2}} \left(2 a^2-4 (a+b \cos (c+d x)) a-b^2+2 (a+b \cos (c+d x))^2\right)}}{7680 a^2 d}+\frac{\sqrt{a+b \cos (c+d x)} \left(\frac{1}{40} \left(10 B \sin (c+d x) a^2+21 A b \sin (c+d x) a\right) \sec ^4(c+d x)+\frac{1}{5} a^2 A \tan (c+d x) \sec ^4(c+d x)+\frac{1}{240} \left(64 A \sin (c+d x) a^2+80 C \sin (c+d x) a^2+170 b B \sin (c+d x) a+93 A b^2 \sin (c+d x)\right) \sec ^3(c+d x)+\frac{\left(360 B \sin (c+d x) a^3+772 A b \sin (c+d x) a^2+1040 b C \sin (c+d x) a^2+590 b^2 B \sin (c+d x) a+15 A b^3 \sin (c+d x)\right) \sec ^2(c+d x)}{960 a}+\frac{\left(1024 A \sin (c+d x) a^4+1280 C \sin (c+d x) a^4+2840 b B \sin (c+d x) a^3+1692 A b^2 \sin (c+d x) a^2+2640 b^2 C \sin (c+d x) a^2+150 b^3 B \sin (c+d x) a-45 A b^4 \sin (c+d x)\right) \sec (c+d x)}{1920 a^2}\right)}{d}","\frac{\tan (c+d x) \sec ^2(c+d x) \left(16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right) \sqrt{a+b \cos (c+d x)}}{240 d}+\frac{\tan (c+d x) \sec (c+d x) \left(360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right) \sqrt{a+b \cos (c+d x)}}{960 a d}-\frac{\tan (c+d x) \left(-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right) \sqrt{a+b \cos (c+d x)}}{1920 a^2 d}-\frac{\left(-256 a^4 (4 A+5 C)-3560 a^3 b B-4 a^2 b^2 (809 A+1180 C)-1330 a b^3 B+15 A b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{1920 a d \sqrt{a+b \cos (c+d x)}}+\frac{\left(-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{1920 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(96 a^5 B+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+40 a^2 b^3 (A+2 C)-10 a b^4 B+3 A b^5\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{128 a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{(2 a B+A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^{3/2}}{8 d}+\frac{A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}",1,"((2*(3088*a^3*A*b^2 + 60*a*A*b^4 + 1440*a^4*b*B + 2360*a^2*b^3*B + 4160*a^3*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(6176*a^4*A*b - 492*a^2*A*b^3 + 135*A*b^5 + 2880*a^5*B + 4360*a^3*b^2*B - 450*a*b^4*B + 8320*a^4*b*C - 240*a^2*b^3*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(-1024*a^4*A*b - 1692*a^2*A*b^3 + 45*A*b^5 - 2840*a^3*b^2*B - 150*a*b^4*B - 1280*a^4*b*C - 2640*a^2*b^3*C)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2)))/(7680*a^2*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]^4*(21*a*A*b*Sin[c + d*x] + 10*a^2*B*Sin[c + d*x]))/40 + (Sec[c + d*x]^3*(64*a^2*A*Sin[c + d*x] + 93*A*b^2*Sin[c + d*x] + 170*a*b*B*Sin[c + d*x] + 80*a^2*C*Sin[c + d*x]))/240 + (Sec[c + d*x]^2*(772*a^2*A*b*Sin[c + d*x] + 15*A*b^3*Sin[c + d*x] + 360*a^3*B*Sin[c + d*x] + 590*a*b^2*B*Sin[c + d*x] + 1040*a^2*b*C*Sin[c + d*x]))/(960*a) + (Sec[c + d*x]*(1024*a^4*A*Sin[c + d*x] + 1692*a^2*A*b^2*Sin[c + d*x] - 45*A*b^4*Sin[c + d*x] + 2840*a^3*b*B*Sin[c + d*x] + 150*a*b^3*B*Sin[c + d*x] + 1280*a^4*C*Sin[c + d*x] + 2640*a^2*b^2*C*Sin[c + d*x]))/(1920*a^2) + (a^2*A*Sec[c + d*x]^4*Tan[c + d*x])/5))/d","C",0
1038,1,259,285,1.3580705,"\int (a+b \cos (c+d x))^{3/2} \left(a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2),x]","\frac{b \sin (c+d x) (a+b \cos (c+d x)) \left(-64 a^2 C+6 b (8 a C+7 b B) \cos (c+d x)+154 a b B+15 b^2 C \cos (2 (c+d x))+65 b^2 C\right)+2 \left(-146 a^3 C+161 a^2 b B+82 a b^2 C+63 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)+2 \left(-105 a^4 C+105 a^3 b B+16 a^2 b^2 C+119 a b^3 B+25 b^4 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 d \sqrt{a+b \cos (c+d x)}}","\frac{2 b \left(-41 a^2 C+56 a b B+25 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 d}-\frac{2 \left(a^2-b^2\right) \left(-41 a^2 C+56 a b B+25 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-146 a^3 C+161 a^2 b B+82 a b^2 C+63 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 b (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d}+\frac{2 b C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}",1,"(2*(105*a^3*b*B + 119*a*b^3*B - 105*a^4*C + 16*a^2*b^2*C + 25*b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + 2*(161*a^2*b*B + 63*b^3*B - 146*a^3*C + 82*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)]) + b*(a + b*Cos[c + d*x])*(154*a*b*B - 64*a^2*C + 65*b^2*C + 6*b*(7*b*B + 8*a*C)*Cos[c + d*x] + 15*b^2*C*Cos[2*(c + d*x)])*Sin[c + d*x])/(105*d*Sqrt[a + b*Cos[c + d*x]])","A",1
1039,1,178,221,1.0226119,"\int \sqrt{a+b \cos (c+d x)} \left(a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2),x]","\frac{2 \left(a^2-b^2\right) (2 a C-5 b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-2 (a+b) \left(17 a^2 C-20 a b B-9 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+2 b \sin (c+d x) (a+b \cos (c+d x)) (a C+5 b B+3 b C \cos (c+d x))}{15 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(a^2-b^2\right) (5 b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-17 a^2 C+20 a b B+9 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 b (5 b B-2 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{2 b C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}",1,"(-2*(a + b)*(-20*a*b*B + 17*a^2*C - 9*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + 2*(a^2 - b^2)*(-5*b*B + 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + 2*b*(a + b*Cos[c + d*x])*(5*b*B + a*C + 3*b*C*Cos[c + d*x])*Sin[c + d*x])/(15*d*Sqrt[a + b*Cos[c + d*x]])","A",1
1040,1,252,344,1.3222722,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{b (a+b \cos (c+d x)) \left(\sin (c+d x) \left(96 a^2 C-112 a b B+140 A b^2+115 b^2 C\right)+3 b (2 (7 b B-6 a C) \sin (2 (c+d x))+5 b C \sin (3 (c+d x)))\right)+4 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b^2 \left(-12 a^2 C+14 a b B+35 A b^2+25 b^2 C\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-\left(48 a^3 C-56 a^2 b B+2 a b^2 (35 A+22 C)-63 b^3 B\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{210 b^4 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \sin (c+d x) \left(24 a^2 C-28 a b B+35 A b^2+25 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{105 b^3 d}+\frac{2 \left(-48 a^3 C+56 a^2 b B-2 a b^2 (35 A+22 C)+63 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 \left(-48 a^4 C+56 a^3 b B-2 a^2 b^2 (35 A+16 C)+49 a b^3 B-5 b^4 (7 A+5 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 (7 b B-6 a C) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{35 b^2 d}+\frac{2 C \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{7 b d}",1,"(4*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(35*A*b^2 + 14*a*b*B - 12*a^2*C + 25*b^2*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - (-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 2*a*b^2*(35*A + 22*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x])*((140*A*b^2 - 112*a*b*B + 96*a^2*C + 115*b^2*C)*Sin[c + d*x] + 3*b*(2*(7*b*B - 6*a*C)*Sin[2*(c + d*x)] + 5*b*C*Sin[3*(c + d*x)])))/(210*b^4*d*Sqrt[a + b*Cos[c + d*x]])","A",1
1041,1,186,258,1.0943697,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(\left(8 a^2 C-10 a b B+15 A b^2+9 b^2 C\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)+b^2 (2 a C+5 b B) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)+2 b \sin (c+d x) (a+b \cos (c+d x)) (-4 a C+5 b B+3 b C \cos (c+d x))}{15 b^3 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(8 a^2 C-10 a b B+15 A b^2+9 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \left(-8 a^3 C+10 a^2 b B-a b^2 (15 A+7 C)+5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 (5 b B-4 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^2 d}+\frac{2 C \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b d}",1,"(2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(5*b*B + 2*a*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + 2*b*(a + b*Cos[c + d*x])*(5*b*B - 4*a*C + 3*b*C*Cos[c + d*x])*Sin[c + d*x])/(15*b^3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
1042,1,160,188,0.7469809,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \left(2 a^2 C-3 a b B+3 A b^2+b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-2 (a+b) (2 a C-3 b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+2 b C \sin (c+d x) (a+b \cos (c+d x))}{3 b^2 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(2 a^2 C-3 a b B+3 A b^2+b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 (3 b B-2 a C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d}",1,"(-2*(a + b)*(-3*b*B + 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + 2*(3*A*b^2 - 3*a*b*B + 2*a^2*C + b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + 2*b*C*(a + b*Cos[c + d*x])*Sin[c + d*x])/(3*b^2*d*Sqrt[a + b*Cos[c + d*x]])","A",1
1043,0,0,189,17.6209167,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/Sqrt[a + b*Cos[c + d*x]],x]","\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","\frac{2 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 (b B-a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/Sqrt[a + b*Cos[c + d*x]], x]","F",-1
1044,1,600,220,13.5107138,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 A \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)} \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right)}{a d (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}+\frac{\cos ^2(c+d x) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(\frac{2 i A b \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{b \cos (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left(2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right)}+\frac{2 (4 a B-3 A b) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{8 a C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}\right)}{2 a d (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}","-\frac{(A b-2 a B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}+\frac{(A+2 C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{a d}-\frac{A \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*A*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*Sin[c + d*x])/(a*d*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^2*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*((8*a*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(-3*A*b + 4*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + ((2*I)*A*b*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2))))/(2*a*d*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x]))","C",0
1045,1,424,303,6.5288302,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\frac{2 \left(8 a^2 (A+2 C)-12 a b B+9 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+4 \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)} ((4 a B-3 A b) \cos (c+d x)+2 a A)+\frac{2 i (3 A b-4 a B) \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \left(b \left(b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)-2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)}{a b \sqrt{-\frac{1}{a+b}}}+\frac{8 a A b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{16 a^2 d}","\frac{\left(4 a^2 (A+2 C)-4 a b B+3 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{(3 A b-4 a B) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a^2 d}+\frac{(3 A b-4 a B) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{(A b-4 a B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 a d}",1,"((8*a*A*b*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(9*A*b^2 - 12*a*b*B + 8*a^2*(A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + ((2*I)*(3*A*b - 4*a*B)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b*Sqrt[-(a + b)^(-1)]) + 4*Sqrt[a + b*Cos[c + d*x]]*(2*a*A + (-3*A*b + 4*a*B)*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(16*a^2*d)","C",1
1046,1,665,405,6.8251208,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\sqrt{a+b \cos (c+d x)} \left(\frac{\sec ^2(c+d x) (6 a B \sin (c+d x)-5 A b \sin (c+d x))}{12 a^2}+\frac{\sec (c+d x) \left(16 a^2 A \sin (c+d x)+24 a^2 C \sin (c+d x)-18 a b B \sin (c+d x)+15 A b^2 \sin (c+d x)\right)}{24 a^3}+\frac{A \tan (c+d x) \sec ^2(c+d x)}{3 a}\right)}{d}+\frac{\frac{2 \left(24 a^2 b B-20 a A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \sin (c+d x) \cos (2 (c+d x)) \left(-16 a^2 A b-24 a^2 b C+18 a b^2 B-15 A b^3\right) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{b \cos (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left(2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right)}+\frac{2 \left(48 a^3 B-40 a^2 A b-72 a^2 b C+54 a b^2 B-45 A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{96 a^3 d}","\frac{\left(8 a^2 (2 A+3 C)-6 a b B+5 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{(5 A b-6 a B) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{12 a^2 d}+\frac{\tan (c+d x) \left(8 a^2 (2 A+3 C)-18 a b B+15 A b^2\right) \sqrt{a+b \cos (c+d x)}}{24 a^3 d}-\frac{\left(8 a^2 (2 A+3 C)-18 a b B+15 A b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{24 a^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{\left(-8 a^3 B+4 a^2 b (A+2 C)-6 a b^2 B+5 A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 a^3 d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{3 a d}",1,"((2*(-20*a*A*b^2 + 24*a^2*b*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(-40*a^2*A*b - 45*A*b^3 + 48*a^3*B + 54*a*b^2*B - 72*a^2*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(-16*a^2*A*b - 15*A*b^3 + 18*a*b^2*B - 24*a^2*b*C)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2)))/(96*a^3*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]^2*(-5*A*b*Sin[c + d*x] + 6*a*B*Sin[c + d*x]))/(12*a^2) + (Sec[c + d*x]*(16*a^2*A*Sin[c + d*x] + 15*A*b^2*Sin[c + d*x] - 18*a*b*B*Sin[c + d*x] + 24*a^2*C*Sin[c + d*x]))/(24*a^3) + (A*Sec[c + d*x]^2*Tan[c + d*x])/(3*a)))/d","C",1
1047,1,328,426,2.3332055,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\frac{30 a^2 b \sin (c+d x) \left(a (a C-b B)+A b^2\right)}{b^2-a^2}+\frac{2 b^2 \left(12 a^3 C-10 a^2 b B+3 a b^2 (5 A+C)-5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{(a-b) (a+b)}+\frac{2 \left(48 a^4 C-40 a^3 b B+6 a^2 b^2 (5 A-4 C)+25 a b^3 B-3 b^4 (5 A+3 C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)}{(a-b) (a+b)}+3 b^2 C \sin (2 (c+d x)) (a+b \cos (c+d x))+2 b (5 b B-9 a C) \sin (c+d x) (a+b \cos (c+d x))}{15 b^4 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \sin (c+d x) \cos ^2(c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \cos (c+d x) \left(6 a^2 C-5 a b B+5 A b^2-b^2 C\right) \sqrt{a+b \cos (c+d x)}}{5 b^2 d \left(a^2-b^2\right)}+\frac{2 \sin (c+d x) \left(-24 a^3 C+20 a^2 b B-3 a b^2 (5 A-3 C)-5 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{15 b^3 d \left(a^2-b^2\right)}+\frac{2 \left(-48 a^3 C+40 a^2 b B-6 a b^2 (5 A+2 C)+5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-48 a^4 C+40 a^3 b B-6 a^2 b^2 (5 A-4 C)-25 a b^3 B+3 b^4 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^4 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"((2*b^2*(-10*a^2*b*B - 5*b^3*B + 12*a^3*C + 3*a*b^2*(5*A + C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/((a - b)*(a + b)) + (2*(-40*a^3*b*B + 25*a*b^3*B + 6*a^2*b^2*(5*A - 4*C) + 48*a^4*C - 3*b^4*(5*A + 3*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/((a - b)*(a + b)) + (30*a^2*b*(A*b^2 + a*(-(b*B) + a*C))*Sin[c + d*x])/(-a^2 + b^2) + 2*b*(5*b*B - 9*a*C)*(a + b*Cos[c + d*x])*Sin[c + d*x] + 3*b^2*C*(a + b*Cos[c + d*x])*Sin[2*(c + d*x)])/(15*b^4*d*Sqrt[a + b*Cos[c + d*x]])","A",1
1048,1,236,280,1.7891208,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 \left(b \sin (c+d x) \left(\frac{a \left(-4 a^2 C+3 a b B-3 A b^2+b^2 C\right)}{b^2-a^2}+b C \cos (c+d x)\right)-\frac{\sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b^2 \left(2 a^2 C-3 a b B+3 A b^2+b^2 C\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(8 a^3 C-6 a^2 b B+a b^2 (3 A-5 C)+3 b^3 B\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{(a-b) (a+b)}\right)}{3 b^3 d \sqrt{a+b \cos (c+d x)}}","\frac{2 a \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(8 a^2 C-6 a b B+3 A b^2+b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-8 a^3 C+6 a^2 b B-a b^2 (3 A-5 C)-3 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^2 d}",1,"(2*(-((Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(3*A*b^2 - 3*a*b*B + 2*a^2*C + b^2*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (-6*a^2*b*B + 3*b^3*B + a*b^2*(3*A - 5*C) + 8*a^3*C)*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])))/((a - b)*(a + b))) + b*((a*(-3*A*b^2 + 3*a*b*B - 4*a^2*C + b^2*C))/(-a^2 + b^2) + b*C*Cos[c + d*x])*Sin[c + d*x]))/(3*b^3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
1049,1,182,219,0.9770637,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 \left(-\left((a+b) \left(2 a^2 C-a b B+A b^2-b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)+\left(a^2-b^2\right) (2 a C-b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+b \sin (c+d x) \left(a (a C-b B)+A b^2\right)\right)}{b^2 d (a-b) (a+b) \sqrt{a+b \cos (c+d x)}}","-\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(2 a^2 C-a b B+A b^2-b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b^2 d \sqrt{a+b \cos (c+d x)}}",1,"(-2*(-((a + b)*(A*b^2 - a*b*B + 2*a^2*C - b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)]) + (a^2 - b^2)*(-(b*B) + 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + b*(A*b^2 + a*(-(b*B) + a*C))*Sin[c + d*x]))/((a - b)*b^2*(a + b)*d*Sqrt[a + b*Cos[c + d*x]])","A",1
1050,0,0,271,33.2446264,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^(3/2),x]","\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(A b^2-a (b B-a C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a b d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{a+b \cos (c+d x)}}",1,"Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^(3/2), x]","F",-1
1051,1,751,313,7.009884,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\cos ^2(c+d x) \sqrt{a+b \cos (c+d x)} \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(\frac{2 A \tan (c+d x)}{a^2}-\frac{4 \left(a^2 b C \sin (c+d x)-a b^2 B \sin (c+d x)+A b^3 \sin (c+d x)\right)}{a^2 \left(a^2-b^2\right) (a+b \cos (c+d x))}\right)}{d (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}+\frac{\cos ^2(c+d x) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(-\frac{2 i \sin (c+d x) \cos (2 (c+d x)) \left(-a^2 A b+2 a^2 b C-2 a b^2 B+3 A b^3\right) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{b \cos (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left(2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right)}+\frac{2 \left(4 a^3 C-4 a^2 b B+4 a A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 \left(4 a^3 B-7 a^2 A b+2 a^2 b C-6 a b^2 B+9 A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}\right)}{2 a^2 d (a-b) (a+b) (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}","-\frac{b \sin (c+d x) \left(-\left(a^2 (A-2 C)\right)-2 a b B+3 A b^2\right)}{a^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\left(-\left(a^2 (A-2 C)\right)-2 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{(3 A b-2 a B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{A \tan (c+d x)}{a d \sqrt{a+b \cos (c+d x)}}+\frac{A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{a+b \cos (c+d x)}}",1,"(Cos[c + d*x]^2*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*((2*(4*a*A*b^2 - 4*a^2*b*B + 4*a^3*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(-7*a^2*A*b + 9*A*b^3 + 4*a^3*B - 6*a*b^2*B + 2*a^2*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(-(a^2*A*b) + 3*A*b^3 - 2*a*b^2*B + 2*a^2*b*C)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2))))/(2*a^2*(a - b)*(a + b)*d*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*((-4*(A*b^3*Sin[c + d*x] - a*b^2*B*Sin[c + d*x] + a^2*b*C*Sin[c + d*x]))/(a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])) + (2*A*Tan[c + d*x])/a^2))/(d*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x]))","C",0
1052,1,723,416,7.2240484,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\sqrt{a+b \cos (c+d x)} \left(\frac{\sec (c+d x) (4 a B \sin (c+d x)-7 A b \sin (c+d x))}{4 a^3}+\frac{A \tan (c+d x) \sec (c+d x)}{2 a^2}+\frac{2 \left(a^2 b^2 C \sin (c+d x)-a b^3 B \sin (c+d x)+A b^4 \sin (c+d x)\right)}{a^3 \left(a^2-b^2\right) (a+b \cos (c+d x))}\right)}{d}-\frac{\frac{2 \left(4 a^3 A b-16 a^3 b C+16 a^2 b^2 B-20 a A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \sin (c+d x) \cos (2 (c+d x)) \left(-4 a^3 b B+7 a^2 A b^2-8 a^2 b^2 C+12 a b^3 B-15 A b^4\right) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{b \cos (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left(2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right)}+\frac{2 \left(8 a^4 A+16 a^4 C-28 a^3 b B+29 a^2 A b^2-24 a^2 b^2 C+36 a b^3 B-45 A b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{16 a^3 d (b-a) (a+b)}","-\frac{(5 A b-4 a B) \tan (c+d x)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{(5 A b-4 a B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{\left(4 a^2 (A+2 C)-12 a b B+15 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^3 d \sqrt{a+b \cos (c+d x)}}+\frac{b \sin (c+d x) \left(4 a^3 B-a^2 (7 A b-8 b C)-12 a b^2 B+15 A b^3\right)}{4 a^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{\left(4 a^3 B-a^2 (7 A b-8 b C)-12 a b^2 B+15 A b^3\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^3 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{A \tan (c+d x) \sec (c+d x)}{2 a d \sqrt{a+b \cos (c+d x)}}",1,"-1/16*((2*(4*a^3*A*b - 20*a*A*b^3 + 16*a^2*b^2*B - 16*a^3*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(8*a^4*A + 29*a^2*A*b^2 - 45*A*b^4 - 28*a^3*b*B + 36*a*b^3*B + 16*a^4*C - 24*a^2*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(7*a^2*A*b^2 - 15*A*b^4 - 4*a^3*b*B + 12*a*b^3*B - 8*a^2*b^2*C)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2)))/(a^3*(-a + b)*(a + b)*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]*(-7*A*b*Sin[c + d*x] + 4*a*B*Sin[c + d*x]))/(4*a^3) + (2*(A*b^4*Sin[c + d*x] - a*b^3*B*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(a^3*(a^2 - b^2)*(a + b*Cos[c + d*x])) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*a^2)))/d","C",0
1053,1,422,622,5.7769667,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{b \left(\frac{10 a^3 \sin (c+d x) \left(a (a C-b B)+A b^2\right)}{a^2-b^2}-\frac{10 a^2 \sin (c+d x) \left(11 a^4 C-8 a^3 b B+5 a^2 b^2 (A-3 C)+12 a b^3 B-9 A b^4\right) (a+b \cos (c+d x))}{\left(a^2-b^2\right)^2}+2 (5 b B-14 a C) \sin (c+d x) (a+b \cos (c+d x))^2+3 b C \sin (2 (c+d x)) (a+b \cos (c+d x))^2\right)+\frac{2 \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{3/2} \left(b^2 \left(32 a^5 C-20 a^4 b B+2 a^3 b^2 (5 A-22 C)+35 a^2 b^3 B-2 a b^4 (15 A+4 C)+5 b^5 B\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(128 a^6 C-80 a^5 b B+4 a^4 b^2 (10 A-53 C)+140 a^3 b^3 B+5 a^2 b^4 (11 C-15 A)-40 a b^5 B+3 b^6 (5 A+3 C)\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{(a-b)^2 (a+b)}}{15 b^5 d (a+b \cos (c+d x))^{3/2}}","-\frac{2 \sin (c+d x) \cos ^3(c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \sin (c+d x) \cos ^2(c+d x) \left(-8 a^4 C+5 a^3 b B-2 a^2 b^2 (A-6 C)-9 a b^3 B+6 A b^4\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \sin (c+d x) \cos (c+d x) \left(-48 a^4 C+30 a^3 b B-a^2 b^2 (15 A-71 C)-50 a b^3 B+b^4 (35 A-3 C)\right) \sqrt{a+b \cos (c+d x)}}{15 b^3 d \left(a^2-b^2\right)^2}+\frac{2 \sin (c+d x) \left(-64 a^5 C+40 a^4 b B-2 a^3 b^2 (10 A-49 C)-65 a^2 b^3 B+2 a b^4 (20 A-7 C)+5 b^5 B\right) \sqrt{a+b \cos (c+d x)}}{15 b^4 d \left(a^2-b^2\right)^2}+\frac{2 \left(-128 a^5 C+80 a^4 b B-4 a^3 b^2 (10 A-29 C)-80 a^2 b^3 B+a b^4 (45 A+17 C)-5 b^5 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^5 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-128 a^6 C+80 a^5 b B-4 a^4 b^2 (10 A-53 C)-140 a^3 b^3 B+5 a^2 b^4 (15 A-11 C)+40 a b^5 B-3 b^6 (5 A+3 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^5 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"((2*((a + b*Cos[c + d*x])/(a + b))^(3/2)*(b^2*(-20*a^4*b*B + 35*a^2*b^3*B + 5*b^5*B + 2*a^3*b^2*(5*A - 22*C) + 32*a^5*C - 2*a*b^4*(15*A + 4*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (-80*a^5*b*B + 140*a^3*b^3*B - 40*a*b^5*B + 4*a^4*b^2*(10*A - 53*C) + 128*a^6*C + 3*b^6*(5*A + 3*C) + 5*a^2*b^4*(-15*A + 11*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])))/((a - b)^2*(a + b)) + b*((10*a^3*(A*b^2 + a*(-(b*B) + a*C))*Sin[c + d*x])/(a^2 - b^2) - (10*a^2*(-9*A*b^4 - 8*a^3*b*B + 12*a*b^3*B + 5*a^2*b^2*(A - 3*C) + 11*a^4*C)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2 + 2*(5*b*B - 14*a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x] + 3*b*C*(a + b*Cos[c + d*x])^2*Sin[2*(c + d*x)]))/(15*b^5*d*(a + b*Cos[c + d*x])^(3/2))","A",1
1054,1,377,453,3.9318586,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(\frac{\left(\frac{a+b \cos (c+d x)}{a+b}\right)^{3/2} \left(b^2 \left(-4 a^4 C+2 a^3 b B+a^2 b^2 (A+7 C)-6 a b^3 B+b^4 (3 A+C)\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-\left(16 a^5 C-8 a^4 b B+2 a^3 b^2 (A-14 C)+15 a^2 b^3 B+2 a b^4 (4 C-3 A)-3 b^5 B\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{(a-b)^2 (a+b)}+\frac{b \sin (c+d x) \left(16 a^6 C-8 a^5 b B+2 a^4 A b^2-25 a^4 b^2 C+16 a^3 b^3 B-10 a^2 A b^4+C \left(b^3-a^2 b\right)^2 \cos (2 (c+d x))+2 a b \cos (c+d x) \left(10 a^4 C-5 a^3 b B+2 a^2 b^2 (A-8 C)+9 a b^3 B+2 b^4 (C-3 A)\right)+b^6 C\right)}{2 \left(a^2-b^2\right)^2}\right)}{3 b^4 d (a+b \cos (c+d x))^{3/2}}","-\frac{2 \sin (c+d x) \cos ^2(c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \sin (c+d x) \left(2 a^2 C-a b B+A b^2-b^2 C\right) \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)}-\frac{2 a \sin (c+d x) \left(a \left(-6 a^3 C+3 a^2 b B+10 a b^2 C-7 b^3 B\right)+4 A b^4\right)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-16 a^4 C+8 a^3 b B-2 a^2 b^2 (A-8 C)-9 a b^3 B+b^4 (3 A+C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^4 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-16 a^5 C+8 a^4 b B-2 a^3 b^2 (A-14 C)-15 a^2 b^3 B+2 a b^4 (3 A-4 C)+3 b^5 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^4 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*((((a + b*Cos[c + d*x])/(a + b))^(3/2)*(b^2*(2*a^3*b*B - 6*a*b^3*B - 4*a^4*C + b^4*(3*A + C) + a^2*b^2*(A + 7*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - (-8*a^4*b*B + 15*a^2*b^3*B - 3*b^5*B + 2*a^3*b^2*(A - 14*C) + 16*a^5*C + 2*a*b^4*(-3*A + 4*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])))/((a - b)^2*(a + b)) + (b*(2*a^4*A*b^2 - 10*a^2*A*b^4 - 8*a^5*b*B + 16*a^3*b^3*B + 16*a^6*C - 25*a^4*b^2*C + b^6*C + 2*a*b*(-5*a^3*b*B + 9*a*b^3*B + 2*a^2*b^2*(A - 8*C) + 10*a^4*C + 2*b^4*(-3*A + C))*Cos[c + d*x] + (-(a^2*b) + b^3)^2*C*Cos[2*(c + d*x)])*Sin[c + d*x])/(2*(a^2 - b^2)^2)))/(3*b^4*d*(a + b*Cos[c + d*x])^(3/2))","A",1
1055,1,323,359,3.1958926,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(\frac{b \sin (c+d x) \left(b \cos (c+d x) \left(-5 a^4 C+2 a^3 b B+a^2 b^2 (A+9 C)-6 a b^3 B+3 A b^4\right)+a \left(-4 a^4 C+a^3 b B+2 a^2 b^2 (A+4 C)-5 a b^3 B+2 A b^4\right)\right)}{\left(a^2-b^2\right)^2}+\frac{(-a-b \cos (c+d x)) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(\left(-8 a^4 C+2 a^3 b B+a^2 b^2 (A+15 C)-6 a b^3 B+3 b^4 (A-C)\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)-b^2 \left(2 a^3 C+a^2 b B-2 a b^2 (2 A+3 C)+3 b^3 B\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)}{(a-b)^2 (a+b)^2}\right)}{3 b^3 d (a+b \cos (c+d x))^{3/2}}","\frac{2 a \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(-8 a^3 C+2 a^2 b B+a b^2 (A+9 C)-3 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \left(-5 a^4 C+2 a^3 b B+a^2 b^2 (A+9 C)-6 a b^3 B+3 A b^4\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-8 a^4 C+2 a^3 b B+a^2 b^2 (A+15 C)-6 a b^3 B+3 b^4 (A-C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*(((-a - b*Cos[c + d*x])*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(-(b^2*(a^2*b*B + 3*b^3*B + 2*a^3*C - 2*a*b^2*(2*A + 3*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)]) + (2*a^3*b*B - 6*a*b^3*B + 3*b^4*(A - C) - 8*a^4*C + a^2*b^2*(A + 15*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])))/((a - b)^2*(a + b)^2) + (b*(a*(2*A*b^4 + a^3*b*B - 5*a*b^3*B - 4*a^4*C + 2*a^2*b^2*(A + 4*C)) + b*(3*A*b^4 + 2*a^3*b*B - 6*a*b^3*B - 5*a^4*C + a^2*b^2*(A + 9*C))*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2))/(3*b^3*d*(a + b*Cos[c + d*x])^(3/2))","A",1
1056,1,278,333,2.5357942,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(\frac{\left(\frac{a+b \cos (c+d x)}{a+b}\right)^{3/2} \left(b^2 \left(a^2 (3 A+C)-4 a b B+b^2 (A+3 C)\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-\left(2 a^3 C+a^2 b B-2 a b^2 (2 A+3 C)+3 b^3 B\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{(a-b)^2 (a+b)}+\frac{b \sin (c+d x) \left(a^4 C+2 a^3 b B-5 a^2 b^2 (A+C)+b \cos (c+d x) \left(2 a^3 C+a^2 b B-2 a b^2 (2 A+3 C)+3 b^3 B\right)+2 a b^3 B+A b^4\right)}{\left(a^2-b^2\right)^2}\right)}{3 b^2 d (a+b \cos (c+d x))^{3/2}}","-\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(-2 a^2 C-a b B+A b^2+3 b^2 C\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \left(2 a^3 C+a^2 b B-2 a b^2 (2 A+3 C)+3 b^3 B\right)}{3 b d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(2 a^3 C+a^2 b B-2 a b^2 (2 A+3 C)+3 b^3 B\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*((((a + b*Cos[c + d*x])/(a + b))^(3/2)*(b^2*(-4*a*b*B + a^2*(3*A + C) + b^2*(A + 3*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - (a^2*b*B + 3*b^3*B + 2*a^3*C - 2*a*b^2*(2*A + 3*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])))/((a - b)^2*(a + b)) + (b*(A*b^4 + 2*a^3*b*B + 2*a*b^3*B + a^4*C - 5*a^2*b^2*(A + C) + b*(a^2*b*B + 3*b^3*B + 2*a^3*C - 2*a*b^2*(2*A + 3*C))*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2))/(3*b^2*d*(a + b*Cos[c + d*x])^(3/2))","A",1
1057,0,0,401,49.9224139,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^(5/2),x]","\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(A b^2-a (b B-a C)\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 A \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \sin (c+d x) \left(a^4 (-C)+4 a^3 b B-a^2 b^2 (7 A+3 C)+3 A b^4\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^4 (-C)+4 a^3 b B-a^2 b^2 (7 A+3 C)+3 A b^4\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^2 b d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(a + b*Cos[c + d*x])^(5/2), x]","F",-1
1058,1,915,461,7.389684,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2),x]","\frac{\left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(\frac{2 \left(12 C a^5-24 b B a^4+36 A b^2 a^3+4 b^2 C a^3+8 b^3 B a^2-20 A b^4 a\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 \left(12 B a^5-33 A b a^4+8 b C a^4-38 b^2 B a^3+86 A b^3 a^2+18 b^4 B a-45 A b^5\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \left(-15 A b^5+6 a B b^4+26 a^2 A b^3-14 a^3 B b^2-3 a^4 A b+8 a^4 C b\right) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{\cos (c+d x) b+b}{a-b}} \cos (2 (c+d x)) \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right) \sin (c+d x)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 (a+b \cos (c+d x)) a-b^2+(a+b \cos (c+d x))^2}{b^2}} \left(2 a^2-4 (a+b \cos (c+d x)) a-b^2+2 (a+b \cos (c+d x))^2\right)}\right) \cos ^2(c+d x)}{6 a^3 (b-a)^2 (a+b)^2 d (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}+\frac{\sqrt{a+b \cos (c+d x)} \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(-\frac{4 \left(A \sin (c+d x) b^3-a B \sin (c+d x) b^2+a^2 C \sin (c+d x) b\right)}{3 a^2 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{4 \left(-6 A \sin (c+d x) b^5+3 a B \sin (c+d x) b^4+10 a^2 A \sin (c+d x) b^3-7 a^3 B \sin (c+d x) b^2+4 a^4 C \sin (c+d x) b\right)}{3 a^3 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{2 A \tan (c+d x)}{a^3}\right) \cos ^2(c+d x)}{d (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}","-\frac{(5 A b-2 a B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^3 d \sqrt{a+b \cos (c+d x)}}-\frac{b \sin (c+d x) \left(-\left(a^2 (3 A-2 C)\right)-2 a b B+5 A b^2\right)}{3 a^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{\left(-\left(a^2 (3 A-2 C)\right)-2 a b B+5 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{b \sin (c+d x) \left(-\left(a^4 (3 A-8 C)\right)-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{\left(-\left(a^4 (3 A-8 C)\right)-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}",1,"(Cos[c + d*x]^2*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*((2*(36*a^3*A*b^2 - 20*a*A*b^4 - 24*a^4*b*B + 8*a^2*b^3*B + 12*a^5*C + 4*a^3*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(-33*a^4*A*b + 86*a^2*A*b^3 - 45*A*b^5 + 12*a^5*B - 38*a^3*b^2*B + 18*a*b^4*B + 8*a^4*b*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(-3*a^4*A*b + 26*a^2*A*b^3 - 15*A*b^5 - 14*a^3*b^2*B + 6*a*b^4*B + 8*a^4*b*C)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2))))/(6*a^3*(-a + b)^2*(a + b)^2*d*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*((-4*(A*b^3*Sin[c + d*x] - a*b^2*B*Sin[c + d*x] + a^2*b*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) - (4*(10*a^2*A*b^3*Sin[c + d*x] - 6*A*b^5*Sin[c + d*x] - 7*a^3*b^2*B*Sin[c + d*x] + 3*a*b^4*B*Sin[c + d*x] + 4*a^4*b*C*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) + (2*A*Tan[c + d*x])/a^3))/(d*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x]))","C",0
1059,1,922,572,7.7362545,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^(5/2),x]","\frac{\frac{2 \left(12 A b a^5-96 b C a^5+144 b^2 B a^4-216 A b^3 a^3+32 b^3 C a^3-80 b^4 B a^2+140 A b^5 a\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 \left(24 A a^6+48 C a^6-132 b B a^5+195 A b^2 a^4-152 b^2 C a^4+344 b^3 B a^3-566 A b^4 a^2+72 b^4 C a^2-180 b^5 B a+315 A b^6\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \left(105 A b^6-60 a B b^5-170 a^2 A b^4+24 a^2 C b^4+104 a^3 B b^3+33 a^4 A b^2-56 a^4 C b^2-12 a^5 B b\right) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{\cos (c+d x) b+b}{a-b}} \cos (2 (c+d x)) \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right) \sin (c+d x)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 (a+b \cos (c+d x)) a-b^2+(a+b \cos (c+d x))^2}{b^2}} \left(2 a^2-4 (a+b \cos (c+d x)) a-b^2+2 (a+b \cos (c+d x))^2\right)}}{48 a^4 (a-b)^2 (a+b)^2 d}+\frac{\sqrt{a+b \cos (c+d x)} \left(\frac{\sec (c+d x) (4 a B \sin (c+d x)-11 A b \sin (c+d x))}{4 a^4}+\frac{2 \left(A \sin (c+d x) b^4-a B \sin (c+d x) b^3+a^2 C \sin (c+d x) b^2\right)}{3 a^3 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{2 \left(-9 A \sin (c+d x) b^6+6 a B \sin (c+d x) b^5+13 a^2 A \sin (c+d x) b^4-3 a^2 C \sin (c+d x) b^4-10 a^3 B \sin (c+d x) b^3+7 a^4 C \sin (c+d x) b^2\right)}{3 a^4 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{A \sec (c+d x) \tan (c+d x)}{2 a^3}\right)}{d}","-\frac{(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac{\left(4 a^2 (A+2 C)-20 a b B+35 A b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^4 d \sqrt{a+b \cos (c+d x)}}+\frac{b \sin (c+d x) \left(12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right)}{12 a^3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{\left(12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{12 a^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{b \sin (c+d x) \left(-12 a^5 B+a^4 b (33 A-56 C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right)}{12 a^4 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{\left(-12 a^5 B+a^4 b (33 A-56 C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{12 a^4 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}",1,"((2*(12*a^5*A*b - 216*a^3*A*b^3 + 140*a*A*b^5 + 144*a^4*b^2*B - 80*a^2*b^4*B - 96*a^5*b*C + 32*a^3*b^3*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(24*a^6*A + 195*a^4*A*b^2 - 566*a^2*A*b^4 + 315*A*b^6 - 132*a^5*b*B + 344*a^3*b^3*B - 180*a*b^5*B + 48*a^6*C - 152*a^4*b^2*C + 72*a^2*b^4*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(33*a^4*A*b^2 - 170*a^2*A*b^4 + 105*A*b^6 - 12*a^5*b*B + 104*a^3*b^3*B - 60*a*b^5*B - 56*a^4*b^2*C + 24*a^2*b^4*C)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2)))/(48*a^4*(a - b)^2*(a + b)^2*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec[c + d*x]*(-11*A*b*Sin[c + d*x] + 4*a*B*Sin[c + d*x]))/(4*a^4) + (2*(A*b^4*Sin[c + d*x] - a*b^3*B*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) + (2*(13*a^2*A*b^4*Sin[c + d*x] - 9*A*b^6*Sin[c + d*x] - 10*a^3*b^3*B*Sin[c + d*x] + 6*a*b^5*B*Sin[c + d*x] + 7*a^4*b^2*C*Sin[c + d*x] - 3*a^2*b^4*C*Sin[c + d*x]))/(3*a^4*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*a^3)))/d","C",0
1060,1,433,449,3.8929057,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{7/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(7/2),x]","\frac{2 \left(\frac{\left(\frac{a+b \cos (c+d x)}{a+b}\right)^{5/2} \left(b^2 \left(a^3 (15 A+7 C)-27 a^2 b B+a b^2 (17 A+25 C)-5 b^3 B\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(-2 a^4 C-3 a^3 b B+a^2 b^2 (23 A+19 C)-29 a b^3 B+3 b^4 (3 A+5 C)\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{(a-b)^3 (a+b)}+\frac{b \sin (c+d x) \left(-2 a^6 C-18 a^5 b B+68 a^4 A b^2+48 a^4 b^2 C-53 a^3 b^3 B+13 a^2 A b^4+35 a^2 b^4 C+b^2 \cos (2 (c+d x)) \left(-2 a^4 C-3 a^3 b B+a^2 b^2 (23 A+19 C)-29 a b^3 B+3 b^4 (3 A+5 C)\right)+2 b \cos (c+d x) \left(-6 a^5 C-9 a^4 b B+2 a^3 b^2 (27 A+25 C)-60 a^2 b^3 B+10 a b^4 (A+2 C)+5 b^5 B\right)-25 a b^5 B+15 A b^6+15 b^6 C\right)}{2 \left(b^2-a^2\right)^3}\right)}{15 b^2 d (a+b \cos (c+d x))^{5/2}}","-\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{5 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{5/2}}+\frac{2 \sin (c+d x) \left(2 a^3 C+3 a^2 b B-2 a b^2 (4 A+5 C)+5 b^3 B\right)}{15 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(2 a^3 C+3 a^2 b B-2 a b^2 (4 A+5 C)+5 b^3 B\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \left(2 a^4 C+3 a^3 b B-a^2 b^2 (23 A+19 C)+29 a b^3 B-3 b^4 (3 A+5 C)\right)}{15 b d \left(a^2-b^2\right)^3 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(2 a^4 C+3 a^3 b B-a^2 b^2 (23 A+19 C)+29 a b^3 B-3 b^4 (3 A+5 C)\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \left(a^2-b^2\right)^3 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*((((a + b*Cos[c + d*x])/(a + b))^(5/2)*(b^2*(-27*a^2*b*B - 5*b^3*B + a^3*(15*A + 7*C) + a*b^2*(17*A + 25*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (-3*a^3*b*B - 29*a*b^3*B - 2*a^4*C + 3*b^4*(3*A + 5*C) + a^2*b^2*(23*A + 19*C))*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])))/((a - b)^3*(a + b)) + (b*(68*a^4*A*b^2 + 13*a^2*A*b^4 + 15*A*b^6 - 18*a^5*b*B - 53*a^3*b^3*B - 25*a*b^5*B - 2*a^6*C + 48*a^4*b^2*C + 35*a^2*b^4*C + 15*b^6*C + 2*b*(-9*a^4*b*B - 60*a^2*b^3*B + 5*b^5*B - 6*a^5*C + 10*a*b^4*(A + 2*C) + 2*a^3*b^2*(27*A + 25*C))*Cos[c + d*x] + b^2*(-3*a^3*b*B - 29*a*b^3*B - 2*a^4*C + 3*b^4*(3*A + 5*C) + a^2*b^2*(23*A + 19*C))*Cos[2*(c + d*x)])*Sin[c + d*x])/(2*(-a^2 + b^2)^3)))/(15*b^2*d*(a + b*Cos[c + d*x])^(5/2))","A",1
1061,1,144,167,0.6545152,"\int \frac{a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{-2 C \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-2 (a+b) (2 a C-3 b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+2 b C \sin (c+d x) (a+b \cos (c+d x))}{3 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 C \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 (3 b B-2 a C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 b C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}",1,"(-2*(a + b)*(-3*b*B + 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 2*(a^2 - b^2)*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + 2*b*C*(a + b*Cos[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
1062,1,90,124,0.2448977,"\int \frac{a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left((b B-2 a C) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+C (a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)}{d \sqrt{a+b \cos (c+d x)}}","\frac{2 (b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 C \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*((a + b)*C*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + (b*B - 2*a*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/(d*Sqrt[a + b*Cos[c + d*x]])","A",1
1063,1,150,180,0.6200462,"\int \frac{a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(C \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+b (2 a C-b B) \sin (c+d x)-\left((a+b) (2 a C-b B) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)\right)}{d (a-b) (a+b) \sqrt{a+b \cos (c+d x)}}","-\frac{2 b (b B-2 a C) \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 (b B-2 a C) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 C \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}",1,"(2*(-((a + b)*(-(b*B) + 2*a*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)]) + (a^2 - b^2)*C*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + b*(-(b*B) + 2*a*C)*Sin[c + d*x]))/((a - b)*(a + b)*d*Sqrt[a + b*Cos[c + d*x]])","A",1
1064,1,193,271,1.6657825,"\int \frac{a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{7/2}} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(7/2),x]","\frac{2 \left(\frac{b \sin (c+d x) \left(7 a^3 C+b \left(5 a^2 C-4 a b B+3 b^2 C\right) \cos (c+d x)-5 a^2 b B+a b^2 C+b^3 B\right)}{\left(a^2-b^2\right)^2}-\frac{\left(\frac{a+b \cos (c+d x)}{a+b}\right)^{3/2} \left(\left(5 a^2 C-4 a b B+3 b^2 C\right) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-(a-b) (2 a C-b B) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)}{(a-b)^2}\right)}{3 d (a+b \cos (c+d x))^{3/2}}","-\frac{2 b \left(-5 a^2 C+4 a b B-3 b^2 C\right) \sin (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 b (b B-2 a C) \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 (b B-2 a C) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-5 a^2 C+4 a b B-3 b^2 C\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*(-((((a + b*Cos[c + d*x])/(a + b))^(3/2)*((-4*a*b*B + 5*a^2*C + 3*b^2*C)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - (a - b)*(-(b*B) + 2*a*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/(a - b)^2) + (b*(-5*a^2*b*B + b^3*B + 7*a^3*C + a*b^2*C + b*(-4*a*b*B + 5*a^2*C + 3*b^2*C)*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2))/(3*d*(a + b*Cos[c + d*x])^(3/2))","A",1
1065,1,143,190,1.0177358,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{60 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (7 a A+5 a C+5 b B)+84 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (9 a B+9 A b+7 b C)+\sin (c+d x) \sqrt{\cos (c+d x)} (7 \cos (c+d x) (36 a B+36 A b+43 b C)+5 (84 a A+18 (a C+b B) \cos (2 (c+d x))+78 a C+78 b B+7 b C \cos (3 (c+d x))))}{630 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (7 a A+5 a C+5 b B)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (9 a B+9 A b+7 b C)}{15 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (9 a B+9 A b+7 b C)}{45 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} (7 a A+5 a C+5 b B)}{21 d}+\frac{2 (a C+b B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}",1,"(84*(9*A*b + 9*a*B + 7*b*C)*EllipticE[(c + d*x)/2, 2] + 60*(7*a*A + 5*b*B + 5*a*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*(36*A*b + 36*a*B + 43*b*C)*Cos[c + d*x] + 5*(84*a*A + 78*b*B + 78*a*C + 18*(b*B + a*C)*Cos[2*(c + d*x)] + 7*b*C*Cos[3*(c + d*x)]))*Sin[c + d*x])/(630*d)","A",1
1066,1,117,154,0.8791894,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (7 a B+7 A b+5 b C)+42 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a A+3 a C+3 b B)+\sin (c+d x) \sqrt{\cos (c+d x)} (42 (a C+b B) \cos (c+d x)+70 a B+70 A b+15 b C \cos (2 (c+d x))+65 b C)}{105 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (7 a B+7 A b+5 b C)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a A+3 a C+3 b B)}{5 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} (7 a B+7 A b+5 b C)}{21 d}+\frac{2 (a C+b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}",1,"(42*(5*a*A + 3*b*B + 3*a*C)*EllipticE[(c + d*x)/2, 2] + 10*(7*A*b + 7*a*B + 5*b*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(70*A*b + 70*a*B + 65*b*C + 42*(b*B + a*C)*Cos[c + d*x] + 15*b*C*Cos[2*(c + d*x)])*Sin[c + d*x])/(105*d)","A",1
1067,1,94,116,0.6075633,"\int \frac{(a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{2 \left(5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a A+a C+b B)+3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a B+5 A b+3 b C)+\sin (c+d x) \sqrt{\cos (c+d x)} (5 a C+5 b B+3 b C \cos (c+d x))\right)}{15 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a (3 A+C)+b B)}{3 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a B+5 A b+3 b C)}{5 d}+\frac{2 (a C+b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(2*(3*(5*A*b + 5*a*B + 3*b*C)*EllipticE[(c + d*x)/2, 2] + 5*(3*a*A + b*B + a*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(5*b*B + 5*a*C + 3*b*C*Cos[c + d*x])*Sin[c + d*x]))/(15*d)","A",1
1068,1,90,107,0.4970484,"\int \frac{(a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a B+3 A b+b C)+E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (-6 a A+6 a C+6 b B)+\frac{2 \sin (c+d x) (3 a A+b C \cos (c+d x))}{\sqrt{\cos (c+d x)}}}{3 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a B+3 A b+b C)}{3 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (b B-a (A-C))}{d}+\frac{2 a A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 b C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"((-6*a*A + 6*b*B + 6*a*C)*EllipticE[(c + d*x)/2, 2] + 2*(3*A*b + 3*a*B + b*C)*EllipticF[(c + d*x)/2, 2] + (2*(3*a*A + b*C*Cos[c + d*x])*Sin[c + d*x])/Sqrt[Cos[c + d*x]])/(3*d)","A",1
1069,1,115,111,0.6220375,"\int \frac{(a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 \left(\sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a A+3 a C+3 b B)-3 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a B+A b-b C)+a A \tan (c+d x)+3 a B \sin (c+d x)+3 A b \sin (c+d x)\right)}{3 d \sqrt{\cos (c+d x)}}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a (A+3 C)+3 b B)}{3 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a B+A b-b C)}{d}+\frac{2 (a B+A b) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(2*(-3*(A*b + a*B - b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + (a*A + 3*b*B + 3*a*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 3*A*b*Sin[c + d*x] + 3*a*B*Sin[c + d*x] + a*A*Tan[c + d*x]))/(3*d*Sqrt[Cos[c + d*x]])","A",1
1070,1,136,152,1.4452092,"\int \frac{(a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{3 \sin (2 (c+d x)) (3 a A+5 a C+5 b B)+10 \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a B+A b+3 b C)-6 \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a A+5 a C+5 b B)+10 (a B+A b) \sin (c+d x)+6 a A \tan (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a B+A b+3 b C)}{3 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a A+5 a C+5 b B)}{5 d}+\frac{2 \sin (c+d x) (3 a A+5 a C+5 b B)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 (a B+A b) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-6*(3*a*A + 5*b*B + 5*a*C)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 10*(A*b + a*B + 3*b*C)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 10*(A*b + a*B)*Sin[c + d*x] + 3*(3*a*A + 5*b*B + 5*a*C)*Sin[2*(c + d*x)] + 6*a*A*Tan[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",1
1071,1,173,190,4.2140858,"\int \frac{(a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a A+7 a C+7 b B)-42 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a B+3 A b+5 b C)+\frac{\sin (c+d x) (21 \cos (c+d x) (13 a B+13 A b+15 b C)+10 \cos (2 (c+d x)) (5 a A+7 a C+7 b B)+110 a A+63 a B \cos (3 (c+d x))+70 a C+63 A b \cos (3 (c+d x))+70 b B+105 b C \cos (3 (c+d x)))}{2 \cos ^{\frac{7}{2}}(c+d x)}}{105 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a A+7 a C+7 b B)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a B+3 A b+5 b C)}{5 d}+\frac{2 \sin (c+d x) (5 a A+7 a C+7 b B)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) (3 a B+3 A b+5 b C)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 (a B+A b) \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(-42*(3*A*b + 3*a*B + 5*b*C)*EllipticE[(c + d*x)/2, 2] + 10*(5*a*A + 7*b*B + 7*a*C)*EllipticF[(c + d*x)/2, 2] + ((110*a*A + 70*b*B + 70*a*C + 21*(13*A*b + 13*a*B + 15*b*C)*Cos[c + d*x] + 10*(5*a*A + 7*b*B + 7*a*C)*Cos[2*(c + d*x)] + 63*A*b*Cos[3*(c + d*x)] + 63*a*B*Cos[3*(c + d*x)] + 105*b*C*Cos[3*(c + d*x)])*Sin[c + d*x])/(2*Cos[c + d*x]^(7/2)))/(105*d)","A",1
1072,1,239,305,1.6687986,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right)+154 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(9 a^2 B+2 a b (9 A+7 C)+7 b^2 B\right)+\frac{1}{12} \sin (c+d x) \sqrt{\cos (c+d x)} \left(154 \cos (c+d x) \left(36 a^2 B+72 a A b+86 a b C+43 b^2 B\right)+5 \left(36 \cos (2 (c+d x)) \left(11 a^2 C+22 a b B+11 A b^2+16 b^2 C\right)+132 a^2 (14 A+13 C)+154 b (2 a C+b B) \cos (3 (c+d x))+3432 a b B+3 b^2 (572 A+531 C)+63 b^2 C \cos (4 (c+d x))\right)\right)}{1155 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right)}{231 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(9 a^2 B+18 a A b+14 a b C+7 b^2 B\right)}{15 d}+\frac{2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(4 a^2 C+22 a b B+11 A b^2+9 b^2 C\right)}{77 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(9 a^2 B+18 a A b+14 a b C+7 b^2 B\right)}{45 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right)}{231 d}+\frac{2 b (4 a C+11 b B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}",1,"(154*(9*a^2*B + 7*b^2*B + 2*a*b*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2] + 10*(110*a*b*B + 11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2] + (Sqrt[Cos[c + d*x]]*(154*(72*a*A*b + 36*a^2*B + 43*b^2*B + 86*a*b*C)*Cos[c + d*x] + 5*(3432*a*b*B + 132*a^2*(14*A + 13*C) + 3*b^2*(572*A + 531*C) + 36*(11*A*b^2 + 22*a*b*B + 11*a^2*C + 16*b^2*C)*Cos[2*(c + d*x)] + 154*b*(b*B + 2*a*C)*Cos[3*(c + d*x)] + 63*b^2*C*Cos[4*(c + d*x)]))*Sin[c + d*x])/12)/(1155*d)","A",1
1073,1,195,251,1.2833644,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{60 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^2 B+2 a b (7 A+5 C)+5 b^2 B\right)+84 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 (5 A+3 C)+18 a b B+b^2 (9 A+7 C)\right)+\sin (c+d x) \sqrt{\cos (c+d x)} \left(7 \cos (c+d x) \left(36 a^2 C+72 a b B+36 A b^2+43 b^2 C\right)+5 \left(84 a^2 B+168 a A b+18 b (2 a C+b B) \cos (2 (c+d x))+156 a b C+78 b^2 B+7 b^2 C \cos (3 (c+d x))\right)\right)}{630 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^2 B+14 a A b+10 a b C+5 b^2 B\right)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 (5 A+3 C)+18 a b B+b^2 (9 A+7 C)\right)}{15 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(4 a^2 C+18 a b B+9 A b^2+7 b^2 C\right)}{45 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(7 a^2 B+14 a A b+10 a b C+5 b^2 B\right)}{21 d}+\frac{2 b (4 a C+9 b B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}",1,"(84*(18*a*b*B + 3*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2] + 60*(7*a^2*B + 5*b^2*B + 2*a*b*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*(36*A*b^2 + 72*a*b*B + 36*a^2*C + 43*b^2*C)*Cos[c + d*x] + 5*(168*a*A*b + 84*a^2*B + 78*b^2*B + 156*a*b*C + 18*b*(b*B + 2*a*C)*Cos[2*(c + d*x)] + 7*b^2*C*Cos[3*(c + d*x)]))*Sin[c + d*x])/(630*d)","A",1
1074,1,160,203,1.3104319,"\int \frac{(a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^2 (3 A+C)+14 a b B+b^2 (7 A+5 C)\right)+42 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 B+2 a b (5 A+3 C)+3 b^2 B\right)+\sin (c+d x) \sqrt{\cos (c+d x)} \left(5 \left(14 a^2 C+28 a b B+14 A b^2+3 b^2 C \cos (2 (c+d x))+13 b^2 C\right)+42 b (2 a C+b B) \cos (c+d x)\right)}{105 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^2 (3 A+C)+14 a b B+b^2 (7 A+5 C)\right)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 B+10 a A b+6 a b C+3 b^2 B\right)}{5 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(4 a^2 C+14 a b B+7 A b^2+5 b^2 C\right)}{21 d}+\frac{2 b (4 a C+7 b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}",1,"(42*(5*a^2*B + 3*b^2*B + 2*a*b*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2] + 10*(14*a*b*B + 7*a^2*(3*A + C) + b^2*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(42*b*(b*B + 2*a*C)*Cos[c + d*x] + 5*(14*A*b^2 + 28*a*b*B + 14*a^2*C + 13*b^2*C + 3*b^2*C*Cos[2*(c + d*x)]))*Sin[c + d*x])/(105*d)","A",1
1075,1,144,189,1.266437,"\int \frac{(a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 B+2 a b (3 A+C)+b^2 B\right)+6 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^2 (A-C)+10 a b B+b^2 (5 A+3 C)\right)+\frac{\sin (c+d x) \left(3 \left(10 a^2 A+b^2 C \cos (2 (c+d x))+b^2 C\right)+10 b (2 a C+b B) \cos (c+d x)\right)}{\sqrt{\cos (c+d x)}}}{15 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 B+2 a b (3 A+C)+b^2 B\right)}{3 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^2 (A-C)+10 a b B+b^2 (5 A+3 C)\right)}{5 d}-\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} (6 a A-2 a C-b B)}{3 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt{\cos (c+d x)}}-\frac{2 b^2 (5 A-C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(6*(10*a*b*B - 5*a^2*(A - C) + b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2] + 10*(3*a^2*B + b^2*B + 2*a*b*(3*A + C))*EllipticF[(c + d*x)/2, 2] + ((10*b*(b*B + 2*a*C)*Cos[c + d*x] + 3*(10*a^2*A + b^2*C + b^2*C*Cos[2*(c + d*x)]))*Sin[c + d*x])/Sqrt[Cos[c + d*x]])/(15*d)","A",1
1076,1,157,180,1.2683527,"\int \frac{(a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (A+3 C)+6 a b B+b^2 (3 A+C)\right)-6 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 B+2 a b (A-C)-b^2 B\right)+2 a^2 A \tan (c+d x)+6 a^2 B \sin (c+d x)+12 a A b \sin (c+d x)+b^2 C \sin (2 (c+d x))}{3 d \sqrt{\cos (c+d x)}}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (A+3 C)+6 a b B+b^2 (3 A+C)\right)}{3 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 B+2 a b (A-C)-b^2 B\right)}{d}+\frac{2 a (3 a B+4 A b) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b^2 (A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(-6*(a^2*B - b^2*B + 2*a*b*(A - C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*(6*a*b*B + b^2*(3*A + C) + a^2*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 12*a*A*b*Sin[c + d*x] + 6*a^2*B*Sin[c + d*x] + b^2*C*Sin[2*(c + d*x)] + 2*a^2*A*Tan[c + d*x])/(3*d*Sqrt[Cos[c + d*x]])","A",1
1077,1,202,200,1.6254309,"\int \frac{(a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{10 \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 B+2 a b (A+3 C)+3 b^2 B\right)-6 \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (3 A+5 C)+10 a b B+5 b^2 (A-C)\right)+9 a^2 A \sin (2 (c+d x))+6 a^2 A \tan (c+d x)+10 a^2 B \sin (c+d x)+15 a^2 C \sin (2 (c+d x))+20 a A b \sin (c+d x)+30 a b B \sin (2 (c+d x))+15 A b^2 \sin (2 (c+d x))}{15 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 B+2 a b (A+3 C)+3 b^2 B\right)}{3 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (3 A+5 C)+10 a b B+5 b^2 (A-C)\right)}{5 d}+\frac{2 \sin (c+d x) \left(a^2 (3 A+5 C)+10 a b B+4 A b^2\right)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a (5 a B+4 A b) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-6*(10*a*b*B + 5*b^2*(A - C) + a^2*(3*A + 5*C))*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 10*(a^2*B + 3*b^2*B + 2*a*b*(A + 3*C))*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 20*a*A*b*Sin[c + d*x] + 10*a^2*B*Sin[c + d*x] + 9*a^2*A*Sin[2*(c + d*x)] + 15*A*b^2*Sin[2*(c + d*x)] + 30*a*b*B*Sin[2*(c + d*x)] + 15*a^2*C*Sin[2*(c + d*x)] + 6*a^2*A*Tan[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",1
1078,1,217,248,4.6075859,"\int \frac{(a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 \left(5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (5 A+7 C)+14 a b B+7 b^2 (A+3 C)\right)-21 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 B+2 a b (3 A+5 C)+5 b^2 B\right)+\frac{5 \sin (c+d x) \left(a^2 (5 A+7 C)+14 a b B+7 A b^2\right)}{\cos ^{\frac{3}{2}}(c+d x)}+\frac{21 \sin (c+d x) \left(3 a^2 B+2 a b (3 A+5 C)+5 b^2 B\right)}{\sqrt{\cos (c+d x)}}+\frac{15 a^2 A \sin (c+d x)}{\cos ^{\frac{7}{2}}(c+d x)}+\frac{21 a (a B+2 A b) \sin (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)}\right)}{105 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (5 A+7 C)+14 a b B+7 b^2 (A+3 C)\right)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 B+6 a A b+10 a b C+5 b^2 B\right)}{5 d}+\frac{2 \sin (c+d x) \left(a^2 (5 A+7 C)+14 a b B+4 A b^2\right)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(3 a^2 B+6 a A b+10 a b C+5 b^2 B\right)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a (7 a B+4 A b) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(2*(-21*(3*a^2*B + 5*b^2*B + 2*a*b*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2] + 5*(14*a*b*B + 7*b^2*(A + 3*C) + a^2*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2] + (15*a^2*A*Sin[c + d*x])/Cos[c + d*x]^(7/2) + (21*a*(2*A*b + a*B)*Sin[c + d*x])/Cos[c + d*x]^(5/2) + (5*(7*A*b^2 + 14*a*b*B + a^2*(5*A + 7*C))*Sin[c + d*x])/Cos[c + d*x]^(3/2) + (21*(3*a^2*B + 5*b^2*B + 2*a*b*(3*A + 5*C))*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(105*d)","A",1
1079,1,266,302,5.279639,"\int \frac{(a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{2 \left(15 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 B+2 a b (5 A+7 C)+7 b^2 B\right)-21 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right)+\frac{15 \sin (c+d x) \left(5 a^2 B+2 a b (5 A+7 C)+7 b^2 B\right)}{\cos ^{\frac{3}{2}}(c+d x)}+\frac{7 \sin (c+d x) \left(a^2 (7 A+9 C)+18 a b B+9 A b^2\right)}{\cos ^{\frac{5}{2}}(c+d x)}+\frac{21 \sin (c+d x) \left(a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right)}{\sqrt{\cos (c+d x)}}+\frac{35 a^2 A \sin (c+d x)}{\cos ^{\frac{9}{2}}(c+d x)}+\frac{45 a (a B+2 A b) \sin (c+d x)}{\cos ^{\frac{7}{2}}(c+d x)}\right)}{315 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 B+10 a A b+14 a b C+7 b^2 B\right)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right)}{15 d}+\frac{2 \sin (c+d x) \left(5 a^2 B+10 a A b+14 a b C+7 b^2 B\right)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(a^2 (7 A+9 C)+18 a b B+4 A b^2\right)}{45 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 a (9 a B+4 A b) \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(2*(-21*(18*a*b*B + 3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2] + 15*(5*a^2*B + 7*b^2*B + 2*a*b*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2] + (35*a^2*A*Sin[c + d*x])/Cos[c + d*x]^(9/2) + (45*a*(2*A*b + a*B)*Sin[c + d*x])/Cos[c + d*x]^(7/2) + (7*(9*A*b^2 + 18*a*b*B + a^2*(7*A + 9*C))*Sin[c + d*x])/Cos[c + d*x]^(5/2) + (15*(5*a^2*B + 7*b^2*B + 2*a*b*(5*A + 7*C))*Sin[c + d*x])/Cos[c + d*x]^(3/2) + (21*(18*a*b*B + 3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(315*d)","A",1
1080,1,285,361,1.9235851,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(77 a^3 B+33 a^2 b (7 A+5 C)+165 a b^2 B+5 b^3 (11 A+9 C)\right)+154 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^3 (5 A+3 C)+27 a^2 b B+3 a b^2 (9 A+7 C)+7 b^3 B\right)+\frac{1}{12} \sin (c+d x) \sqrt{\cos (c+d x)} \left(154 \cos (c+d x) \left(36 a^3 C+108 a^2 b B+3 a b^2 (36 A+43 C)+43 b^3 B\right)+5 \left(1848 a^3 B+36 b \cos (2 (c+d x)) \left(33 a^2 C+33 a b B+11 A b^2+16 b^2 C\right)+396 a^2 b (14 A+13 C)+154 b^2 (3 a C+b B) \cos (3 (c+d x))+5148 a b^2 B+3 b^3 (572 A+531 C)+63 b^3 C \cos (4 (c+d x))\right)\right)}{1155 d}","\frac{2 b \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(24 a^2 C+143 a b B+99 A b^2+81 b^2 C\right)}{693 d}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(77 a^3 B+33 a^2 b (7 A+5 C)+165 a b^2 B+5 b^3 (11 A+9 C)\right)}{231 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^3 (5 A+3 C)+27 a^2 b B+3 a b^2 (9 A+7 C)+7 b^3 B\right)}{15 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(24 a^3 C+242 a^2 b B+33 a b^2 (9 A+7 C)+77 b^3 B\right)}{495 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(77 a^3 B+33 a^2 b (7 A+5 C)+165 a b^2 B+5 b^3 (11 A+9 C)\right)}{231 d}+\frac{2 (6 a C+11 b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{99 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}",1,"(154*(27*a^2*b*B + 7*b^3*B + 3*a^3*(5*A + 3*C) + 3*a*b^2*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2] + 10*(77*a^3*B + 165*a*b^2*B + 33*a^2*b*(7*A + 5*C) + 5*b^3*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2] + (Sqrt[Cos[c + d*x]]*(154*(108*a^2*b*B + 43*b^3*B + 36*a^3*C + 3*a*b^2*(36*A + 43*C))*Cos[c + d*x] + 5*(1848*a^3*B + 5148*a*b^2*B + 396*a^2*b*(14*A + 13*C) + 3*b^3*(572*A + 531*C) + 36*b*(11*A*b^2 + 33*a*b*B + 33*a^2*C + 16*b^2*C)*Cos[2*(c + d*x)] + 154*b^2*(b*B + 3*a*C)*Cos[3*(c + d*x)] + 63*b^3*C*Cos[4*(c + d*x)]))*Sin[c + d*x])/12)/(1155*d)","A",1
1081,1,230,296,2.079867,"\int \frac{(a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{60 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right)+84 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right)+\sin (c+d x) \sqrt{\cos (c+d x)} \left(7 b \cos (c+d x) \left(108 a^2 C+108 a b B+36 A b^2+43 b^2 C\right)+5 \left(84 a^3 C+252 a^2 b B+18 a b^2 (14 A+13 C)+18 b^2 (3 a C+b B) \cos (2 (c+d x))+78 b^3 B+7 b^3 C \cos (3 (c+d x))\right)\right)}{630 d}","\frac{2 b \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right)}{315 d}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right)}{15 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(8 a^3 C+54 a^2 b B+9 a b^2 (7 A+5 C)+15 b^3 B\right)}{63 d}+\frac{2 (2 a C+3 b B) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{21 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}",1,"(84*(15*a^3*B + 27*a*b^2*B + 9*a^2*b*(5*A + 3*C) + b^3*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2] + 60*(21*a^2*b*B + 5*b^3*B + 7*a^3*(3*A + C) + 3*a*b^2*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*b*(36*A*b^2 + 108*a*b*B + 108*a^2*C + 43*b^2*C)*Cos[c + d*x] + 5*(252*a^2*b*B + 78*b^3*B + 84*a^3*C + 18*a*b^2*(14*A + 13*C) + 18*b^2*(b*B + 3*a*C)*Cos[2*(c + d*x)] + 7*b^3*C*Cos[3*(c + d*x)]))*Sin[c + d*x])/(630*d)","A",1
1082,1,212,279,1.8618666,"\int \frac{(a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{20 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(21 a^3 B+21 a^2 b (3 A+C)+21 a b^2 B+b^3 (7 A+5 C)\right)-84 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^3 (A-C)-15 a^2 b B-3 a b^2 (5 A+3 C)-3 b^3 B\right)+\frac{\sin (c+d x) \left(420 a^3 A+5 b \cos (c+d x) \left(84 a^2 C+84 a b B+28 A b^2+29 b^2 C\right)+42 b^2 (3 a C+b B) \cos (2 (c+d x))+126 a b^2 C+42 b^3 B+15 b^3 C \cos (3 (c+d x))\right)}{\sqrt{\cos (c+d x)}}}{210 d}","\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} \left(-6 a^2 (7 A-3 C)+21 a b B+b^2 (7 A+5 C)\right)}{21 d}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(21 a^3 B+21 a^2 b (3 A+C)+21 a b^2 B+b^3 (7 A+5 C)\right)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^3 (A-C)+15 a^2 b B+3 a b^2 (5 A+3 C)+3 b^3 B\right)}{5 d}-\frac{2 b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (35 a A-11 a C-7 b B)}{35 d}-\frac{2 b (7 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{d \sqrt{\cos (c+d x)}}",1,"(-84*(-15*a^2*b*B - 3*b^3*B + 5*a^3*(A - C) - 3*a*b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2] + 20*(21*a^3*B + 21*a*b^2*B + 21*a^2*b*(3*A + C) + b^3*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2] + ((420*a^3*A + 42*b^3*B + 126*a*b^2*C + 5*b*(28*A*b^2 + 84*a*b*B + 84*a^2*C + 29*b^2*C)*Cos[c + d*x] + 42*b^2*(b*B + 3*a*C)*Cos[2*(c + d*x)] + 15*b^3*C*Cos[3*(c + d*x)])*Sin[c + d*x])/Sqrt[Cos[c + d*x]])/(210*d)","A",1
1083,1,186,271,2.4001749,"\int \frac{(a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (A+3 C)+9 a^2 b B+3 a b^2 (3 A+C)+b^3 B\right)+2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-15 a^3 B-45 a^2 b (A-C)+45 a b^2 B+3 b^3 (5 A+3 C)\right)+\frac{5 \left(2 a^3 A \tan (c+d x)+b^2 (3 a C+b B) \sin (2 (c+d x))\right)+6 \sin (c+d x) \left(5 a^2 (a B+3 A b)+b^3 C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}}}{15 d}","-\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} \left(6 a^2 B+3 a b (5 A-C)-b^2 B\right)}{3 d}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (A+3 C)+9 a^2 b B+3 a b^2 (3 A+C)+b^3 B\right)}{3 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^3 B+15 a^2 b (A-C)-15 a b^2 B-b^3 (5 A+3 C)\right)}{5 d}-\frac{2 b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (15 a B+35 A b-3 b C)}{15 d}+\frac{2 (a B+2 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(2*(-15*a^3*B + 45*a*b^2*B - 45*a^2*b*(A - C) + 3*b^3*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2] + 10*(9*a^2*b*B + b^3*B + 3*a*b^2*(3*A + C) + a^3*(A + 3*C))*EllipticF[(c + d*x)/2, 2] + (6*(5*a^2*(3*A*b + a*B) + b^3*C*Cos[c + d*x]^2)*Sin[c + d*x] + 5*(b^2*(b*B + 3*a*C)*Sin[2*(c + d*x)] + 2*a^3*A*Tan[c + d*x]))/Sqrt[Cos[c + d*x]])/(15*d)","A",1
1084,1,248,273,2.2332468,"\int \frac{(a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{9 a^3 A \sin (2 (c+d x))+6 a^3 A \tan (c+d x)+10 a^3 B \sin (c+d x)+15 a^3 C \sin (2 (c+d x))+30 a^2 A b \sin (c+d x)+45 a^2 b B \sin (2 (c+d x))+10 \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 B+3 a^2 b (A+3 C)+9 a b^2 B+b^3 (3 A+C)\right)-6 \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (3 A+5 C)+15 a^2 b B+15 a b^2 (A-C)-5 b^3 B\right)+45 a A b^2 \sin (2 (c+d x))+10 b^3 C \sin (c+d x) \cos ^2(c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 a \sin (c+d x) \left(3 a^2 (3 A+5 C)+35 a b B+24 A b^2\right)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 B+3 a^2 b (A+3 C)+9 a b^2 B+b^3 (3 A+C)\right)}{3 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (3 A+5 C)+15 a^2 b B+15 a b^2 (A-C)-5 b^3 B\right)}{5 d}-\frac{2 b^2 \sin (c+d x) \sqrt{\cos (c+d x)} (5 a B+9 A b-5 b C)}{15 d}+\frac{2 (5 a B+6 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-6*(15*a^2*b*B - 5*b^3*B + 15*a*b^2*(A - C) + a^3*(3*A + 5*C))*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 10*(a^3*B + 9*a*b^2*B + b^3*(3*A + C) + 3*a^2*b*(A + 3*C))*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 30*a^2*A*b*Sin[c + d*x] + 10*a^3*B*Sin[c + d*x] + 10*b^3*C*Cos[c + d*x]^2*Sin[c + d*x] + 9*a^3*A*Sin[2*(c + d*x)] + 45*a*A*b^2*Sin[2*(c + d*x)] + 45*a^2*b*B*Sin[2*(c + d*x)] + 15*a^3*C*Sin[2*(c + d*x)] + 6*a^3*A*Tan[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",1
1085,1,251,294,4.9617792,"\int \frac{(a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{2 \left(\frac{15 a^3 A \sin (c+d x)}{\cos ^{\frac{7}{2}}(c+d x)}+\frac{5 a \sin (c+d x) \left(a^2 (5 A+7 C)+21 a b B+21 A b^2\right)}{\cos ^{\frac{3}{2}}(c+d x)}+\frac{21 a^2 (a B+3 A b) \sin (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)}+5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right)-21 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^3 B+3 a^2 b (3 A+5 C)+15 a b^2 B+5 b^3 (A-C)\right)+\frac{21 \sin (c+d x) \left(3 a^3 B+3 a^2 b (3 A+5 C)+15 a b^2 B+5 A b^3\right)}{\sqrt{\cos (c+d x)}}\right)}{105 d}","\frac{2 a \sin (c+d x) \left(5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^3 B+3 a^2 b (3 A+5 C)+15 a b^2 B+5 b^3 (A-C)\right)}{5 d}+\frac{2 \sin (c+d x) \left(21 a^3 B+21 a^2 b (3 A+5 C)+98 a b^2 B+24 A b^3\right)}{35 d \sqrt{\cos (c+d x)}}+\frac{2 (7 a B+6 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(2*(-21*(3*a^3*B + 15*a*b^2*B + 5*b^3*(A - C) + 3*a^2*b*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2] + 5*(21*a^2*b*B + 21*b^3*B + 21*a*b^2*(A + 3*C) + a^3*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2] + (15*a^3*A*Sin[c + d*x])/Cos[c + d*x]^(7/2) + (21*a^2*(3*A*b + a*B)*Sin[c + d*x])/Cos[c + d*x]^(5/2) + (5*a*(21*A*b^2 + 21*a*b*B + a^2*(5*A + 7*C))*Sin[c + d*x])/Cos[c + d*x]^(3/2) + (21*(5*A*b^3 + 3*a^3*B + 15*a*b^2*B + 3*a^2*b*(3*A + 5*C))*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(105*d)","A",1
1086,1,414,357,6.9130041,"\int \frac{(a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(25 a^3 B+75 a^2 A b+105 a^2 b C+105 a b^2 B+35 A b^3+105 b^3 C\right)+2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-49 a^3 A-63 a^3 C-189 a^2 b B-189 a A b^2-315 a b^2 C-105 b^3 B\right)}{105 d}+\frac{\sqrt{\cos (c+d x)} \left(\frac{2}{9} a^3 A \tan (c+d x) \sec ^4(c+d x)+\frac{2}{45} \sec ^3(c+d x) \left(7 a^3 A \sin (c+d x)+9 a^3 C \sin (c+d x)+27 a^2 b B \sin (c+d x)+27 a A b^2 \sin (c+d x)\right)+\frac{2}{21} \sec ^2(c+d x) \left(5 a^3 B \sin (c+d x)+15 a^2 A b \sin (c+d x)+21 a^2 b C \sin (c+d x)+21 a b^2 B \sin (c+d x)+7 A b^3 \sin (c+d x)\right)+\frac{2}{15} \sec (c+d x) \left(7 a^3 A \sin (c+d x)+9 a^3 C \sin (c+d x)+27 a^2 b B \sin (c+d x)+27 a A b^2 \sin (c+d x)+45 a b^2 C \sin (c+d x)+15 b^3 B \sin (c+d x)\right)+\frac{2}{7} \sec ^4(c+d x) \left(a^3 B \sin (c+d x)+3 a^2 A b \sin (c+d x)\right)\right)}{d}","\frac{2 a \sin (c+d x) \left(7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^3 B+3 a^2 b (5 A+7 C)+21 a b^2 B+7 b^3 (A+3 C)\right)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right)}{15 d}+\frac{2 \sin (c+d x) \left(15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right)}{63 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (3 a B+2 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(2*(-49*a^3*A - 189*a*A*b^2 - 189*a^2*b*B - 105*b^3*B - 63*a^3*C - 315*a*b^2*C)*EllipticE[(c + d*x)/2, 2] + 2*(75*a^2*A*b + 35*A*b^3 + 25*a^3*B + 105*a*b^2*B + 105*a^2*b*C + 105*b^3*C)*EllipticF[(c + d*x)/2, 2])/(105*d) + (Sqrt[Cos[c + d*x]]*((2*Sec[c + d*x]^4*(3*a^2*A*b*Sin[c + d*x] + a^3*B*Sin[c + d*x]))/7 + (2*Sec[c + d*x]^3*(7*a^3*A*Sin[c + d*x] + 27*a*A*b^2*Sin[c + d*x] + 27*a^2*b*B*Sin[c + d*x] + 9*a^3*C*Sin[c + d*x]))/45 + (2*Sec[c + d*x]^2*(15*a^2*A*b*Sin[c + d*x] + 7*A*b^3*Sin[c + d*x] + 5*a^3*B*Sin[c + d*x] + 21*a*b^2*B*Sin[c + d*x] + 21*a^2*b*C*Sin[c + d*x]))/21 + (2*Sec[c + d*x]*(7*a^3*A*Sin[c + d*x] + 27*a*A*b^2*Sin[c + d*x] + 27*a^2*b*B*Sin[c + d*x] + 15*b^3*B*Sin[c + d*x] + 9*a^3*C*Sin[c + d*x] + 45*a*b^2*C*Sin[c + d*x]))/15 + (2*a^3*A*Sec[c + d*x]^4*Tan[c + d*x])/9))/d","A",1
1087,1,381,477,3.5025817,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(154 \cos (c+d x) \left(936 a^4 C+3744 a^3 b B+156 a^2 b^2 (36 A+43 C)+4472 a b^3 B+b^4 (1118 A+1171 C)\right)+5 \left(77 b^2 \cos (3 (c+d x)) \left(312 a^2 C+208 a b B+52 A b^2+89 b^2 C\right)+1872 b \cos (2 (c+d x)) \left(22 a^3 C+33 a^2 b B+2 a b^2 (11 A+16 C)+8 b^3 B\right)+78 \left(616 a^4 B+176 a^3 b (14 A+13 C)+3432 a^2 b^2 B+4 a b^3 (572 A+531 C)+531 b^4 B\right)+1638 b^3 (4 a C+b B) \cos (4 (c+d x))+693 b^4 C \cos (5 (c+d x))\right)\right)+48 \left(65 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right)+77 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(39 a^4 (5 A+3 C)+468 a^3 b B+78 a^2 b^2 (9 A+7 C)+364 a b^3 B+7 b^4 (13 A+11 C)\right)\right)}{360360 d}","\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right) (a+b \cos (c+d x))^2}{1287 d}+\frac{2 b \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right)}{9009 d}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right)}{231 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(39 a^4 (5 A+3 C)+468 a^3 b B+78 a^2 b^2 (9 A+7 C)+364 a b^3 B+7 b^4 (13 A+11 C)\right)}{195 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(192 a^4 C+3458 a^3 b B+11 a^2 b^2 (637 A+491 C)+4004 a b^3 B+77 b^4 (13 A+11 C)\right)}{6435 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right)}{231 d}+\frac{2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{143 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}",1,"(48*(77*(468*a^3*b*B + 364*a*b^3*B + 39*a^4*(5*A + 3*C) + 78*a^2*b^2*(9*A + 7*C) + 7*b^4*(13*A + 11*C))*EllipticE[(c + d*x)/2, 2] + 65*(77*a^4*B + 330*a^2*b^2*B + 45*b^4*B + 44*a^3*b*(7*A + 5*C) + 20*a*b^3*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2]) + Sqrt[Cos[c + d*x]]*(154*(3744*a^3*b*B + 4472*a*b^3*B + 936*a^4*C + 156*a^2*b^2*(36*A + 43*C) + b^4*(1118*A + 1171*C))*Cos[c + d*x] + 5*(78*(616*a^4*B + 3432*a^2*b^2*B + 531*b^4*B + 176*a^3*b*(14*A + 13*C) + 4*a*b^3*(572*A + 531*C)) + 1872*b*(33*a^2*b*B + 8*b^3*B + 22*a^3*C + 2*a*b^2*(11*A + 16*C))*Cos[2*(c + d*x)] + 77*b^2*(52*A*b^2 + 208*a*b*B + 312*a^2*C + 89*b^2*C)*Cos[3*(c + d*x)] + 1638*b^3*(b*B + 4*a*C)*Cos[4*(c + d*x)] + 693*b^4*C*Cos[5*(c + d*x)]))*Sin[c + d*x])/(360360*d)","A",1
1088,1,319,404,2.4454153,"\int \frac{(a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(77 a^4 (3 A+C)+308 a^3 b B+66 a^2 b^2 (7 A+5 C)+220 a b^3 B+5 b^4 (11 A+9 C)\right)+154 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right)+\frac{1}{12} \sin (c+d x) \sqrt{\cos (c+d x)} \left(154 b \cos (c+d x) \left(144 a^3 C+216 a^2 b B+4 a b^2 (36 A+43 C)+43 b^3 B\right)+5 \left(1848 a^4 C+7392 a^3 b B+36 b^2 \cos (2 (c+d x)) \left(66 a^2 C+44 a b B+11 A b^2+16 b^2 C\right)+792 a^2 b^2 (14 A+13 C)+154 b^3 (4 a C+b B) \cos (3 (c+d x))+6864 a b^3 B+3 b^4 (572 A+531 C)+63 b^4 C \cos (4 (c+d x))\right)\right)}{1155 d}","\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right) (a+b \cos (c+d x))^2}{231 d}+\frac{2 b \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right)}{3465 d}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(77 a^4 (3 A+C)+308 a^3 b B+66 a^2 b^2 (7 A+5 C)+220 a b^3 B+5 b^4 (11 A+9 C)\right)}{231 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right)}{15 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(64 a^4 C+682 a^3 b B+9 a^2 b^2 (143 A+101 C)+660 a b^3 B+15 b^4 (11 A+9 C)\right)}{693 d}+\frac{2 (8 a C+11 b B) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3}{99 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}",1,"(154*(15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 12*a^3*b*(5*A + 3*C) + 4*a*b^3*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2] + 10*(308*a^3*b*B + 220*a*b^3*B + 77*a^4*(3*A + C) + 66*a^2*b^2*(7*A + 5*C) + 5*b^4*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2] + (Sqrt[Cos[c + d*x]]*(154*b*(216*a^2*b*B + 43*b^3*B + 144*a^3*C + 4*a*b^2*(36*A + 43*C))*Cos[c + d*x] + 5*(7392*a^3*b*B + 6864*a*b^3*B + 1848*a^4*C + 792*a^2*b^2*(14*A + 13*C) + 3*b^4*(572*A + 531*C) + 36*b^2*(11*A*b^2 + 44*a*b*B + 66*a^2*C + 16*b^2*C)*Cos[2*(c + d*x)] + 154*b^3*(b*B + 4*a*C)*Cos[3*(c + d*x)] + 63*b^4*C*Cos[4*(c + d*x)]))*Sin[c + d*x])/12)/(1155*d)","A",1
1089,1,275,379,4.3554252,"\int \frac{(a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right)-14 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(15 a^4 (A-C)-60 a^3 b B-18 a^2 b^2 (5 A+3 C)-36 a b^3 B-b^4 (9 A+7 C)\right)+\frac{1}{12} \sqrt{\cos (c+d x)} \left(35 \left(72 a^4 A \tan (c+d x)+b^4 C \sin (4 (c+d x))\right)+14 b^2 \sin (2 (c+d x)) \left(108 a^2 C+72 a b B+18 A b^2+19 b^2 C\right)+30 b \sin (c+d x) \left(112 a^3 C+168 a^2 b B+4 a b^2 (28 A+23 C)+23 b^3 B\right)+90 b^3 (4 a C+b B) \sin (3 (c+d x))\right)}{105 d}","\frac{2 b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(-\left(a^2 (315 A-123 C)\right)+162 a b B+7 b^2 (9 A+7 C)\right)}{315 d}+\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} \left(-\left(a^3 (126 A-62 C)\right)+117 a^2 b B+12 a b^2 (7 A+5 C)+15 b^3 B\right)}{63 d}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-15 a^4 (A-C)+60 a^3 b B+18 a^2 b^2 (5 A+3 C)+36 a b^3 B+b^4 (9 A+7 C)\right)}{15 d}-\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{21 d}-\frac{2 b (9 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt{\cos (c+d x)}}",1,"(-14*(-60*a^3*b*B - 36*a*b^3*B + 15*a^4*(A - C) - 18*a^2*b^2*(5*A + 3*C) - b^4*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2] + 10*(21*a^4*B + 42*a^2*b^2*B + 5*b^4*B + 28*a^3*b*(3*A + C) + 4*a*b^3*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2] + (Sqrt[Cos[c + d*x]]*(30*b*(168*a^2*b*B + 23*b^3*B + 112*a^3*C + 4*a*b^2*(28*A + 23*C))*Sin[c + d*x] + 14*b^2*(18*A*b^2 + 72*a*b*B + 108*a^2*C + 19*b^2*C)*Sin[2*(c + d*x)] + 90*b^3*(b*B + 4*a*C)*Sin[3*(c + d*x)] + 35*(b^4*C*Sin[4*(c + d*x)] + 72*a^4*A*Tan[c + d*x])))/12)/(105*d)","A",1
1090,1,257,373,2.6167561,"\int \frac{(a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^4 (A+3 C)+84 a^3 b B+42 a^2 b^2 (3 A+C)+28 a b^3 B+b^4 (7 A+5 C)\right)-42 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^4 B+20 a^3 b (A-C)-30 a^2 b^2 B-4 a b^3 (5 A+3 C)-3 b^4 B\right)+\frac{168 \sin (c+d x) \left(5 a^3 (a B+4 A b)+b^3 (4 a C+b B) \cos ^2(c+d x)\right)+5 \left(56 a^4 A \tan (c+d x)+b^2 \sin (2 (c+d x)) \left(168 a^2 C+112 a b B+28 A b^2+23 b^2 C\right)+6 b^4 C \sin (3 (c+d x)) \cos (c+d x)\right)}{4 \sqrt{\cos (c+d x)}}}{105 d}","-\frac{2 b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(105 a^2 B+350 a A b-54 a b C-21 b^2 B\right)}{105 d}-\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} \left(42 a^3 B+3 a^2 b (49 A-13 C)-28 a b^2 B-b^3 (7 A+5 C)\right)}{21 d}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^4 (A+3 C)+84 a^3 b B+42 a^2 b^2 (3 A+C)+28 a b^3 B+b^4 (7 A+5 C)\right)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^4 B+20 a^3 b (A-C)-30 a^2 b^2 B-4 a b^3 (5 A+3 C)-3 b^4 B\right)}{5 d}-\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} (7 a B+21 A b-b C) (a+b \cos (c+d x))^2}{7 d}+\frac{2 (3 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{3 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-42*(5*a^4*B - 30*a^2*b^2*B - 3*b^4*B + 20*a^3*b*(A - C) - 4*a*b^3*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2] + 10*(84*a^3*b*B + 28*a*b^3*B + 42*a^2*b^2*(3*A + C) + 7*a^4*(A + 3*C) + b^4*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2] + (168*(5*a^3*(4*A*b + a*B) + b^3*(b*B + 4*a*C)*Cos[c + d*x]^2)*Sin[c + d*x] + 5*(b^2*(28*A*b^2 + 112*a*b*B + 168*a^2*C + 23*b^2*C)*Sin[2*(c + d*x)] + 6*b^4*C*Cos[c + d*x]*Sin[3*(c + d*x)] + 56*a^4*A*Tan[c + d*x]))/(4*Sqrt[Cos[c + d*x]]))/(105*d)","A",1
1091,1,316,386,2.3847251,"\int \frac{(a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^4 B+20 a^3 A b+60 a^3 b C+90 a^2 b^2 B+60 a A b^3+20 a b^3 C+5 b^4 B\right)+2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-9 a^4 A-15 a^4 C-60 a^3 b B-90 a^2 A b^2+90 a^2 b^2 C+60 a b^3 B+15 A b^4+9 b^4 C\right)}{15 d}+\frac{\sqrt{\cos (c+d x)} \left(\frac{2}{5} a^4 A \tan (c+d x) \sec ^2(c+d x)+\frac{2}{3} \sec ^2(c+d x) \left(a^4 B \sin (c+d x)+4 a^3 A b \sin (c+d x)\right)+\frac{2}{5} \sec (c+d x) \left(3 a^4 A \sin (c+d x)+5 a^4 C \sin (c+d x)+20 a^3 b B \sin (c+d x)+30 a^2 A b^2 \sin (c+d x)\right)+\frac{2}{3} b^3 (4 a C+b B) \sin (c+d x)+\frac{1}{5} b^4 C \sin (2 (c+d x))\right)}{d}","-\frac{2 b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(3 a^2 (3 A+5 C)+50 a b B+b^2 (59 A-3 C)\right)}{15 d}+\frac{2 \sin (c+d x) \left(a^2 (3 A+5 C)+15 a b B+16 A b^2\right) (a+b \cos (c+d x))^2}{5 d \sqrt{\cos (c+d x)}}-\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} \left(6 a^3 (3 A+5 C)+105 a^2 b B+4 a b^2 (33 A-5 C)-5 b^3 B\right)}{15 d}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 B+4 a^3 b (A+3 C)+18 a^2 b^2 B+4 a b^3 (3 A+C)+b^4 B\right)}{3 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (3 A+5 C)+20 a^3 b B+30 a^2 b^2 (A-C)-20 a b^3 B-b^4 (5 A+3 C)\right)}{5 d}+\frac{2 (5 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(2*(-9*a^4*A - 90*a^2*A*b^2 + 15*A*b^4 - 60*a^3*b*B + 60*a*b^3*B - 15*a^4*C + 90*a^2*b^2*C + 9*b^4*C)*EllipticE[(c + d*x)/2, 2] + 2*(20*a^3*A*b + 60*a*A*b^3 + 5*a^4*B + 90*a^2*b^2*B + 5*b^4*B + 60*a^3*b*C + 20*a*b^3*C)*EllipticF[(c + d*x)/2, 2])/(15*d) + (Sqrt[Cos[c + d*x]]*((2*b^3*(b*B + 4*a*C)*Sin[c + d*x])/3 + (2*Sec[c + d*x]^2*(4*a^3*A*b*Sin[c + d*x] + a^4*B*Sin[c + d*x]))/3 + (2*Sec[c + d*x]*(3*a^4*A*Sin[c + d*x] + 30*a^2*A*b^2*Sin[c + d*x] + 20*a^3*b*B*Sin[c + d*x] + 5*a^4*C*Sin[c + d*x]))/5 + (b^4*C*Sin[2*(c + d*x)])/5 + (2*a^4*A*Sec[c + d*x]^2*Tan[c + d*x])/5))/d","A",1
1092,1,271,383,5.2383696,"\int \frac{(a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right)-42 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^4 B+4 a^3 b (3 A+5 C)+30 a^2 b^2 B+20 a b^3 (A-C)-5 b^4 B\right)+\frac{5 \left(6 a^4 A \tan (c+d x)+a^2 \sin (2 (c+d x)) \left(a^2 (5 A+7 C)+28 a b B+42 A b^2\right)\right)+14 \sin (c+d x) \left(3 a^3 (a B+4 A b)+3 a \cos ^2(c+d x) \left(3 a^3 B+4 a^2 b (3 A+5 C)+30 a b^2 B+20 A b^3\right)+5 b^4 C \cos ^3(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)}}{105 d}","\frac{2 \sin (c+d x) \left(5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right) (a+b \cos (c+d x))^2}{105 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b^2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right)}{105 d}+\frac{2 a \sin (c+d x) \left(63 a^3 B+a^2 (202 A b+350 b C)+413 a b^2 B+192 A b^3\right)}{105 d \sqrt{\cos (c+d x)}}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^4 B+4 a^3 b (3 A+5 C)+30 a^2 b^2 B+20 a b^3 (A-C)-5 b^4 B\right)}{5 d}+\frac{2 (7 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(-42*(3*a^4*B + 30*a^2*b^2*B - 5*b^4*B + 20*a*b^3*(A - C) + 4*a^3*b*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2] + 10*(28*a^3*b*B + 84*a*b^3*B + 7*b^4*(3*A + C) + 42*a^2*b^2*(A + 3*C) + a^4*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2] + (14*(3*a^3*(4*A*b + a*B) + 3*a*(20*A*b^3 + 3*a^3*B + 30*a*b^2*B + 4*a^2*b*(3*A + 5*C))*Cos[c + d*x]^2 + 5*b^4*C*Cos[c + d*x]^3)*Sin[c + d*x] + 5*(a^2*(42*A*b^2 + 28*a*b*B + a^2*(5*A + 7*C))*Sin[2*(c + d*x)] + 6*a^4*A*Tan[c + d*x]))/Cos[c + d*x]^(5/2))/(105*d)","A",1
1093,1,463,401,7.2327422,"\int \frac{(a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(25 a^4 B+100 a^3 A b+140 a^3 b C+210 a^2 b^2 B+140 a A b^3+420 a b^3 C+105 b^4 B\right)+2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-49 a^4 A-63 a^4 C-252 a^3 b B-378 a^2 A b^2-630 a^2 b^2 C-420 a b^3 B-105 A b^4+105 b^4 C\right)}{105 d}+\frac{\sqrt{\cos (c+d x)} \left(\frac{2}{9} a^4 A \tan (c+d x) \sec ^4(c+d x)+\frac{2}{7} \sec ^4(c+d x) \left(a^4 B \sin (c+d x)+4 a^3 A b \sin (c+d x)\right)+\frac{2}{45} \sec ^3(c+d x) \left(7 a^4 A \sin (c+d x)+9 a^4 C \sin (c+d x)+36 a^3 b B \sin (c+d x)+54 a^2 A b^2 \sin (c+d x)\right)+\frac{2}{21} \sec ^2(c+d x) \left(5 a^4 B \sin (c+d x)+20 a^3 A b \sin (c+d x)+28 a^3 b C \sin (c+d x)+42 a^2 b^2 B \sin (c+d x)+28 a A b^3 \sin (c+d x)\right)+\frac{2}{15} \sec (c+d x) \left(7 a^4 A \sin (c+d x)+9 a^4 C \sin (c+d x)+36 a^3 b B \sin (c+d x)+54 a^2 A b^2 \sin (c+d x)+90 a^2 b^2 C \sin (c+d x)+60 a b^3 B \sin (c+d x)+15 A b^4 \sin (c+d x)\right)\right)}{d}","\frac{2 \sin (c+d x) \left(7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right) (a+b \cos (c+d x))^2}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \left(75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right)}{315 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^4 B+4 a^3 b (5 A+7 C)+42 a^2 b^2 B+28 a b^3 (A+3 C)+21 b^4 B\right)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (7 A+9 C)+36 a^3 b B+18 a^2 b^2 (3 A+5 C)+60 a b^3 B+15 b^4 (A-C)\right)}{15 d}+\frac{2 \sin (c+d x) \left(21 a^4 (7 A+9 C)+756 a^3 b B+7 a^2 b^2 (155 A+261 C)+1098 a b^3 B+192 A b^4\right)}{315 d \sqrt{\cos (c+d x)}}+\frac{2 (9 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(2*(-49*a^4*A - 378*a^2*A*b^2 - 105*A*b^4 - 252*a^3*b*B - 420*a*b^3*B - 63*a^4*C - 630*a^2*b^2*C + 105*b^4*C)*EllipticE[(c + d*x)/2, 2] + 2*(100*a^3*A*b + 140*a*A*b^3 + 25*a^4*B + 210*a^2*b^2*B + 105*b^4*B + 140*a^3*b*C + 420*a*b^3*C)*EllipticF[(c + d*x)/2, 2])/(105*d) + (Sqrt[Cos[c + d*x]]*((2*Sec[c + d*x]^4*(4*a^3*A*b*Sin[c + d*x] + a^4*B*Sin[c + d*x]))/7 + (2*Sec[c + d*x]^3*(7*a^4*A*Sin[c + d*x] + 54*a^2*A*b^2*Sin[c + d*x] + 36*a^3*b*B*Sin[c + d*x] + 9*a^4*C*Sin[c + d*x]))/45 + (2*Sec[c + d*x]^2*(20*a^3*A*b*Sin[c + d*x] + 28*a*A*b^3*Sin[c + d*x] + 5*a^4*B*Sin[c + d*x] + 42*a^2*b^2*B*Sin[c + d*x] + 28*a^3*b*C*Sin[c + d*x]))/21 + (2*Sec[c + d*x]*(7*a^4*A*Sin[c + d*x] + 54*a^2*A*b^2*Sin[c + d*x] + 15*A*b^4*Sin[c + d*x] + 36*a^3*b*B*Sin[c + d*x] + 60*a*b^3*B*Sin[c + d*x] + 9*a^4*C*Sin[c + d*x] + 90*a^2*b^2*C*Sin[c + d*x]))/15 + (2*a^4*A*Sec[c + d*x]^4*Tan[c + d*x])/9))/d","A",1
1094,1,381,475,6.3737269,"\int \frac{(a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","\frac{10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right)-154 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right)+\frac{45 \left(14 a^4 A \tan (c+d x)+a^2 \sin (2 (c+d x)) \left(a^2 (9 A+11 C)+44 a b B+66 A b^2\right)\right)+2 \sin (c+d x) \left(385 a^3 (a B+4 A b)+77 a \cos ^2(c+d x) \left(7 a^3 B+4 a^2 b (7 A+9 C)+54 a b^2 B+36 A b^3\right)+231 \cos ^4(c+d x) \left(7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right)+15 \cos ^3(c+d x) \left(5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 A b^4\right)\right)}{3 \cos ^{\frac{9}{2}}(c+d x)}}{1155 d}","\frac{2 \sin (c+d x) \left(3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right) (a+b \cos (c+d x))^2}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \left(539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right)}{3465 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right)}{231 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right)}{15 d}+\frac{2 \sin (c+d x) \left(15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right)}{693 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \cos ^{\frac{11}{2}}(c+d x)}",1,"(-154*(7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 12*a*b^3*(3*A + 5*C) + 4*a^3*b*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2] + 10*(220*a^3*b*B + 308*a*b^3*B + 77*b^4*(A + 3*C) + 66*a^2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*EllipticF[(c + d*x)/2, 2] + (2*(385*a^3*(4*A*b + a*B) + 77*a*(36*A*b^3 + 7*a^3*B + 54*a*b^2*B + 4*a^2*b*(7*A + 9*C))*Cos[c + d*x]^2 + 15*(77*A*b^4 + 220*a^3*b*B + 308*a*b^3*B + 66*a^2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*Cos[c + d*x]^3 + 231*(7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 12*a*b^3*(3*A + 5*C) + 4*a^3*b*(7*A + 9*C))*Cos[c + d*x]^4)*Sin[c + d*x] + 45*(a^2*(66*A*b^2 + 44*a*b*B + a^2*(9*A + 11*C))*Sin[2*(c + d*x)] + 14*a^4*A*Tan[c + d*x]))/(3*Cos[c + d*x]^(9/2)))/(1155*d)","A",1
1095,1,335,285,2.7037481,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(70 a^2 C+42 b (b B-a C) \cos (c+d x)-70 a b B+70 A b^2+15 b^2 C \cos (2 (c+d x))+65 b^2 C\right)+\frac{4 \left(-28 a^2 C+28 a b B+35 A b^2+25 b^2 C\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}-\frac{2 \left(35 a^3 C-35 a^2 b B+a b^2 (35 A+13 C)-63 b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}-\frac{42 \sin (c+d x) \left(5 a^3 C-5 a^2 b B+a b^2 (5 A+3 C)-3 b^3 B\right) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}}{210 b^3 d}","-\frac{2 a^3 \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a+b)}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right)}{21 b^3 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^3 C+5 a^2 b B-a b^2 (5 A+3 C)+3 b^3 B\right)}{5 b^4 d}-\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-21 a^4 C+21 a^3 b B-7 a^2 b^2 (3 A+C)+7 a b^3 B-b^4 (7 A+5 C)\right)}{21 b^5 d}+\frac{2 (b B-a C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b^2 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 b d}",1,"((-2*(-35*a^2*b*B - 63*b^3*B + 35*a^3*C + a*b^2*(35*A + 13*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (4*(35*A*b^2 + 28*a*b*B - 28*a^2*C + 25*b^2*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + 2*Sqrt[Cos[c + d*x]]*(70*A*b^2 - 70*a*b*B + 70*a^2*C + 65*b^2*C + 42*b*(b*B - a*C)*Cos[c + d*x] + 15*b^2*C*Cos[2*(c + d*x)])*Sin[c + d*x] - (42*(-5*a^2*b*B - 3*b^3*B + 5*a^3*C + a*b^2*(5*A + 3*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/(210*b^3*d)","A",1
1096,1,272,210,2.26748,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{\frac{2 b^2 \left(5 a^2 C-5 a b B+15 A b^2+9 b^2 C\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{6 \sin (c+d x) \left(5 a^2 C-5 a b B+5 A b^2+3 b^2 C\right) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a \sqrt{\sin ^2(c+d x)}}+4 b^2 \sin (c+d x) \sqrt{\cos (c+d x)} (-5 a C+5 b B+3 b C \cos (c+d x))+2 b^2 (4 a C+5 b B) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{30 b^4 d}","\frac{2 a^2 \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a+b)}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 C-5 a b B+5 A b^2+3 b^2 C\right)}{5 b^3 d}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-3 a^3 C+3 a^2 b B-a b^2 (3 A+C)+b^3 B\right)}{3 b^4 d}+\frac{2 (b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b d}",1,"((2*b^2*(15*A*b^2 - 5*a*b*B + 5*a^2*C + 9*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + 2*b^2*(5*b*B + 4*a*C)*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)) + 4*b^2*Sqrt[Cos[c + d*x]]*(5*b*B - 5*a*C + 3*b*C*Cos[c + d*x])*Sin[c + d*x] + (6*(5*A*b^2 - 5*a*b*B + 5*a^2*C + 3*b^2*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*Sqrt[Sin[c + d*x]^2]))/(30*b^4*d)","A",1
1097,1,214,147,1.3126741,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\frac{\frac{6 (b B-a C) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}+\frac{4 (3 A+C) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}-\frac{2 (a C-3 b B) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+4 C \sin (c+d x) \sqrt{\cos (c+d x)}}{6 b d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(b^2 (3 A+C)-3 a (b B-a C)\right)}{3 b^3 d}-\frac{2 a \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a+b)}+\frac{2 (b B-a C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d}",1,"((-2*(-3*b*B + a*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (4*(3*A + C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + 4*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x] + (6*(b*B - a*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/(6*b*d)","A",1
1098,1,173,97,1.582636,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])),x]","\frac{\frac{C \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}+\frac{(2 A+C) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{B \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{b}}{d}","\frac{2 \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}+\frac{2 (b B-a C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(((2*A + C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (B*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)))/b + (C*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/d","A",1
1099,1,212,118,1.2482916,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])),x]","\frac{-\frac{2 A \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}+\frac{2 (2 a B-3 A b) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}-\frac{4 a (A-C) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{b (a+b)}+\frac{4 A \sin (c+d x)}{\sqrt{\cos (c+d x)}}}{2 a d}","-\frac{2 \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d (a+b)}-\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{2 A \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}+\frac{2 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"((2*(-3*A*b + 2*a*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) - (4*a*(A - C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(b*(a + b)) + (4*A*Sin[c + d*x])/Sqrt[Cos[c + d*x]] - (2*A*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/(2*a*d)","A",1
1100,1,264,158,2.4764227,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])),x]","\frac{\frac{2 a \left(2 a^2 (A+3 C)-9 a b B+9 A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{6 (A b-a B) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{b \sqrt{\sin ^2(c+d x)}}+\frac{a \left(8 a A b-6 a^2 B\right) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{b}+\frac{4 a^2 A \sin (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)}+\frac{12 a (a B-A b) \sin (c+d x)}{\sqrt{\cos (c+d x)}}}{6 a^3 d}","\frac{2 \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a+b)}+\frac{2 (A b-a B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{2 (A b-a B) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}+\frac{2 A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{2 A \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}",1,"((2*a*(9*A*b^2 - 9*a*b*B + 2*a^2*(A + 3*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (a*(8*a*A*b - 6*a^2*B)*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)))/b + (4*a^2*A*Sin[c + d*x])/Cos[c + d*x]^(3/2) + (12*a*(-(A*b) + a*B)*Sin[c + d*x])/Sqrt[Cos[c + d*x]] + (6*(A*b - a*B)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(b*Sqrt[Sin[c + d*x]^2]))/(6*a^3*d)","A",1
1101,1,332,234,4.4709316,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])),x]","-\frac{\frac{4 a \left(3 a^2 (3 A+5 C)-20 a b B+20 A b^2\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{b (a+b)}-\frac{2 \left(3 \sin (2 (c+d x)) \left(a^2 (3 A+5 C)-5 a b B+5 A b^2\right)+6 a^2 A \tan (c+d x)+10 a (a B-A b) \sin (c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)}+\frac{6 \sin (c+d x) \left(a^2 (3 A+5 C)-5 a b B+5 A b^2\right) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}+\frac{2 \left(-10 a^3 B+a^2 b (19 A+45 C)-45 a b^2 B+45 A b^3\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{30 a^3 d}","-\frac{2 b \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a+b)}-\frac{2 (A b-a B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{2 (A b-a B) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (3 A+5 C)-5 a b B+5 A b^2\right)}{5 a^3 d}+\frac{2 \sin (c+d x) \left(a^2 (3 A+5 C)-5 a b B+5 A b^2\right)}{5 a^3 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x)}{5 a d \cos ^{\frac{5}{2}}(c+d x)}",1,"-1/30*((2*(45*A*b^3 - 10*a^3*B - 45*a*b^2*B + a^2*b*(19*A + 45*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (4*a*(20*A*b^2 - 20*a*b*B + 3*a^2*(3*A + 5*C))*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(b*(a + b)) + (6*(5*A*b^2 - 5*a*b*B + a^2*(3*A + 5*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]) - (2*(10*a*(-(A*b) + a*B)*Sin[c + d*x] + 3*(5*A*b^2 - 5*a*b*B + a^2*(3*A + 5*C))*Sin[2*(c + d*x)] + 6*a^2*A*Tan[c + d*x]))/Cos[c + d*x]^(3/2))/(a^3*d)","A",1
1102,1,416,318,4.8411678,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*(a + b*Cos[c + d*x])),x]","\frac{\frac{4 a \left(-63 a^3 B+4 a^2 b (22 A+35 C)-140 a b^2 B+140 A b^3\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{b (a+b)}+\frac{2 \left(5 \left(6 a^3 A \tan (c+d x)+a \sin (2 (c+d x)) \left(a^2 (5 A+7 C)-7 a b B+7 A b^2\right)\right)+42 \sin (c+d x) \left(a^2 (a B-A b)+\cos ^2(c+d x) \left(3 a^3 B-a^2 b (3 A+5 C)+5 a b^2 B-5 A b^3\right)\right)\right)}{\cos ^{\frac{5}{2}}(c+d x)}-\frac{42 \sin (c+d x) \left(3 a^3 B-a^2 b (3 A+5 C)+5 a b^2 B-5 A b^3\right) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}+\frac{2 \left(10 a^4 (5 A+7 C)-133 a^3 b B+7 a^2 b^2 (19 A+45 C)-315 a b^3 B+315 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{210 a^4 d}","\frac{2 b^2 \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d (a+b)}-\frac{2 (A b-a B) \sin (c+d x)}{5 a^2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (5 A+7 C)-7 a b B+7 A b^2\right)}{21 a^3 d}+\frac{2 \sin (c+d x) \left(a^2 (5 A+7 C)-7 a b B+7 A b^2\right)}{21 a^3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right)}{5 a^4 d}-\frac{2 \sin (c+d x) \left(-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right)}{5 a^4 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x)}{7 a d \cos ^{\frac{7}{2}}(c+d x)}",1,"((2*(315*A*b^4 - 133*a^3*b*B - 315*a*b^3*B + 10*a^4*(5*A + 7*C) + 7*a^2*b^2*(19*A + 45*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (4*a*(140*A*b^3 - 63*a^3*B - 140*a*b^2*B + 4*a^2*b*(22*A + 35*C))*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(b*(a + b)) - (42*(-5*A*b^3 + 3*a^3*B + 5*a*b^2*B - a^2*b*(3*A + 5*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]) + (2*(42*(a^2*(-(A*b) + a*B) + (-5*A*b^3 + 3*a^3*B + 5*a*b^2*B - a^2*b*(3*A + 5*C))*Cos[c + d*x]^2)*Sin[c + d*x] + 5*(a*(7*A*b^2 - 7*a*b*B + a^2*(5*A + 7*C))*Sin[2*(c + d*x)] + 6*a^3*A*Tan[c + d*x])))/Cos[c + d*x]^(5/2))/(210*a^4*d)","A",1
1103,1,404,445,5.7373505,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{4 \sqrt{\cos (c+d x)} \left(-\frac{15 a^2 \sin (c+d x) \left(a (a C-b B)+A b^2\right)}{\left(a^2-b^2\right) (a+b \cos (c+d x))}+10 (b B-2 a C) \sin (c+d x)+3 b C \sin (2 (c+d x))\right)+\frac{\frac{8 \left(14 a^3 C-10 a^2 b B+a b^2 (15 A+C)-5 b^3 B\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{2 \left(35 a^4 C-25 a^3 b B+a^2 b^2 (15 A-32 C)+40 a b^3 B-6 b^4 (5 A+3 C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{6 \sin (c+d x) \left(35 a^4 C-25 a^3 b B+3 a^2 b^2 (5 A-8 C)+20 a b^3 B-2 b^4 (5 A+3 C)\right) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}}{(a-b) (a+b)}}{60 b^3 d}","-\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(7 a^2 C-5 a b B+5 A b^2-2 b^2 C\right)}{5 b^2 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(-7 a^3 C+5 a^2 b B-a b^2 (3 A-4 C)-2 b^3 B\right)}{3 b^3 d \left(a^2-b^2\right)}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-35 a^4 C+25 a^3 b B-3 a^2 b^2 (5 A-8 C)-20 a b^3 B+2 b^4 (5 A+3 C)\right)}{5 b^4 d \left(a^2-b^2\right)}-\frac{a^2 \left(-7 a^4 C+5 a^3 b B-3 a^2 b^2 (A-3 C)-7 a b^3 B+5 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a-b) (a+b)^2}+\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-21 a^5 C+15 a^4 b B-a^3 b^2 (9 A-20 C)-16 a^2 b^3 B+4 a b^4 (3 A+C)-2 b^5 B\right)}{3 b^5 d \left(a^2-b^2\right)}",1,"(((2*(-25*a^3*b*B + 40*a*b^3*B + a^2*b^2*(15*A - 32*C) + 35*a^4*C - 6*b^4*(5*A + 3*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(-10*a^2*b*B - 5*b^3*B + 14*a^3*C + a*b^2*(15*A + C))*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (6*(-25*a^3*b*B + 20*a*b^3*B + 3*a^2*b^2*(5*A - 8*C) + 35*a^4*C - 2*b^4*(5*A + 3*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b)) + 4*Sqrt[Cos[c + d*x]]*(10*(b*B - 2*a*C)*Sin[c + d*x] - (15*a^2*(A*b^2 + a*(-(b*B) + a*C))*Sin[c + d*x])/((a^2 - b^2)*(a + b*Cos[c + d*x])) + 3*b*C*Sin[2*(c + d*x)]))/(60*b^3*d)","A",1
1104,1,339,343,3.7966116,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)} \left(\frac{3 a \left(a (a C-b B)+A b^2\right)}{\left(a^2-b^2\right) (a+b \cos (c+d x))}+2 C\right)-\frac{\frac{8 \left(2 a^2 C-3 a b B+3 A b^2+b^2 C\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{2 \left(5 a^3 C-3 a^2 b B-a b^2 (3 A+8 C)+6 b^3 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{6 \sin (c+d x) \left(5 a^3 C-3 a^2 b B+a b^2 (A-4 C)+2 b^3 B\right) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}}{(a-b) (a+b)}}{12 b^2 d}","-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(5 a^2 C-3 a b B+3 A b^2-2 b^2 C\right)}{3 b^2 d \left(a^2-b^2\right)}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^3 C+3 a^2 b B-a b^2 (A-4 C)-2 b^3 B\right)}{b^3 d \left(a^2-b^2\right)}-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-15 a^4 C+9 a^3 b B-a^2 b^2 (3 A-16 C)-12 a b^3 B+2 b^4 (3 A+C)\right)}{3 b^4 d \left(a^2-b^2\right)}+\frac{a \left(-5 a^4 C+3 a^3 b B-a^2 b^2 (A-7 C)-5 a b^3 B+3 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a-b) (a+b)^2}",1,"(4*Sqrt[Cos[c + d*x]]*(2*C + (3*a*(A*b^2 + a*(-(b*B) + a*C)))/((a^2 - b^2)*(a + b*Cos[c + d*x])))*Sin[c + d*x] - ((2*(-3*a^2*b*B + 6*b^3*B + 5*a^3*C - a*b^2*(3*A + 8*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(3*A*b^2 - 3*a*b*B + 2*a^2*C + b^2*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (6*(-3*a^2*b*B + 2*b^3*B + a*b^2*(A - 4*C) + 5*a^3*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b)))/(12*b^2*d)","A",1
1105,1,300,251,4.0648565,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","-\frac{\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)} \left(a (a C-b B)+A b^2\right)}{\left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\frac{2 \left(a^2 C+a b B-A b^2-2 b^2 C\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{2 \sin (c+d x) \left(3 a^2 C-a b B+A b^2-2 b^2 C\right) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}+\frac{8 (a (A+C)-b B) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}}{(b-a) (a+b)}}{4 b d}","\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 C-a b B+A b^2-2 b^2 C\right)}{b^2 d \left(a^2-b^2\right)}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-3 a^3 C+a^2 b B+a b^2 (A+4 C)-2 b^3 B\right)}{b^3 d \left(a^2-b^2\right)}-\frac{\left(-3 a^4 C+a^3 b B+a^2 b^2 (A+5 C)-3 a b^3 B+A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}",1,"-1/4*((4*(A*b^2 + a*(-(b*B) + a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*(a + b*Cos[c + d*x])) + ((2*(-(A*b^2) + a*b*B + a^2*C - 2*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(-(b*B) + a*(A + C))*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (2*(A*b^2 - a*b*B + 3*a^2*C - 2*b^2*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/((-a + b)*(a + b)))/(b*d)","A",1
1106,1,297,243,3.4419871,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2),x]","\frac{\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)} \left(a (a C-b B)+A b^2\right)}{\left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\frac{2 \left(a^2 (4 A+C)-a b B-3 A b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}-\frac{2 \sin (c+d x) \left(a (a C-b B)+A b^2\right) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}-\frac{8 a (-a B+A b+b C) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{b (a+b)}}{(a-b) (a+b)}}{4 a d}","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (-C)-a b B+A b^2+2 b^2 C\right)}{b^2 d \left(a^2-b^2\right)}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(A b^2-a (b B-a C)\right)}{a b d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\left(a^4 C+a^3 b B-3 a^2 b^2 (A+C)+a b^3 B+A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b^2 d (a-b) (a+b)^2}",1,"((4*(A*b^2 + a*(-(b*B) + a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*(a + b*Cos[c + d*x])) + ((2*(-3*A*b^2 - a*b*B + a^2*(4*A + C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) - (8*a*(A*b - a*B + b*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(b*(a + b)) - (2*(A*b^2 + a*(-(b*B) + a*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b)))/(4*a*d)","A",1
1107,1,351,306,4.4312492,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2),x]","\frac{\frac{4 \sin (c+d x) \left(b \cos (c+d x) \left(a^2 (2 A-C)+a b B-3 A b^2\right)+2 a A \left(a^2-b^2\right)\right)}{\left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}-\frac{-\frac{8 a \left(a^2 (A-C)+a b B-2 A b^2\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{b (a+b)}-\frac{2 \sin (c+d x) \left(a^2 (2 A-C)+a b B-3 A b^2\right) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}+\frac{2 \left(4 a^3 B-a^2 b (10 A+C)-3 a b^2 B+9 A b^3\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{(b-a) (a+b)}}{4 a^2 d}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(A b^2-a (b B-a C)\right)}{a b d \left(a^2-b^2\right)}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-\left(a^2 (2 A-C)\right)-a b B+3 A b^2\right)}{a^2 d \left(a^2-b^2\right)}-\frac{\sin (c+d x) \left(-\left(a^2 (2 A-C)\right)-a b B+3 A b^2\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}+\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}+\frac{\left(a^4 (-C)+3 a^3 b B-a^2 b^2 (5 A+C)-a b^3 B+3 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 b d (a-b) (a+b)^2}",1,"((4*(2*a*A*(a^2 - b^2) + b*(-3*A*b^2 + a*b*B + a^2*(2*A - C))*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])) - ((2*(9*A*b^3 + 4*a^3*B - 3*a*b^2*B - a^2*b*(10*A + C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) - (8*a*(-2*A*b^2 + a*b*B + a^2*(A - C))*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(b*(a + b)) - (2*(-3*A*b^2 + a*b*B + a^2*(2*A - C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/((-a + b)*(a + b)))/(4*a^2*d)","A",1
1108,1,472,392,7.1672714,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2),x]","\frac{\sqrt{\cos (c+d x)} \left(\frac{2 \sec (c+d x) (a B \sin (c+d x)-2 A b \sin (c+d x))}{a^3}+\frac{2 A \tan (c+d x) \sec (c+d x)}{3 a^2}+\frac{a^2 b^2 C \sin (c+d x)-a b^3 B \sin (c+d x)+A b^4 \sin (c+d x)}{a^3 \left(a^2-b^2\right) (a+b \cos (c+d x))}\right)}{d}+\frac{\frac{2 \sin (c+d x) \cos (2 (c+d x)) \left(-6 a^3 b B+12 a^2 A b^2-3 a^2 b^2 C+9 a b^3 B-15 A b^4\right) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{1-\cos ^2(c+d x)} \left(2 \cos ^2(c+d x)-1\right)}+\frac{\left(-12 a^4 B+28 a^3 A b-12 a^3 b C+24 a^2 b^2 B-40 a A b^3\right) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{b}+\frac{2 \left(4 a^4 A+12 a^4 C-30 a^3 b B+44 a^2 A b^2-9 a^2 b^2 C+27 a b^3 B-45 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{12 a^3 d (a-b) (a+b)}","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-\left(a^2 (2 A-3 C)\right)-3 a b B+5 A b^2\right)}{3 a^2 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{\sin (c+d x) \left(-\left(a^2 (2 A-3 C)\right)-3 a b B+5 A b^2\right)}{3 a^2 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(2 a^3 B-a^2 b (4 A-C)-3 a b^2 B+5 A b^3\right)}{a^3 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \left(2 a^3 B-a^2 b (4 A-C)-3 a b^2 B+5 A b^3\right)}{a^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{\left(-3 a^4 C+5 a^3 b B-a^2 b^2 (7 A-C)-3 a b^3 B+5 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}",1,"((2*(4*a^4*A + 44*a^2*A*b^2 - 45*A*b^4 - 30*a^3*b*B + 27*a*b^3*B + 12*a^4*C - 9*a^2*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + ((28*a^3*A*b - 40*a*A*b^3 - 12*a^4*B + 24*a^2*b^2*B - 12*a^3*b*C)*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)))/b + (2*(12*a^2*A*b^2 - 15*A*b^4 - 6*a^3*b*B + 9*a*b^3*B - 3*a^2*b^2*C)*Cos[2*(c + d*x)]*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[1 - Cos[c + d*x]^2]*(-1 + 2*Cos[c + d*x]^2)))/(12*a^3*(a - b)*(a + b)*d) + (Sqrt[Cos[c + d*x]]*((2*Sec[c + d*x]*(-2*A*b*Sin[c + d*x] + a*B*Sin[c + d*x]))/a^3 + (A*b^4*Sin[c + d*x] - a*b^3*B*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x])/(a^3*(a^2 - b^2)*(a + b*Cos[c + d*x])) + (2*A*Sec[c + d*x]*Tan[c + d*x])/(3*a^2)))/d","A",1
1109,1,551,654,7.694045,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{4 \sqrt{\cos (c+d x)} \left(\frac{30 a^3 \sin (c+d x) \left(a (a C-b B)+A b^2\right)}{\left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{15 a^2 \sin (c+d x) \left(15 a^4 C-11 a^3 b B+7 a^2 b^2 (A-3 C)+17 a b^3 B-13 A b^4\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+40 (b B-3 a C) \sin (c+d x)+12 b C \sin (2 (c+d x))\right)+\frac{\frac{16 \left(63 a^5 C-35 a^4 b B+3 a^3 b^2 (5 A-32 C)+70 a^2 b^3 B-12 a b^4 (5 A+C)+10 b^5 B\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{2 \left(315 a^6 C-175 a^5 b B+3 a^4 b^2 (25 A-211 C)+365 a^3 b^3 B-21 a^2 b^4 (5 A-16 C)-280 a b^5 B+24 b^6 (5 A+3 C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{6 \sin (c+d x) \left(315 a^6 C-175 a^5 b B+3 a^4 b^2 (25 A-187 C)+325 a^3 b^3 B+a^2 b^4 (192 C-145 A)-120 a b^5 B+8 b^6 (5 A+3 C)\right) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{240 b^4 d}","-\frac{\sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(-9 a^4 C+5 a^3 b B-a^2 b^2 (A-15 C)-11 a b^3 B+7 A b^4\right)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(-63 a^4 C+35 a^3 b B-a^2 b^2 (15 A-101 C)-65 a b^3 B+b^4 (45 A-8 C)\right)}{20 b^3 d \left(a^2-b^2\right)^2}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(-63 a^5 C+35 a^4 b B-15 a^3 b^2 (A-7 C)-61 a^2 b^3 B+3 a b^4 (11 A-8 C)+8 b^5 B\right)}{12 b^4 d \left(a^2-b^2\right)^2}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-315 a^6 C+175 a^5 b B-3 a^4 b^2 (25 A-187 C)-325 a^3 b^3 B+a^2 b^4 (145 A-192 C)+120 a b^5 B-8 b^6 (5 A+3 C)\right)}{20 b^5 d \left(a^2-b^2\right)^2}+\frac{a^2 \left(63 a^6 C-35 a^5 b B+15 a^4 b^2 (A-10 C)+86 a^3 b^3 B-a^2 b^4 (38 A-99 C)-63 a b^5 B+35 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^6 d (a-b)^2 (a+b)^3}+\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-189 a^7 C+105 a^6 b B-9 a^5 b^2 (5 A-43 C)-223 a^4 b^3 B+3 a^3 b^4 (33 A-64 C)+128 a^2 b^5 B-24 a b^6 (3 A+C)+8 b^7 B\right)}{12 b^6 d \left(a^2-b^2\right)^2}",1,"(((2*(-175*a^5*b*B + 365*a^3*b^3*B - 280*a*b^5*B + 3*a^4*b^2*(25*A - 211*C) - 21*a^2*b^4*(5*A - 16*C) + 315*a^6*C + 24*b^6*(5*A + 3*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (16*(-35*a^4*b*B + 70*a^2*b^3*B + 10*b^5*B + 3*a^3*b^2*(5*A - 32*C) + 63*a^5*C - 12*a*b^4*(5*A + C))*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (6*(-175*a^5*b*B + 325*a^3*b^3*B - 120*a*b^5*B + 3*a^4*b^2*(25*A - 187*C) + 315*a^6*C + 8*b^6*(5*A + 3*C) + a^2*b^4*(-145*A + 192*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2) + 4*Sqrt[Cos[c + d*x]]*(40*(b*B - 3*a*C)*Sin[c + d*x] + (30*a^3*(A*b^2 + a*(-(b*B) + a*C))*Sin[c + d*x])/((a^2 - b^2)*(a + b*Cos[c + d*x])^2) - (15*a^2*(-13*A*b^4 - 11*a^3*b*B + 17*a*b^3*B + 7*a^2*b^2*(A - 3*C) + 15*a^4*C)*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])) + 12*b*C*Sin[2*(c + d*x)]))/(240*b^4*d)","A",1
1110,1,520,536,6.2381212,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)} \left(35 a^6 C-15 a^5 b B+3 a^4 A b^2-57 a^4 b^2 C+33 a^3 b^3 B-21 a^2 A b^4+4 C \left(b^3-a^2 b\right)^2 \cos (2 (c+d x))+a b \cos (c+d x) \left(49 a^4 C-21 a^3 b B+a^2 b^2 (9 A-83 C)+39 a b^3 B+b^4 (16 C-27 A)\right)+4 b^6 C\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{-\frac{16 \left(-7 a^4 C+3 a^3 b B+a^2 b^2 (3 A+14 C)-12 a b^3 B+2 b^4 (3 A+C)\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{2 \left(35 a^5 C-15 a^4 b B+a^3 b^2 (3 A-73 C)+21 a^2 b^3 B+a b^4 (15 A+56 C)-24 b^5 B\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{6 \sin (c+d x) \left(35 a^5 C-15 a^4 b B+a^3 b^2 (3 A-65 C)+29 a^2 b^3 B+3 a b^4 (8 C-3 A)-8 b^5 B\right) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{48 b^3 d}","-\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right)}{12 b^3 d \left(a^2-b^2\right)^2}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right)}{4 b^4 d \left(a^2-b^2\right)^2}-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-105 a^6 C+45 a^5 b B-a^4 b^2 (9 A-223 C)-99 a^3 b^3 B+a^2 b^4 (15 A-128 C)+72 a b^5 B-8 b^6 (3 A+C)\right)}{12 b^5 d \left(a^2-b^2\right)^2}-\frac{a \left(35 a^6 C-15 a^5 b B+a^4 b^2 (3 A-86 C)+38 a^3 b^3 B-3 a^2 b^4 (2 A-21 C)-35 a b^5 B+15 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^5 d (a-b)^2 (a+b)^3}",1,"((4*Sqrt[Cos[c + d*x]]*(3*a^4*A*b^2 - 21*a^2*A*b^4 - 15*a^5*b*B + 33*a^3*b^3*B + 35*a^6*C - 57*a^4*b^2*C + 4*b^6*C + a*b*(-21*a^3*b*B + 39*a*b^3*B + a^2*b^2*(9*A - 83*C) + 49*a^4*C + b^4*(-27*A + 16*C))*Cos[c + d*x] + 4*(-(a^2*b) + b^3)^2*C*Cos[2*(c + d*x)])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) - ((2*(-15*a^4*b*B + 21*a^2*b^3*B - 24*b^5*B + a^3*b^2*(3*A - 73*C) + 35*a^5*C + a*b^4*(15*A + 56*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) - (16*(3*a^3*b*B - 12*a*b^3*B - 7*a^4*C + 2*b^4*(3*A + C) + a^2*b^2*(3*A + 14*C))*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (6*(-15*a^4*b*B + 29*a^2*b^3*B - 8*b^5*B + a^3*b^2*(3*A - 65*C) + 35*a^5*C + 3*a*b^4*(-3*A + 8*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(48*b^3*d)","A",1
1111,1,437,423,6.4779848,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(b \cos (c+d x) \left(-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right)+a \left(-5 a^4 C+a^3 b B+a^2 b^2 (3 A+11 C)-7 a b^3 B+3 A b^4\right)\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\frac{8 \left(a^3 C+a^2 b B-a b^2 (3 A+4 C)+2 b^3 B\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{\left(5 a^4 C-a^3 b B+a^2 b^2 (5 A-7 C)-5 a b^3 B+b^4 (A+8 C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{\sin (c+d x) \left(15 a^4 C-3 a^3 b B-a^2 b^2 (A+29 C)+9 a b^3 B+b^4 (8 C-5 A)\right) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{8 b^2 d}","-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-15 a^4 C+3 a^3 b B+a^2 b^2 (A+29 C)-9 a b^3 B+b^4 (5 A-8 C)\right)}{4 b^3 d \left(a^2-b^2\right)^2}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(-5 a^4 C+a^3 b B+a^2 b^2 (3 A+11 C)-7 a b^3 B+3 A b^4\right)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-15 a^5 C+3 a^4 b B+a^3 b^2 (A+33 C)-5 a^2 b^3 B-a b^4 (7 A+24 C)+8 b^5 B\right)}{4 b^4 d \left(a^2-b^2\right)^2}+\frac{\left(15 a^6 C-3 a^5 b B-a^4 b^2 (A+38 C)+6 a^3 b^3 B+5 a^2 b^4 (2 A+7 C)-15 a b^5 B+3 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d (a-b)^2 (a+b)^3}",1,"((2*Sqrt[Cos[c + d*x]]*(a*(3*A*b^4 + a^3*b*B - 7*a*b^3*B - 5*a^4*C + a^2*b^2*(3*A + 11*C)) + b*(5*A*b^4 + 3*a^3*b*B - 9*a*b^3*B - 7*a^4*C + a^2*b^2*(A + 13*C))*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) + (((-(a^3*b*B) - 5*a*b^3*B + a^2*b^2*(5*A - 7*C) + 5*a^4*C + b^4*(A + 8*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(a^2*b*B + 2*b^3*B + a^3*C - a*b^2*(3*A + 4*C))*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + ((-3*a^3*b*B + 9*a*b^3*B + 15*a^4*C + b^4*(-5*A + 8*C) - a^2*b^2*(A + 29*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(8*b^2*d)","A",1
1112,1,425,418,4.5722478,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(b \cos (c+d x) \left(3 a^4 C+a^3 b B-a^2 b^2 (5 A+9 C)+5 a b^3 B-A b^4\right)+a \left(a^4 C+3 a^3 b B-7 a^2 b^2 (A+C)+3 a b^3 B+A b^4\right)\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{-\frac{8 a \left(a^2 (2 A+C)-3 a b B+b^2 (A+2 C)\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{\left(a^4 C-5 a^3 b B+a^2 b^2 (9 A+5 C)-a b^3 B-3 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{\sin (c+d x) \left(3 a^4 C+a^3 b B-a^2 b^2 (5 A+9 C)+5 a b^3 B-A b^4\right) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{8 a b d}","-\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^4 C+a^3 b B+a^2 b^2 (3 A-5 C)-7 a b^3 B+b^4 (3 A+8 C)\right)}{4 b^3 d \left(a^2-b^2\right)^2}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right)}{4 a b^2 d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right)}{4 a b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(-3 a^6 C-a^5 b B-3 a^4 b^2 (A-2 C)+10 a^3 b^3 B-5 a^2 b^4 (2 A+3 C)+3 a b^5 B+A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^3 d (a-b)^2 (a+b)^3}",1,"((2*Sqrt[Cos[c + d*x]]*(a*(A*b^4 + 3*a^3*b*B + 3*a*b^3*B + a^4*C - 7*a^2*b^2*(A + C)) + b*(-(A*b^4) + a^3*b*B + 5*a*b^3*B + 3*a^4*C - a^2*b^2*(5*A + 9*C))*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) - (((-3*A*b^4 - 5*a^3*b*B - a*b^3*B + a^4*C + a^2*b^2*(9*A + 5*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) - (8*a*(-3*a*b*B + a^2*(2*A + C) + b^2*(A + 2*C))*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + ((-(A*b^4) + a^3*b*B + 5*a*b^3*B + 3*a^4*C - a^2*b^2*(5*A + 9*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(8*a*b*d)","A",1
1113,1,439,413,6.2519719,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3),x]","\frac{\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)} \left(b \cos (c+d x) \left(a^4 C-5 a^3 b B+a^2 b^2 (9 A+5 C)-a b^3 B-3 A b^4\right)+a \left(3 a^4 C-7 a^3 b B+a^2 b^2 (11 A+3 C)+a b^3 B-5 A b^4\right)\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\frac{16 a \left(2 a^3 B-a^2 b (4 A+3 C)+a b^2 B+A b^3\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{b (a+b)}+\frac{2 \left(a^4 (16 A+5 C)-9 a^3 b B+a^2 b^2 (C-19 A)+3 a b^3 B+9 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}-\frac{2 \sin (c+d x) \left(a^4 C-5 a^3 b B+a^2 b^2 (9 A+5 C)-a b^3 B-3 A b^4\right) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{16 a^2 d}","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 C+3 a^3 b B-7 a^2 b^2 (A+C)+3 a b^3 B+A b^4\right)}{4 a b^2 d \left(a^2-b^2\right)^2}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (-C)+5 a^3 b B-a^2 b^2 (9 A+5 C)+a b^3 B+3 A b^4\right)}{4 a^2 b d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(a^4 (-C)+5 a^3 b B-a^2 b^2 (9 A+5 C)+a b^3 B+3 A b^4\right)}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(a^6 (-C)-3 a^5 b B+5 a^4 b^2 (3 A+2 C)-10 a^3 b^3 B-3 a^2 b^4 (2 A-C)+a b^5 B+3 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b^2 d (a-b)^2 (a+b)^3}",1,"((4*Sqrt[Cos[c + d*x]]*(a*(-5*A*b^4 - 7*a^3*b*B + a*b^3*B + 3*a^4*C + a^2*b^2*(11*A + 3*C)) + b*(-3*A*b^4 - 5*a^3*b*B - a*b^3*B + a^4*C + a^2*b^2*(9*A + 5*C))*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) + ((2*(9*A*b^4 - 9*a^3*b*B + 3*a*b^3*B + a^2*b^2*(-19*A + C) + a^4*(16*A + 5*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (16*a*(A*b^3 + 2*a^3*B + a*b^2*B - a^2*b*(4*A + 3*C))*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(b*(a + b)) - (2*(-3*A*b^4 - 5*a^3*b*B - a*b^3*B + a^4*C + a^2*b^2*(9*A + 5*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(16*a^2*d)","A",1
1114,1,506,502,7.2307451,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3),x]","\frac{\frac{2 \sqrt{\cos (c+d x)} \left(16 A \left(a^3-a b^2\right)^2 \tan (c+d x)+b^2 \sin (2 (c+d x)) \left(a^4 (8 A-5 C)+9 a^3 b B-a^2 b^2 (29 A+C)-3 a b^3 B+15 A b^4\right)+2 a b \sin (c+d x) \left(a^4 (16 A-7 C)+11 a^3 b B+a^2 b^2 (C-47 A)-5 a b^3 B+25 A b^4\right)\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{\frac{16 a \left(2 a^4 (A-C)+4 a^3 b B-a^2 b^2 (10 A+C)-a b^3 B+5 A b^4\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{b (a+b)}+\frac{2 \sin (c+d x) \left(a^4 (8 A-5 C)+9 a^3 b B-a^2 b^2 (29 A+C)-3 a b^3 B+15 A b^4\right) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}-\frac{2 \left(16 a^5 B-a^4 b (56 A+9 C)-19 a^3 b^2 B+a^2 b^3 (95 A+3 C)+9 a b^4 B-45 A b^5\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{(a-b)^2 (a+b)^2}}{16 a^3 d}","\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-3 a^4 C+7 a^3 b B-a^2 b^2 (11 A+3 C)-a b^3 B+5 A b^4\right)}{4 a^2 b d \left(a^2-b^2\right)^2}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (8 A-5 C)+9 a^3 b B-a^2 b^2 (29 A+C)-3 a b^3 B+15 A b^4\right)}{4 a^3 d \left(a^2-b^2\right)^2}+\frac{\sin (c+d x) \left(a^4 (8 A-5 C)+9 a^3 b B-a^2 b^2 (29 A+C)-3 a b^3 B+15 A b^4\right)}{4 a^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x) \left(-3 a^4 C+7 a^3 b B-a^2 b^2 (11 A+3 C)-a b^3 B+5 A b^4\right)}{4 a^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}-\frac{\left(3 a^6 C-15 a^5 b B+5 a^4 b^2 (7 A+2 C)+6 a^3 b^3 B-a^2 b^4 (38 A+C)-3 a b^5 B+15 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 b d (a-b)^2 (a+b)^3}",1,"(-(((-2*(-45*A*b^5 + 16*a^5*B - 19*a^3*b^2*B + 9*a*b^4*B + a^2*b^3*(95*A + 3*C) - a^4*b*(56*A + 9*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (16*a*(5*A*b^4 + 4*a^3*b*B - a*b^3*B + 2*a^4*(A - C) - a^2*b^2*(10*A + C))*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(b*(a + b)) + (2*(15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B + a^4*(8*A - 5*C) - a^2*b^2*(29*A + C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2)) + (2*Sqrt[Cos[c + d*x]]*(2*a*b*(25*A*b^4 + 11*a^3*b*B - 5*a*b^3*B + a^4*(16*A - 7*C) + a^2*b^2*(-47*A + C))*Sin[c + d*x] + b^2*(15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B + a^4*(8*A - 5*C) - a^2*b^2*(29*A + C))*Sin[2*(c + d*x)] + 16*A*(a^3 - a*b^2)^2*Tan[c + d*x]))/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2))/(16*a^3*d)","A",1
1115,1,672,609,7.7637603,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^3),x]","\frac{\sqrt{\cos (c+d x)} \left(\frac{2 \sec (c+d x) (a B \sin (c+d x)-3 A b \sin (c+d x))}{a^4}+\frac{2 A \tan (c+d x) \sec (c+d x)}{3 a^3}+\frac{a^2 b^2 C \sin (c+d x)-a b^3 B \sin (c+d x)+A b^4 \sin (c+d x)}{2 a^3 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{9 a^4 b^2 C \sin (c+d x)-13 a^3 b^3 B \sin (c+d x)+17 a^2 A b^4 \sin (c+d x)-3 a^2 b^4 C \sin (c+d x)+7 a b^5 B \sin (c+d x)-11 A b^6 \sin (c+d x)}{4 a^4 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}\right)}{d}+\frac{\frac{2 \sin (c+d x) \cos (2 (c+d x)) \left(-24 a^5 b B+72 a^4 A b^2-27 a^4 b^2 C+87 a^3 b^3 B-195 a^2 A b^4+9 a^2 b^4 C-45 a b^5 B+105 A b^6\right) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{1-\cos ^2(c+d x)} \left(2 \cos ^2(c+d x)-1\right)}+\frac{\left(-48 a^6 B+160 a^5 A b-96 a^5 b C+240 a^4 b^2 B-512 a^3 A b^3+24 a^3 b^3 C-120 a^2 b^4 B+280 a A b^5\right) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{b}+\frac{2 \left(16 a^6 A+48 a^6 C-168 a^5 b B+328 a^4 A b^2-57 a^4 b^2 C+285 a^3 b^3 B-641 a^2 A b^4+27 a^2 b^4 C-135 a b^5 B+315 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{48 a^4 d (a-b)^2 (a+b)^2}","\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right)}{12 a^3 d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x) \left(-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right)}{4 a^2 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \left(a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right)}{12 a^3 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x)}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right)}{4 a^4 d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x) \left(-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right)}{4 a^4 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}+\frac{\left(15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}",1,"((2*(16*a^6*A + 328*a^4*A*b^2 - 641*a^2*A*b^4 + 315*A*b^6 - 168*a^5*b*B + 285*a^3*b^3*B - 135*a*b^5*B + 48*a^6*C - 57*a^4*b^2*C + 27*a^2*b^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + ((160*a^5*A*b - 512*a^3*A*b^3 + 280*a*A*b^5 - 48*a^6*B + 240*a^4*b^2*B - 120*a^2*b^4*B - 96*a^5*b*C + 24*a^3*b^3*C)*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)))/b + (2*(72*a^4*A*b^2 - 195*a^2*A*b^4 + 105*A*b^6 - 24*a^5*b*B + 87*a^3*b^3*B - 45*a*b^5*B - 27*a^4*b^2*C + 9*a^2*b^4*C)*Cos[2*(c + d*x)]*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[1 - Cos[c + d*x]^2]*(-1 + 2*Cos[c + d*x]^2)))/(48*a^4*(a - b)^2*(a + b)^2*d) + (Sqrt[Cos[c + d*x]]*((2*Sec[c + d*x]*(-3*A*b*Sin[c + d*x] + a*B*Sin[c + d*x]))/a^4 + (A*b^4*Sin[c + d*x] - a*b^3*B*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x])/(2*a^3*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) + (17*a^2*A*b^4*Sin[c + d*x] - 11*A*b^6*Sin[c + d*x] - 13*a^3*b^3*B*Sin[c + d*x] + 7*a*b^5*B*Sin[c + d*x] + 9*a^4*b^2*C*Sin[c + d*x] - 3*a^2*b^4*C*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) + (2*A*Sec[c + d*x]*Tan[c + d*x])/(3*a^3)))/d","A",0
1116,1,1242,586,6.4261818,"\int \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{-\frac{4 a \left(-C a^2+18 b B a+24 A b^2+16 b^2 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(24 B b^2+48 a A b+28 a C b\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-3 C a^2+6 b B a+24 A b^2+16 b^2 C\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{48 b d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{(6 b B+a C) \sin (c+d x)}{12 b}+\frac{1}{6} C \sin (2 (c+d x))\right)}{d}","\frac{\sqrt{a+b} \cot (c+d x) \left(a^3 (-C)+2 a^2 b B-4 a b^2 (2 A+C)-8 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^3 d}+\frac{\sin (c+d x) \left(3 a (2 b B-a C)+8 b^2 (3 A+2 C)\right) \sqrt{a+b \cos (c+d x)}}{24 b^2 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \cot (c+d x) \left((a+2 b) (-3 a C+6 b B+8 b C)+24 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^2 d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(3 a (2 b B-a C)+8 b^2 (3 A+2 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^2 d}+\frac{(2 b B-a C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 b d}",1,"((-4*a*(24*A*b^2 + 18*a*b*B - a^2*C + 16*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(48*a*A*b + 24*b^2*B + 28*a*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(24*A*b^2 + 6*a*b*B - 3*a^2*C + 16*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(48*b*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(((6*b*B + a*C)*Sin[c + d*x])/(12*b) + (C*Sin[2*(c + d*x)])/6))/d","C",0
1117,1,1183,483,6.3418098,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{C \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{2 d}+\frac{-\frac{4 a (8 a A+4 b B+3 a C) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a (8 A b+4 C b+8 a B) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 (4 b B+a C) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{8 d}","-\frac{\sqrt{a+b} \cot (c+d x) \left(a^2 (-C)+4 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}+\frac{\sqrt{a+b} \cot (c+d x) (a C+8 A b+2 b (2 B+C)) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}+\frac{(a C+4 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b d \sqrt{\cos (c+d x)}}-\frac{(a-b) \sqrt{a+b} (a C+4 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}",1,"(C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + ((-4*a*(8*a*A + 4*b*B + 3*a*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(8*A*b + 8*a*B + 4*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(4*b*B + a*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(8*d)","C",1
1118,1,1176,449,20.7855723,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{2 A \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{-\frac{4 a (2 a B+b C) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a (-2 a A+2 b B+2 a C) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 (b C-2 A b) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{2 d}","\frac{\sqrt{a+b} \cot (c+d x) (2 A b-a (2 A-2 B-C)) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}-\frac{(2 A-C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{(a-b) \sqrt{a+b} (2 A-C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} (a C+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d}",1,"(2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + ((-4*a*(2*a*B + b*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-2*a*A + 2*b*B + 2*a*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-2*A*b + b*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(2*d)","C",1
1119,1,1240,407,6.3863655,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{-\frac{4 a \left(A a^2+3 C a^2-A b^2\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-3 B a^2-A b a+3 b C a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-A b^2-3 a B b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{3 a d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \sec (c+d x) (A b \sin (c+d x)+3 a B \sin (c+d x))}{3 a}+\frac{2}{3} A \sec (c+d x) \tan (c+d x)\right)}{d}","\frac{2 (a-b) \sqrt{a+b} (3 a B+A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d}-\frac{2 \sqrt{a+b} \cot (c+d x) (b (A-3 B)-a (A-3 B+3 C)) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}",1,"((-4*a*(a^2*A - A*b^2 + 3*a^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-(a*A*b) - 3*a^2*B + 3*a*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-(A*b^2) - 3*a*b*B)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(3*a*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]*(A*b*Sin[c + d*x] + 3*a*B*Sin[c + d*x]))/(3*a) + (2*A*Sec[c + d*x]*Tan[c + d*x])/3))/d","C",1
1120,1,1340,360,6.498594,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 (A b \sin (c+d x)+5 a B \sin (c+d x)) \sec ^2(c+d x)}{15 a}+\frac{2}{5} A \tan (c+d x) \sec ^2(c+d x)+\frac{2 \left(9 A \sin (c+d x) a^2+15 C \sin (c+d x) a^2+5 b B \sin (c+d x) a-2 A b^2 \sin (c+d x)\right) \sec (c+d x)}{15 a^2}\right)}{d}-\frac{-\frac{4 a \left(-5 B a^3+2 A b a^2+5 b^2 B a-2 A b^3\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(9 A a^3+15 C a^3+5 b B a^2-2 A b^2 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-2 A b^3+5 a B b^2+9 a^2 A b+15 a^2 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{15 a^2 d}","-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) (a (9 A-5 B+15 C)+2 A b) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^2 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-3 a^2 (3 A+5 C)-5 a b B+2 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d}+\frac{2 (5 a B+A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"-1/15*((-4*a*(2*a^2*A*b - 2*A*b^3 - 5*a^3*B + 5*a*b^2*B)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(9*a^3*A - 2*a*A*b^2 + 5*a^2*b*B + 15*a^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(9*a^2*A*b - 2*A*b^3 + 5*a*b^2*B + 15*a^2*b*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a^2*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^2*(A*b*Sin[c + d*x] + 5*a*B*Sin[c + d*x]))/(15*a) + (2*Sec[c + d*x]*(9*a^2*A*Sin[c + d*x] - 2*A*b^2*Sin[c + d*x] + 5*a*b*B*Sin[c + d*x] + 15*a^2*C*Sin[c + d*x]))/(15*a^2) + (2*A*Sec[c + d*x]^2*Tan[c + d*x])/5))/d","C",1
1121,1,1464,447,6.6929992,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{-\frac{4 a \left(25 A a^4+35 C a^4-14 b B a^3-17 A b^2 a^2-35 b^2 C a^2+14 b^3 B a-8 A b^4\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-63 B a^4-19 A b a^3-35 b C a^3+14 b^2 B a^2-8 A b^3 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-8 A b^4+14 a B b^3-19 a^2 A b^2-35 a^2 C b^2-63 a^3 B b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{105 a^3 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 (A b \sin (c+d x)+7 a B \sin (c+d x)) \sec ^3(c+d x)}{35 a}+\frac{2}{7} A \tan (c+d x) \sec ^3(c+d x)+\frac{2 \left(25 A \sin (c+d x) a^2+35 C \sin (c+d x) a^2+7 b B \sin (c+d x) a-4 A b^2 \sin (c+d x)\right) \sec ^2(c+d x)}{105 a^2}+\frac{2 \left(63 B \sin (c+d x) a^3+19 A b \sin (c+d x) a^2+35 b C \sin (c+d x) a^2-14 b^2 B \sin (c+d x) a+8 A b^3 \sin (c+d x)\right) \sec (c+d x)}{105 a^3}\right)}{d}","-\frac{2 \sin (c+d x) \left(-5 a^2 (5 A+7 C)-7 a b B+4 A b^2\right) \sqrt{a+b \cos (c+d x)}}{105 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(a^2 (25 A-63 B+35 C)+2 a b (3 A-7 B)+8 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(63 a^3 B+a^2 b (19 A+35 C)-14 a b^2 B+8 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d}+\frac{2 (7 a B+A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"((-4*a*(25*a^4*A - 17*a^2*A*b^2 - 8*A*b^4 - 14*a^3*b*B + 14*a*b^3*B + 35*a^4*C - 35*a^2*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-19*a^3*A*b - 8*a*A*b^3 - 63*a^4*B + 14*a^2*b^2*B - 35*a^3*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-19*a^2*A*b^2 - 8*A*b^4 - 63*a^3*b*B + 14*a*b^3*B - 35*a^2*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(105*a^3*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^3*(A*b*Sin[c + d*x] + 7*a*B*Sin[c + d*x]))/(35*a) + (2*Sec[c + d*x]^2*(25*a^2*A*Sin[c + d*x] - 4*A*b^2*Sin[c + d*x] + 7*a*b*B*Sin[c + d*x] + 35*a^2*C*Sin[c + d*x]))/(105*a^2) + (2*Sec[c + d*x]*(19*a^2*A*b*Sin[c + d*x] + 8*A*b^3*Sin[c + d*x] + 63*a^3*B*Sin[c + d*x] - 14*a*b^2*B*Sin[c + d*x] + 35*a^2*b*C*Sin[c + d*x]))/(105*a^3) + (2*A*Sec[c + d*x]^3*Tan[c + d*x])/7))/d","C",0
1122,1,1317,704,6.5626067,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{\left(3 C a^2+56 b B a+48 A b^2+42 b^2 C\right) \sin (c+d x)}{96 b}+\frac{1}{48} (8 b B+9 a C) \sin (2 (c+d x))+\frac{1}{16} b C \sin (3 (c+d x))\right)}{d}-\frac{-\frac{4 a \left(3 C a^3-136 b B a^2-336 A b^2 a-228 b^2 C a-128 b^3 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-192 A b^3-144 C b^3-416 a B b^2-384 a^2 A b-228 a^2 C b\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(9 C a^3-24 b B a^2-240 A b^2 a-156 b^2 C a-128 b^3 B\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{384 b d}","\frac{\sin (c+d x) \left(-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{192 b^2 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \cot (c+d x) \left(9 a^3 C-6 a^2 b (4 B+C)-4 a b^2 (60 A+28 B+39 C)-8 b^3 (12 A+16 B+9 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b^2 d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 a b^2 d}+\frac{\sqrt{a+b} \cot (c+d x) \left(-3 a^4 C+8 a^3 b B-24 a^2 b^2 (2 A+C)-96 a b^3 B-16 b^4 (4 A+3 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^3 d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right) \sqrt{a+b \cos (c+d x)}}{32 b d}+\frac{(8 b B-3 a C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{24 b d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}",1,"-1/384*((-4*a*(-336*a*A*b^2 - 136*a^2*b*B - 128*b^3*B + 3*a^3*C - 228*a*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-384*a^2*A*b - 192*A*b^3 - 416*a*b^2*B - 228*a^2*b*C - 144*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-240*a*A*b^2 - 24*a^2*b*B - 128*b^3*B + 9*a^3*C - 156*a*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(b*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(((48*A*b^2 + 56*a*b*B + 3*a^2*C + 42*b^2*C)*Sin[c + d*x])/(96*b) + ((8*b*B + 9*a*C)*Sin[2*(c + d*x)])/48 + (b*C*Sin[3*(c + d*x)])/16))/d","C",0
1123,1,1250,587,6.5966721,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{-\frac{4 a \left(48 A a^2+17 C a^2+42 b B a+24 A b^2+16 b^2 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(48 B a^2+96 A b a+52 b C a+24 b^2 B\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(3 C a^2+30 b B a+24 A b^2+16 b^2 C\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{48 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{1}{12} (6 b B+7 a C) \sin (c+d x)+\frac{1}{6} b C \sin (2 (c+d x))\right)}{d}","\frac{\sin (c+d x) \left(3 a^2 C+30 a b B+24 A b^2+16 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{24 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \cot (c+d x) \left(3 a^2 C+2 a b (24 A+15 B+7 C)+4 b^2 (6 A+3 B+4 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(3 a^2 C+30 a b B+24 A b^2+16 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b d}-\frac{\sqrt{a+b} \cot (c+d x) \left(a^3 (-C)+6 a^2 b B+12 a b^2 (2 A+C)+8 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^2 d}+\frac{(a C+2 b B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}",1,"((-4*a*(48*a^2*A + 24*A*b^2 + 42*a*b*B + 17*a^2*C + 16*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(96*a*A*b + 48*a^2*B + 24*b^2*B + 52*a*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(48*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(((6*b*B + 7*a*C)*Sin[c + d*x])/12 + (b*C*Sin[2*(c + d*x)])/6))/d","C",0
1124,1,1232,535,6.5668645,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{\frac{4 a \left(-8 B a^2-8 A b a-7 b C a-4 b^2 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+4 a \left(8 A a^2-8 C a^2-16 b B a-8 A b^2-4 b^2 C\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)-2 \left(-4 B b^2+8 a A b-5 a C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{8 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{1}{2} b C \sin (c+d x)+2 a A \tan (c+d x)\right)}{d}","-\frac{\sqrt{a+b} \cot (c+d x) \left(3 a^2 C+12 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}-\frac{\sin (c+d x) (8 a A-5 a C-4 b B) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \cot (c+d x) (a (8 A-8 B-5 C)-2 b (8 A+2 B+C)) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) (8 a A-5 a C-4 b B) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a d}-\frac{b (4 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt{\cos (c+d x)}}",1,"((4*a*(-8*a*A*b - 8*a^2*B - 4*b^2*B - 7*a*b*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) + 4*a*(8*a^2*A - 8*A*b^2 - 16*a*b*B - 8*a^2*C - 4*b^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) - 2*(8*a*A*b - 4*b^2*B - 5*a*b*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(8*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((b*C*Sin[c + d*x])/2 + 2*a*A*Tan[c + d*x]))/d","C",0
1125,1,1260,528,6.5637447,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{-\frac{4 a \left(2 A a^2+6 C a^2+6 b B a-2 A b^2+3 b^2 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-6 B a^2-8 A b a+12 b C a+6 b^2 B\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-8 A b^2+3 C b^2-6 a B b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{6 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{3} \sec (c+d x) (4 A b \sin (c+d x)+3 a B \sin (c+d x))+\frac{2}{3} a A \sec (c+d x) \tan (c+d x)\right)}{d}","\frac{\sqrt{a+b} \cot (c+d x) \left(2 a^2 (A-3 B+3 C)-a b (8 A-3 (4 B+C))+6 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}-\frac{\sin (c+d x) (6 a B+8 A b-3 b C) \sqrt{a+b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) (6 a B+8 A b-3 b C) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}+\frac{2 (a B+A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{\sqrt{a+b} (3 a C+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}",1,"((-4*a*(2*a^2*A - 2*A*b^2 + 6*a*b*B + 6*a^2*C + 3*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-8*a*A*b - 6*a^2*B + 6*b^2*B + 12*a*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-8*A*b^2 - 6*a*b*B + 3*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(6*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]*(4*A*b*Sin[c + d*x] + 3*a*B*Sin[c + d*x]))/3 + (2*a*A*Sec[c + d*x]*Tan[c + d*x])/3))/d","C",1
1126,1,1353,490,6.6733917,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{15} (6 A b \sin (c+d x)+5 a B \sin (c+d x)) \sec ^2(c+d x)+\frac{2}{5} a A \tan (c+d x) \sec ^2(c+d x)+\frac{2 \left(9 A \sin (c+d x) a^2+15 C \sin (c+d x) a^2+20 b B \sin (c+d x) a+3 A b^2 \sin (c+d x)\right) \sec (c+d x)}{15 a}\right)}{d}-\frac{-\frac{4 a \left(-5 B a^3-3 A b a^2-15 b C a^2+5 b^2 B a+3 A b^3\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(9 A a^3+15 C a^3+20 b B a^2+3 A b^2 a-15 b^2 C a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(3 A b^3+20 a B b^2+9 a^2 A b+15 a^2 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{15 a d}","-\frac{2 \sqrt{a+b} \cot (c+d x) \left(a^2 (9 A-5 B+15 C)-2 a b (6 A-10 B+15 C)+3 b^2 (A-5 B)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(3 a^2 (3 A+5 C)+20 a b B+3 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^2 d}+\frac{2 (5 a B+3 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 b C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}",1,"-1/15*((-4*a*(-3*a^2*A*b + 3*A*b^3 - 5*a^3*B + 5*a*b^2*B - 15*a^2*b*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(9*a^3*A + 3*a*A*b^2 + 20*a^2*b*B + 15*a^3*C - 15*a*b^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(9*a^2*A*b + 3*A*b^3 + 20*a*b^2*B + 15*a^2*b*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^2*(6*A*b*Sin[c + d*x] + 5*a*B*Sin[c + d*x]))/15 + (2*Sec[c + d*x]*(9*a^2*A*Sin[c + d*x] + 3*A*b^2*Sin[c + d*x] + 20*a*b*B*Sin[c + d*x] + 15*a^2*C*Sin[c + d*x]))/(15*a) + (2*a*A*Sec[c + d*x]^2*Tan[c + d*x])/5))/d","C",0
1127,1,1463,450,6.8124729,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{-\frac{4 a \left(25 A a^4+35 C a^4+21 b B a^3-31 A b^2 a^2-35 b^2 C a^2-21 b^3 B a+6 A b^4\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-63 B a^4-82 A b a^3-140 b C a^3-21 b^2 B a^2+6 A b^3 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(6 A b^4-21 a B b^3-82 a^2 A b^2-140 a^2 C b^2-63 a^3 B b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{105 a^2 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{35} (8 A b \sin (c+d x)+7 a B \sin (c+d x)) \sec ^3(c+d x)+\frac{2}{7} a A \tan (c+d x) \sec ^3(c+d x)+\frac{2 \left(25 A \sin (c+d x) a^2+35 C \sin (c+d x) a^2+42 b B \sin (c+d x) a+3 A b^2 \sin (c+d x)\right) \sec ^2(c+d x)}{105 a}+\frac{2 \left(63 B \sin (c+d x) a^3+82 A b \sin (c+d x) a^2+140 b C \sin (c+d x) a^2+21 b^2 B \sin (c+d x) a-6 A b^3 \sin (c+d x)\right) \sec (c+d x)}{105 a^2}\right)}{d}","\frac{2 \sin (c+d x) \left(5 a^2 (5 A+7 C)+42 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)}}{105 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-\left(a^2 (25 A-63 B+35 C)\right)+3 a b (19 A-7 B+35 C)+6 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-63 a^3 B-2 a^2 b (41 A+70 C)-21 a b^2 B+6 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d}+\frac{2 (7 a B+3 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"((-4*a*(25*a^4*A - 31*a^2*A*b^2 + 6*A*b^4 + 21*a^3*b*B - 21*a*b^3*B + 35*a^4*C - 35*a^2*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-82*a^3*A*b + 6*a*A*b^3 - 63*a^4*B - 21*a^2*b^2*B - 140*a^3*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-82*a^2*A*b^2 + 6*A*b^4 - 63*a^3*b*B - 21*a*b^3*B - 140*a^2*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(105*a^2*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^3*(8*A*b*Sin[c + d*x] + 7*a*B*Sin[c + d*x]))/35 + (2*Sec[c + d*x]^2*(25*a^2*A*Sin[c + d*x] + 3*A*b^2*Sin[c + d*x] + 42*a*b*B*Sin[c + d*x] + 35*a^2*C*Sin[c + d*x]))/(105*a) + (2*Sec[c + d*x]*(82*a^2*A*b*Sin[c + d*x] - 6*A*b^3*Sin[c + d*x] + 63*a^3*B*Sin[c + d*x] + 21*a*b^2*B*Sin[c + d*x] + 140*a^2*b*C*Sin[c + d*x]))/(105*a^2) + (2*a*A*Sec[c + d*x]^3*Tan[c + d*x])/7))/d","C",0
1128,1,1614,550,6.9464619,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{63} (10 A b \sin (c+d x)+9 a B \sin (c+d x)) \sec ^4(c+d x)+\frac{2}{9} a A \tan (c+d x) \sec ^4(c+d x)+\frac{2 \left(49 A \sin (c+d x) a^2+63 C \sin (c+d x) a^2+72 b B \sin (c+d x) a+3 A b^2 \sin (c+d x)\right) \sec ^3(c+d x)}{315 a}+\frac{2 \left(75 B \sin (c+d x) a^3+88 A b \sin (c+d x) a^2+126 b C \sin (c+d x) a^2+9 b^2 B \sin (c+d x) a-4 A b^3 \sin (c+d x)\right) \sec ^2(c+d x)}{315 a^2}+\frac{2 \left(147 A \sin (c+d x) a^4+189 C \sin (c+d x) a^4+246 b B \sin (c+d x) a^3+33 A b^2 \sin (c+d x) a^2+63 b^2 C \sin (c+d x) a^2-18 b^3 B \sin (c+d x) a+8 A b^4 \sin (c+d x)\right) \sec (c+d x)}{315 a^3}\right)}{d}-\frac{-\frac{4 a \left(-75 B a^5-39 A b a^4-63 b C a^4+93 b^2 B a^3+31 A b^3 a^2+63 b^3 C a^2-18 b^4 B a+8 A b^5\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(147 A a^5+189 C a^5+246 b B a^4+33 A b^2 a^3+63 b^2 C a^3-18 b^3 B a^2+8 A b^4 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(8 A b^5-18 a B b^4+33 a^2 A b^3+63 a^2 C b^3+246 a^3 B b^2+147 a^4 A b+189 a^4 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{315 a^3 d}","\frac{2 \sin (c+d x) \left(7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)}}{315 a d \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 \sin (c+d x) \left(-75 a^3 B-2 a^2 b (44 A+63 C)-9 a b^2 B+4 A b^3\right) \sqrt{a+b \cos (c+d x)}}{315 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-3 a^3 (49 A-25 B+63 C)+3 a^2 b (13 A-57 B+21 C)+6 a b^2 (A-3 B)+8 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(21 a^4 (7 A+9 C)+246 a^3 b B+3 a^2 b^2 (11 A+21 C)-18 a b^3 B+8 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^4 d}+\frac{2 (3 a B+A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"-1/315*((-4*a*(-39*a^4*A*b + 31*a^2*A*b^3 + 8*A*b^5 - 75*a^5*B + 93*a^3*b^2*B - 18*a*b^4*B - 63*a^4*b*C + 63*a^2*b^3*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(147*a^5*A + 33*a^3*A*b^2 + 8*a*A*b^4 + 246*a^4*b*B - 18*a^2*b^3*B + 189*a^5*C + 63*a^3*b^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(147*a^4*A*b + 33*a^2*A*b^3 + 8*A*b^5 + 246*a^3*b^2*B - 18*a*b^4*B + 189*a^4*b*C + 63*a^2*b^3*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a^3*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^4*(10*A*b*Sin[c + d*x] + 9*a*B*Sin[c + d*x]))/63 + (2*Sec[c + d*x]^3*(49*a^2*A*Sin[c + d*x] + 3*A*b^2*Sin[c + d*x] + 72*a*b*B*Sin[c + d*x] + 63*a^2*C*Sin[c + d*x]))/(315*a) + (2*Sec[c + d*x]^2*(88*a^2*A*b*Sin[c + d*x] - 4*A*b^3*Sin[c + d*x] + 75*a^3*B*Sin[c + d*x] + 9*a*b^2*B*Sin[c + d*x] + 126*a^2*b*C*Sin[c + d*x]))/(315*a^2) + (2*Sec[c + d*x]*(147*a^4*A*Sin[c + d*x] + 33*a^2*A*b^2*Sin[c + d*x] + 8*A*b^4*Sin[c + d*x] + 246*a^3*b*B*Sin[c + d*x] - 18*a*b^3*B*Sin[c + d*x] + 189*a^4*C*Sin[c + d*x] + 63*a^2*b^2*C*Sin[c + d*x]))/(315*a^3) + (2*a*A*Sec[c + d*x]^4*Tan[c + d*x])/9))/d","C",0
1129,1,1410,834,6.7311601,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{1}{40} C \sin (4 (c+d x)) b^2+\frac{1}{160} (10 b B+21 a C) \sin (3 (c+d x)) b+\frac{1}{480} \left(93 C a^2+170 b B a+80 A b^2+88 b^2 C\right) \sin (2 (c+d x))+\frac{\left(15 C a^3+590 b B a^2+1040 A b^2 a+898 b^2 C a+420 b^3 B\right) \sin (c+d x)}{960 b}\right)}{d}-\frac{-\frac{4 a \left(15 C a^4-1330 b B a^3-4720 A b^2 a^2-3236 b^2 C a^2-3560 b^3 B a-1280 A b^4-1024 b^4 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-1440 B b^4-6080 a A b^3-4624 a C b^3-6440 a^2 B b^2-3840 a^3 A b-2292 a^3 C b\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(45 C a^4-150 b B a^3-2640 A b^2 a^2-1692 b^2 C a^2-2840 b^3 B a-1280 A b^4-1024 b^4 C\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{3840 b d}","\frac{C \sqrt{\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d}+\frac{(10 b B-3 a C) \sqrt{\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{40 b d}+\frac{\left(-15 C a^2+50 b B a+80 A b^2+64 b^2 C\right) \sqrt{\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{240 b d}+\frac{\left(-15 C a^3+50 b B a^2+4 b^2 (60 A+43 C) a+120 b^3 B\right) \sqrt{\cos (c+d x)} \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{320 b d}+\frac{\left(-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d \sqrt{\cos (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right) \cot (c+d x) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 a b^2 d}-\frac{\sqrt{a+b} \left(45 C a^4-30 b (5 B+C) a^3-4 b^2 (660 A+295 B+423 C) a^2-8 b^3 (260 A+355 B+193 C) a-16 b^4 (80 A+45 B+64 C)\right) \cot (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 b^2 d}+\frac{\sqrt{a+b} \left(-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right) \cot (c+d x) \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{128 b^3 d}",1,"-1/3840*((-4*a*(-4720*a^2*A*b^2 - 1280*A*b^4 - 1330*a^3*b*B - 3560*a*b^3*B + 15*a^4*C - 3236*a^2*b^2*C - 1024*b^4*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-3840*a^3*A*b - 6080*a*A*b^3 - 6440*a^2*b^2*B - 1440*b^4*B - 2292*a^3*b*C - 4624*a*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-2640*a^2*A*b^2 - 1280*A*b^4 - 150*a^3*b*B - 2840*a*b^3*B + 45*a^4*C - 1692*a^2*b^2*C - 1024*b^4*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(b*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(((1040*a*A*b^2 + 590*a^2*b*B + 420*b^3*B + 15*a^3*C + 898*a*b^2*C)*Sin[c + d*x])/(960*b) + ((80*A*b^2 + 170*a*b*B + 93*a^2*C + 88*b^2*C)*Sin[2*(c + d*x)])/480 + (b*(10*b*B + 21*a*C)*Sin[3*(c + d*x)])/160 + (b^2*C*Sin[4*(c + d*x)])/40))/d","C",0
1130,1,1326,700,6.9081649,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{-\frac{4 a \left(384 A a^3+133 C a^3+472 b B a^2+528 A b^2 a+356 b^2 C a+128 b^3 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(384 B a^3+1152 A b a^2+644 b C a^2+608 b^2 B a+192 A b^3+144 b^3 C\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(15 C a^3+264 b B a^2+432 A b^2 a+284 b^2 C a+128 b^3 B\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{384 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{1}{16} C \sin (3 (c+d x)) b^2+\frac{1}{48} (8 b B+17 a C) \sin (2 (c+d x)) b+\frac{1}{96} \left(59 C a^2+104 b B a+48 A b^2+42 b^2 C\right) \sin (c+d x)\right)}{d}","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{32 d}+\frac{\sin (c+d x) \left(15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{192 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \cot (c+d x) \left(15 a^3 C+2 a^2 b (192 A+132 B+59 C)+4 a b^2 (108 A+52 B+71 C)+8 b^3 (12 A+16 B+9 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 a b d}-\frac{\sqrt{a+b} \cot (c+d x) \left(-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d}+\frac{(5 a C+8 b B) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{24 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 d}",1,"((-4*a*(384*a^3*A + 528*a*A*b^2 + 472*a^2*b*B + 128*b^3*B + 133*a^3*C + 356*a*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(1152*a^2*A*b + 192*A*b^3 + 384*a^3*B + 608*a*b^2*B + 644*a^2*b*C + 144*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(432*a*A*b^2 + 264*a^2*b*B + 128*b^3*B + 15*a^3*C + 284*a*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(384*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(((48*A*b^2 + 104*a*b*B + 59*a^2*C + 42*b^2*C)*Sin[c + d*x])/96 + (b*(8*b*B + 17*a*C)*Sin[2*(c + d*x)])/48 + (b^2*C*Sin[3*(c + d*x)])/16))/d","C",0
1131,1,1302,647,6.8672947,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{\frac{4 a \left(-48 B a^3-96 A b a^2-59 b C a^2-66 b^2 B a-24 A b^3-16 b^3 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+4 a \left(48 A a^3-48 C a^3-144 b B a^2-144 A b^2 a-76 b^2 C a-24 b^3 B\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)-2 \left(-24 A b^3-16 C b^3-54 a B b^2+48 a^2 A b-33 a^2 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{48 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(2 A \tan (c+d x) a^2+\frac{1}{12} b (6 b B+13 a C) \sin (c+d x)+\frac{1}{6} b^2 C \sin (2 (c+d x))\right)}{d}","\frac{\sin (c+d x) \left(-\left(a^2 (48 A-33 C)\right)+54 a b B+8 b^2 (3 A+2 C)\right) \sqrt{a+b \cos (c+d x)}}{24 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \cot (c+d x) \left(a^2 (48 A-48 B-33 C)-2 a b (72 A+27 B+13 C)-4 b^2 (6 A+3 B+4 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(-\left(a^2 (48 A-33 C)\right)+54 a b B+8 b^2 (3 A+2 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a d}-\frac{\sqrt{a+b} \cot (c+d x) \left(5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b d}-\frac{b \sin (c+d x) \sqrt{\cos (c+d x)} (8 a A-3 a C-2 b B) \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{b (6 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt{\cos (c+d x)}}",1,"((4*a*(-96*a^2*A*b - 24*A*b^3 - 48*a^3*B - 66*a*b^2*B - 59*a^2*b*C - 16*b^3*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) + 4*a*(48*a^3*A - 144*a*A*b^2 - 144*a^2*b*B - 24*b^3*B - 48*a^3*C - 76*a*b^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) - 2*(48*a^2*A*b - 24*A*b^3 - 54*a*b^2*B - 33*a^2*b*C - 16*b^3*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(48*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((b*(6*b*B + 13*a*C)*Sin[c + d*x])/12 + (b^2*C*Sin[2*(c + d*x)])/6 + 2*a^2*A*Tan[c + d*x]))/d","C",0
1132,1,1316,622,6.8049416,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{-\frac{4 a \left(8 A a^3+24 C a^3+48 b B a^2+16 A b^2 a+33 b^2 C a+12 b^3 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-24 B a^3-56 A b a^2+72 b C a^2+72 b^2 B a+24 A b^3+12 b^3 C\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(12 B b^3-56 a A b^2+27 a C b^2-24 a^2 B b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{24 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{3} A \sec (c+d x) \tan (c+d x) a^2+\frac{1}{2} b^2 C \sin (c+d x)+\frac{2}{3} \sec (c+d x) \left(3 B \sin (c+d x) a^2+7 A b \sin (c+d x) a\right)\right)}{d}","-\frac{\sin (c+d x) \left(24 a^2 B+a b (56 A-27 C)-12 b^2 B\right) \sqrt{a+b \cos (c+d x)}}{12 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \cot (c+d x) \left(-8 a^2 (A-3 B+3 C)+a b (56 A-72 B-27 C)-6 b^2 (12 A+2 B+C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{12 d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(24 a^2 B+a b (56 A-27 C)-12 b^2 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{12 a d}-\frac{\sqrt{a+b} \cot (c+d x) \left(15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}-\frac{b \sin (c+d x) \sqrt{\cos (c+d x)} (4 a B+8 A b-b C) \sqrt{a+b \cos (c+d x)}}{2 d}+\frac{2 (3 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"((-4*a*(8*a^3*A + 16*a*A*b^2 + 48*a^2*b*B + 12*b^3*B + 24*a^3*C + 33*a*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-56*a^2*A*b + 24*A*b^3 - 24*a^3*B + 72*a*b^2*B + 72*a^2*b*C + 12*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-56*a*A*b^2 - 24*a^2*b*B + 12*b^3*B + 27*a*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(24*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((b^2*C*Sin[c + d*x])/2 + (2*Sec[c + d*x]*(7*a*A*b*Sin[c + d*x] + 3*a^2*B*Sin[c + d*x]))/3 + (2*a^2*A*Sec[c + d*x]*Tan[c + d*x])/3))/d","C",0
1133,1,1370,643,6.8452731,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{\frac{4 a \left(-10 B a^3-16 A b a^2-60 b C a^2-20 b^2 B a+16 A b^3-15 b^3 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+4 a \left(18 A a^3+30 C a^3+70 b B a^2+46 A b^2 a-90 b^2 C a-30 b^3 B\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)-2 \left(46 A b^3-15 C b^3+70 a B b^2+18 a^2 A b+30 a^2 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{30 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{15} \left(5 B \sin (c+d x) a^2+11 A b \sin (c+d x) a\right) \sec ^2(c+d x)+\frac{2}{5} a^2 A \tan (c+d x) \sec ^2(c+d x)+\frac{2}{15} \left(9 A \sin (c+d x) a^2+15 C \sin (c+d x) a^2+35 b B \sin (c+d x) a+23 A b^2 \sin (c+d x)\right) \sec (c+d x)\right)}{d}","\frac{2 \sin (c+d x) \left(a^2 (3 A+5 C)+10 a b B+5 A b^2\right) \sqrt{a+b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x) \left(6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right) \sqrt{a+b \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d}+\frac{\sqrt{a+b} \cot (c+d x) \left(-2 a^3 (9 A-5 B+15 C)+2 a^2 b (17 A-35 B+45 C)-a b^2 (46 A-15 (6 B+C))+30 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d}+\frac{2 (a B+A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{b \sqrt{a+b} (5 a C+2 b B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}",1,"((4*a*(-16*a^2*A*b + 16*A*b^3 - 10*a^3*B - 20*a*b^2*B - 60*a^2*b*C - 15*b^3*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) + 4*a*(18*a^3*A + 46*a*A*b^2 + 70*a^2*b*B - 30*b^3*B + 30*a^3*C - 90*a*b^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) - 2*(18*a^2*A*b + 46*A*b^3 + 70*a*b^2*B + 30*a^2*b*C - 15*b^3*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(30*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^2*(11*a*A*b*Sin[c + d*x] + 5*a^2*B*Sin[c + d*x]))/15 + (2*Sec[c + d*x]*(9*a^2*A*Sin[c + d*x] + 23*A*b^2*Sin[c + d*x] + 35*a*b*B*Sin[c + d*x] + 15*a^2*C*Sin[c + d*x]))/15 + (2*a^2*A*Sec[c + d*x]^2*Tan[c + d*x])/5))/d","C",0
1134,1,1472,580,6.957682,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{-\frac{4 a \left(25 A a^4+35 C a^4+56 b B a^3-10 A b^2 a^2+70 b^2 C a^2-56 b^3 B a-15 A b^4\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-63 B a^4-145 A b a^3-245 b C a^3-161 b^2 B a^2-15 A b^3 a+105 b^3 C a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-15 A b^4-161 a B b^3-145 a^2 A b^2-245 a^2 C b^2-63 a^3 B b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{105 a d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{35} \left(7 B \sin (c+d x) a^2+15 A b \sin (c+d x) a\right) \sec ^3(c+d x)+\frac{2}{7} a^2 A \tan (c+d x) \sec ^3(c+d x)+\frac{2}{105} \left(25 A \sin (c+d x) a^2+35 C \sin (c+d x) a^2+77 b B \sin (c+d x) a+45 A b^2 \sin (c+d x)\right) \sec ^2(c+d x)+\frac{2 \left(63 B \sin (c+d x) a^3+145 A b \sin (c+d x) a^2+245 b C \sin (c+d x) a^2+161 b^2 B \sin (c+d x) a+15 A b^3 \sin (c+d x)\right) \sec (c+d x)}{105 a}\right)}{d}","\frac{2 \sin (c+d x) \left(5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right) \sqrt{a+b \cos (c+d x)}}{105 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \sqrt{a+b} \cot (c+d x) \left(-\left(a^3 (25 A-63 B+35 C)\right)+a^2 b (145 A-119 B+245 C)-a b^2 (135 A-161 B+315 C)+15 b^3 (A-7 B)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(63 a^3 B+5 a^2 b (29 A+49 C)+161 a b^2 B+15 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d}+\frac{2 (7 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}-\frac{2 b^2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}",1,"((-4*a*(25*a^4*A - 10*a^2*A*b^2 - 15*A*b^4 + 56*a^3*b*B - 56*a*b^3*B + 35*a^4*C + 70*a^2*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-145*a^3*A*b - 15*a*A*b^3 - 63*a^4*B - 161*a^2*b^2*B - 245*a^3*b*C + 105*a*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-145*a^2*A*b^2 - 15*A*b^4 - 63*a^3*b*B - 161*a*b^3*B - 245*a^2*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(105*a*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^3*(15*a*A*b*Sin[c + d*x] + 7*a^2*B*Sin[c + d*x]))/35 + (2*Sec[c + d*x]^2*(25*a^2*A*Sin[c + d*x] + 45*A*b^2*Sin[c + d*x] + 77*a*b*B*Sin[c + d*x] + 35*a^2*C*Sin[c + d*x]))/105 + (2*Sec[c + d*x]*(145*a^2*A*b*Sin[c + d*x] + 15*A*b^3*Sin[c + d*x] + 63*a^3*B*Sin[c + d*x] + 161*a*b^2*B*Sin[c + d*x] + 245*a^2*b*C*Sin[c + d*x]))/(105*a) + (2*a^2*A*Sec[c + d*x]^3*Tan[c + d*x])/7))/d","C",0
1135,1,1616,552,7.087587,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{63} \left(9 B \sin (c+d x) a^2+19 A b \sin (c+d x) a\right) \sec ^4(c+d x)+\frac{2}{9} a^2 A \tan (c+d x) \sec ^4(c+d x)+\frac{2}{315} \left(49 A \sin (c+d x) a^2+63 C \sin (c+d x) a^2+135 b B \sin (c+d x) a+75 A b^2 \sin (c+d x)\right) \sec ^3(c+d x)+\frac{2 \left(75 B \sin (c+d x) a^3+163 A b \sin (c+d x) a^2+231 b C \sin (c+d x) a^2+135 b^2 B \sin (c+d x) a+5 A b^3 \sin (c+d x)\right) \sec ^2(c+d x)}{315 a}+\frac{2 \left(147 A \sin (c+d x) a^4+189 C \sin (c+d x) a^4+435 b B \sin (c+d x) a^3+279 A b^2 \sin (c+d x) a^2+483 b^2 C \sin (c+d x) a^2+45 b^3 B \sin (c+d x) a-10 A b^4 \sin (c+d x)\right) \sec (c+d x)}{315 a^2}\right)}{d}-\frac{-\frac{4 a \left(-75 B a^5-114 A b a^4-168 b C a^4+30 b^2 B a^3+124 A b^3 a^2+168 b^3 C a^2+45 b^4 B a-10 A b^5\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(147 A a^5+189 C a^5+435 b B a^4+279 A b^2 a^3+483 b^2 C a^3+45 b^3 B a^2-10 A b^4 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-10 A b^5+45 a B b^4+279 a^2 A b^3+483 a^2 C b^3+435 a^3 B b^2+147 a^4 A b+189 a^4 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{315 a^2 d}","\frac{2 \sin (c+d x) \left(7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right) \sqrt{a+b \cos (c+d x)}}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(75 a^3 B+a^2 b (163 A+231 C)+135 a b^2 B+5 A b^3\right) \sqrt{a+b \cos (c+d x)}}{315 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(3 a^3 (49 A-25 B+63 C)-6 a^2 b (19 A-60 B+28 C)+15 a b^2 (11 A-3 B+21 C)+10 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^2 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-21 a^4 (7 A+9 C)-435 a^3 b B-3 a^2 b^2 (93 A+161 C)-45 a b^3 B+10 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d}+\frac{2 (9 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"-1/315*((-4*a*(-114*a^4*A*b + 124*a^2*A*b^3 - 10*A*b^5 - 75*a^5*B + 30*a^3*b^2*B + 45*a*b^4*B - 168*a^4*b*C + 168*a^2*b^3*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(147*a^5*A + 279*a^3*A*b^2 - 10*a*A*b^4 + 435*a^4*b*B + 45*a^2*b^3*B + 189*a^5*C + 483*a^3*b^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(147*a^4*A*b + 279*a^2*A*b^3 - 10*A*b^5 + 435*a^3*b^2*B + 45*a*b^4*B + 189*a^4*b*C + 483*a^2*b^3*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a^2*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^4*(19*a*A*b*Sin[c + d*x] + 9*a^2*B*Sin[c + d*x]))/63 + (2*Sec[c + d*x]^3*(49*a^2*A*Sin[c + d*x] + 75*A*b^2*Sin[c + d*x] + 135*a*b*B*Sin[c + d*x] + 63*a^2*C*Sin[c + d*x]))/315 + (2*Sec[c + d*x]^2*(163*a^2*A*b*Sin[c + d*x] + 5*A*b^3*Sin[c + d*x] + 75*a^3*B*Sin[c + d*x] + 135*a*b^2*B*Sin[c + d*x] + 231*a^2*b*C*Sin[c + d*x]))/(315*a) + (2*Sec[c + d*x]*(147*a^4*A*Sin[c + d*x] + 279*a^2*A*b^2*Sin[c + d*x] - 10*A*b^4*Sin[c + d*x] + 435*a^3*b*B*Sin[c + d*x] + 45*a*b^3*B*Sin[c + d*x] + 189*a^4*C*Sin[c + d*x] + 483*a^2*b^2*C*Sin[c + d*x]))/(315*a^2) + (2*a^2*A*Sec[c + d*x]^4*Tan[c + d*x])/9))/d","C",0
1136,1,1241,593,6.5378064,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{-\frac{4 a \left(5 C a^2-6 b B a+24 A b^2+16 b^2 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(24 B b^2+4 a C b\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(15 C a^2-18 b B a+24 A b^2+16 b^2 C\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{48 b^2 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{(6 b B-5 a C) \sin (c+d x)}{12 b^2}+\frac{C \sin (2 (c+d x))}{6 b}\right)}{d}","\frac{\sin (c+d x) \left(15 a^2 C-18 a b B+24 A b^2+16 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{24 b^3 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \cot (c+d x) \left(15 a^2 C-18 a b B-10 a b C+24 A b^2+12 b^2 B+16 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^3 d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(15 a^2 C-18 a b B+24 A b^2+16 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^3 d}-\frac{\sqrt{a+b} \cot (c+d x) \left(-5 a^3 C+6 a^2 b B-4 a b^2 (2 A+C)+8 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^4 d}+\frac{(6 b B-5 a C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{12 b^2 d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d}",1,"((-4*a*(24*A*b^2 - 6*a*b*B + 5*a^2*C + 16*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(24*b^2*B + 4*a*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(24*A*b^2 - 18*a*b*B + 15*a^2*C + 16*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(48*b^2*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(((6*b*B - 5*a*C)*Sin[c + d*x])/(12*b^2) + (C*Sin[2*(c + d*x)])/(6*b)))/d","C",0
1137,1,1182,485,13.133019,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{C \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{2 b d}+\frac{-\frac{4 a (4 b B-a C) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a (8 A b+4 C b) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 (4 b B-3 a C) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{8 b d}","-\frac{\sqrt{a+b} \cot (c+d x) \left(3 a^2 C-4 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d}+\frac{(4 b B-3 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b^2 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} (3 a C-2 b (2 B+C)) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}-\frac{(a-b) \sqrt{a+b} (4 b B-3 a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b^2 d}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 b d}",1,"(C*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b*d) + ((-4*a*(4*b*B - a*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(8*A*b + 4*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(4*b*B - 3*a*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(8*b*d)","C",0
1138,1,1117,401,19.1161763,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{-\frac{4 a (2 A+C) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-8 a B \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 C \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{2 d}","\frac{\sqrt{a+b} (a C+2 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d}-\frac{\sqrt{a+b} (2 b B-a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}-\frac{C (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d}",1,"((-4*a*(2*A + C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 8*a*B*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*C*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(2*d)","C",1
1139,1,1169,347,19.0511188,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{2 A \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}-\frac{-\frac{4 a (A b-a B) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a (a A-a C) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 A b \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{a d}","\frac{2 A (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d}-\frac{2 \sqrt{a+b} (A-B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d}",1,"(2*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((-4*a*(A*b - a*B)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(a*A - a*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*A*b*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a*d)","C",1
1140,1,1244,293,6.5087641,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{-\frac{4 a \left(A a^2+3 C a^2-3 b B a+2 A b^2\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(2 a A b-3 a^2 B\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(2 A b^2-3 a b B\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{3 a^2 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \sec (c+d x) (3 a B \sin (c+d x)-2 A b \sin (c+d x))}{3 a^2}+\frac{2 A \sec (c+d x) \tan (c+d x)}{3 a}\right)}{d}","-\frac{2 (a-b) \sqrt{a+b} (2 A b-3 a B) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d}+\frac{2 \sqrt{a+b} \cot (c+d x) (a (A-3 B+3 C)+2 A b) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 a d \cos ^{\frac{3}{2}}(c+d x)}",1,"((-4*a*(a^2*A + 2*A*b^2 - 3*a*b*B + 3*a^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(2*a*A*b - 3*a^2*B)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(2*A*b^2 - 3*a*b*B)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(3*a^2*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]*(-2*A*b*Sin[c + d*x] + 3*a*B*Sin[c + d*x]))/(3*a^2) + (2*A*Sec[c + d*x]*Tan[c + d*x])/(3*a)))/d","C",1
1141,1,1351,372,6.6090566,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 (5 a B \sin (c+d x)-4 A b \sin (c+d x)) \sec ^2(c+d x)}{15 a^2}+\frac{2 A \tan (c+d x) \sec ^2(c+d x)}{5 a}+\frac{2 \left(9 A \sin (c+d x) a^2+15 C \sin (c+d x) a^2-10 b B \sin (c+d x) a+8 A b^2 \sin (c+d x)\right) \sec (c+d x)}{15 a^3}\right)}{d}-\frac{-\frac{4 a \left(-5 B a^3+7 A b a^2+15 b C a^2-10 b^2 B a+8 A b^3\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(9 A a^3+15 C a^3-10 b B a^2+8 A b^2 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(8 A b^3-10 a B b^2+9 a^2 A b+15 a^2 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{15 a^3 d}","-\frac{2 (4 A b-5 a B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(3 a^2 (3 A+5 C)-10 a b B+8 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^4 d}-\frac{2 \sqrt{a+b} \cot (c+d x) \left(a^2 (9 A-5 B+15 C)-2 a b (A+5 B)+8 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 a d \cos ^{\frac{5}{2}}(c+d x)}",1,"-1/15*((-4*a*(7*a^2*A*b + 8*A*b^3 - 5*a^3*B - 10*a*b^2*B + 15*a^2*b*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(9*a^3*A + 8*a*A*b^2 - 10*a^2*b*B + 15*a^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(9*a^2*A*b + 8*A*b^3 - 10*a*b^2*B + 15*a^2*b*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a^3*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^2*(-4*A*b*Sin[c + d*x] + 5*a*B*Sin[c + d*x]))/(15*a^2) + (2*Sec[c + d*x]*(9*a^2*A*Sin[c + d*x] + 8*A*b^2*Sin[c + d*x] - 10*a*b*B*Sin[c + d*x] + 15*a^2*C*Sin[c + d*x]))/(15*a^3) + (2*A*Sec[c + d*x]^2*Tan[c + d*x])/(5*a)))/d","C",0
1142,1,1468,466,6.7284347,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{9}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{-\frac{4 a \left(25 A a^4+35 C a^4-49 b B a^3+32 A b^2 a^2+70 b^2 C a^2-56 b^3 B a+48 A b^4\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-63 B a^4+44 A b a^3+70 b C a^3-56 b^2 B a^2+48 A b^3 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(48 A b^4-56 a B b^3+44 a^2 A b^2+70 a^2 C b^2-63 a^3 B b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{105 a^4 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 (7 a B \sin (c+d x)-6 A b \sin (c+d x)) \sec ^3(c+d x)}{35 a^2}+\frac{2 A \tan (c+d x) \sec ^3(c+d x)}{7 a}+\frac{2 \left(25 A \sin (c+d x) a^2+35 C \sin (c+d x) a^2-28 b B \sin (c+d x) a+24 A b^2 \sin (c+d x)\right) \sec ^2(c+d x)}{105 a^3}+\frac{2 \left(63 B \sin (c+d x) a^3-44 A b \sin (c+d x) a^2-70 b C \sin (c+d x) a^2+56 b^2 B \sin (c+d x) a-48 A b^3 \sin (c+d x)\right) \sec (c+d x)}{105 a^4}\right)}{d}","-\frac{2 (6 A b-7 a B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 a^2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(5 a^2 (5 A+7 C)-28 a b B+24 A b^2\right) \sqrt{a+b \cos (c+d x)}}{105 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-63 a^3 B+a^2 (44 A b+70 b C)-56 a b^2 B+48 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^5 d}+\frac{2 \sqrt{a+b} \cot (c+d x) \left(a^3 (25 A-63 B+35 C)+2 a^2 b (22 A+7 (B+5 C))-4 a b^2 (3 A+14 B)+48 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d}+\frac{2 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 a d \cos ^{\frac{7}{2}}(c+d x)}",1,"((-4*a*(25*a^4*A + 32*a^2*A*b^2 + 48*A*b^4 - 49*a^3*b*B - 56*a*b^3*B + 35*a^4*C + 70*a^2*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(44*a^3*A*b + 48*a*A*b^3 - 63*a^4*B - 56*a^2*b^2*B + 70*a^3*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(44*a^2*A*b^2 + 48*A*b^4 - 63*a^3*b*B - 56*a*b^3*B + 70*a^2*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(105*a^4*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^3*(-6*A*b*Sin[c + d*x] + 7*a*B*Sin[c + d*x]))/(35*a^2) + (2*Sec[c + d*x]^2*(25*a^2*A*Sin[c + d*x] + 24*A*b^2*Sin[c + d*x] - 28*a*b*B*Sin[c + d*x] + 35*a^2*C*Sin[c + d*x]))/(105*a^3) + (2*Sec[c + d*x]*(-44*a^2*A*b*Sin[c + d*x] - 48*A*b^3*Sin[c + d*x] + 63*a^3*B*Sin[c + d*x] + 56*a*b^2*B*Sin[c + d*x] - 70*a^2*b*C*Sin[c + d*x]))/(105*a^4) + (2*A*Sec[c + d*x]^3*Tan[c + d*x])/(7*a)))/d","C",0
1143,1,1175,473,6.1935188,"\int \frac{\sqrt{\cos (c+d x)} \left(a A+(A b+a B) \cos (c+d x)+b B \cos ^2(c+d x)\right)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(a*A + (A*b + a*B)*Cos[c + d*x] + b*B*Cos[c + d*x]^2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{B \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{2 d}+\frac{-\frac{4 a (4 A b+3 a B) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a (8 a A+4 b B) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 (4 A b+a B) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{8 d}","-\frac{\sqrt{a+b} \left(a^2 (-B)+4 a A b+4 b^2 B\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}+\frac{(a B+4 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} (B (a+2 b)+4 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}-\frac{(a-b) \sqrt{a+b} (a B+4 A b) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b d}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}",1,"(B*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + ((-4*a*(4*A*b + 3*a*B)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(8*a*A + 4*b*B)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(4*A*b + a*B)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(8*d)","C",1
1144,1,160,256,4.7446483,"\int \frac{a+a \cos (c+d x)+2 b \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(a + a*Cos[c + d*x] + 2*b*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{\sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{a+b \cos (c+d x)} \left(\frac{2 E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)}{\sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}}}+\frac{\left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}}}\right)}{d \sqrt{\cos (c+d x)}}","\frac{2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{4 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}",1,"(Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[a + b*Cos[c + d*x]]*((2*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (Sec[(c + d*x)/2]*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2]))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]))/(d*Sqrt[Cos[c + d*x]])","A",1
1145,1,1322,660,6.7564971,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{C \sin (c+d x)}{2 b^2}-\frac{2 \left(C \sin (c+d x) a^3-b B \sin (c+d x) a^2+A b^2 \sin (c+d x) a\right)}{b^2 \left(b^2-a^2\right) (a+b \cos (c+d x))}\right)}{d}-\frac{-\frac{4 a \left(5 C a^3-4 b B a^2-5 b^2 C a+4 b^3 B\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(8 A b^3+4 C b^3-8 a B b^2+4 a^2 C b\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(15 C a^3-12 b B a^2+8 A b^2 a-7 b^2 C a+4 b^3 B\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{8 (a-b) b^2 (a+b) d}","-\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(5 a^2 C-4 a b B+4 A b^2-b^2 C\right) \sqrt{a+b \cos (c+d x)}}{2 b^2 d \left(a^2-b^2\right)}-\frac{\sqrt{a+b} \cot (c+d x) \left(15 a^2 C-12 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^4 d}-\frac{\cot (c+d x) \left(15 a^2 C-a b (12 B-5 C)+8 A b^2-2 b^2 (2 B+C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{a+b}}+\frac{\sin (c+d x) \left(-15 a^3 C+12 a^2 b B-a b^2 (8 A-7 C)-4 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{4 b^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{\cot (c+d x) \left(-15 a^3 C+12 a^2 b B-a b^2 (8 A-7 C)-4 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b^3 d \sqrt{a+b}}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((C*Sin[c + d*x])/(2*b^2) - (2*(a*A*b^2*Sin[c + d*x] - a^2*b*B*Sin[c + d*x] + a^3*C*Sin[c + d*x]))/(b^2*(-a^2 + b^2)*(a + b*Cos[c + d*x]))))/d - ((-4*a*(-4*a^2*b*B + 4*b^3*B + 5*a^3*C - 5*a*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(8*A*b^3 - 8*a*b^2*B + 4*a^2*b*C + 4*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(8*a*A*b^2 - 12*a^2*b*B + 4*b^3*B + 15*a^3*C - 7*a*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(8*(a - b)*b^2*(a + b)*d)","C",0
1146,1,1256,535,6.5053833,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 \sqrt{\cos (c+d x)} \left(C \sin (c+d x) a^2-b B \sin (c+d x) a+A b^2 \sin (c+d x)\right)}{b \left(b^2-a^2\right) d \sqrt{a+b \cos (c+d x)}}+\frac{-\frac{4 a \left(a^2 C-b^2 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-2 B b^2+2 a A b+2 a C b\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(3 C a^2-2 b B a+2 A b^2-b^2 C\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{2 (a-b) b (a+b) d}","\frac{\sin (c+d x) \left(3 a^2 C-2 a b B+2 A b^2-b^2 C\right) \sqrt{a+b \cos (c+d x)}}{b^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{\cot (c+d x) \left(3 a^2 C-2 a b B+2 A b^2-b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b}}+\frac{\cot (c+d x) \left(2 A b^2-a (b (2 B-C)-3 a C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b}}-\frac{\sqrt{a+b} (2 b B-3 a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d}",1,"(2*Sqrt[Cos[c + d*x]]*(A*b^2*Sin[c + d*x] - a*b*B*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(b*(-a^2 + b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((-4*a*(a^2*C - b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(2*a*A*b - 2*b^2*B + 2*a*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(2*(a - b)*b*(a + b)*d)","C",0
1147,1,1245,436,6.5087561,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{2 \sqrt{\cos (c+d x)} \left(C \sin (c+d x) a^2-b B \sin (c+d x) a+A b^2 \sin (c+d x)\right)}{a \left(a^2-b^2\right) d \sqrt{a+b \cos (c+d x)}}+\frac{-\frac{4 a \left(a^2 A-A b^2\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(B a^2-A b a-b C a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-C a^2+b B a-A b^2\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{a (a-b) (a+b) d}","-\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \cot (c+d x) \left(A b^2-a (b B-a C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 b d \sqrt{a+b}}+\frac{2 \cot (c+d x) (-a C+A b+b B) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{a+b}}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d}",1,"(2*Sqrt[Cos[c + d*x]]*(A*b^2*Sin[c + d*x] - a*b*B*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((-4*a*(a^2*A - A*b^2)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-(a*A*b) + a^2*B - a*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-(A*b^2) + a*b*B - a^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a*(a - b)*(a + b)*d)","C",1
1148,1,1306,322,6.6855782,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{-\frac{4 a \left(-B a^3+2 A b a^2+b^2 B a-2 A b^3\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(A a^3-C a^3+b B a^2-2 A b^2 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-2 A b^3+a B b^2+a^2 A b-a^2 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{a^2 (b-a) (a+b) d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 A \tan (c+d x)}{a^2}-\frac{2 \left(A \sin (c+d x) b^3-a B \sin (c+d x) b^2+a^2 C \sin (c+d x) b\right)}{a^2 \left(a^2-b^2\right) (a+b \cos (c+d x))}\right)}{d}","\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{2 \cot (c+d x) (a (A-B-C)+2 A b) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d \sqrt{a+b}}-\frac{2 \cot (c+d x) \left(-\left(a^2 (A-C)\right)-a b B+2 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^3 d \sqrt{a+b}}",1,"((-4*a*(2*a^2*A*b - 2*A*b^3 - a^3*B + a*b^2*B)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(a^3*A - 2*a*A*b^2 + a^2*b*B - a^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(a^2*A*b - 2*A*b^3 + a*b^2*B - a^2*b*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a^2*(-a + b)*(a + b)*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((-2*(A*b^3*Sin[c + d*x] - a*b^2*B*Sin[c + d*x] + a^2*b*C*Sin[c + d*x]))/(a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])) + (2*A*Tan[c + d*x])/a^2))/d","C",1
1149,1,1402,424,6.9205884,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{-\frac{4 a \left(A a^4+3 C a^4-6 b B a^3+7 A b^2 a^2-3 b^2 C a^2+6 b^3 B a-8 A b^4\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-3 B a^4+5 A b a^3-3 b C a^3+6 b^2 B a^2-8 A b^3 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-8 A b^4+6 a B b^3+5 a^2 A b^2-3 a^2 C b^2-3 a^3 B b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{3 a^3 (a-b) (a+b) d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \sec (c+d x) (3 a B \sin (c+d x)-5 A b \sin (c+d x))}{3 a^3}+\frac{2 \left(A \sin (c+d x) b^4-a B \sin (c+d x) b^3+a^2 C \sin (c+d x) b^2\right)}{a^3 \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{2 A \sec (c+d x) \tan (c+d x)}{3 a^2}\right)}{d}","-\frac{2 \sin (c+d x) \left(-\left(a^2 (A-3 C)\right)-3 a b B+4 A b^2\right) \sqrt{a+b \cos (c+d x)}}{3 a^2 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}+\frac{2 \cot (c+d x) \left(a^2 (A-3 B+3 C)+6 a b (A-B)+8 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b}}+\frac{2 \cot (c+d x) \left(3 a^3 B-a^2 (5 A b-3 b C)-6 a b^2 B+8 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b}}",1,"((-4*a*(a^4*A + 7*a^2*A*b^2 - 8*A*b^4 - 6*a^3*b*B + 6*a*b^3*B + 3*a^4*C - 3*a^2*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(5*a^3*A*b - 8*a*A*b^3 - 3*a^4*B + 6*a^2*b^2*B - 3*a^3*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(5*a^2*A*b^2 - 8*A*b^4 - 3*a^3*b*B + 6*a*b^3*B - 3*a^2*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(3*a^3*(a - b)*(a + b)*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]*(-5*A*b*Sin[c + d*x] + 3*a*B*Sin[c + d*x]))/(3*a^3) + (2*(A*b^4*Sin[c + d*x] - a*b^3*B*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(a^3*(a^2 - b^2)*(a + b*Cos[c + d*x])) + (2*A*Sec[c + d*x]*Tan[c + d*x])/(3*a^2)))/d","C",0
1150,1,1511,545,7.1239713,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{-\frac{4 a \left(-5 B a^5+12 A b a^4+30 b C a^4-35 b^2 B a^3+36 A b^3 a^2-30 b^3 C a^2+40 b^4 B a-48 A b^5\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(9 A a^5+15 C a^5-25 b B a^4+24 A b^2 a^3-30 b^2 C a^3+40 b^3 B a^2-48 A b^4 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-48 A b^5+40 a B b^4+24 a^2 A b^3-30 a^2 C b^3-25 a^3 B b^2+9 a^4 A b+15 a^4 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{15 a^4 (b-a) (a+b) d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 (5 a B \sin (c+d x)-9 A b \sin (c+d x)) \sec ^2(c+d x)}{15 a^3}+\frac{2 A \tan (c+d x) \sec ^2(c+d x)}{5 a^2}+\frac{2 \left(9 A \sin (c+d x) a^2+15 C \sin (c+d x) a^2-25 b B \sin (c+d x) a+33 A b^2 \sin (c+d x)\right) \sec (c+d x)}{15 a^4}-\frac{2 \left(A \sin (c+d x) b^5-a B \sin (c+d x) b^4+a^2 C \sin (c+d x) b^3\right)}{a^4 \left(a^2-b^2\right) (a+b \cos (c+d x))}\right)}{d}","-\frac{2 \sin (c+d x) \left(-\left(a^2 (A-5 C)\right)-5 a b B+6 A b^2\right) \sqrt{a+b \cos (c+d x)}}{5 a^2 d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \left(5 a^3 B-a^2 (9 A b-15 b C)-20 a b^2 B+24 A b^3\right) \sqrt{a+b \cos (c+d x)}}{15 a^3 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 \cot (c+d x) \left(a^3 (9 A-5 B+15 C)+6 a^2 b (2 A-5 B+5 C)+4 a b^2 (9 A-10 B)+48 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^4 d \sqrt{a+b}}-\frac{2 \cot (c+d x) \left(-3 a^4 (3 A+5 C)+25 a^3 b B-6 a^2 b^2 (4 A-5 C)-40 a b^3 B+48 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^5 d \sqrt{a+b}}",1,"((-4*a*(12*a^4*A*b + 36*a^2*A*b^3 - 48*A*b^5 - 5*a^5*B - 35*a^3*b^2*B + 40*a*b^4*B + 30*a^4*b*C - 30*a^2*b^3*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(9*a^5*A + 24*a^3*A*b^2 - 48*a*A*b^4 - 25*a^4*b*B + 40*a^2*b^3*B + 15*a^5*C - 30*a^3*b^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(9*a^4*A*b + 24*a^2*A*b^3 - 48*A*b^5 - 25*a^3*b^2*B + 40*a*b^4*B + 15*a^4*b*C - 30*a^2*b^3*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(15*a^4*(-a + b)*(a + b)*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^2*(-9*A*b*Sin[c + d*x] + 5*a*B*Sin[c + d*x]))/(15*a^3) + (2*Sec[c + d*x]*(9*a^2*A*Sin[c + d*x] + 33*A*b^2*Sin[c + d*x] - 25*a*b*B*Sin[c + d*x] + 15*a^2*C*Sin[c + d*x]))/(15*a^4) - (2*(A*b^5*Sin[c + d*x] - a*b^4*B*Sin[c + d*x] + a^2*b^3*C*Sin[c + d*x]))/(a^4*(a^2 - b^2)*(a + b*Cos[c + d*x])) + (2*A*Sec[c + d*x]^2*Tan[c + d*x])/(5*a^2)))/d","C",0
1151,1,1448,723,6.8533196,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \left(-6 C \sin (c+d x) a^4+3 b B \sin (c+d x) a^3+10 b^2 C \sin (c+d x) a^2-7 b^3 B \sin (c+d x) a+4 A b^4 \sin (c+d x)\right)}{3 b^2 \left(b^2-a^2\right)^2 (a+b \cos (c+d x))}-\frac{2 \left(C \sin (c+d x) a^3-b B \sin (c+d x) a^2+A b^2 \sin (c+d x) a\right)}{3 b^2 \left(b^2-a^2\right) (a+b \cos (c+d x))^2}\right)}{d}+\frac{-\frac{4 a \left(5 C a^4-2 b B a^3+2 A b^2 a^2-8 b^2 C a^2+2 b^3 B a-2 A b^4+3 b^4 C\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(6 B b^4-8 a A b^3-12 a C b^3+2 a^2 B b^2+4 a^3 C b\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(15 C a^4-6 b B a^3-26 b^2 C a^2+14 b^3 B a-8 A b^4+3 b^4 C\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{6 (a-b)^2 b^2 (a+b)^2 d}","-\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(-5 a^4 C+2 a^3 b B+a^2 b^2 (A+9 C)-6 a b^3 B+3 A b^4\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{\sin (c+d x) \left(-15 a^4 C+6 a^3 b B+26 a^2 b^2 C-14 a b^3 B+8 A b^4-3 b^4 C\right) \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}-\frac{\cot (c+d x) \left(-15 a^4 C+a^3 b (6 B-5 C)+a^2 b^2 (2 B+21 C)-a b^3 (2 A+3 (4 B-C))+6 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^3 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{\cot (c+d x) \left(-15 a^4 C+6 a^3 b B+26 a^2 b^2 C-14 a b^3 B+8 A b^4-3 b^4 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^3 d (a-b) (a+b)^{3/2}}-\frac{\sqrt{a+b} (2 b B-5 a C) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^4 d}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((-2*(a*A*b^2*Sin[c + d*x] - a^2*b*B*Sin[c + d*x] + a^3*C*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (2*(4*A*b^4*Sin[c + d*x] + 3*a^3*b*B*Sin[c + d*x] - 7*a*b^3*B*Sin[c + d*x] - 6*a^4*C*Sin[c + d*x] + 10*a^2*b^2*C*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)^2*(a + b*Cos[c + d*x]))))/d + ((-4*a*(2*a^2*A*b^2 - 2*A*b^4 - 2*a^3*b*B + 2*a*b^3*B + 5*a^4*C - 8*a^2*b^2*C + 3*b^4*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-8*a*A*b^3 + 2*a^2*b^2*B + 6*b^4*B + 4*a^3*b*C - 12*a*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-8*A*b^4 - 6*a^3*b*B + 14*a*b^3*B + 15*a^4*C - 26*a^2*b^2*C + 3*b^4*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(6*(a - b)^2*b^2*(a + b)^2*d)","C",0
1152,1,1441,589,6.7385219,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \left(C \sin (c+d x) a^2-b B \sin (c+d x) a+A b^2 \sin (c+d x)\right)}{3 b \left(b^2-a^2\right) (a+b \cos (c+d x))^2}+\frac{2 \left(3 C \sin (c+d x) a^4-3 A b^2 \sin (c+d x) a^2-7 b^2 C \sin (c+d x) a^2+4 b^3 B \sin (c+d x) a-A b^4 \sin (c+d x)\right)}{3 a b \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}\right)}{d}-\frac{-\frac{4 a \left(C a^4-b B a^3+A b^2 a^2-b^2 C a^2+b^3 B a-A b^4\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-3 A b a^3-b C a^3+4 b^2 B a^2-A b^3 a-3 b^3 C a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(3 C a^4-3 A b^2 a^2-7 b^2 C a^2+4 b^3 B a-A b^4\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{3 a (a-b)^2 b (a+b)^2 d}","-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \sin (c+d x) \left(-3 a^4 C+a^2 b^2 (3 A+7 C)-4 a b^3 B+A b^4\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{2 \cot (c+d x) \left(-3 a^4 C+a^2 b^2 (3 A+7 C)-4 a b^3 B+A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 b^2 d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{2 \cot (c+d x) \left(3 a^3 C+a^2 b C-a b^2 (3 A+B+6 C)+b^3 (A+3 B)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^2 d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*(A*b^2*Sin[c + d*x] - a*b*B*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(3*b*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (2*(-3*a^2*A*b^2*Sin[c + d*x] - A*b^4*Sin[c + d*x] + 4*a*b^3*B*Sin[c + d*x] + 3*a^4*C*Sin[c + d*x] - 7*a^2*b^2*C*Sin[c + d*x]))/(3*a*b*(a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/d - ((-4*a*(a^2*A*b^2 - A*b^4 - a^3*b*B + a*b^3*B + a^4*C - a^2*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-3*a^3*A*b - a*A*b^3 + 4*a^2*b^2*B - a^3*b*C - 3*a*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-3*a^2*A*b^2 - A*b^4 + 4*a*b^3*B + 3*a^4*C - 7*a^2*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(3*a*(a - b)^2*b*(a + b)^2*d)","C",0
1153,1,1440,457,6.7565306,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \left(C \sin (c+d x) a^2-b B \sin (c+d x) a+A b^2 \sin (c+d x)\right)}{3 a \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{2 \left(2 A \sin (c+d x) b^4+a B \sin (c+d x) b^3-6 a^2 A \sin (c+d x) b^2-4 a^2 C \sin (c+d x) b^2+3 a^3 B \sin (c+d x) b\right)}{3 a^2 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}\right)}{d}+\frac{-\frac{4 a \left(3 A a^4+C a^4-b B a^3-5 A b^2 a^2-b^2 C a^2+b^3 B a+2 A b^4\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(3 B a^4-6 A b a^3-4 b C a^3+b^2 B a^2+2 A b^3 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(2 A b^4+a B b^3-6 a^2 A b^2-4 a^2 C b^2+3 a^3 B b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{3 a^2 (a-b)^2 (a+b)^2 d}","\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \cot (c+d x) \left(-\left(a^2 (3 A+3 B+C)\right)+a b (3 A+B+3 C)+2 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{2 \sin (c+d x) \left(3 a^3 B-2 a^2 b (3 A+2 C)+a b^2 B+2 A b^3\right)}{3 a d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \cot (c+d x) \left(-3 a^3 B+6 a^2 A b+4 a^2 b C-a b^2 B-2 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d (a-b) (a+b)^{3/2}}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*(A*b^2*Sin[c + d*x] - a*b*B*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(3*a*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) - (2*(-6*a^2*A*b^2*Sin[c + d*x] + 2*A*b^4*Sin[c + d*x] + 3*a^3*b*B*Sin[c + d*x] + a*b^3*B*Sin[c + d*x] - 4*a^2*b^2*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/d + ((-4*a*(3*a^4*A - 5*a^2*A*b^2 + 2*A*b^4 - a^3*b*B + a*b^3*B + a^4*C - a^2*b^2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-6*a^3*A*b + 2*a*A*b^3 + 3*a^4*B + a^2*b^2*B - 4*a^3*b*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-6*a^2*A*b^2 + 2*A*b^4 + 3*a^3*b*B + a*b^3*B - 4*a^2*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(3*a^2*(a - b)^2*(a + b)^2*d)","C",0
1154,1,1516,495,6.9920495,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(5/2)),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(-\frac{2 \left(A \sin (c+d x) b^3-a B \sin (c+d x) b^2+a^2 C \sin (c+d x) b\right)}{3 a^2 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{2 \left(-5 A \sin (c+d x) b^5+2 a B \sin (c+d x) b^4+9 a^2 A \sin (c+d x) b^3+a^2 C \sin (c+d x) b^3-6 a^3 B \sin (c+d x) b^2+3 a^4 C \sin (c+d x) b\right)}{3 a^3 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{2 A \tan (c+d x)}{a^3}\right)}{d}-\frac{-\frac{4 a \left(-3 B a^5+9 A b a^4+b C a^4+5 b^2 B a^3-17 A b^3 a^2-b^3 C a^2-2 b^4 B a+8 A b^5\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(3 A a^5-3 C a^5+6 b B a^4-15 A b^2 a^3-b^2 C a^3-2 b^3 B a^2+8 A b^4 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(8 A b^5-2 a B b^4-15 a^2 A b^3-a^2 C b^3+6 a^3 B b^2+3 a^4 A b-3 a^4 C b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{3 a^3 (a-b)^2 (a+b)^2 d}","\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}+\frac{2 \cot (c+d x) \left(-3 a^3 (A-B-C)-a^2 b (9 A+3 B+C)+2 a b^2 (3 A-B)+8 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{2 \sin (c+d x) \left(-2 a^4 C+5 a^3 b B-2 a^2 b^2 (4 A+C)-a b^3 B+4 A b^4\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{2 \cot (c+d x) \left(3 a^4 (A-C)+6 a^3 b B-a^2 b^2 (15 A+C)-2 a b^3 B+8 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \left(a^2-b^2\right)}",1,"-1/3*((-4*a*(9*a^4*A*b - 17*a^2*A*b^3 + 8*A*b^5 - 3*a^5*B + 5*a^3*b^2*B - 2*a*b^4*B + a^4*b*C - a^2*b^3*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(3*a^5*A - 15*a^3*A*b^2 + 8*a*A*b^4 + 6*a^4*b*B - 2*a^2*b^3*B - 3*a^5*C - a^3*b^2*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(3*a^4*A*b - 15*a^2*A*b^3 + 8*A*b^5 + 6*a^3*b^2*B - 2*a*b^4*B - 3*a^4*b*C - a^2*b^3*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a^3*(a - b)^2*(a + b)^2*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((-2*(A*b^3*Sin[c + d*x] - a*b^2*B*Sin[c + d*x] + a^2*b*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) - (2*(9*a^2*A*b^3*Sin[c + d*x] - 5*A*b^5*Sin[c + d*x] - 6*a^3*b^2*B*Sin[c + d*x] + 2*a*b^4*B*Sin[c + d*x] + 3*a^4*b*C*Sin[c + d*x] + a^2*b^3*C*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) + (2*A*Tan[c + d*x])/a^3))/d","C",0
1155,1,1601,620,7.1942591,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(5/2)),x]","\frac{-\frac{4 a \left(A a^6+3 C a^6-9 b B a^5+15 A b^2 a^4-5 b^2 C a^4+17 b^3 B a^3-32 A b^4 a^2+2 b^4 C a^2-8 b^5 B a+16 A b^6\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-3 B a^6+8 A b a^5-6 b C a^5+15 b^2 B a^4-28 A b^3 a^3+2 b^3 C a^3-8 b^4 B a^2+16 A b^5 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(16 A b^6-8 a B b^5-28 a^2 A b^4+2 a^2 C b^4+15 a^3 B b^3+8 a^4 A b^2-6 a^4 C b^2-3 a^5 B b\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{3 a^4 (a-b)^2 (a+b)^2 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \sec (c+d x) (3 a B \sin (c+d x)-8 A b \sin (c+d x))}{3 a^4}+\frac{2 \left(A \sin (c+d x) b^4-a B \sin (c+d x) b^3+a^2 C \sin (c+d x) b^2\right)}{3 a^3 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{2 \left(-8 A \sin (c+d x) b^6+5 a B \sin (c+d x) b^5+12 a^2 A \sin (c+d x) b^4-2 a^2 C \sin (c+d x) b^4-9 a^3 B \sin (c+d x) b^3+6 a^4 C \sin (c+d x) b^2\right)}{3 a^4 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{2 A \sec (c+d x) \tan (c+d x)}{3 a^3}\right)}{d}","\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}+\frac{2 \sin (c+d x) \left(a^4 (A-5 C)+8 a^3 b B-a^2 b^2 (13 A-C)-4 a b^3 B+8 A b^4\right) \sqrt{a+b \cos (c+d x)}}{3 a^3 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(4 a^4 C-7 a^3 b B+10 a^2 A b^2+3 a b^3 B-6 A b^4\right)}{3 a^2 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}-\frac{2 \cot (c+d x) \left(-\left(a^4 (A-3 B+3 C)\right)-3 a^3 b (3 A-3 B-C)-2 a^2 b^2 (8 A+3 B-C)+4 a b^3 (3 A-2 B)+16 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{2 \cot (c+d x) \left(-3 a^5 B+a^4 (8 A b-6 b C)+15 a^3 b^2 B-2 a^2 b^3 (14 A-C)-8 a b^4 B+16 A b^5\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^5 d \sqrt{a+b} \left(a^2-b^2\right)}",1,"((-4*a*(a^6*A + 15*a^4*A*b^2 - 32*a^2*A*b^4 + 16*A*b^6 - 9*a^5*b*B + 17*a^3*b^3*B - 8*a*b^5*B + 3*a^6*C - 5*a^4*b^2*C + 2*a^2*b^4*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(8*a^5*A*b - 28*a^3*A*b^3 + 16*a*A*b^5 - 3*a^6*B + 15*a^4*b^2*B - 8*a^2*b^4*B - 6*a^5*b*C + 2*a^3*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(8*a^4*A*b^2 - 28*a^2*A*b^4 + 16*A*b^6 - 3*a^5*b*B + 15*a^3*b^3*B - 8*a*b^5*B - 6*a^4*b^2*C + 2*a^2*b^4*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(3*a^4*(a - b)^2*(a + b)^2*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]*(-8*A*b*Sin[c + d*x] + 3*a*B*Sin[c + d*x]))/(3*a^4) + (2*(A*b^4*Sin[c + d*x] - a*b^3*B*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) + (2*(12*a^2*A*b^4*Sin[c + d*x] - 8*A*b^6*Sin[c + d*x] - 9*a^3*b^3*B*Sin[c + d*x] + 5*a*b^5*B*Sin[c + d*x] + 6*a^4*b^2*C*Sin[c + d*x] - 2*a^2*b^4*C*Sin[c + d*x]))/(3*a^4*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) + (2*A*Sec[c + d*x]*Tan[c + d*x])/(3*a^3)))/d","C",0
1156,1,268,367,3.220715,"\int \cos ^m(c+d x) (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^m*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sin (c+d x) \cos ^{m+1}(c+d x) \left(\cos (c+d x) \left(\cos (c+d x) \left(b \cos (c+d x) \left(-\frac{(2 a C+b B) \, _2F_1\left(\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};\cos ^2(c+d x)\right)}{m+4}-\frac{b C \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+5}{2};\frac{m+7}{2};\cos ^2(c+d x)\right)}{m+5}\right)-\frac{\left(a (a C+2 b B)+A b^2\right) \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\cos ^2(c+d x)\right)}{m+3}\right)-\frac{a (a B+2 A b) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{m+2}\right)-\frac{a^2 A \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{m+1}\right)}{d \sqrt{\sin ^2(c+d x)}}","-\frac{\sin (c+d x) \cos ^{m+1}(c+d x) \left(a^2 (m+4) (A (m+2)+C (m+1))+2 a b B \left(m^2+5 m+4\right)+b^2 (m+1) (A (m+4)+C (m+3))\right) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) (m+4) \sqrt{\sin ^2(c+d x)}}-\frac{\sin (c+d x) \cos ^{m+2}(c+d x) \left(a^2 B (m+3)+2 a b (A (m+3)+C (m+2))+b^2 B (m+2)\right) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{\sin (c+d x) \cos ^{m+1}(c+d x) \left(2 a^2 C+2 a b B (m+4)+A b^2 (m+4)+b^2 C (m+3)\right)}{d (m+2) (m+4)}+\frac{b \sin (c+d x) (2 a C+b B (m+4)) \cos ^{m+2}(c+d x)}{d (m+3) (m+4)}+\frac{C \sin (c+d x) \cos ^{m+1}(c+d x) (a+b \cos (c+d x))^2}{d (m+4)}",1,"(Cos[c + d*x]^(1 + m)*(-((a^2*A*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2])/(1 + m)) + Cos[c + d*x]*(-((a*(2*A*b + a*B)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2])/(2 + m)) + Cos[c + d*x]*(-(((A*b^2 + a*(2*b*B + a*C))*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, Cos[c + d*x]^2])/(3 + m)) + b*Cos[c + d*x]*(-(((b*B + 2*a*C)*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, Cos[c + d*x]^2])/(4 + m)) - (b*C*Cos[c + d*x]*Hypergeometric2F1[1/2, (5 + m)/2, (7 + m)/2, Cos[c + d*x]^2])/(5 + m)))))*Sin[c + d*x])/(d*Sqrt[Sin[c + d*x]^2])","A",1
1157,1,205,235,1.7606184,"\int \cos ^m(c+d x) (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^m*(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sin (c+d x) \cos ^{m+1}(c+d x) \left(\cos (c+d x) \left(\cos (c+d x) \left(-\frac{(a C+b B) \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\cos ^2(c+d x)\right)}{m+3}-\frac{b C \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};\cos ^2(c+d x)\right)}{m+4}\right)-\frac{(a B+A b) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{m+2}\right)-\frac{a A \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{m+1}\right)}{d \sqrt{\sin ^2(c+d x)}}","-\frac{\sin (c+d x) \cos ^{m+1}(c+d x) (a A (m+2)+(m+1) (a C+b B)) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) \sqrt{\sin ^2(c+d x)}}-\frac{\sin (c+d x) \cos ^{m+2}(c+d x) (a B (m+3)+A b (m+3)+b C (m+2)) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{(a C+b B) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2)}+\frac{b C \sin (c+d x) \cos ^{m+2}(c+d x)}{d (m+3)}",1,"(Cos[c + d*x]^(1 + m)*(-((a*A*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2])/(1 + m)) + Cos[c + d*x]*(-(((A*b + a*B)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2])/(2 + m)) + Cos[c + d*x]*(-(((b*B + a*C)*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, Cos[c + d*x]^2])/(3 + m)) - (b*C*Cos[c + d*x]*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, Cos[c + d*x]^2])/(4 + m))))*Sin[c + d*x])/(d*Sqrt[Sin[c + d*x]^2])","A",1
1158,1,15557,372,30.1273837,"\int \frac{\cos ^m(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^m*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]","\text{Result too large to show}","\frac{a \sin (c+d x) \left(A b^2-a (b B-a C)\right) \cos ^{m-1}(c+d x) \cos ^2(c+d x)^{\frac{1-m}{2}} F_1\left(\frac{1}{2};\frac{1-m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{b^2 d \left(a^2-b^2\right)}-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right) \cos ^m(c+d x) \cos ^2(c+d x)^{-m/2} F_1\left(\frac{1}{2};-\frac{m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{b d \left(a^2-b^2\right)}-\frac{(b B-a C) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{b^2 d (m+1) \sqrt{\sin ^2(c+d x)}}-\frac{C \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{b d (m+2) \sqrt{\sin ^2(c+d x)}}",1,"Result too large to show","B",0
1159,1,12349,564,45.6345572,"\int \frac{\cos ^m(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^m*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^2,x]","\text{Result too large to show}","\frac{\sin (c+d x) \cos ^{m+1}(c+d x) \left(a^2 (-C) (m+1)+a b B m+b^2 (C-A m)\right) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{b^2 d (m+1) \left(a^2-b^2\right) \sqrt{\sin ^2(c+d x)}}+\frac{(m+1) \sin (c+d x) \left(A b^2-a (b B-a C)\right) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{a b d (m+2) \left(a^2-b^2\right) \sqrt{\sin ^2(c+d x)}}+\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right) \cos ^{m+1}(c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \cos ^{m-1}(c+d x) \cos ^2(c+d x)^{\frac{1-m}{2}} \left(a^4 (-C) (m+1)+a^3 b B m+a^2 b^2 (A (-m)+A+C (m+2))-a b^3 B (m+1)+A b^4 m\right) F_1\left(\frac{1}{2};\frac{1-m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{b^2 d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x) \cos ^m(c+d x) \cos ^2(c+d x)^{-m/2} \left(a^4 (-C) (m+1)+a^3 b B m+a^2 b^2 (A (-m)+A+C (m+2))-a b^3 B (m+1)+A b^4 m\right) F_1\left(\frac{1}{2};-\frac{m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{a b d \left(a^2-b^2\right)^2}",1,"Result too large to show","B",0
1160,1,392,205,4.2052747,"\int (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{a \csc (c) e^{-i d x} (\cos (c+d x)+1) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(7 \sqrt{2} \left(-1+e^{2 i c}\right) (3 A+5 C) e^{2 i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-\frac{\left(-1+e^{2 i c}\right) e^{-i (c-d x)} \sqrt{\sec (c+d x)} \left(A \left(21 e^{i (c+d x)}-85 e^{2 i (c+d x)}+189 e^{3 i (c+d x)}+85 e^{4 i (c+d x)}+231 e^{5 i (c+d x)}+25 e^{6 i (c+d x)}+63 e^{7 i (c+d x)}-25\right)+35 C \left(3 e^{i (c+d x)}+e^{2 i (c+d x)}+3 e^{3 i (c+d x)}-1\right) \left(1+e^{2 i (c+d x)}\right)^2\right)}{\left(1+e^{2 i (c+d x)}\right)^3}+10 (5 A+7 C) \sin (c) e^{i d x} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 a (5 A+7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 a (3 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}",1,"(a*(1 + Cos[c + d*x])*Csc[c]*Sec[(c + d*x)/2]^2*(7*Sqrt[2]*(3*A + 5*C)*E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] - ((-1 + E^((2*I)*c))*(35*C*(1 + E^((2*I)*(c + d*x)))^2*(-1 + 3*E^(I*(c + d*x)) + E^((2*I)*(c + d*x)) + 3*E^((3*I)*(c + d*x))) + A*(-25 + 21*E^(I*(c + d*x)) - 85*E^((2*I)*(c + d*x)) + 189*E^((3*I)*(c + d*x)) + 85*E^((4*I)*(c + d*x)) + 231*E^((5*I)*(c + d*x)) + 25*E^((6*I)*(c + d*x)) + 63*E^((7*I)*(c + d*x))))*Sqrt[Sec[c + d*x]])/(E^(I*(c - d*x))*(1 + E^((2*I)*(c + d*x)))^3) + 10*(5*A + 7*C)*E^(I*d*x)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]*Sin[c]))/(210*d*E^(I*d*x))","C",1
1161,1,277,172,1.8268729,"\int (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{a e^{-i c} \left(-1+e^{2 i c}\right) \csc (c) (\cos (c+d x)+1) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left((3 A+5 C) e^{i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{5/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-5 i (A+3 C) \left(1+e^{2 i (c+d x)}\right)^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-3 A e^{i (c+d x)}-24 A e^{3 i (c+d x)}-5 A e^{4 i (c+d x)}-9 A e^{5 i (c+d x)}+5 A-15 C e^{i (c+d x)}-30 C e^{3 i (c+d x)}-15 C e^{5 i (c+d x)}\right)}{30 d \left(1+e^{2 i (c+d x)}\right)^2}","\frac{2 a (3 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}",1,"(a*(-1 + E^((2*I)*c))*(1 + Cos[c + d*x])*Csc[c]*(5*A - 3*A*E^(I*(c + d*x)) - 15*C*E^(I*(c + d*x)) - 24*A*E^((3*I)*(c + d*x)) - 30*C*E^((3*I)*(c + d*x)) - 5*A*E^((4*I)*(c + d*x)) - 9*A*E^((5*I)*(c + d*x)) - 15*C*E^((5*I)*(c + d*x)) - (5*I)*(A + 3*C)*(1 + E^((2*I)*(c + d*x)))^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (3*A + 5*C)*E^(I*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(5/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[(c + d*x)/2]^2*Sqrt[Sec[c + d*x]])/(30*d*E^(I*c)*(1 + E^((2*I)*(c + d*x)))^2)","C",1
1162,1,173,135,1.1915521,"\int (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{a e^{-i d x} \sec ^{\frac{3}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left(i (A-C) \left(1+e^{2 i (c+d x)}\right)^{3/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+2 (A+3 C) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 A \sin (c+d x)+3 A \sin (2 (c+d x))-3 i A \cos (2 (c+d x))-3 i A+3 i C \cos (2 (c+d x))+3 i C\right)}{3 d}","\frac{2 a (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}",1,"(a*Sec[c + d*x]^(3/2)*(Cos[d*x] + I*Sin[d*x])*((-3*I)*A + (3*I)*C - (3*I)*A*Cos[2*(c + d*x)] + (3*I)*C*Cos[2*(c + d*x)] + 2*(A + 3*C)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + I*(A - C)*(1 + E^((2*I)*(c + d*x)))^(3/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 2*A*Sin[c + d*x] + 3*A*Sin[2*(c + d*x)]))/(3*d*E^(I*d*x))","C",1
1163,1,169,135,1.1624941,"\int (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{a e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(2 i (A-C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+2 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 A \sin (c+d x)-6 i A \cos (c+d x)+C \sin (2 (c+d x))+6 i C \cos (c+d x)\right)}{3 d}","\frac{2 a (3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a C \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(a*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*((-6*I)*A*Cos[c + d*x] + (6*I)*C*Cos[c + d*x] + 2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (2*I)*(A - C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 6*A*Sin[c + d*x] + C*Sin[2*(c + d*x)]))/(3*d*E^(I*d*x))","C",1
1164,1,169,141,1.5354241,"\int (a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{a e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-2 i (5 A+3 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (6 i (5 A+3 C)+10 C \sin (c+d x)+3 C \sin (2 (c+d x)))+10 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d}","\frac{2 a (3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a C \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a C \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(a*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(10*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (2*I)*(5*A + 3*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((6*I)*(5*A + 3*C) + 10*C*Sin[c + d*x] + 3*C*Sin[2*(c + d*x)])))/(15*d*E^(I*d*x))","C",1
1165,1,188,174,2.0535216,"\int \frac{(a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{a e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-28 i (5 A+3 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (5 (28 A+23 C) \sin (c+d x)+84 i (5 A+3 C)+42 C \sin (2 (c+d x))+15 C \sin (3 (c+d x)))+20 (7 A+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 a (7 A+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a C \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a C \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(20*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (28*I)*(5*A + 3*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((84*I)*(5*A + 3*C) + 5*(28*A + 23*C)*Sin[c + d*x] + 42*C*Sin[2*(c + d*x)] + 15*C*Sin[3*(c + d*x)])))/(210*d*E^(I*d*x))","C",1
1166,1,204,205,2.7043298,"\int \frac{(a+a \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{a e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-56 i (9 A+7 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (30 (28 A+23 C) \sin (c+d x)+14 (18 A+19 C) \sin (2 (c+d x))+1512 i A+90 C \sin (3 (c+d x))+35 C \sin (4 (c+d x))+1176 i C)+120 (7 A+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{1260 d}","\frac{2 a (9 A+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (7 A+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (9 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a C \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a C \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(a*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(120*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (56*I)*(9*A + 7*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((1512*I)*A + (1176*I)*C + 30*(28*A + 23*C)*Sin[c + d*x] + 14*(18*A + 19*C)*Sin[2*(c + d*x)] + 90*C*Sin[3*(c + d*x)] + 35*C*Sin[4*(c + d*x)])))/(1260*d*E^(I*d*x))","C",1
1167,1,655,270,6.8641229,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\sqrt{\sec (c+d x)} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^2 \left(\frac{4 (2 A+3 C) \csc (c) \cos (d x)}{15 d}+\frac{\sec (c) \sec ^2(c+d x) (90 A \sin (c)+112 A \sin (d x)+63 C \sin (d x))}{630 d}+\frac{\sec (c) \sec (c+d x) (112 A \sin (c)+150 A \sin (d x)+63 C \sin (c)+210 C \sin (d x))}{630 d}+\frac{(5 A+7 C) \tan (c)}{21 d}+\frac{A \sec (c) \sin (d x) \sec ^4(c+d x)}{18 d}+\frac{\sec (c) \sec ^3(c+d x) (7 A \sin (c)+18 A \sin (d x))}{126 d}\right)+\frac{4 \sqrt{2} A \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^2}{45 d}+\frac{5 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a \cos (c+d x)+a)^2}{21 d}+\frac{2 \sqrt{2} C \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^2}{15 d}+\frac{C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a \cos (c+d x)+a)^2}{3 d}","\frac{2 a^2 (19 A+21 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d}+\frac{4 a^2 (5 A+7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{16 a^2 (2 A+3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^2 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{16 a^2 (2 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{8 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}",1,"(4*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(a + a*Cos[c + d*x])^2*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^4)/(45*d*E^(I*d*x)) + (2*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(a + a*Cos[c + d*x])^2*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^4)/(15*d*E^(I*d*x)) + (5*A*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^4*Sqrt[Sec[c + d*x]])/(21*d) + (C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^4*Sqrt[Sec[c + d*x]])/(3*d) + (a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*Sqrt[Sec[c + d*x]]*((4*(2*A + 3*C)*Cos[d*x]*Csc[c])/(15*d) + (A*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(18*d) + (Sec[c]*Sec[c + d*x]^3*(7*A*Sin[c] + 18*A*Sin[d*x]))/(126*d) + (Sec[c]*Sec[c + d*x]^2*(90*A*Sin[c] + 112*A*Sin[d*x] + 63*C*Sin[d*x]))/(630*d) + (Sec[c]*Sec[c + d*x]*(112*A*Sin[c] + 63*C*Sin[c] + 150*A*Sin[d*x] + 210*C*Sin[d*x]))/(630*d) + ((5*A + 7*C)*Tan[c])/(21*d))","C",1
1168,1,399,237,4.3809703,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{a^2 \csc (c) e^{-i d x} (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(7 \sqrt{2} \left(-1+e^{2 i c}\right) (3 A+5 C) e^{2 i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-\frac{\left(-1+e^{2 i c}\right) e^{-i (c-d x)} \sqrt{\sec (c+d x)} \left(6 A \left(7 e^{i (c+d x)}-20 e^{2 i (c+d x)}+63 e^{3 i (c+d x)}+20 e^{4 i (c+d x)}+77 e^{5 i (c+d x)}+10 e^{6 i (c+d x)}+21 e^{7 i (c+d x)}-10\right)+35 C \left(6 e^{i (c+d x)}+e^{2 i (c+d x)}+6 e^{3 i (c+d x)}-1\right) \left(1+e^{2 i (c+d x)}\right)^2\right)}{2 \left(1+e^{2 i (c+d x)}\right)^3}+20 (3 A+7 C) \sin (c) e^{i d x} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 a^2 (33 A+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^2 (3 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{8 a^2 (3 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2}{7 d}",1,"(a^2*(1 + Cos[c + d*x])^2*Csc[c]*Sec[(c + d*x)/2]^4*(7*Sqrt[2]*(3*A + 5*C)*E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] - ((-1 + E^((2*I)*c))*(35*C*(1 + E^((2*I)*(c + d*x)))^2*(-1 + 6*E^(I*(c + d*x)) + E^((2*I)*(c + d*x)) + 6*E^((3*I)*(c + d*x))) + 6*A*(-10 + 7*E^(I*(c + d*x)) - 20*E^((2*I)*(c + d*x)) + 63*E^((3*I)*(c + d*x)) + 20*E^((4*I)*(c + d*x)) + 77*E^((5*I)*(c + d*x)) + 10*E^((6*I)*(c + d*x)) + 21*E^((7*I)*(c + d*x))))*Sqrt[Sec[c + d*x]])/(2*E^(I*(c - d*x))*(1 + E^((2*I)*(c + d*x)))^3) + 20*(3*A + 7*C)*E^(I*d*x)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]*Sin[c]))/(210*d*E^(I*d*x))","C",1
1169,1,301,196,2.8254054,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{1}{15} a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(\frac{\tan (c) \sqrt{\sec (c+d x)} \left(3 \cot (c) \csc (c) \cos (d x) (16 A-5 C \cos (2 c)+5 C)+6 A \csc (c) \sin (d x) \sec ^2(c+d x)+2 A \sec (c+d x) (10 \csc (c) \sin (d x)+3)+20 A+30 C \cos (c) \cot (c) \sin (d x)\right)}{4 d}-\frac{i \sqrt{2} \left(5 \left(-1+e^{2 i c}\right) (A+3 C) e^{i (c+d x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+12 A \left(-1+e^{2 i c}\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+12 A \sqrt{1+e^{2 i (c+d x)}}\right)}{\left(-1+e^{2 i c}\right) d \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}}}\right)","\frac{2 a^2 (17 A+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^2 (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{8 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{15 d}-\frac{16 a^2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{5 d}",1,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(((-I)*Sqrt[2]*(12*A*Sqrt[1 + E^((2*I)*(c + d*x))] + 12*A*(-1 + E^((2*I)*c))*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 5*(A + 3*C)*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*(-1 + E^((2*I)*c))*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]) + (Sqrt[Sec[c + d*x]]*(20*A + 3*(16*A + 5*C - 5*C*Cos[2*c])*Cos[d*x]*Cot[c]*Csc[c] + 30*C*Cos[c]*Cot[c]*Sin[d*x] + 6*A*Csc[c]*Sec[c + d*x]^2*Sin[d*x] + 2*A*Sec[c + d*x]*(3 + 10*Csc[c]*Sin[d*x]))*Tan[c])/(4*d)))/15","C",1
1170,1,191,196,1.7030193,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{a^2 e^{-i d x} \sec ^{\frac{3}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left(4 i (A-C) \left(1+e^{2 i (c+d x)}\right)^{3/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+16 (A+C) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+4 A \sin (c+d x)+12 A \sin (2 (c+d x))-12 i A \cos (2 (c+d x))-12 i A+C \sin (c+d x)+C \sin (3 (c+d x))+12 i C \cos (2 (c+d x))+12 i C\right)}{6 d}","-\frac{2 a^2 (5 A-C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{8 a^2 (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^2 (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{8 A \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{3 d}",1,"(a^2*Sec[c + d*x]^(3/2)*(Cos[d*x] + I*Sin[d*x])*((-12*I)*A + (12*I)*C - (12*I)*A*Cos[2*(c + d*x)] + (12*I)*C*Cos[2*(c + d*x)] + 16*(A + C)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + (4*I)*(A - C)*(1 + E^((2*I)*(c + d*x)))^(3/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 4*A*Sin[c + d*x] + C*Sin[c + d*x] + 12*A*Sin[2*(c + d*x)] + C*Sin[3*(c + d*x)]))/(6*d*E^(I*d*x))","C",1
1171,1,281,200,2.4474241,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{1}{15} a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(\frac{\sqrt{\sec (c+d x)} (3 (20 A-31 C) \csc (c) \cos (d x)-3 (20 A+33 C) \csc (c) \cos (2 c+d x)+40 C \sin (2 (c+d x))+6 C \sin (3 (c+d x)))}{16 d}+\frac{i \sqrt{2} \left(-5 \left(-1+e^{2 i c}\right) (3 A+C) e^{i (c+d x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+12 \left(-1+e^{2 i c}\right) C \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+12 C \sqrt{1+e^{2 i (c+d x)}}\right)}{\left(-1+e^{2 i c}\right) d \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}}}\right)","-\frac{2 a^2 (15 A-7 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}-\frac{2 (5 A-C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{5 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{16 a^2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2}{d}",1,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*((I*Sqrt[2]*(12*C*Sqrt[1 + E^((2*I)*(c + d*x))] + 12*C*(-1 + E^((2*I)*c))*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 5*(3*A + C)*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*(-1 + E^((2*I)*c))*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]) + (Sqrt[Sec[c + d*x]]*(3*(20*A - 31*C)*Cos[d*x]*Csc[c] - 3*(20*A + 33*C)*Cos[2*c + d*x]*Csc[c] + 40*C*Sin[2*(c + d*x)] + 6*C*Sin[3*(c + d*x)]))/(16*d)))/15","C",1
1172,1,189,204,1.9848173,"\int (a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{a^2 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-56 i (5 A+3 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (5 (28 A+51 C) \sin (c+d x)+840 i A+84 C \sin (2 (c+d x))+15 C \sin (3 (c+d x))+504 i C)+80 (7 A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 a^2 (35 A+33 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{8 a^2 (7 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{35 d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \sqrt{\sec (c+d x)}}",1,"(a^2*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(80*(7*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (56*I)*(5*A + 3*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((840*I)*A + (504*I)*C + 5*(28*A + 51*C)*Sin[c + d*x] + 84*C*Sin[2*(c + d*x)] + 15*C*Sin[3*(c + d*x)])))/(210*d*E^(I*d*x))","C",1
1173,1,206,237,2.6715156,"\int \frac{(a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{a^2 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-448 i (3 A+2 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (60 (28 A+23 C) \sin (c+d x)+14 (18 A+37 C) \sin (2 (c+d x))+4032 i A+180 C \sin (3 (c+d x))+35 C \sin (4 (c+d x))+2688 i C)+240 (7 A+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{1260 d}","\frac{2 a^2 (21 A+19 C) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (7 A+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{16 a^2 (3 A+2 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{8 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a^2*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(240*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (448*I)*(3*A + 2*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((4032*I)*A + (2688*I)*C + 60*(28*A + 23*C)*Sin[c + d*x] + 14*(18*A + 37*C)*Sin[2*(c + d*x)] + 180*C*Sin[3*(c + d*x)] + 35*C*Sin[4*(c + d*x)])))/(1260*d*E^(I*d*x))","C",1
1174,1,228,270,2.9359024,"\int \frac{(a+a \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{a^2 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-2464 i (9 A+7 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (30 (1122 A+941 C) \sin (c+d x)+616 (18 A+19 C) \sin (2 (c+d x))+1980 A \sin (3 (c+d x))+66528 i A+4545 C \sin (3 (c+d x))+1540 C \sin (4 (c+d x))+315 C \sin (5 (c+d x))+51744 i C)+960 (33 A+25 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{27720 d}","\frac{4 a^2 (9 A+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (99 A+89 C) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a^2 (33 A+25 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{8 a^2 (33 A+25 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^2 (9 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{8 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{99 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{11 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a^2*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(960*(33*A + 25*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (2464*I)*(9*A + 7*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((66528*I)*A + (51744*I)*C + 30*(1122*A + 941*C)*Sin[c + d*x] + 616*(18*A + 19*C)*Sin[2*(c + d*x)] + 1980*A*Sin[3*(c + d*x)] + 4545*C*Sin[3*(c + d*x)] + 1540*C*Sin[4*(c + d*x)] + 315*C*Sin[5*(c + d*x)])))/(27720*d*E^(I*d*x))","C",1
1175,1,697,319,7.0107985,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{13}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2),x]","\sqrt{\sec (c+d x)} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \left(\frac{(5 A+7 C) \csc (c) \cos (d x)}{10 d}+\frac{\sec (c) \sec ^3(c+d x) (77 A \sin (c)+126 A \sin (d x)+33 C \sin (d x))}{924 d}+\frac{\sec (c) \sec ^2(c+d x) (630 A \sin (c)+770 A \sin (d x)+165 C \sin (c)+693 C \sin (d x))}{4620 d}+\frac{\sec (c) \sec (c+d x) (770 A \sin (c)+1050 A \sin (d x)+693 C \sin (c)+1430 C \sin (d x))}{4620 d}+\frac{(105 A+143 C) \tan (c)}{462 d}+\frac{A \sec (c) \sin (d x) \sec ^5(c+d x)}{44 d}+\frac{\sec (c) \sec ^4(c+d x) (3 A \sin (c)+11 A \sin (d x))}{132 d}\right)+\frac{A \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3}{6 \sqrt{2} d}+\frac{5 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a \cos (c+d x)+a)^3}{22 d}+\frac{7 C \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3}{30 \sqrt{2} d}+\frac{13 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a \cos (c+d x)+a)^3}{42 d}","\frac{8 a^3 (35 A+44 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{385 d}+\frac{4 a^3 (105 A+143 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{231 d}+\frac{4 a^3 (5 A+7 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 (35 A+33 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{231 d}+\frac{4 a^3 (105 A+143 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}-\frac{4 a^3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{33 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}",1,"(A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(a + a*Cos[c + d*x])^3*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6)/(6*Sqrt[2]*d*E^(I*d*x)) + (7*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(a + a*Cos[c + d*x])^3*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6)/(30*Sqrt[2]*d*E^(I*d*x)) + (5*A*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^6*Sqrt[Sec[c + d*x]])/(22*d) + (13*C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^6*Sqrt[Sec[c + d*x]])/(42*d) + (a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*Sqrt[Sec[c + d*x]]*(((5*A + 7*C)*Cos[d*x]*Csc[c])/(10*d) + (A*Sec[c]*Sec[c + d*x]^5*Sin[d*x])/(44*d) + (Sec[c]*Sec[c + d*x]^4*(3*A*Sin[c] + 11*A*Sin[d*x]))/(132*d) + (Sec[c]*Sec[c + d*x]^3*(77*A*Sin[c] + 126*A*Sin[d*x] + 33*C*Sin[d*x]))/(924*d) + (Sec[c]*Sec[c + d*x]^2*(630*A*Sin[c] + 165*C*Sin[c] + 770*A*Sin[d*x] + 693*C*Sin[d*x]))/(4620*d) + (Sec[c]*Sec[c + d*x]*(770*A*Sin[c] + 693*C*Sin[c] + 1050*A*Sin[d*x] + 1430*C*Sin[d*x]))/(4620*d) + ((105*A + 143*C)*Tan[c])/(462*d))","C",0
1176,1,655,286,6.9074973,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\sqrt{\sec (c+d x)} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \left(\frac{(17 A+27 C) \csc (c) \cos (d x)}{30 d}+\frac{\sec (c) \sec ^2(c+d x) (135 A \sin (c)+238 A \sin (d x)+63 C \sin (d x))}{1260 d}+\frac{\sec (c) \sec (c+d x) (238 A \sin (c)+330 A \sin (d x)+63 C \sin (c)+315 C \sin (d x))}{1260 d}+\frac{(22 A+21 C) \tan (c)}{84 d}+\frac{A \sec (c) \sin (d x) \sec ^4(c+d x)}{36 d}+\frac{\sec (c) \sec ^3(c+d x) (7 A \sin (c)+27 A \sin (d x))}{252 d}\right)+\frac{17 A \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3}{90 \sqrt{2} d}+\frac{11 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a \cos (c+d x)+a)^3}{42 d}+\frac{3 C \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3}{10 \sqrt{2} d}+\frac{C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a \cos (c+d x)+a)^3}{2 d}","\frac{8 a^3 (16 A+21 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^3 (17 A+27 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 (73 A+63 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{315 d}+\frac{4 a^3 (11 A+21 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (17 A+27 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^3}{9 d}",1,"(17*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(a + a*Cos[c + d*x])^3*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6)/(90*Sqrt[2]*d*E^(I*d*x)) + (3*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(a + a*Cos[c + d*x])^3*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6)/(10*Sqrt[2]*d*E^(I*d*x)) + (11*A*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^6*Sqrt[Sec[c + d*x]])/(42*d) + (C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^6*Sqrt[Sec[c + d*x]])/(2*d) + (a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*Sqrt[Sec[c + d*x]]*(((17*A + 27*C)*Cos[d*x]*Csc[c])/(30*d) + (A*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(36*d) + (Sec[c]*Sec[c + d*x]^3*(7*A*Sin[c] + 27*A*Sin[d*x]))/(252*d) + (Sec[c]*Sec[c + d*x]^2*(135*A*Sin[c] + 238*A*Sin[d*x] + 63*C*Sin[d*x]))/(1260*d) + (Sec[c]*Sec[c + d*x]*(238*A*Sin[c] + 63*C*Sin[c] + 330*A*Sin[d*x] + 315*C*Sin[d*x]))/(1260*d) + ((22*A + 21*C)*Tan[c])/(84*d))","C",1
1177,1,302,253,4.6104594,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{a^3 \csc (c) \sec (c) e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(14 \left(-1+e^{4 i c}\right) (7 A+5 C) e^{-i (c-d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\frac{1}{4} \sin (2 c) \sec ^3(c+d x) \left(-168 i (7 A+5 C) \cos (2 (c+d x))+80 (13 A+35 C) \cos ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+380 A \sin (c+d x)+840 A \sin (2 (c+d x))+260 A \sin (3 (c+d x))+294 A \sin (4 (c+d x))-294 i A \cos (4 (c+d x))-882 i A+70 C \sin (c+d x)+630 C \sin (2 (c+d x))+70 C \sin (3 (c+d x))+315 C \sin (4 (c+d x))-210 i C \cos (4 (c+d x))-630 i C\right)\right)}{210 d}","\frac{8 a^3 (53 A+70 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d}+\frac{2 (7 A+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d}+\frac{4 a^3 (13 A+35 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{12 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{35 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^3}{7 d}",1,"(a^3*Csc[c]*Sec[c]*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*((14*(7*A + 5*C)*(-1 + E^((4*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^(I*(c - d*x)) + (Sec[c + d*x]^3*Sin[2*c]*((-882*I)*A - (630*I)*C - (168*I)*(7*A + 5*C)*Cos[2*(c + d*x)] - (294*I)*A*Cos[4*(c + d*x)] - (210*I)*C*Cos[4*(c + d*x)] + 80*(13*A + 35*C)*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] + 380*A*Sin[c + d*x] + 70*C*Sin[c + d*x] + 840*A*Sin[2*(c + d*x)] + 630*C*Sin[2*(c + d*x)] + 260*A*Sin[3*(c + d*x)] + 70*C*Sin[3*(c + d*x)] + 294*A*Sin[4*(c + d*x)] + 315*C*Sin[4*(c + d*x)]))/4))/(210*d*E^(I*d*x))","C",1
1178,1,279,253,3.1818019,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{a^3 \csc (c) \sec (c) e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(4 \left(-1+e^{4 i c}\right) (9 A-5 C) e^{-i (c-d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\frac{1}{2} \sin (2 c) \sec ^2(c+d x) \left(-36 i (9 A-5 C) \cos (c+d x)+80 (3 A+5 C) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+132 A \sin (c+d x)+60 A \sin (2 (c+d x))+108 A \sin (3 (c+d x))-108 i A \cos (3 (c+d x))+30 C \sin (c+d x)+10 C \sin (2 (c+d x))+30 C \sin (3 (c+d x))+5 C \sin (4 (c+d x))+60 i C \cos (3 (c+d x))\right)\right)}{60 d}","-\frac{4 a^3 (21 A+5 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 (11 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{5 d}+\frac{4 a^3 (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (9 A-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{5 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{5 d}",1,"(a^3*Csc[c]*Sec[c]*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*((4*(9*A - 5*C)*(-1 + E^((4*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^(I*(c - d*x)) + (Sec[c + d*x]^2*Sin[2*c]*((-36*I)*(9*A - 5*C)*Cos[c + d*x] - (108*I)*A*Cos[3*(c + d*x)] + (60*I)*C*Cos[3*(c + d*x)] + 80*(3*A + 5*C)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 132*A*Sin[c + d*x] + 30*C*Sin[c + d*x] + 60*A*Sin[2*(c + d*x)] + 10*C*Sin[2*(c + d*x)] + 108*A*Sin[3*(c + d*x)] + 30*C*Sin[3*(c + d*x)] + 5*C*Sin[4*(c + d*x)]))/2))/(60*d*E^(I*d*x))","C",1
1179,1,221,251,2.3412762,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{a^3 e^{-i d x} \sec ^{\frac{3}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left(8 i (5 A-9 C) \left(1+e^{2 i (c+d x)}\right)^{3/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+80 (5 A+3 C) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+40 A \sin (c+d x)+180 A \sin (2 (c+d x))-120 i A \cos (2 (c+d x))-120 i A+30 C \sin (c+d x)+6 C \sin (2 (c+d x))+30 C \sin (3 (c+d x))+3 C \sin (4 (c+d x))+216 i C \cos (2 (c+d x))+216 i C\right)}{60 d}","-\frac{8 a^3 (10 A-3 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}-\frac{2 (35 A-3 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (5 A-9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 A \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)^2}{a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{3 d}",1,"(a^3*Sec[c + d*x]^(3/2)*(Cos[d*x] + I*Sin[d*x])*((-120*I)*A + (216*I)*C - (120*I)*A*Cos[2*(c + d*x)] + (216*I)*C*Cos[2*(c + d*x)] + 80*(5*A + 3*C)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + (8*I)*(5*A - 9*C)*(1 + E^((2*I)*(c + d*x)))^(3/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 40*A*Sin[c + d*x] + 30*C*Sin[c + d*x] + 180*A*Sin[2*(c + d*x)] + 6*C*Sin[2*(c + d*x)] + 30*C*Sin[3*(c + d*x)] + 3*C*Sin[4*(c + d*x)]))/(60*d*E^(I*d*x))","C",1
1180,1,218,257,1.9331859,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{a^3 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-112 i (5 A+7 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+80 (35 A+13 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+840 A \sin (c+d x)+140 A \sin (2 (c+d x))+1680 i A \cos (c+d x)+126 C \sin (c+d x)+550 C \sin (2 (c+d x))+126 C \sin (3 (c+d x))+15 C \sin (4 (c+d x))+2352 i C \cos (c+d x)\right)}{420 d}","-\frac{4 a^3 (35 A-41 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}-\frac{2 (35 A-11 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{35 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (35 A+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{2 (7 A-C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{7 a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^3}{d}",1,"(a^3*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*((1680*I)*A*Cos[c + d*x] + (2352*I)*C*Cos[c + d*x] + 80*(35*A + 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (112*I)*(5*A + 7*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 840*A*Sin[c + d*x] + 126*C*Sin[c + d*x] + 140*A*Sin[2*(c + d*x)] + 550*C*Sin[2*(c + d*x)] + 126*C*Sin[3*(c + d*x)] + 15*C*Sin[4*(c + d*x)]))/(420*d*E^(I*d*x))","C",1
1181,1,206,253,2.6901259,"\int (a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{a^3 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-112 i (27 A+17 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (30 (84 A+97 C) \sin (c+d x)+14 (18 A+73 C) \sin (2 (c+d x))+9072 i A+270 C \sin (3 (c+d x))+35 C \sin (4 (c+d x))+5712 i C)+240 (21 A+11 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{1260 d}","\frac{8 a^3 (21 A+16 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{2 (63 A+73 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{315 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (21 A+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (27 A+17 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \sqrt{\sec (c+d x)}}",1,"(a^3*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(240*(21*A + 11*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (112*I)*(27*A + 17*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((9072*I)*A + (5712*I)*C + 30*(84*A + 97*C)*Sin[c + d*x] + 14*(18*A + 73*C)*Sin[2*(c + d*x)] + 270*C*Sin[3*(c + d*x)] + 35*C*Sin[4*(c + d*x)])))/(1260*d*E^(I*d*x))","C",1
1182,1,228,286,2.8427682,"\int \frac{(a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{a^3 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-2464 i (7 A+5 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (10 (2354 A+1953 C) \sin (c+d x)+308 (18 A+25 C) \sin (2 (c+d x))+660 A \sin (3 (c+d x))+51744 i A+2835 C \sin (3 (c+d x))+770 C \sin (4 (c+d x))+105 C \sin (5 (c+d x))+36960 i C)+160 (143 A+105 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{9240 d}","\frac{8 a^3 (44 A+35 C) \sin (c+d x)}{385 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (143 A+105 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 (33 A+35 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{231 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (143 A+105 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{33 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a^3*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(160*(143*A + 105*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (2464*I)*(7*A + 5*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((51744*I)*A + (36960*I)*C + 10*(2354*A + 1953*C)*Sin[c + d*x] + 308*(18*A + 25*C)*Sin[2*(c + d*x)] + 660*A*Sin[3*(c + d*x)] + 2835*C*Sin[3*(c + d*x)] + 770*C*Sin[4*(c + d*x)] + 105*C*Sin[5*(c + d*x)])))/(9240*d*E^(I*d*x))","C",1
1183,1,250,319,3.2989475,"\int \frac{(a+a \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{a^3 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-4928 i (221 A+175 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (780 (2134 A+1811 C) \sin (c+d x)+77 (7592 A+7825 C) \sin (2 (c+d x))+154440 A \sin (3 (c+d x))+20020 A \sin (4 (c+d x))+3267264 i A+251550 C \sin (3 (c+d x))+90860 C \sin (4 (c+d x))+24570 C \sin (5 (c+d x))+3465 C \sin (6 (c+d x))+2587200 i C)+12480 (121 A+95 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{720720 d}","\frac{4 a^3 (221 A+175 C) \sin (c+d x)}{585 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{40 a^3 (143 A+118 C) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (121 A+95 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 (143 A+145 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{1287 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (121 A+95 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (221 A+175 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{195 d}+\frac{12 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{143 a d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{13 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a^3*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(12480*(121*A + 95*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (4928*I)*(221*A + 175*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((3267264*I)*A + (2587200*I)*C + 780*(2134*A + 1811*C)*Sin[c + d*x] + 77*(7592*A + 7825*C)*Sin[2*(c + d*x)] + 154440*A*Sin[3*(c + d*x)] + 251550*C*Sin[3*(c + d*x)] + 20020*A*Sin[4*(c + d*x)] + 90860*C*Sin[4*(c + d*x)] + 24570*C*Sin[5*(c + d*x)] + 3465*C*Sin[6*(c + d*x)])))/(720720*d*E^(I*d*x))","C",1
1184,1,685,232,7.4945014,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x]),x]","\frac{\cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\sec (c+d x)} \left(-\frac{2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}+\frac{3 (7 A+5 C) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos (d x)}{5 d}-\frac{2 \tan \left(\frac{c}{2}\right) \sec (c) (5 A \cos (c)+2 A+3 C \cos (c))}{3 d}+\frac{4 A \sec (c) \sin (d x) \sec ^2(c+d x)}{5 d}+\frac{4 \sec (c) \sec (c+d x) (3 A \sin (c)-5 A \sin (d x))}{15 d}\right)}{a \cos (c+d x)+a}+\frac{7 A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 \sqrt{2} d (a \cos (c+d x)+a)}-\frac{5 A \sin (c) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sqrt{\cos (c+d x)} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d (a \cos (c+d x)+a)}+\frac{C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{2} d (a \cos (c+d x)+a)}-\frac{C \sin (c) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sqrt{\cos (c+d x)} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d (a \cos (c+d x)+a)}","\frac{(7 A+5 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 a d}-\frac{(5 A+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}+\frac{3 (7 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 a d}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}",1,"(7*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2])/(5*Sqrt[2]*d*E^(I*d*x)*(a + a*Cos[c + d*x])) + (C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2])/(Sqrt[2]*d*E^(I*d*x)*(a + a*Cos[c + d*x])) - (5*A*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*Sin[c])/(3*d*(a + a*Cos[c + d*x])) - (C*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*Sin[c])/(d*(a + a*Cos[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Sec[c + d*x]]*((3*(7*A + 5*C)*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (4*A*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(5*d) + (4*Sec[c]*Sec[c + d*x]*(3*A*Sin[c] - 5*A*Sin[d*x]))/(15*d) - (2*(2*A + 5*A*Cos[c] + 3*C*Cos[c])*Sec[c]*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])","C",1
1185,1,651,190,7.3107839,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x]),x]","\frac{\cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\sec (c+d x)} \left(\frac{2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}-\frac{(3 A+C) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos (d x)}{d}+\frac{2 \tan \left(\frac{c}{2}\right) \sec (c) (5 A \cos (c)+2 A+3 C \cos (c))}{3 d}+\frac{4 A \sec (c) \sin (d x) \sec (c+d x)}{3 d}\right)}{a \cos (c+d x)+a}-\frac{A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{2} d (a \cos (c+d x)+a)}+\frac{5 A \sin (c) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sqrt{\cos (c+d x)} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d (a \cos (c+d x)+a)}-\frac{C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 \sqrt{2} d (a \cos (c+d x)+a)}+\frac{C \sin (c) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sqrt{\cos (c+d x)} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d (a \cos (c+d x)+a)}","\frac{(5 A+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}-\frac{(3 A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{(3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"-((A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2])/(Sqrt[2]*d*E^(I*d*x)*(a + a*Cos[c + d*x]))) - (C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2])/(3*Sqrt[2]*d*E^(I*d*x)*(a + a*Cos[c + d*x])) + (5*A*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*Sin[c])/(3*d*(a + a*Cos[c + d*x])) + (C*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*Sin[c])/(d*(a + a*Cos[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Sec[c + d*x]]*(-(((3*A + C)*Cos[d*x]*Csc[c/2]*Sec[c/2])/d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (4*A*Sec[c]*Sec[c + d*x]*Sin[d*x])/(3*d) + (2*(2*A + 5*A*Cos[c] + 3*C*Cos[c])*Sec[c]*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])","C",1
1186,1,396,153,2.4668291,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x]),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(6 \sqrt{\sec (c+d x)} \left(2 (3 A+C) \csc (c) \cos (d x)-2 (A+C) \tan \left(\frac{1}{2} (c+d x)\right)\right)+6 \sqrt{2} A \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)-12 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sqrt{2} C \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)+12 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{6 a d (\cos (c+d x)+1)}","\frac{(3 A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{(A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \cos (c+d x)+a)}-\frac{(A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(Cos[(c + d*x)/2]^2*((6*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + (2*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) - 12*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] + 12*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] + 6*Sqrt[Sec[c + d*x]]*(2*(3*A + C)*Cos[d*x]*Csc[c] - 2*(A + C)*Tan[(c + d*x)/2])))/(6*a*d*(1 + Cos[c + d*x]))","C",1
1187,1,421,123,3.5233545,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{a+a \cos (c+d x)} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x]),x]","-\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(\frac{6 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right) \left((A+2 C) \cos \left(\frac{1}{2} (c-d x)\right)+C \cos \left(\frac{1}{2} (3 c+d x)\right)\right)}{\sqrt{\sec (c+d x)}}+2 \sqrt{2} A \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)-12 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 \sqrt{2} C \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)+12 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{6 a d (\cos (c+d x)+1)}","-\frac{(A+C) \sin (c+d x)}{d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)}+\frac{(A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"-1/6*(Cos[(c + d*x)/2]^2*((2*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + (6*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + (6*((A + 2*C)*Cos[(c - d*x)/2] + C*Cos[(3*c + d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2])/Sqrt[Sec[c + d*x]] - 12*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] + 12*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]))/(a*d*(1 + Cos[c + d*x]))","C",1
1188,1,439,162,4.255401,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x)) \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right) \left((6 A+13 C) \cos \left(\frac{1}{2} (c-d x)\right)+C \left(2 \sin (c) \sin \left(\frac{3}{2} (c+d x)\right)+5 \cos \left(\frac{1}{2} (3 c+d x)\right)\right)\right)}{\sqrt{\sec (c+d x)}}+2 \sqrt{2} A \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)+12 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 \sqrt{2} C \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)+20 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{6 a d (\cos (c+d x)+1)}","\frac{(3 A+5 C) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{(A+C) \sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{(3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{(A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(Cos[(c + d*x)/2]^2*((2*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + (6*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + 12*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] + 20*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] + (Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]*((6*A + 13*C)*Cos[(c - d*x)/2] + C*(5*Cos[(3*c + d*x)/2] + 2*Sin[c]*Sin[(3*(c + d*x))/2])))/Sqrt[Sec[c + d*x]]))/(6*a*d*(1 + Cos[c + d*x]))","C",1
1189,1,458,199,3.108345,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)),x]","-\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(\frac{2 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right) \left((60 A+83 C) \cos \left(\frac{1}{2} (c-d x)\right)+(30 A+43 C) \cos \left(\frac{1}{2} (3 c+d x)\right)+C \sin (c) \left(7 \sin \left(\frac{3}{2} (c+d x)\right)-3 \sin \left(\frac{5}{2} (c+d x)\right)\right)\right)}{\sqrt{\sec (c+d x)}}+60 \sqrt{2} A \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)+120 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+84 \sqrt{2} C \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)+200 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{60 a d (\cos (c+d x)+1)}","\frac{(5 A+7 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(3 A+5 C) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{(A+C) \sin (c+d x)}{d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)}-\frac{(3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}",1,"-1/60*(Cos[(c + d*x)/2]^2*((60*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + (84*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + 120*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] + 200*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] + (2*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]*((60*A + 83*C)*Cos[(c - d*x)/2] + (30*A + 43*C)*Cos[(3*c + d*x)/2] + C*Sin[c]*(7*Sin[(3*(c + d*x))/2] - 3*Sin[(5*(c + d*x))/2])))/Sqrt[Sec[c + d*x]]))/(a*d*(1 + Cos[c + d*x]))","C",1
1190,1,542,232,4.1036685,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(\sqrt{\sec (c+d x)} \left(20 (14 A+27 C) \sin (2 c) \cos (2 d x)-84 (20 A+33 C) \cos (c) \sin (d x)+20 (14 A+27 C) \cos (2 c) \sin (2 d x)-840 (A+C) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)+21 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos (d x) ((20 A+33 C) \cos (2 c)+40 A+51 C)-840 (A+C) \tan \left(\frac{c}{2}\right)-84 C \sin (3 c) \cos (3 d x)+30 C \sin (4 c) \cos (4 d x)-84 C \cos (3 c) \sin (3 d x)+30 C \cos (4 c) \sin (4 d x)\right)+420 \sqrt{2} A \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)+1400 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+588 \sqrt{2} C \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)+1800 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{420 a d (\cos (c+d x)+1)}","-\frac{(5 A+7 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{(7 A+9 C) \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}+\frac{5 (7 A+9 C) \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}-\frac{(A+C) \sin (c+d x)}{d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{5 (7 A+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a d}-\frac{3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}",1,"(Cos[(c + d*x)/2]^2*((420*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + (588*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + 1400*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] + 1800*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] + Sqrt[Sec[c + d*x]]*(21*(40*A + 51*C + (20*A + 33*C)*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2] + 20*(14*A + 27*C)*Cos[2*d*x]*Sin[2*c] - 84*C*Cos[3*d*x]*Sin[3*c] + 30*C*Cos[4*d*x]*Sin[4*c] - 840*(A + C)*Sec[c/2]*Sec[(c + d*x)/2]*Sin[(d*x)/2] - 84*(20*A + 33*C)*Cos[c]*Sin[d*x] + 20*(14*A + 27*C)*Cos[2*c]*Sin[2*d*x] - 84*C*Cos[3*c]*Sin[3*d*x] + 30*C*Cos[4*c]*Sin[4*d*x] - 840*(A + C)*Tan[c/2])))/(420*a*d*(1 + Cos[c + d*x]))","C",1
1191,1,734,229,7.7561103,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^2,x]","\frac{\cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\sec (c+d x)} \left(\frac{2 \sec \left(\frac{c}{2}\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{3 d}+\frac{2 (A+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{8 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(4 A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{3 d}-\frac{2 (7 A+C) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos (d x)}{d}+\frac{8 \tan \left(\frac{c}{2}\right) \sec (c) (5 A \cos (c)+A+C \cos (c))}{3 d}+\frac{8 A \sec (c) \sin (d x) \sec (c+d x)}{3 d}\right)}{(a \cos (c+d x)+a)^2}-\frac{7 \sqrt{2} A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (a \cos (c+d x)+a)^2}+\frac{20 A \sin (c) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sqrt{\cos (c+d x)} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d (a \cos (c+d x)+a)^2}-\frac{\sqrt{2} C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (a \cos (c+d x)+a)^2}+\frac{4 C \sin (c) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sqrt{\cos (c+d x)} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d (a \cos (c+d x)+a)^2}","\frac{2 (5 A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d}-\frac{(7 A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}-\frac{(7 A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{2 (5 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(7 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(-7*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2])/(3*d*E^(I*d*x)*(a + a*Cos[c + d*x])^2) - (Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2])/(3*d*E^(I*d*x)*(a + a*Cos[c + d*x])^2) + (20*A*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*Sin[c])/(3*d*(a + a*Cos[c + d*x])^2) + (4*C*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*Sin[c])/(3*d*(a + a*Cos[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Sec[c + d*x]]*((-2*(7*A + C)*Cos[d*x]*Csc[c/2]*Sec[c/2])/d + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(4*A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) + (8*A*Sec[c]*Sec[c + d*x]*Sin[d*x])/(3*d) + (8*(A + 5*A*Cos[c] + C*Cos[c])*Sec[c]*Tan[c/2])/(3*d) + (2*(A + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])^2","C",1
1192,1,275,195,1.4813418,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^2,x]","\frac{e^{-2 i (c+d x)} \left(1+e^{i (c+d x)}\right) \sqrt{\sec (c+d x)} \left(i \left(4 A e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-19 A e^{i (c+d x)}-29 A e^{2 i (c+d x)}-31 A e^{3 i (c+d x)}-12 A e^{4 i (c+d x)}-5 A-C e^{i (c+d x)}+C e^{2 i (c+d x)}-C e^{3 i (c+d x)}+C\right)-(5 A-C) \left(1+e^{i (c+d x)}\right)^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{12 a^2 d (\cos (c+d x)+1)^2}","-\frac{(5 A-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d (\cos (c+d x)+1)}-\frac{(5 A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{4 A \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}-\frac{4 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d (a \cos (c+d x)+a)^2}",1,"((1 + E^(I*(c + d*x)))*(-((5*A - C)*(1 + E^(I*(c + d*x)))^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]) + I*(-5*A + C - 19*A*E^(I*(c + d*x)) - C*E^(I*(c + d*x)) - 29*A*E^((2*I)*(c + d*x)) + C*E^((2*I)*(c + d*x)) - 31*A*E^((3*I)*(c + d*x)) - C*E^((3*I)*(c + d*x)) - 12*A*E^((4*I)*(c + d*x)) + 4*A*E^(I*(c + d*x))*(1 + E^(I*(c + d*x)))^3*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))*Sqrt[Sec[c + d*x]])/(12*a^2*d*E^((2*I)*(c + d*x))*(1 + Cos[c + d*x])^2)","C",1
1193,1,450,165,5.0612894,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^2} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^2,x]","\frac{\cos ^4\left(\frac{1}{2} (c+d x)\right) \left(-\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \left((7 A-5 C) \cos \left(\frac{1}{2} (c-d x)\right)+2 (A-2 C) \cos \left(\frac{1}{2} (3 c+d x)\right)+3 (A-C) \cos \left(\frac{1}{2} (c+3 d x)\right)\right)}{2 \sqrt{\sec (c+d x)}}-2 \sqrt{2} A \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)+8 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sqrt{2} C \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)+8 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 a^2 d (\cos (c+d x)+1)^2}","-\frac{(A-C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1) \sqrt{\sec (c+d x)}}+\frac{2 (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]^4*((-2*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + (2*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) - (((7*A - 5*C)*Cos[(c - d*x)/2] + 2*(A - 2*C)*Cos[(3*c + d*x)/2] + 3*(A - C)*Cos[(c + 3*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^3)/(2*Sqrt[Sec[c + d*x]]) + 8*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] + 8*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]))/(3*a^2*d*(1 + Cos[c + d*x])^2)","C",1
1194,1,267,166,1.4921191,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2 \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]),x]","\frac{e^{-3 i (c+d x)} \left(1+e^{i (c+d x)}\right) \sqrt{\sec (c+d x)} \left(i \left(1+e^{2 i (c+d x)}\right) \left(C \left(16 e^{i (c+d x)}+20 e^{2 i (c+d x)}+9 e^{3 i (c+d x)}+3\right)-A e^{i (c+d x)} \left(-1+e^{i (c+d x)}\right)\right)+(A-5 C) e^{i (c+d x)} \left(1+e^{i (c+d x)}\right)^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-4 i C e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)\right)}{12 a^2 d (\cos (c+d x)+1)^2}","\frac{(A-5 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1) \sqrt{\sec (c+d x)}}+\frac{(A-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{4 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+C) \sin (c+d x)}{3 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"((1 + E^(I*(c + d*x)))*(I*(1 + E^((2*I)*(c + d*x)))*(-(A*E^(I*(c + d*x))*(-1 + E^(I*(c + d*x)))) + C*(3 + 16*E^(I*(c + d*x)) + 20*E^((2*I)*(c + d*x)) + 9*E^((3*I)*(c + d*x)))) + (A - 5*C)*E^(I*(c + d*x))*(1 + E^(I*(c + d*x)))^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (4*I)*C*E^((2*I)*(c + d*x))*(1 + E^(I*(c + d*x)))^3*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sqrt[Sec[c + d*x]])/(12*a^2*d*E^((3*I)*(c + d*x))*(1 + Cos[c + d*x])^2)","C",1
1195,1,762,201,6.8469262,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)),x]","\frac{\cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\sec (c+d x)} \left(\frac{2 \sec \left(\frac{c}{2}\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{3 d}+\frac{2 (A+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{8 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(2 A \sin \left(\frac{d x}{2}\right)+5 C \sin \left(\frac{d x}{2}\right)\right)}{3 d}+\frac{2 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos (d x) (A+2 C \cos (2 c)+5 C)}{d}-\frac{8 (2 A+5 C) \tan \left(\frac{c}{2}\right)}{3 d}+\frac{4 C \sin (2 c) \cos (2 d x)}{3 d}-\frac{16 C \cos (c) \sin (d x)}{d}+\frac{4 C \cos (2 c) \sin (2 d x)}{3 d}\right)}{(a \cos (c+d x)+a)^2}+\frac{\sqrt{2} A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (a \cos (c+d x)+a)^2}+\frac{4 A \sin (c) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sqrt{\cos (c+d x)} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d (a \cos (c+d x)+a)^2}+\frac{7 \sqrt{2} C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (a \cos (c+d x)+a)^2}+\frac{20 C \sin (c) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sqrt{\cos (c+d x)} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d (a \cos (c+d x)+a)^2}","\frac{2 (A+5 C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{(A+7 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1) \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+C) \sin (c+d x)}{3 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"(Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2])/(3*d*E^(I*d*x)*(a + a*Cos[c + d*x])^2) + (7*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2])/(3*d*E^(I*d*x)*(a + a*Cos[c + d*x])^2) + (4*A*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*Sin[c])/(3*d*(a + a*Cos[c + d*x])^2) + (20*C*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*Sin[c])/(3*d*(a + a*Cos[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Sec[c + d*x]]*((2*(A + 5*C + 2*C*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2])/d + (4*C*Cos[2*d*x]*Sin[2*c])/(3*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(2*A*Sin[(d*x)/2] + 5*C*Sin[(d*x)/2]))/(3*d) - (16*C*Cos[c]*Sin[d*x])/d + (4*C*Cos[2*c]*Sin[2*d*x])/(3*d) - (8*(2*A + 5*C)*Tan[c/2])/(3*d) + (2*(A + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])^2","C",1
1196,1,813,236,6.9432337,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2 \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)),x]","-\frac{4 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2}-\frac{56 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (c+d x) a+a)^2}-\frac{10 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^2}-\frac{10 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^2}+\frac{\sqrt{\sec (c+d x)} \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (A+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(7 A \sin \left(\frac{d x}{2}\right)+13 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{(20 \cos (2 c) A+60 A+151 C+73 C \cos (2 c)) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{10 d}-\frac{8 C \cos (2 d x) \sin (2 c)}{3 d}+\frac{2 C \cos (3 d x) \sin (3 c)}{5 d}+\frac{2 (20 A+73 C) \cos (c) \sin (d x)}{5 d}-\frac{8 C \cos (2 c) \sin (2 d x)}{3 d}+\frac{2 C \cos (3 c) \sin (3 d x)}{5 d}+\frac{4 (7 A+13 C) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^2}","\frac{4 (5 A+14 C) \sin (c+d x)}{15 a^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 (A+3 C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{(A+3 C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1) \sec ^{\frac{5}{2}}(c+d x)}-\frac{5 (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{4 (5 A+14 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{(A+C) \sin (c+d x)}{3 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"(-4*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2])/(3*d*E^(I*d*x)*(a + a*Cos[c + d*x])^2) - (56*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2])/(15*d*E^(I*d*x)*(a + a*Cos[c + d*x])^2) - (10*A*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*Sin[c])/(3*d*(a + a*Cos[c + d*x])^2) - (10*C*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*Sin[c])/(d*(a + a*Cos[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Sec[c + d*x]]*(-1/10*((60*A + 151*C + 20*A*Cos[2*c] + 73*C*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2])/d - (8*C*Cos[2*d*x]*Sin[2*c])/(3*d) + (2*C*Cos[3*d*x]*Sin[3*c])/(5*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(7*A*Sin[(d*x)/2] + 13*C*Sin[(d*x)/2]))/(3*d) + (2*(20*A + 73*C)*Cos[c]*Sin[d*x])/(5*d) - (8*C*Cos[2*c]*Sin[2*d*x])/(3*d) + (2*C*Cos[3*c]*Sin[3*d*x])/(5*d) + (4*(7*A + 13*C)*Tan[c/2])/(3*d) - (2*(A + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])^2","C",1
1197,1,822,282,8.0355767,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^3,x]","-\frac{119 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (c+d x) a+a)^3}-\frac{3 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (\cos (c+d x) a+a)^3}+\frac{22 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^3}+\frac{2 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^3}+\frac{\sqrt{\sec (c+d x)} \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{2 (A+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(13 A \sin \left(\frac{d x}{2}\right)+3 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 (13 A+3 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(29 A \sin \left(\frac{d x}{2}\right)+3 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (119 A+9 C) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}+\frac{16 A \sec (c) \sec (c+d x) \sin (d x)}{3 d}+\frac{4 (33 \cos (c) A+4 A+3 C \cos (c)) \sec (c) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^3}","\frac{(11 A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 a^3 d}-\frac{(119 A+9 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{(119 A+9 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{30 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(11 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{(119 A+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d (a \cos (c+d x)+a)^2}",1,"(-119*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2])/(15*d*E^(I*d*x)*(a + a*Cos[c + d*x])^3) - (3*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2])/(5*d*E^(I*d*x)*(a + a*Cos[c + d*x])^3) + (22*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*Sin[c])/(d*(a + a*Cos[c + d*x])^3) + (2*C*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*Sin[c])/(d*(a + a*Cos[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sqrt[Sec[c + d*x]]*((-2*(119*A + 9*C)*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(13*A*Sin[(d*x)/2] + 3*C*Sin[(d*x)/2]))/(15*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(29*A*Sin[(d*x)/2] + 3*C*Sin[(d*x)/2]))/(3*d) + (16*A*Sec[c]*Sec[c + d*x]*Sin[d*x])/(3*d) + (4*(4*A + 33*A*Cos[c] + 3*C*Cos[c])*Sec[c]*Tan[c/2])/(3*d) + (4*(13*A + 3*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (2*(A + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",1
1198,1,359,259,5.5184909,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^3,x]","-\frac{e^{-i d x} \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(\cos \left(\frac{1}{2} (c+3 d x)\right)+i \sin \left(\frac{1}{2} (c+3 d x)\right)\right) \left(-i (49 A-C) e^{-2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)^5 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+2 i (2 (541 A-4 C) \cos (c+d x)+18 (29 A-C) \cos (2 (c+d x))+161 i A \sin (c+d x)+148 i A \sin (2 (c+d x))+41 i A \sin (3 (c+d x))+106 A \cos (3 (c+d x))+642 A+i C \sin (c+d x)+8 i C \sin (2 (c+d x))+i C \sin (3 (c+d x))-4 C \cos (3 (c+d x))-18 C)+160 (13 A-C) \cos ^5\left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-i \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{120 a^3 d (\cos (c+d x)+1)^3}","\frac{(49 A-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{(13 A-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(13 A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(49 A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{2 (4 A-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d (a \cos (c+d x)+a)^3}",1,"-1/120*(Cos[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(((-I)*(49*A - C)*(1 + E^(I*(c + d*x)))^5*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^((2*I)*(c + d*x)) + 160*(13*A - C)*Cos[(c + d*x)/2]^5*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[(c + d*x)/2] - I*Sin[(c + d*x)/2]) + (2*I)*(642*A - 18*C + 2*(541*A - 4*C)*Cos[c + d*x] + 18*(29*A - C)*Cos[2*(c + d*x)] + 106*A*Cos[3*(c + d*x)] - 4*C*Cos[3*(c + d*x)] + (161*I)*A*Sin[c + d*x] + I*C*Sin[c + d*x] + (148*I)*A*Sin[2*(c + d*x)] + (8*I)*C*Sin[2*(c + d*x)] + (41*I)*A*Sin[3*(c + d*x)] + I*C*Sin[3*(c + d*x)]))*(Cos[(c + 3*d*x)/2] + I*Sin[(c + 3*d*x)/2]))/(a^3*d*E^(I*d*x)*(1 + Cos[c + d*x])^3)","C",1
1199,1,792,224,7.0001748,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^3} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^3,x]","\frac{\cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\sec (c+d x)} \left(\frac{2 \sec \left(\frac{c}{2}\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{5 d}+\frac{2 (A+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 \sec \left(\frac{c}{2}\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \left(3 A \sin \left(\frac{d x}{2}\right)-7 C \sin \left(\frac{d x}{2}\right)\right)}{15 d}+\frac{4 (3 A-7 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(3 A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{3 d}-\frac{2 (9 A-C) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos (d x)}{5 d}+\frac{4 (3 A+C) \tan \left(\frac{c}{2}\right)}{3 d}\right)}{(a \cos (c+d x)+a)^3}-\frac{3 \sqrt{2} A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (a \cos (c+d x)+a)^3}+\frac{2 A \sin (c) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sqrt{\cos (c+d x)} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d (a \cos (c+d x)+a)^3}+\frac{\sqrt{2} C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (a \cos (c+d x)+a)^3}+\frac{2 C \sin (c) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sqrt{\cos (c+d x)} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d (a \cos (c+d x)+a)^3}","-\frac{(9 A-C) \sin (c+d x)}{10 d \sqrt{\sec (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(9 A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{2 (3 A-2 C) \sin (c+d x)}{15 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^3}",1,"(-3*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2])/(5*d*E^(I*d*x)*(a + a*Cos[c + d*x])^3) + (Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2])/(15*d*E^(I*d*x)*(a + a*Cos[c + d*x])^3) + (2*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*Sin[c])/(d*(a + a*Cos[c + d*x])^3) + (2*C*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*Sin[c])/(3*d*(a + a*Cos[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sqrt[Sec[c + d*x]]*((-2*(9*A - C)*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(3*A*Sin[(d*x)/2] - 7*C*Sin[(d*x)/2]))/(15*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(3*A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) + (4*(3*A + C)*Tan[c/2])/(3*d) + (4*(3*A - 7*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (2*(A + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",1
1200,1,787,220,7.0679019,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]),x]","\frac{\cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\sec (c+d x)} \left(-\frac{2 \sec \left(\frac{c}{2}\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{5 d}-\frac{2 (A+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{8 \sec \left(\frac{c}{2}\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+6 C \sin \left(\frac{d x}{2}\right)\right)}{15 d}+\frac{8 (A+6 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-9 C \sin \left(\frac{d x}{2}\right)\right)}{3 d}-\frac{2 (A-9 C) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos (d x)}{5 d}+\frac{4 (A-9 C) \tan \left(\frac{c}{2}\right)}{3 d}\right)}{(a \cos (c+d x)+a)^3}-\frac{\sqrt{2} A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (a \cos (c+d x)+a)^3}+\frac{2 A \sin (c) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sqrt{\cos (c+d x)} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d (a \cos (c+d x)+a)^3}+\frac{3 \sqrt{2} C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (a \cos (c+d x)+a)^3}+\frac{2 C \sin (c) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sqrt{\cos (c+d x)} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d (a \cos (c+d x)+a)^3}","-\frac{(A-9 C) \sin (c+d x)}{10 d \sqrt{\sec (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(A-9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A+C) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}+\frac{2 (2 A-3 C) \sin (c+d x)}{15 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2}",1,"-1/15*(Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2])/(d*E^(I*d*x)*(a + a*Cos[c + d*x])^3) + (3*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2])/(5*d*E^(I*d*x)*(a + a*Cos[c + d*x])^3) + (2*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*Sin[c])/(3*d*(a + a*Cos[c + d*x])^3) + (2*C*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*Sin[c])/(d*(a + a*Cos[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sqrt[Sec[c + d*x]]*((-2*(A - 9*C)*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - 9*C*Sin[(d*x)/2]))/(3*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + 6*C*Sin[(d*x)/2]))/(15*d) + (4*(A - 9*C)*Tan[c/2])/(3*d) + (8*(A + 6*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (2*(A + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",1
1201,1,813,218,7.0304469,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)),x]","\frac{\sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (c+d x) a+a)^3}-\frac{49 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (c+d x) a+a)^3}+\frac{2 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^3}-\frac{26 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (c+d x) a+a)^3}+\frac{\sqrt{\sec (c+d x)} \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{2 (A+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(7 A \sin \left(\frac{d x}{2}\right)+17 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 (7 A+17 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+23 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (-A+39 C+10 C \cos (2 c)) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}+\frac{16 C \cos (c) \sin (d x)}{d}+\frac{4 (A+23 C) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (c+d x) a+a)^3}","\frac{(A-13 C) \sin (c+d x)}{6 d \sqrt{\sec (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A-13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(A-49 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{2 (A-4 C) \sin (c+d x)}{15 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x)}{5 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^3}",1,"(Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2])/(15*d*E^(I*d*x)*(a + a*Cos[c + d*x])^3) - (49*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2])/(15*d*E^(I*d*x)*(a + a*Cos[c + d*x])^3) + (2*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*Sin[c])/(3*d*(a + a*Cos[c + d*x])^3) - (26*C*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*Sin[c])/(3*d*(a + a*Cos[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sqrt[Sec[c + d*x]]*((-2*(-A + 39*C + 10*C*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(7*A*Sin[(d*x)/2] + 17*C*Sin[(d*x)/2]))/(15*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + 23*C*Sin[(d*x)/2]))/(3*d) + (16*C*Cos[c]*Sin[d*x])/d + (4*(A + 23*C)*Tan[c/2])/(3*d) - (4*(7*A + 17*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (2*(A + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",1
1202,1,573,249,4.4421788,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)),x]","\frac{\cos ^6\left(\frac{1}{2} (c+d x)\right) \left(\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \left((156 A+1961 C) \cos \left(\frac{1}{2} (c-d x)\right)+(114 A+1609 C) \cos \left(\frac{1}{2} (3 c+d x)\right)+90 A \cos \left(\frac{1}{2} (c+3 d x)\right)+45 A \cos \left(\frac{1}{2} (5 c+3 d x)\right)+27 A \cos \left(\frac{1}{2} (3 c+5 d x)\right)+1165 C \cos \left(\frac{1}{2} (c+3 d x)\right)+620 C \cos \left(\frac{1}{2} (5 c+3 d x)\right)+292 C \cos \left(\frac{1}{2} (3 c+5 d x)\right)+65 C \cos \left(\frac{1}{2} (7 c+5 d x)\right)+5 C \cos \left(\frac{1}{2} (5 c+7 d x)\right)-5 C \cos \left(\frac{1}{2} (9 c+7 d x)\right)\right)}{8 \sqrt{\sec (c+d x)}}+18 \sqrt{2} A \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)+60 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+238 \sqrt{2} C \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)+660 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 a^3 d (\cos (c+d x)+1)^3}","\frac{(A+11 C) \sin (c+d x)}{2 a^3 d \sqrt{\sec (c+d x)}}-\frac{(9 A+119 C) \sin (c+d x)}{30 d \sec ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{(9 A+119 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A+C) \sin (c+d x)}{5 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^3}-\frac{2 C \sin (c+d x)}{3 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]^6*((18*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + (238*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + (((156*A + 1961*C)*Cos[(c - d*x)/2] + (114*A + 1609*C)*Cos[(3*c + d*x)/2] + 90*A*Cos[(c + 3*d*x)/2] + 1165*C*Cos[(c + 3*d*x)/2] + 45*A*Cos[(5*c + 3*d*x)/2] + 620*C*Cos[(5*c + 3*d*x)/2] + 27*A*Cos[(3*c + 5*d*x)/2] + 292*C*Cos[(3*c + 5*d*x)/2] + 65*C*Cos[(7*c + 5*d*x)/2] + 5*C*Cos[(5*c + 7*d*x)/2] - 5*C*Cos[(9*c + 7*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^5)/(8*Sqrt[Sec[c + d*x]]) + 60*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] + 660*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]))/(15*a^3*d*(1 + Cos[c + d*x])^3)","C",0
1203,1,623,290,5.7645179,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)),x]","-\frac{\cos ^6\left(\frac{1}{2} (c+d x)\right) \left(\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \left(2 (806 A+3795 C) \cos \left(\frac{1}{2} (c-d x)\right)+2 (664 A+3135 C) \cos \left(\frac{1}{2} (3 c+d x)\right)+940 A \cos \left(\frac{1}{2} (c+3 d x)\right)+530 A \cos \left(\frac{1}{2} (5 c+3 d x)\right)+234 A \cos \left(\frac{1}{2} (3 c+5 d x)\right)+60 A \cos \left(\frac{1}{2} (7 c+5 d x)\right)+4500 C \cos \left(\frac{1}{2} (c+3 d x)\right)+2430 C \cos \left(\frac{1}{2} (5 c+3 d x)\right)+1110 C \cos \left(\frac{1}{2} (3 c+5 d x)\right)+276 C \cos \left(\frac{1}{2} (7 c+5 d x)\right)+15 C \cos \left(\frac{1}{2} (5 c+7 d x)\right)-15 C \cos \left(\frac{1}{2} (9 c+7 d x)\right)-3 C \cos \left(\frac{1}{2} (7 c+9 d x)\right)+3 C \cos \left(\frac{1}{2} (11 c+9 d x)\right)\right)}{16 \sqrt{\sec (c+d x)}}+98 \sqrt{2} A \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)+260 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+462 \sqrt{2} C \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)+1260 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 a^3 d (\cos (c+d x)+1)^3}","\frac{7 (7 A+33 C) \sin (c+d x)}{30 a^3 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(13 A+63 C) \sin (c+d x)}{6 a^3 d \sqrt{\sec (c+d x)}}-\frac{(13 A+63 C) \sin (c+d x)}{10 d \sec ^{\frac{5}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(13 A+63 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{7 (7 A+33 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{2 (A+6 C) \sin (c+d x)}{15 a d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x)}{5 d \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^3}",1,"-1/15*(Cos[(c + d*x)/2]^6*((98*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + (462*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + ((2*(806*A + 3795*C)*Cos[(c - d*x)/2] + 2*(664*A + 3135*C)*Cos[(3*c + d*x)/2] + 940*A*Cos[(c + 3*d*x)/2] + 4500*C*Cos[(c + 3*d*x)/2] + 530*A*Cos[(5*c + 3*d*x)/2] + 2430*C*Cos[(5*c + 3*d*x)/2] + 234*A*Cos[(3*c + 5*d*x)/2] + 1110*C*Cos[(3*c + 5*d*x)/2] + 60*A*Cos[(7*c + 5*d*x)/2] + 276*C*Cos[(7*c + 5*d*x)/2] + 15*C*Cos[(5*c + 7*d*x)/2] - 15*C*Cos[(9*c + 7*d*x)/2] - 3*C*Cos[(7*c + 9*d*x)/2] + 3*C*Cos[(11*c + 9*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^5)/(16*Sqrt[Sec[c + d*x]]) + 260*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] + 1260*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]))/(a^3*d*(1 + Cos[c + d*x])^3)","C",0
1204,1,124,213,0.7771855,"\int \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} (2 (88 A+63 C) \cos (c+d x)+11 (16 A+21 C) \cos (2 (c+d x))+32 A \cos (3 (c+d x))+32 A \cos (4 (c+d x))+214 A+42 C \cos (3 (c+d x))+42 C \cos (4 (c+d x))+189 C)}{315 d}","\frac{2 a (16 A+21 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a (16 A+21 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a (16 A+21 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{9 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(214*A + 189*C + 2*(88*A + 63*C)*Cos[c + d*x] + 11*(16*A + 21*C)*Cos[2*(c + d*x)] + 32*A*Cos[3*(c + d*x)] + 42*C*Cos[3*(c + d*x)] + 32*A*Cos[4*(c + d*x)] + 42*C*Cos[4*(c + d*x)])*Sec[c + d*x]^(9/2)*Tan[(c + d*x)/2])/(315*d)","A",1
1205,1,101,168,0.5675496,"\int \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} (3 (36 A+35 C) \cos (c+d x)+(24 A+35 C) \cos (2 (c+d x))+24 A \cos (3 (c+d x))+54 A+35 C \cos (3 (c+d x))+35 C)}{105 d}","\frac{2 a (24 A+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a (24 A+35 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{7 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(54*A + 35*C + 3*(36*A + 35*C)*Cos[c + d*x] + (24*A + 35*C)*Cos[2*(c + d*x)] + 24*A*Cos[3*(c + d*x)] + 35*C*Cos[3*(c + d*x)])*Sec[c + d*x]^(7/2)*Tan[(c + d*x)/2])/(105*d)","A",1
1206,1,73,123,0.3156636,"\int \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} ((8 A+15 C) \cos (2 (c+d x))+8 A \cos (c+d x)+14 A+15 C)}{15 d}","\frac{2 a (8 A+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(14*A + 15*C + 8*A*Cos[c + d*x] + (8*A + 15*C)*Cos[2*(c + d*x)])*Sec[c + d*x]^(5/2)*Tan[(c + d*x)/2])/(15*d)","A",1
1207,1,90,136,0.2585986,"\int \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(2 A \sin \left(\frac{3}{2} (c+d x)\right)+3 \sqrt{2} C \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{3}{2}}(c+d x)\right)}{3 d}","\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 \sqrt{a} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^(3/2)*(3*Sqrt[2]*C*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(3/2) + 2*A*Sin[(3*(c + d*x))/2]))/(3*d)","A",1
1208,1,100,137,0.2882187,"\int \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) (2 A+C \cos (c+d x))+\sqrt{2} C \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}\right)}{2 d}","-\frac{a (2 A-C) \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{d}+\frac{\sqrt{a} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(Sqrt[2]*C*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*(2*A + C*Cos[c + d*x])*Sin[(c + d*x)/2]))/(2*d)","A",1
1209,1,118,144,0.3676194,"\int \sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(\sqrt{2} (8 A+3 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 C \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{3}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}\right)}{8 d}","\frac{\sqrt{a} (8 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}+\frac{a C \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(Sqrt[2]*(8*A + 3*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*C*Sqrt[Cos[c + d*x]]*(2*Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(8*d)","A",1
1210,1,134,189,0.5632287,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} (8 A+5 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+\left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (24 A+10 C \cos (c+d x)+4 C \cos (2 (c+d x))+19 C)\right)}{48 d}","\frac{\sqrt{a} (8 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a (8 A+5 C) \sin (c+d x)}{8 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{a C \sin (c+d x)}{12 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(3*Sqrt[2]*(8*A + 5*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + (24*A + 19*C + 10*C*Cos[c + d*x] + 4*C*Cos[2*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(48*d)","A",1
1211,1,151,234,0.592477,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} (48 A+35 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+\left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (2 (48 A+53 C) \cos (c+d x)+144 A+28 C \cos (2 (c+d x))+12 C \cos (3 (c+d x))+133 C)\right)}{384 d}","\frac{a (48 A+35 C) \sin (c+d x)}{96 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (48 A+35 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a (48 A+35 C) \sin (c+d x)}{64 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a C \sin (c+d x)}{24 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(3*Sqrt[2]*(48*A + 35*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + (144*A + 133*C + 2*(48*A + 53*C)*Cos[c + d*x] + 28*C*Cos[2*(c + d*x)] + 12*C*Cos[3*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(384*d)","A",1
1212,1,146,266,0.9478976,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{13}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} ((4228 A+4147 C) \cos (c+d x)+2 (728 A+737 C) \cos (2 (c+d x))+1456 A \cos (3 (c+d x))+224 A \cos (4 (c+d x))+224 A \cos (5 (c+d x))+1652 A+1859 C \cos (3 (c+d x))+286 C \cos (4 (c+d x))+286 C \cos (5 (c+d x))+1188 C)}{2310 d}","\frac{2 a^2 (28 A+33 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{231 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (112 A+143 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{385 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a^2 (112 A+143 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{1155 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (112 A+143 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{1155 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{33 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(1652*A + 1188*C + (4228*A + 4147*C)*Cos[c + d*x] + 2*(728*A + 737*C)*Cos[2*(c + d*x)] + 1456*A*Cos[3*(c + d*x)] + 1859*C*Cos[3*(c + d*x)] + 224*A*Cos[4*(c + d*x)] + 286*C*Cos[4*(c + d*x)] + 224*A*Cos[5*(c + d*x)] + 286*C*Cos[5*(c + d*x)])*Sec[c + d*x]^(11/2)*Tan[(c + d*x)/2])/(2310*d)","A",1
1213,1,123,219,0.8285055,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} ((748 A+567 C) \cos (c+d x)+(748 A+882 C) \cos (2 (c+d x))+136 A \cos (3 (c+d x))+136 A \cos (4 (c+d x))+752 A+189 C \cos (3 (c+d x))+189 C \cos (4 (c+d x))+693 C)}{630 d}","\frac{2 a^2 (52 A+63 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (136 A+189 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (136 A+189 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{21 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(752*A + 693*C + (748*A + 567*C)*Cos[c + d*x] + (748*A + 882*C)*Cos[2*(c + d*x)] + 136*A*Cos[3*(c + d*x)] + 189*C*Cos[3*(c + d*x)] + 136*A*Cos[4*(c + d*x)] + 189*C*Cos[4*(c + d*x)])*Sec[c + d*x]^(9/2)*Tan[(c + d*x)/2])/(630*d)","A",1
1214,1,102,172,0.6020648,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} ((468 A+525 C) \cos (c+d x)+2 (52 A+35 C) \cos (2 (c+d x))+104 A \cos (3 (c+d x))+164 A+175 C \cos (3 (c+d x))+70 C)}{210 d}","\frac{2 a^2 (4 A+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (104 A+175 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}+\frac{6 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{35 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(164*A + 70*C + (468*A + 525*C)*Cos[c + d*x] + 2*(52*A + 35*C)*Cos[2*(c + d*x)] + 104*A*Cos[3*(c + d*x)] + 175*C*Cos[3*(c + d*x)])*Sec[c + d*x]^(7/2)*Tan[(c + d*x)/2])/(210*d)","A",1
1215,1,121,183,0.760499,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) ((6 A+5 C) \cos (2 (c+d x))+6 A \cos (c+d x)+8 A+5 C)+5 \sqrt{2} C \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{5}{2}}(c+d x)\right)}{5 d}","\frac{2 a^{3/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a^2 (4 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{5 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^(5/2)*(5*Sqrt[2]*C*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(5/2) + (8*A + 5*C + 6*A*Cos[c + d*x] + (6*A + 5*C)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(5*d)","A",1
1216,1,116,181,0.5579043,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (20 A \cos (c+d x)+4 A+3 C \cos (2 (c+d x))+3 C)+9 \sqrt{2} C \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{3}{2}}(c+d x)\right)}{6 d}","\frac{3 a^{3/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a^2 (8 A-3 C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^(3/2)*(9*Sqrt[2]*C*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(3/2) + (4*A + 3*C + 20*A*Cos[c + d*x] + 3*C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(6*d)","A",1
1217,1,119,195,0.6572927,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(\sqrt{2} (8 A+7 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+2 \sin \left(\frac{1}{2} (c+d x)\right) (8 A+7 C \cos (c+d x)+C \cos (2 (c+d x))+C)\right)}{8 d}","\frac{a^{3/2} (8 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^2 (8 A-5 C) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a (4 A-C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}{d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(Sqrt[2]*(8*A + 7*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*(8*A + C + 7*C*Cos[c + d*x] + C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(8*d)","A",1
1218,1,133,191,0.7197301,"\int (a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{a \sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} (24 A+11 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} (24 A+22 C \cos (c+d x)+4 C \cos (2 (c+d x))+37 C)\right)}{48 d}","\frac{a^{3/2} (24 A+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 (24 A+19 C) \sin (c+d x)}{24 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}}",1,"(a*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(3*Sqrt[2]*(24*A + 11*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(24*A + 37*C + 22*C*Cos[c + d*x] + 4*C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
1219,1,150,238,0.6942258,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(\sqrt{2} (112 A+75 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+\left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) ((32 A+62 C) \cos (c+d x)+112 A+20 C \cos (2 (c+d x))+4 C \cos (3 (c+d x))+95 C)\right)}{128 d}","\frac{a^{3/2} (112 A+75 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (16 A+13 C) \sin (c+d x)}{32 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (112 A+75 C) \sin (c+d x)}{64 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{8 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(Sqrt[2]*(112*A + 75*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + (112*A + 95*C + (32*A + 62*C)*Cos[c + d*x] + 20*C*Cos[2*(c + d*x)] + 4*C*Cos[3*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(128*d)","A",1
1220,1,169,285,1.078597,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(15 \sqrt{2} (176 A+133 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+\left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (2 (880 A+1007 C) \cos (c+d x)+4 (80 A+181 C) \cos (2 (c+d x))+2960 A+228 C \cos (3 (c+d x))+48 C \cos (4 (c+d x))+2671 C)\right)}{3840 d}","\frac{a^{3/2} (176 A+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^2 (176 A+133 C) \sin (c+d x)}{192 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (80 A+67 C) \sin (c+d x)}{240 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (176 A+133 C) \sin (c+d x)}{128 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{3 a C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{40 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(15*Sqrt[2]*(176*A + 133*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + (2960*A + 2671*C + 2*(880*A + 1007*C)*Cos[c + d*x] + 4*(80*A + 181*C)*Cos[2*(c + d*x)] + 228*C*Cos[3*(c + d*x)] + 48*C*Cos[4*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(3840*d)","A",1
1221,1,171,313,1.0669052,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{15}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(15/2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{13}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} (1120 (347 A+286 C) \cos (c+d x)+14 (30334 A+32747 C) \cos (2 (c+d x))+125520 A \cos (3 (c+d x))+125520 A \cos (4 (c+d x))+16736 A \cos (5 (c+d x))+16736 A \cos (6 (c+d x))+343612 A+141570 C \cos (3 (c+d x))+156585 C \cos (4 (c+d x))+20878 C \cos (5 (c+d x))+20878 C \cos (6 (c+d x))+322751 C)}{180180 d}","\frac{2 a^3 (2224 A+2717 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{9009 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (8368 A+10439 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{15015 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a^3 (8368 A+10439 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{45045 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^3 (8368 A+10439 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{45045 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (136 A+143 C) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{1287 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{13}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}+\frac{10 a A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(343612*A + 322751*C + 1120*(347*A + 286*C)*Cos[c + d*x] + 14*(30334*A + 32747*C)*Cos[2*(c + d*x)] + 125520*A*Cos[3*(c + d*x)] + 141570*C*Cos[3*(c + d*x)] + 125520*A*Cos[4*(c + d*x)] + 156585*C*Cos[4*(c + d*x)] + 16736*A*Cos[5*(c + d*x)] + 20878*C*Cos[5*(c + d*x)] + 16736*A*Cos[6*(c + d*x)] + 20878*C*Cos[6*(c + d*x)])*Sec[c + d*x]^(13/2)*Tan[(c + d*x)/2])/(180180*d)","A",1
1222,1,149,266,1.1957138,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{13}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} (2 (5014 A+4983 C) \cos (c+d x)+52 (71 A+66 C) \cos (2 (c+d x))+3692 A \cos (3 (c+d x))+568 A \cos (4 (c+d x))+568 A \cos (5 (c+d x))+3628 A+4587 C \cos (3 (c+d x))+759 C \cos (4 (c+d x))+759 C \cos (5 (c+d x))+2673 C)}{2772 d}","\frac{2 a^3 (232 A+297 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (568 A+759 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^3 (568 A+759 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (32 A+33 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{231 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d}+\frac{10 a A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{99 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(3628*A + 2673*C + 2*(5014*A + 4983*C)*Cos[c + d*x] + 52*(71*A + 66*C)*Cos[2*(c + d*x)] + 3692*A*Cos[3*(c + d*x)] + 4587*C*Cos[3*(c + d*x)] + 568*A*Cos[4*(c + d*x)] + 759*C*Cos[4*(c + d*x)] + 568*A*Cos[5*(c + d*x)] + 759*C*Cos[5*(c + d*x)])*Sec[c + d*x]^(11/2)*Tan[(c + d*x)/2])/(2772*d)","A",1
1223,1,127,219,1.0608073,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} (4 (698 A+441 C) \cos (c+d x)+4 (803 A+966 C) \cos (2 (c+d x))+584 A \cos (3 (c+d x))+584 A \cos (4 (c+d x))+2908 A+588 C \cos (3 (c+d x))+903 C \cos (4 (c+d x))+2961 C)}{1260 d}","\frac{2 a^3 (8 A+11 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (584 A+903 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (64 A+63 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{9 d}+\frac{10 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{63 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(2908*A + 2961*C + 4*(698*A + 441*C)*Cos[c + d*x] + 4*(803*A + 966*C)*Cos[2*(c + d*x)] + 584*A*Cos[3*(c + d*x)] + 588*C*Cos[3*(c + d*x)] + 584*A*Cos[4*(c + d*x)] + 903*C*Cos[4*(c + d*x)])*Sec[c + d*x]^(9/2)*Tan[(c + d*x)/2])/(1260*d)","A",1
1224,1,151,230,1.528926,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(4 \sin \left(\frac{1}{2} (c+d x)\right) ((93 A+84 C) \cos (c+d x)+(23 A+7 C) \cos (2 (c+d x))+23 A \cos (3 (c+d x))+29 A+28 C \cos (3 (c+d x))+7 C)+84 \sqrt{2} C \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{7}{2}}(c+d x)\right)}{84 d}","\frac{2 a^{5/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a^3 (32 A+49 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{21 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (8 A+7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{21 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^(7/2)*(84*Sqrt[2]*C*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(7/2) + 4*(29*A + 7*C + (93*A + 84*C)*Cos[c + d*x] + (23*A + 7*C)*Cos[2*(c + d*x)] + 23*A*Cos[3*(c + d*x)] + 28*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(84*d)","A",1
1225,1,141,230,1.0660847,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) ((112 A+45 C) \cos (c+d x)+4 (43 A+15 C) \cos (2 (c+d x))+196 A+15 C \cos (3 (c+d x))+60 C)+300 \sqrt{2} C \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{5}{2}}(c+d x)\right)}{120 d}","\frac{5 a^{5/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a^3 (64 A+15 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (8 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^(5/2)*(300*Sqrt[2]*C*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(5/2) + 2*(196*A + 60*C + (112*A + 45*C)*Cos[c + d*x] + 4*(43*A + 15*C)*Cos[2*(c + d*x)] + 15*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(120*d)","A",1
1226,1,141,238,0.8927968,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(6 \sqrt{2} (8 A+19 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{3}{2}}(c+d x)+2 \sin \left(\frac{1}{2} (c+d x)\right) ((128 A+9 C) \cos (c+d x)+16 A+33 C \cos (2 (c+d x))+3 C \cos (3 (c+d x))+33 C)\right)}{48 d}","\frac{a^{5/2} (8 A+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^3 (56 A-27 C) \sin (c+d x)}{12 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A-C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{3 d}+\frac{10 a A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}{3 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^(3/2)*(6*Sqrt[2]*(8*A + 19*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(3/2) + 2*(16*A + 33*C + (128*A + 9*C)*Cos[c + d*x] + 33*C*Cos[2*(c + d*x)] + 3*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
1227,1,142,242,1.0652798,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(15 \sqrt{2} (8 A+5 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+2 \sin \left(\frac{1}{2} (c+d x)\right) (3 (8 A+27 C) \cos (c+d x)+48 A+17 C \cos (2 (c+d x))+2 C \cos (3 (c+d x))+17 C)\right)}{48 d}","\frac{5 a^{5/2} (8 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}-\frac{a^3 (24 A-49 C) \sin (c+d x)}{24 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A-3 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sqrt{\sec (c+d x)}}-\frac{a (6 A-C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}{d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(15*Sqrt[2]*(8*A + 5*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*(48*A + 17*C + 3*(8*A + 27*C)*Cos[c + d*x] + 17*C*Cos[2*(c + d*x)] + 2*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
1228,1,153,238,0.7755371,"\int (a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} (304 A+163 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+\left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) ((96 A+362 C) \cos (c+d x)+528 A+92 C \cos (2 (c+d x))+12 C \cos (3 (c+d x))+581 C)\right)}{384 d}","\frac{a^{5/2} (304 A+163 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^3 (432 A+299 C) \sin (c+d x)}{192 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (16 A+17 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{32 d \sqrt{\sec (c+d x)}}+\frac{5 a C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{24 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{4 d \sqrt{\sec (c+d x)}}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(3*Sqrt[2]*(304*A + 163*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + (528*A + 581*C + (96*A + 362*C)*Cos[c + d*x] + 92*C*Cos[2*(c + d*x)] + 12*C*Cos[3*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(384*d)","A",1
1229,1,170,285,1.2524906,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(15 \sqrt{2} (400 A+283 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+\left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) ((2720 A+3874 C) \cos (c+d x)+4 (80 A+331 C) \cos (2 (c+d x))+6320 A+348 C \cos (3 (c+d x))+48 C \cos (4 (c+d x))+5521 C)\right)}{3840 d}","\frac{a^{5/2} (400 A+283 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^3 (1040 A+787 C) \sin (c+d x)}{960 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (400 A+283 C) \sin (c+d x)}{128 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (80 A+79 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{240 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{a C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{8 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(15*Sqrt[2]*(400*A + 283*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + (6320*A + 5521*C + (2720*A + 3874*C)*Cos[c + d*x] + 4*(80*A + 331*C)*Cos[2*(c + d*x)] + 348*C*Cos[3*(c + d*x)] + 48*C*Cos[4*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(3840*d)","A",1
1230,1,192,332,1.3147884,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} (1304 A+1015 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+\left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) ((2896 A+3234 C) \cos (c+d x)+4 (184 A+315 C) \cos (2 (c+d x))+96 A \cos (3 (c+d x))+4648 A+428 C \cos (3 (c+d x))+112 C \cos (4 (c+d x))+16 C \cos (5 (c+d x))+4193 C)\right)}{3072 d}","\frac{a^{5/2} (1304 A+1015 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{512 d}+\frac{a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (136 A+109 C) \sin (c+d x)}{192 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (24 A+23 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{96 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{12 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(3*Sqrt[2]*(1304*A + 1015*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + (4648*A + 4193*C + (2896*A + 3234*C)*Cos[c + d*x] + 4*(184*A + 315*C)*Cos[2*(c + d*x)] + 96*A*Cos[3*(c + d*x)] + 428*C*Cos[3*(c + d*x)] + 112*C*Cos[4*(c + d*x)] + 16*C*Cos[5*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(3072*d)","A",1
1231,1,271,289,9.0282219,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 e^{-\frac{1}{2} i (c+d x)} \cos \left(\frac{1}{2} (c+d x)\right) \left(-315 i (A+C) \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)-\frac{1}{4} \sin \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \left(\cos \left(\frac{1}{2} (c+d x)\right)+i \sin \left(\frac{1}{2} (c+d x)\right)\right) (2 (107 A+63 C) \cos (c+d x)-8 (157 A+168 C) \cos (2 (c+d x))+58 A \cos (3 (c+d x))-257 A \cos (4 (c+d x))-1279 A+42 C \cos (3 (c+d x))-273 C \cos (4 (c+d x))-1071 C)\right)}{315 d \sqrt{a (\cos (c+d x)+1)}}","\frac{2 (19 A+21 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (29 A+21 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (257 A+273 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}}-\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}",1,"(2*Cos[(c + d*x)/2]*((-315*I)*(A + C)*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - ((-1279*A - 1071*C + 2*(107*A + 63*C)*Cos[c + d*x] - 8*(157*A + 168*C)*Cos[2*(c + d*x)] + 58*A*Cos[3*(c + d*x)] + 42*C*Cos[3*(c + d*x)] - 257*A*Cos[4*(c + d*x)] - 273*C*Cos[4*(c + d*x)])*Sec[c + d*x]^(9/2)*(Cos[(c + d*x)/2] + I*Sin[(c + d*x)/2])*Sin[(c + d*x)/2])/4))/(315*d*E^((I/2)*(c + d*x))*Sqrt[a*(1 + Cos[c + d*x])])","C",1
1232,1,2480,244,10.6661693,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2))/Sqrt[a + a*Cos[c + d*x]],x]","\text{Result too large to show}","\frac{2 (31 A+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (43 A+35 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}-\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}",1,"(2*Cos[c/2 + (d*x)/2]*Sqrt[(1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]*(-1/3*(C*Sin[c/2 + (d*x)/2])/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2) + ((A + C)*Csc[c/2 + (d*x)/2]^9*(363825*Sin[c/2 + (d*x)/2]^2 - 4729725*Sin[c/2 + (d*x)/2]^4 + 26785605*Sin[c/2 + (d*x)/2]^6 - 86790165*Sin[c/2 + (d*x)/2]^8 + 177677808*Sin[c/2 + (d*x)/2]^10 - 239283044*Sin[c/2 + (d*x)/2]^12 + 52080*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12 + 560*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12 + 213120160*Sin[c/2 + (d*x)/2]^14 - 168280*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14 - 2240*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14 - 121497024*Sin[c/2 + (d*x)/2]^16 + 212520*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^16 + 3360*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^16 + 40125184*Sin[c/2 + (d*x)/2]^18 - 124320*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^18 - 2240*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^18 - 5840384*Sin[c/2 + (d*x)/2]^20 + 28000*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^20 + 560*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^20 + 363825*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 5336100*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^2*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 34636140*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^4*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 131060160*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^6*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 320535600*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^8*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 530671680*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^10*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 604296000*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^12*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 468948480*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^14*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 237726720*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^16*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 70963200*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^18*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 9461760*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^20*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 1120*Cos[(c + d*x)/2]^6*HypergeometricPFQ[{2, 2, 2, 11/2}, {1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12*(-6 + 5*Sin[c/2 + (d*x)/2]^2) + 280*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 11/2}, {1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12*(103 - 164*Sin[c/2 + (d*x)/2]^2 + 70*Sin[c/2 + (d*x)/2]^4)))/(40425*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(9/2)*(-1 + 2*Sin[c/2 + (d*x)/2]^2)) + (C*((5*Sin[c/2 + (d*x)/2])/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2) + 2*((3*Sin[c/2 + (d*x)/2])/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2) + 4*(Sin[c/2 + (d*x)/2]/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2) + (2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]))))/105))/(d*Sqrt[a*(1 + Cos[c + d*x])])","C",0
1233,1,1757,201,7.709512,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\frac{1}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}} \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)} \left(-\frac{(A+C) \left(440 \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right) \sin ^{16}\left(\frac{c}{2}+\frac{d x}{2}\right)+69120 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^{16}\left(\frac{c}{2}+\frac{d x}{2}\right)-42048 \sin ^{16}\left(\frac{c}{2}+\frac{d x}{2}\right)-1500 \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right) \sin ^{14}\left(\frac{c}{2}+\frac{d x}{2}\right)-414720 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^{14}\left(\frac{c}{2}+\frac{d x}{2}\right)+226656 \sin ^{14}\left(\frac{c}{2}+\frac{d x}{2}\right)+1770 \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right) \sin ^{12}\left(\frac{c}{2}+\frac{d x}{2}\right)+1080000 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^{12}\left(\frac{c}{2}+\frac{d x}{2}\right)-518760 \sin ^{12}\left(\frac{c}{2}+\frac{d x}{2}\right)-710 \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right) \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)-40 \cos ^6\left(\frac{1}{2} (c+d x)\right) \, _4F_3\left(2,2,2,\frac{9}{2};1,1,\frac{11}{2};\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right) \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)+60 \cos ^4\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(2,2,\frac{9}{2};1,\frac{11}{2};\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right) \left(4 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-5\right) \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)-1598400 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)+655812 \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)+1458000 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)-486630 \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)-833760 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)+210105 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)+291060 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-48825 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-56700 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+4725 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+4725 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \csc ^7\left(\frac{c}{2}+\frac{d x}{2}\right)}{675 \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{7/2} \left(2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1\right)}+\frac{1}{30} C \left(\frac{3 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{5/2}}+4 \left(\frac{2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}+\frac{\sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{3/2}}\right)\right)-\frac{C \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{5/2}}\right)}{d \sqrt{a (\cos (c+d x)+1)}}","\frac{2 (13 A+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}-\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}",1,"(2*Cos[c/2 + (d*x)/2]*Sqrt[(1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]*(-1/2*(C*Sin[c/2 + (d*x)/2])/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2) - ((A + C)*Csc[c/2 + (d*x)/2]^7*(4725*Sin[c/2 + (d*x)/2]^2 - 48825*Sin[c/2 + (d*x)/2]^4 + 210105*Sin[c/2 + (d*x)/2]^6 - 486630*Sin[c/2 + (d*x)/2]^8 + 655812*Sin[c/2 + (d*x)/2]^10 - 710*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^10 - 40*Cos[(c + d*x)/2]^6*HypergeometricPFQ[{2, 2, 2, 9/2}, {1, 1, 11/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^10 - 518760*Sin[c/2 + (d*x)/2]^12 + 1770*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12 + 226656*Sin[c/2 + (d*x)/2]^14 - 1500*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14 - 42048*Sin[c/2 + (d*x)/2]^16 + 440*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^16 + 4725*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 56700*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^2*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 291060*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^4*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 833760*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^6*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 1458000*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^8*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 1598400*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^10*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 1080000*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^12*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 414720*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^14*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 69120*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^16*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 60*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 9/2}, {1, 11/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^10*(-5 + 4*Sin[c/2 + (d*x)/2]^2)))/(675*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2)*(-1 + 2*Sin[c/2 + (d*x)/2]^2)) + (C*((3*Sin[c/2 + (d*x)/2])/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2) + 4*(Sin[c/2 + (d*x)/2]/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2) + (2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])))/30))/(d*Sqrt[a*(1 + Cos[c + d*x])])","C",0
1234,1,576,156,6.7831074,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \sqrt{\frac{1}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}} \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)} \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{(A+C) \csc ^5\left(\frac{c}{2}+\frac{d x}{2}\right) \left(-12 \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right) \cos ^4\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(2,2,\frac{7}{2};1,\frac{9}{2};-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}\right)-12 \left(3 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-7 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+4\right) \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right) \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}\right)+7 \sqrt{-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}} \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3 \left(8 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-20 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+15\right) \left(\left(3-7 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sqrt{-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}-3 \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right) \tanh ^{-1}\left(\sqrt{-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}\right)\right)\right)}{63 \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{7/2}}-\frac{4 C \sin ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{3/2}}\right)}{d \sqrt{a (\cos (c+d x)+1)}}","\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}",1,"(2*Cos[c/2 + (d*x)/2]*Sqrt[(1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]*((-4*C*Sin[c/2 + (d*x)/2]^3)/(3*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) + ((A + C)*Csc[c/2 + (d*x)/2]^5*(-12*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 7/2}, {1, 9/2}, -(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]*Sin[c/2 + (d*x)/2]^8 - 12*Hypergeometric2F1[2, 7/2, 9/2, -(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]*Sin[c/2 + (d*x)/2]^8*(4 - 7*Sin[c/2 + (d*x)/2]^2 + 3*Sin[c/2 + (d*x)/2]^4) + 7*Sqrt[-(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]*(1 - 2*Sin[c/2 + (d*x)/2]^2)^3*(15 - 20*Sin[c/2 + (d*x)/2]^2 + 8*Sin[c/2 + (d*x)/2]^4)*((3 - 7*Sin[c/2 + (d*x)/2]^2)*Sqrt[-(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))] - 3*ArcTanh[Sqrt[-(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]]*(1 - 2*Sin[c/2 + (d*x)/2]^2))))/(63*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2))))/(d*Sqrt[a*(1 + Cos[c + d*x])])","C",0
1235,1,251,175,3.6225811,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{(A+C) \csc ^3\left(\frac{1}{2} (c+d x)\right) \left(5 \cos ^2(c+d x) (\cos (c+d x)+2) \left(-\cos (c+d x)+\cos (c+d x) \sqrt{2-2 \sec (c+d x)} \tanh ^{-1}\left(\sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right) (-\sec (c+d x))}\right)+1\right)-\sin ^4\left(\frac{1}{2} (c+d x)\right) \sin ^2(c+d x) \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};-\sec (c+d x) \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}{10 \cos ^{\frac{5}{2}}(c+d x)}+\sqrt{2} C \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)-\frac{2 C \sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)}{d \sqrt{a (\cos (c+d x)+1)}}","-\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(2*Cos[(c + d*x)/2]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(Sqrt[2]*C*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] - (2*C*Sin[(c + d*x)/2])/Sqrt[Cos[c + d*x]] + ((A + C)*Csc[(c + d*x)/2]^3*(5*Cos[c + d*x]^2*(2 + Cos[c + d*x])*(1 - Cos[c + d*x] + ArcTanh[Sqrt[-(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]]*Cos[c + d*x]*Sqrt[2 - 2*Sec[c + d*x]]) - Hypergeometric2F1[2, 5/2, 7/2, -(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^4*Sin[c + d*x]^2))/(10*Cos[c + d*x]^(5/2))))/(d*Sqrt[a*(1 + Cos[c + d*x])])","C",0
1236,1,124,173,0.3271657,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(2 (A+C) \tan ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)-\sqrt{2} C \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 C \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)}\right)}{d \sqrt{a (\cos (c+d x)+1)}}","\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{C \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-(Sqrt[2]*C*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]) + 2*(A + C)*ArcTan[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]] + 2*C*Sqrt[Cos[c + d*x]]*Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
1237,1,496,223,1.222308,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","-\frac{i e^{-3 i (c+d x)} \left(1+e^{i (c+d x)}\right) \sqrt{\sec (c+d x)} \left((8 A+7 C) e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)-8 \sqrt{2} A e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{-1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)-8 A e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)+2 C e^{i (c+d x)}-3 C e^{2 i (c+d x)}+3 C e^{3 i (c+d x)}-2 C e^{4 i (c+d x)}+C e^{5 i (c+d x)}+\sqrt{2} C e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)-7 \sqrt{2} C e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{-1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)-7 C e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)-C\right)}{16 d \sqrt{a (\cos (c+d x)+1)}}","\frac{(8 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{C \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{C \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"((-1/16*I)*(1 + E^(I*(c + d*x)))*(-C + 2*C*E^(I*(c + d*x)) - 3*C*E^((2*I)*(c + d*x)) + 3*C*E^((3*I)*(c + d*x)) - 2*C*E^((4*I)*(c + d*x)) + C*E^((5*I)*(c + d*x)) + (8*A + 7*C)*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))] + Sqrt[2]*C*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - 8*Sqrt[2]*A*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[(-1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - 7*Sqrt[2]*C*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[(-1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - 8*A*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]] - 7*C*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Sqrt[Sec[c + d*x]])/(d*E^((3*I)*(c + d*x))*Sqrt[a*(1 + Cos[c + d*x])])","C",1
1238,1,439,266,1.6059053,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)),x]","\frac{i e^{-4 i (c+d x)} \left(1+e^{i (c+d x)}\right) \sqrt{\sec (c+d x)} \left(3 (8 A+9 C) e^{3 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)+48 \sqrt{2} (A+C) e^{3 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+24 A e^{2 i (c+d x)}-24 A e^{3 i (c+d x)}+24 A e^{4 i (c+d x)}-24 A e^{5 i (c+d x)}-24 A e^{3 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)-3 C e^{i (c+d x)}+28 C e^{2 i (c+d x)}-29 C e^{3 i (c+d x)}+29 C e^{4 i (c+d x)}-28 C e^{5 i (c+d x)}+3 C e^{6 i (c+d x)}-2 C e^{7 i (c+d x)}-27 C e^{3 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)+2 C\right)}{96 d \sqrt{a (\cos (c+d x)+1)}}","-\frac{(8 A+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{(8 A+7 C) \sin (c+d x)}{8 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{C \sin (c+d x)}{12 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x)}{3 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"((I/96)*(1 + E^(I*(c + d*x)))*(2*C - 3*C*E^(I*(c + d*x)) + 24*A*E^((2*I)*(c + d*x)) + 28*C*E^((2*I)*(c + d*x)) - 24*A*E^((3*I)*(c + d*x)) - 29*C*E^((3*I)*(c + d*x)) + 24*A*E^((4*I)*(c + d*x)) + 29*C*E^((4*I)*(c + d*x)) - 24*A*E^((5*I)*(c + d*x)) - 28*C*E^((5*I)*(c + d*x)) + 3*C*E^((6*I)*(c + d*x)) - 2*C*E^((7*I)*(c + d*x)) + 3*(8*A + 9*C)*E^((3*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))] + 48*Sqrt[2]*(A + C)*E^((3*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - 24*A*E^((3*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]] - 27*C*E^((3*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Sqrt[Sec[c + d*x]])/(d*E^((4*I)*(c + d*x))*Sqrt[a*(1 + Cos[c + d*x])])","C",1
1239,1,3121,315,10.0703824,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2))/(a + a*Cos[c + d*x])^(3/2),x]","\text{Result too large to show}","\frac{(19 A+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(11 A+7 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{14 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(67 A+35 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{70 a d \sqrt{a \cos (c+d x)+a}}+\frac{(397 A+245 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{210 a d \sqrt{a \cos (c+d x)+a}}-\frac{(1201 A+665 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{210 a d \sqrt{a \cos (c+d x)+a}}",1,"(2*Cos[c/2 + (d*x)/2]^3*Sqrt[(1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]*((4*C*Sin[c/2 + (d*x)/2])/(7*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2)) - ((A + C)*(1 - 2*Sin[c/2 + (d*x)/2]))/(28*(1 + Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2)) + ((A + C)*(1 + 2*Sin[c/2 + (d*x)/2]))/(28*(1 - Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2)) - ((A + C)*(315*ArcTan[(1 - 2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]] + (5 + 3*Sin[c/2 + (d*x)/2])/((1 - Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2)) - (11 + 17*Sin[c/2 + (d*x)/2])/((1 - Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) + (61 + 71*Sin[c/2 + (d*x)/2])/((1 - Sin[c/2 + (d*x)/2])*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) + (193*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])/(1 - Sin[c/2 + (d*x)/2])))/70 + ((A + C)*(315*ArcTan[(1 + 2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]] + (5 - 3*Sin[c/2 + (d*x)/2])/((1 + Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2)) - (11 - 17*Sin[c/2 + (d*x)/2])/((1 + Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) + (61 - 71*Sin[c/2 + (d*x)/2])/((1 + Sin[c/2 + (d*x)/2])*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) + (193*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])/(1 + Sin[c/2 + (d*x)/2])))/70 - ((-A + 7*C)*Csc[c/2 + (d*x)/2]^9*(363825*Sin[c/2 + (d*x)/2]^2 - 4729725*Sin[c/2 + (d*x)/2]^4 + 26785605*Sin[c/2 + (d*x)/2]^6 - 86790165*Sin[c/2 + (d*x)/2]^8 + 177677808*Sin[c/2 + (d*x)/2]^10 - 239283044*Sin[c/2 + (d*x)/2]^12 + 52080*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12 + 560*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12 + 213120160*Sin[c/2 + (d*x)/2]^14 - 168280*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14 - 2240*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14 - 121497024*Sin[c/2 + (d*x)/2]^16 + 212520*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^16 + 3360*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^16 + 40125184*Sin[c/2 + (d*x)/2]^18 - 124320*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^18 - 2240*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^18 - 5840384*Sin[c/2 + (d*x)/2]^20 + 28000*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^20 + 560*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^20 + 363825*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 5336100*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^2*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 34636140*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^4*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 131060160*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^6*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 320535600*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^8*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 530671680*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^10*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 604296000*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^12*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 468948480*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^14*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 237726720*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^16*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 70963200*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^18*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 9461760*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^20*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 1120*Cos[(c + d*x)/2]^6*HypergeometricPFQ[{2, 2, 2, 11/2}, {1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12*(-6 + 5*Sin[c/2 + (d*x)/2]^2) + 280*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 11/2}, {1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12*(103 - 164*Sin[c/2 + (d*x)/2]^2 + 70*Sin[c/2 + (d*x)/2]^4)))/(80850*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(9/2)*(-1 + 2*Sin[c/2 + (d*x)/2]^2)) + (8*C*((3*Sin[c/2 + (d*x)/2])/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2) + 4*(Sin[c/2 + (d*x)/2]/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2) + (2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])))/35))/(d*(a*(1 + Cos[c + d*x]))^(3/2))","C",0
1240,1,2280,268,7.7978815,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x])^(3/2),x]","\text{Result too large to show}","-\frac{(15 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(9 A+5 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(13 A+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}+\frac{(49 A+25 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a d \sqrt{a \cos (c+d x)+a}}",1,"(2*Cos[c/2 + (d*x)/2]^3*Sqrt[(1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]*((4*C*Sin[c/2 + (d*x)/2])/(5*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2)) - ((A + C)*(1 - 2*Sin[c/2 + (d*x)/2]))/(20*(1 + Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2)) + ((A + C)*(1 + 2*Sin[c/2 + (d*x)/2]))/(20*(1 - Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2)) + (16*C*(Sin[c/2 + (d*x)/2]/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2) + (2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]))/15 - ((A + C)*(-105*ArcTan[(1 - 2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]] + (4 + 3*Sin[c/2 + (d*x)/2])/((1 - Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) - (19 + 29*Sin[c/2 + (d*x)/2])/((1 - Sin[c/2 + (d*x)/2])*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) - (67*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])/(1 - Sin[c/2 + (d*x)/2])))/30 + ((A + C)*(-105*ArcTan[(1 + 2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]] + (4 - 3*Sin[c/2 + (d*x)/2])/((1 + Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) - (19 - 29*Sin[c/2 + (d*x)/2])/((1 + Sin[c/2 + (d*x)/2])*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) - (67*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])/(1 + Sin[c/2 + (d*x)/2])))/30 + ((-A + 7*C)*Csc[c/2 + (d*x)/2]^7*(4725*Sin[c/2 + (d*x)/2]^2 - 48825*Sin[c/2 + (d*x)/2]^4 + 210105*Sin[c/2 + (d*x)/2]^6 - 486630*Sin[c/2 + (d*x)/2]^8 + 655812*Sin[c/2 + (d*x)/2]^10 - 710*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^10 - 40*Cos[(c + d*x)/2]^6*HypergeometricPFQ[{2, 2, 2, 9/2}, {1, 1, 11/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^10 - 518760*Sin[c/2 + (d*x)/2]^12 + 1770*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12 + 226656*Sin[c/2 + (d*x)/2]^14 - 1500*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14 - 42048*Sin[c/2 + (d*x)/2]^16 + 440*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^16 + 4725*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 56700*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^2*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 291060*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^4*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 833760*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^6*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 1458000*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^8*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 1598400*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^10*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 1080000*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^12*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 414720*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^14*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 69120*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^16*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 60*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 9/2}, {1, 11/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^10*(-5 + 4*Sin[c/2 + (d*x)/2]^2)))/(1350*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2)*(-1 + 2*Sin[c/2 + (d*x)/2]^2))))/(d*(a*(1 + Cos[c + d*x]))^(3/2))","C",0
1241,1,1055,221,6.8501202,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{2 \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\frac{1}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}} \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)} \left(\frac{(A-7 C) \left(-12 \cos ^4\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(2,2,\frac{7}{2};1,\frac{9}{2};-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}\right) \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)-12 \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}\right) \left(3 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-7 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+4\right) \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)+7 \sqrt{-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}} \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3 \left(8 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-20 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+15\right) \left(\left(3-7 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sqrt{-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}-3 \tanh ^{-1}\left(\sqrt{-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}\right) \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)\right)\right) \csc ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{126 \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{7/2}}-\frac{1}{2} (A+C) \left(5 \tan ^{-1}\left(\frac{1-2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}\right)+\frac{3 \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}{1-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)}+\frac{\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1}{\left(1-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}\right)+\frac{1}{2} (A+C) \left(5 \tan ^{-1}\left(\frac{2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1}{\sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}\right)+\frac{3 \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}{\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1}+\frac{1-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1\right) \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}\right)+\frac{8 C \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}+\frac{4 C \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{3/2}}+\frac{(A+C) \left(2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1\right)}{12 \left(1-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{3/2}}-\frac{(A+C) \left(1-2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{12 \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1\right) \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{3/2}}\right)}{d (a (\cos (c+d x)+1))^{3/2}}","\frac{(11 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(7 A+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(19 A+3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 a d \sqrt{a \cos (c+d x)+a}}",1,"(2*Cos[c/2 + (d*x)/2]^3*Sqrt[(1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]*((4*C*Sin[c/2 + (d*x)/2])/(3*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) - ((A + C)*(1 - 2*Sin[c/2 + (d*x)/2]))/(12*(1 + Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) + ((A + C)*(1 + 2*Sin[c/2 + (d*x)/2]))/(12*(1 - Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) + (8*C*Sin[c/2 + (d*x)/2])/(3*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) - ((A + C)*(5*ArcTan[(1 - 2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]] + (1 + Sin[c/2 + (d*x)/2])/((1 - Sin[c/2 + (d*x)/2])*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) + (3*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])/(1 - Sin[c/2 + (d*x)/2])))/2 + ((A + C)*(5*ArcTan[(1 + 2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]] + (1 - Sin[c/2 + (d*x)/2])/((1 + Sin[c/2 + (d*x)/2])*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) + (3*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])/(1 + Sin[c/2 + (d*x)/2])))/2 + ((A - 7*C)*Csc[c/2 + (d*x)/2]^5*(-12*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 7/2}, {1, 9/2}, -(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]*Sin[c/2 + (d*x)/2]^8 - 12*Hypergeometric2F1[2, 7/2, 9/2, -(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]*Sin[c/2 + (d*x)/2]^8*(4 - 7*Sin[c/2 + (d*x)/2]^2 + 3*Sin[c/2 + (d*x)/2]^4) + 7*Sqrt[-(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]*(1 - 2*Sin[c/2 + (d*x)/2]^2)^3*(15 - 20*Sin[c/2 + (d*x)/2]^2 + 8*Sin[c/2 + (d*x)/2]^4)*((3 - 7*Sin[c/2 + (d*x)/2]^2)*Sqrt[-(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))] - 3*ArcTanh[Sqrt[-(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]]*(1 - 2*Sin[c/2 + (d*x)/2]^2))))/(126*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2))))/(d*(a*(1 + Cos[c + d*x]))^(3/2))","C",0
1242,1,460,172,4.7924153,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{2 \cos ^3\left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{(A-7 C) \csc ^3\left(\frac{1}{2} (c+d x)\right) \left(5 (4 \cos (c+d x)+\cos (2 (c+d x))+1) \left(-\cos (c+d x)+\cos (c+d x) \sqrt{2-2 \sec (c+d x)} \tanh ^{-1}\left(\sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right) (-\sec (c+d x))}\right)+1\right)-2 \sin ^4\left(\frac{1}{2} (c+d x)\right) \sin (c+d x) \tan (c+d x) \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};-\sec (c+d x) \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}{40 \cos ^{\frac{3}{2}}(c+d x)}+\frac{(A+C) \left(2 \sin \left(\frac{1}{2} (c+d x)\right)-1\right)}{4 \sqrt{\cos (c+d x)} \left(\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)\right)^2}-\frac{(A+C) \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+1\right)}{4 \left(\sin \left(\frac{1}{2} (c+d x)\right)-1\right) \sqrt{\cos (c+d x)}}-\frac{(A+C) \sqrt{\cos (c+d x)}}{\sin \left(\frac{1}{2} (c+d x)\right)-1}-\frac{(A+C) \sqrt{\cos (c+d x)}}{\sin \left(\frac{1}{2} (c+d x)\right)+1}+\frac{3}{2} (A+C) \tan ^{-1}\left(\frac{1-2 \sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)-\frac{3}{2} (A+C) \tan ^{-1}\left(\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)+1}{\sqrt{\cos (c+d x)}}\right)+\frac{4 C \sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)}{d (a (\cos (c+d x)+1))^{3/2}}","-\frac{(7 A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(2*Cos[(c + d*x)/2]^3*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((3*(A + C)*ArcTan[(1 - 2*Sin[(c + d*x)/2])/Sqrt[Cos[c + d*x]]])/2 - (3*(A + C)*ArcTan[(1 + 2*Sin[(c + d*x)/2])/Sqrt[Cos[c + d*x]]])/2 - ((A + C)*Sqrt[Cos[c + d*x]])/(-1 + Sin[(c + d*x)/2]) + (4*C*Sin[(c + d*x)/2])/Sqrt[Cos[c + d*x]] - ((A + C)*Sqrt[Cos[c + d*x]])/(1 + Sin[(c + d*x)/2]) + ((A + C)*(-1 + 2*Sin[(c + d*x)/2]))/(4*Sqrt[Cos[c + d*x]]*(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^2) - ((A + C)*(1 + 2*Sin[(c + d*x)/2]))/(4*Sqrt[Cos[c + d*x]]*(-1 + Sin[(c + d*x)/2])) + ((A - 7*C)*Csc[(c + d*x)/2]^3*(5*(1 + 4*Cos[c + d*x] + Cos[2*(c + d*x)])*(1 - Cos[c + d*x] + ArcTanh[Sqrt[-(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]]*Cos[c + d*x]*Sqrt[2 - 2*Sec[c + d*x]]) - 2*Hypergeometric2F1[2, 5/2, 7/2, -(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^4*Sin[c + d*x]*Tan[c + d*x]))/(40*Cos[c + d*x]^(3/2))))/(d*(a*(1 + Cos[c + d*x]))^(3/2))","C",0
1243,1,245,185,1.6621249,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^(3/2),x]","-\frac{i \cos ^3\left(\frac{1}{2} (c+d x)\right) \left(i (A+C) \left(\sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)\right) \sqrt{\sec (c+d x)} \sec ^2\left(\frac{1}{2} (c+d x)\right)+\sqrt{2} e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(-\sqrt{2} (3 A-5 C) \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+4 C \sinh ^{-1}\left(e^{i (c+d x)}\right)-4 C \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)\right)}{2 d (a (\cos (c+d x)+1))^{3/2}}","\frac{(3 A-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A+C) \sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"((-1/2*I)*Cos[(c + d*x)/2]^3*((Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(4*C*ArcSinh[E^(I*(c + d*x))] - Sqrt[2]*(3*A - 5*C)*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - 4*C*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))/E^((I/2)*(c + d*x)) + I*(A + C)*Sec[(c + d*x)/2]^2*Sqrt[Sec[c + d*x]]*(Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2])))/(d*(a*(1 + Cos[c + d*x]))^(3/2))","C",1
1244,1,251,228,1.7133837,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]),x]","\frac{\cos ^3\left(\frac{1}{2} (c+d x)\right) \left(\left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\sec (c+d x)} \sec ^2\left(\frac{1}{2} (c+d x)\right) (A+2 C \cos (c+d x)+3 C)+i \sqrt{2} e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\sqrt{2} (A+9 C) \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+6 C \sinh ^{-1}\left(e^{i (c+d x)}\right)-6 C \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)\right)}{2 d (a (\cos (c+d x)+1))^{3/2}}","\frac{(A+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{3 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A+C) \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}+\frac{(A+3 C) \sin (c+d x)}{2 a d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]^3*((I*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(6*C*ArcSinh[E^(I*(c + d*x))] + Sqrt[2]*(A + 9*C)*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - 6*C*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))/E^((I/2)*(c + d*x)) + (A + 3*C + 2*C*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sqrt[Sec[c + d*x]]*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(2*d*(a*(1 + Cos[c + d*x]))^(3/2))","C",1
1245,1,385,285,6.3389067,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)),x]","-\frac{i e^{\frac{1}{2} i (c+d x)} \cos \left(\frac{1}{2} (c+d x)\right) \left(\frac{\sqrt{2} e^{-2 i (c+d x)} \left(e^{i (c+d x)}-e^{2 i (c+d x)}+e^{3 i (c+d x)}-1\right) \left(C \left(-3 e^{i (c+d x)}-12 e^{2 i (c+d x)}-3 e^{3 i (c+d x)}+e^{4 i (c+d x)}+1\right)-4 A e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}+\sqrt{2} (8 A+19 C) \left(1+e^{i (c+d x)}\right)^2 \sinh ^{-1}\left(e^{i (c+d x)}\right)+4 (5 A+13 C) \left(1+e^{i (c+d x)}\right)^2 \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)-\sqrt{2} (8 A+19 C) \left(1+e^{i (c+d x)}\right)^2 \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{16 d \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{3/2} \left(1+e^{2 i (c+d x)}\right)^{3/2} (a (\cos (c+d x)+1))^{3/2}}","\frac{(8 A+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{(5 A+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(A+2 C) \sin (c+d x)}{2 a d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(2 A+7 C) \sin (c+d x)}{4 a d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"((-1/16*I)*E^((I/2)*(c + d*x))*((Sqrt[2]*(-1 + E^(I*(c + d*x)) - E^((2*I)*(c + d*x)) + E^((3*I)*(c + d*x)))*(-4*A*E^((2*I)*(c + d*x)) + C*(1 - 3*E^(I*(c + d*x)) - 12*E^((2*I)*(c + d*x)) - 3*E^((3*I)*(c + d*x)) + E^((4*I)*(c + d*x)))))/(E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]) + Sqrt[2]*(8*A + 19*C)*(1 + E^(I*(c + d*x)))^2*ArcSinh[E^(I*(c + d*x))] + 4*(5*A + 13*C)*(1 + E^(I*(c + d*x)))^2*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - Sqrt[2]*(8*A + 19*C)*(1 + E^(I*(c + d*x)))^2*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Cos[(c + d*x)/2])/(d*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(3/2)*(1 + E^((2*I)*(c + d*x)))^(3/2)*(a*(1 + Cos[c + d*x]))^(3/2))","C",1
1246,1,261,315,8.0864946,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\cos ^5\left(\frac{1}{2} (c+d x)\right) \left(\tan \left(\frac{1}{2} (c+d x)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) (50 (521 A+153 C) \cos (c+d x)+108 (157 A+45 C) \cos (2 (c+d x))+9110 A \cos (3 (c+d x))+2671 A \cos (4 (c+d x))+15053 A+2550 C \cos (3 (c+d x))+735 C \cos (4 (c+d x))+4125 C)-240 i (283 A+75 C) e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)\right)}{960 d (a (\cos (c+d x)+1))^{5/2}}","-\frac{(283 A+75 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(157 A+45 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{80 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(787 A+195 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{240 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(2671 A+735 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{240 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(21 A+5 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(Cos[(c + d*x)/2]^5*(((-240*I)*(283*A + 75*C)*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/E^((I/2)*(c + d*x)) + (15053*A + 4125*C + 50*(521*A + 153*C)*Cos[c + d*x] + 108*(157*A + 45*C)*Cos[2*(c + d*x)] + 9110*A*Cos[3*(c + d*x)] + 2550*C*Cos[3*(c + d*x)] + 2671*A*Cos[4*(c + d*x)] + 735*C*Cos[4*(c + d*x)])*Sec[(c + d*x)/2]^3*Sec[c + d*x]^(5/2)*Tan[(c + d*x)/2]))/(960*d*(a*(1 + Cos[c + d*x]))^(5/2))","C",1
1247,1,243,266,3.6520146,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{i \cos ^5\left(\frac{1}{2} (c+d x)\right) \left(3 (163 A+19 C) e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+\frac{1}{8} i \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \sec ^3\left(\frac{1}{2} (c+d x)\right) ((1537 A+81 C) \cos (c+d x)+2 (503 A+39 C) \cos (2 (c+d x))+299 A \cos (3 (c+d x))+878 A+27 C \cos (3 (c+d x))+78 C)\right)}{12 d (a (\cos (c+d x)+1))^{5/2}}","\frac{(163 A+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{5 (19 A+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{48 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(299 A+27 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{48 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(17 A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"((I/12)*Cos[(c + d*x)/2]^5*((3*(163*A + 19*C)*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/E^((I/2)*(c + d*x)) + (I/8)*(878*A + 78*C + (1537*A + 81*C)*Cos[c + d*x] + 2*(503*A + 39*C)*Cos[2*(c + d*x)] + 299*A*Cos[3*(c + d*x)] + 27*C*Cos[3*(c + d*x)])*Sec[(c + d*x)/2]^3*Sec[c + d*x]^(3/2)*Tan[(c + d*x)/2]))/(d*(a*(1 + Cos[c + d*x]))^(5/2))","C",1
1248,1,213,219,2.2445669,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\cos ^5\left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{4} \tan \left(\frac{1}{2} (c+d x)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} (10 (17 A+C) \cos (c+d x)+(49 A+C) \cos (2 (c+d x))+113 A+C)-5 i (15 A-C) e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)\right)}{4 d (a (\cos (c+d x)+1))^{5/2}}","-\frac{5 (15 A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(49 A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(13 A-3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(Cos[(c + d*x)/2]^5*(((-5*I)*(15*A - C)*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/E^((I/2)*(c + d*x)) + ((113*A + C + 10*(17*A + C)*Cos[c + d*x] + (49*A + C)*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^3*Sqrt[Sec[c + d*x]]*Tan[(c + d*x)/2])/4))/(4*d*(a*(1 + Cos[c + d*x]))^(5/2))","C",1
1249,1,216,174,1.7862923,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^(5/2),x]","\frac{i \cos ^5\left(\frac{1}{2} (c+d x)\right) \left((19 A+3 C) e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)-\frac{1}{4} i \left(\sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} ((9 A-7 C) \cos (c+d x)+13 A-3 C)\right)}{4 d (a (\cos (c+d x)+1))^{5/2}}","\frac{(19 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(9 A-7 C) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}",1,"((I/4)*Cos[(c + d*x)/2]^5*(((19*A + 3*C)*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/E^((I/2)*(c + d*x)) - (I/4)*(13*A - 3*C + (9*A - 7*C)*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Sqrt[Sec[c + d*x]]*(Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2])))/(d*(a*(1 + Cos[c + d*x]))^(5/2))","C",1
1250,1,262,232,2.4275694,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]),x]","\frac{\cos ^5\left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{2} \left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} ((A-15 C) \cos (c+d x)+5 A-11 C)-i \sqrt{2} e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(-\sqrt{2} (5 A-43 C) \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+32 C \sinh ^{-1}\left(e^{i (c+d x)}\right)-32 C \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)\right)}{8 d (a (\cos (c+d x)+1))^{5/2}}","\frac{(5 A-43 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(A+C) \sin (c+d x)}{4 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}+\frac{(5 A-11 C) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]^5*(((-I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(32*C*ArcSinh[E^(I*(c + d*x))] - Sqrt[2]*(5*A - 43*C)*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - 32*C*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))/E^((I/2)*(c + d*x)) + ((5*A - 11*C + (A - 15*C)*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Sqrt[Sec[c + d*x]]*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2]))/2))/(8*d*(a*(1 + Cos[c + d*x]))^(5/2))","C",1
1251,1,274,277,2.7973596,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)),x]","\frac{\cos ^5\left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{2} \left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\sec (c+d x)} \sec ^4\left(\frac{1}{2} (c+d x)\right) ((7 A+55 C) \cos (c+d x)+3 A+8 C \cos (2 (c+d x))+43 C)+i \sqrt{2} e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\sqrt{2} (3 A+115 C) \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+80 C \sinh ^{-1}\left(e^{i (c+d x)}\right)-80 C \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)\right)}{8 d (a (\cos (c+d x)+1))^{5/2}}","\frac{(3 A+115 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{5 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}+\frac{(3 A+35 C) \sin (c+d x)}{16 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{(A-15 C) \sin (c+d x)}{16 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x)}{4 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}",1,"(Cos[(c + d*x)/2]^5*((I*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(80*C*ArcSinh[E^(I*(c + d*x))] + Sqrt[2]*(3*A + 115*C)*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - 80*C*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))/E^((I/2)*(c + d*x)) + ((3*A + 43*C + (7*A + 55*C)*Cos[c + d*x] + 8*C*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^4*Sqrt[Sec[c + d*x]]*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2]))/2))/(8*d*(a*(1 + Cos[c + d*x]))^(5/2))","C",1
1252,1,968,334,7.2598778,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{\sec \left(\frac{c}{2}\right) \left(-A \sin \left(\frac{d x}{2}\right)-C \sin \left(\frac{d x}{2}\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{(A+C) \tan \left(\frac{c}{2}\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{\sec \left(\frac{c}{2}\right) \left(19 A \sin \left(\frac{d x}{2}\right)+35 C \sin \left(\frac{d x}{2}\right)\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}+\frac{(19 A+35 C) \tan \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}-\frac{3 (5 A+3 C) \cos \left(\frac{d x}{2}\right) \sin \left(\frac{c}{2}\right)}{2 d}-\frac{10 C \cos \left(\frac{3 d x}{2}\right) \sin \left(\frac{3 c}{2}\right)}{d}+\frac{C \cos \left(\frac{5 d x}{2}\right) \sin \left(\frac{5 c}{2}\right)}{d}-\frac{3 (5 A+3 C) \cos \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)}{2 d}-\frac{10 C \cos \left(\frac{3 c}{2}\right) \sin \left(\frac{3 d x}{2}\right)}{d}+\frac{C \cos \left(\frac{5 c}{2}\right) \sin \left(\frac{5 d x}{2}\right)}{d}\right) \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{(a (\cos (c+d x)+1))^{5/2}}-\frac{11 i A e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right) \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d (a (\cos (c+d x)+1))^{5/2}}-\frac{63 i C e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right) \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d (a (\cos (c+d x)+1))^{5/2}}+\frac{4 i \sqrt{2} A e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(-\sinh ^{-1}\left(e^{i (c+d x)}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{-1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+\tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right) \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{5/2}}+\frac{39 i C e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(-\sinh ^{-1}\left(e^{i (c+d x)}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{-1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+\tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right) \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{2} d (a (\cos (c+d x)+1))^{5/2}}","\frac{(8 A+39 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{5/2} d}-\frac{(43 A+219 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(7 A+31 C) \sin (c+d x)}{16 a^2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(11 A+63 C) \sin (c+d x)}{16 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(3 A+19 C) \sin (c+d x)}{16 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x)}{4 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}",1,"(((-11*I)/4)*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])]*Cos[c/2 + (d*x)/2]^5)/(d*E^((I/2)*(c + d*x))*(a*(1 + Cos[c + d*x]))^(5/2)) - (((63*I)/4)*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])]*Cos[c/2 + (d*x)/2]^5)/(d*E^((I/2)*(c + d*x))*(a*(1 + Cos[c + d*x]))^(5/2)) + ((4*I)*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(-ArcSinh[E^(I*(c + d*x))] + Sqrt[2]*ArcTanh[(-1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] + ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Cos[c/2 + (d*x)/2]^5)/(d*E^((I/2)*(c + d*x))*(a*(1 + Cos[c + d*x]))^(5/2)) + ((39*I)*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(-ArcSinh[E^(I*(c + d*x))] + Sqrt[2]*ArcTanh[(-1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] + ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Cos[c/2 + (d*x)/2]^5)/(Sqrt[2]*d*E^((I/2)*(c + d*x))*(a*(1 + Cos[c + d*x]))^(5/2)) + (Cos[c/2 + (d*x)/2]^5*Sqrt[Sec[c + d*x]]*((-3*(5*A + 3*C)*Cos[(d*x)/2]*Sin[c/2])/(2*d) - (10*C*Cos[(3*d*x)/2]*Sin[(3*c)/2])/d + (C*Cos[(5*d*x)/2]*Sin[(5*c)/2])/d - (3*(5*A + 3*C)*Cos[c/2]*Sin[(d*x)/2])/(2*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]^4*(-(A*Sin[(d*x)/2]) - C*Sin[(d*x)/2]))/(2*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]^2*(19*A*Sin[(d*x)/2] + 35*C*Sin[(d*x)/2]))/(4*d) - (10*C*Cos[(3*c)/2]*Sin[(3*d*x)/2])/d + (C*Cos[(5*c)/2]*Sin[(5*d*x)/2])/d + ((19*A + 35*C)*Sec[c/2 + (d*x)/2]*Tan[c/2])/(4*d) - ((A + C)*Sec[c/2 + (d*x)/2]^3*Tan[c/2])/(2*d)))/(a*(1 + Cos[c + d*x]))^(5/2)","C",1
1253,1,97,151,0.3405724,"\int \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{\sec ^{\frac{5}{2}}(c+d x) \left(21 B \sin (c+d x)+9 B \sin (3 (c+d x))-36 B \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+10 C \sin (2 (c+d x))+20 C \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{30 d}","\frac{2 B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{6 B \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{6 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}",1,"(Sec[c + d*x]^(5/2)*(-36*B*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 20*C*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 21*B*Sin[c + d*x] + 10*C*Sin[2*(c + d*x)] + 9*B*Sin[3*(c + d*x)]))/(30*d)","A",1
1254,1,85,123,0.2264977,"\int \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{\sec ^{\frac{3}{2}}(c+d x) \left(2 \sin (c+d x) (B+3 C \cos (c+d x))+2 B \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 C \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d}","\frac{2 B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 C \sin (c+d x) \sqrt{\sec (c+d x)}}{d}-\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(Sec[c + d*x]^(3/2)*(-6*C*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 2*B*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 2*(B + 3*C*Cos[c + d*x])*Sin[c + d*x]))/(3*d)","A",1
1255,1,71,97,0.115837,"\int \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{2 \sqrt{\sec (c+d x)} \left(B \sin (c+d x)-B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d}","\frac{2 B \sin (c+d x) \sqrt{\sec (c+d x)}}{d}-\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*Sqrt[Sec[c + d*x]]*(-(B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]) + C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + B*Sin[c + d*x]))/d","A",1
1256,1,52,75,0.0741152,"\int \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d}","\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*Sqrt[Cos[c + d*x]]*(C*EllipticE[(c + d*x)/2, 2] + B*EllipticF[(c + d*x)/2, 2])*Sqrt[Sec[c + d*x]])/d","A",1
1257,1,76,101,0.1340261,"\int \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(6 B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \left(\sin (2 (c+d x))+2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)\right)}{3 d}","\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 C \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}",1,"(Sqrt[Sec[c + d*x]]*(6*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + C*(2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + Sin[2*(c + d*x)])))/(3*d)","A",1
1258,1,88,127,0.3448645,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) (5 B+3 C \cos (c+d x))+10 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+18 C \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d}","\frac{2 B \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 C \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{6 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",1,"(Sqrt[Sec[c + d*x]]*(18*C*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 10*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (5*B + 3*C*Cos[c + d*x])*Sin[2*(c + d*x)]))/(15*d)","A",1
1259,1,99,151,0.5368665,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) (42 B \cos (c+d x)+15 C \cos (2 (c+d x))+65 C)+252 B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+100 C \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 B \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{6 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 C \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{10 C \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{10 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}",1,"(Sqrt[Sec[c + d*x]]*(252*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 100*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (65*C + 42*B*Cos[c + d*x] + 15*C*Cos[2*(c + d*x)])*Sin[2*(c + d*x)]))/(210*d)","A",1
1260,1,112,163,1.2070946,"\int \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{\sec ^{\frac{5}{2}}(c+d x) \left(2 \sin (c+d x) (3 (3 A+5 C) \cos (2 (c+d x))+15 (A+C)+10 B \cos (c+d x))-12 (3 A+5 C) \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+20 B \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{30 d}","\frac{2 (3 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{2 (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}",1,"(Sec[c + d*x]^(5/2)*(-12*(3*A + 5*C)*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 20*B*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 2*(15*(A + C) + 10*B*Cos[c + d*x] + 3*(3*A + 5*C)*Cos[2*(c + d*x)])*Sin[c + d*x]))/(30*d)","A",1
1261,1,89,127,0.3121243,"\int \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{\sec ^{\frac{3}{2}}(c+d x) \left(2 \sin (c+d x) (A+3 B \cos (c+d x))+2 (A+3 C) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 B \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d}","\frac{2 (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 B \sin (c+d x) \sqrt{\sec (c+d x)}}{d}-\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(Sec[c + d*x]^(3/2)*(-6*B*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 2*(A + 3*C)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 2*(A + 3*B*Cos[c + d*x])*Sin[c + d*x]))/(3*d)","A",1
1262,1,75,101,0.1904221,"\int \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{2 \sqrt{\sec (c+d x)} \left(-\left((A-C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)+A \sin (c+d x)+B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d}","-\frac{2 (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*Sqrt[Sec[c + d*x]]*(-((A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]) + B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + A*Sin[c + d*x]))/d","A",1
1263,1,80,105,0.1810783,"\int \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(2 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (2 (c+d x))\right)}{3 d}","\frac{2 (3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 C \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Sec[c + d*x]]*(6*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + C*Sin[2*(c + d*x)]))/(3*d)","A",1
1264,1,94,133,0.2937485,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(6 (5 A+3 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (2 (c+d x)) (5 B+3 C \cos (c+d x))+10 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d}","\frac{2 (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 C \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(6*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 10*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (5*B + 3*C*Cos[c + d*x])*Sin[2*(c + d*x)]))/(15*d)","A",1
1265,1,108,163,0.6616447,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) (70 A+42 B \cos (c+d x)+15 C \cos (2 (c+d x))+65 C)+20 (7 A+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+252 B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 (7 A+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 B \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{6 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 C \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(252*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 20*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (70*A + 65*C + 42*B*Cos[c + d*x] + 15*C*Cos[2*(c + d*x)])*Sin[2*(c + d*x)]))/(210*d)","A",1
1266,1,172,217,2.2354639,"\int (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{a \sec ^{\frac{7}{2}}(c+d x) \left(40 (5 A+7 (B+C)) \cos ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-168 (3 A+3 B+5 C) \cos ^{\frac{7}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) (21 (13 A+13 B+15 C) \cos (c+d x)+10 (5 A+7 (B+C)) \cos (2 (c+d x))+63 A \cos (3 (c+d x))+110 A+63 B \cos (3 (c+d x))+70 B+105 C \cos (3 (c+d x))+70 C)\right)}{420 d}","\frac{2 a (5 A+7 (B+C)) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 a (3 A+3 B+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a (5 A+7 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a (3 A+3 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (A+B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}",1,"(a*Sec[c + d*x]^(7/2)*(-168*(3*A + 3*B + 5*C)*Cos[c + d*x]^(7/2)*EllipticE[(c + d*x)/2, 2] + 40*(5*A + 7*(B + C))*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] + 2*(110*A + 70*B + 70*C + 21*(13*A + 13*B + 15*C)*Cos[c + d*x] + 10*(5*A + 7*(B + C))*Cos[2*(c + d*x)] + 63*A*Cos[3*(c + d*x)] + 63*B*Cos[3*(c + d*x)] + 105*C*Cos[3*(c + d*x)])*Sin[c + d*x]))/(420*d)","A",1
1267,1,147,179,1.0038523,"\int (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{a \sqrt{\cos (c+d x)} (\cos (c+d x)+1) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(5 (A+B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-3 (3 A+5 (B+C)) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{\sin (c+d x) (3 (3 A+5 (B+C)) \cos (2 (c+d x))+10 (A+B) \cos (c+d x)+15 (A+B+C))}{2 \cos ^{\frac{5}{2}}(c+d x)}\right)}{15 d}","\frac{2 a (3 A+5 (B+C)) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a (A+B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (3 A+5 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (A+B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}",1,"(a*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sqrt[Sec[c + d*x]]*(-3*(3*A + 5*(B + C))*EllipticE[(c + d*x)/2, 2] + 5*(A + B + 3*C)*EllipticF[(c + d*x)/2, 2] + ((15*(A + B + C) + 10*(A + B)*Cos[c + d*x] + 3*(3*A + 5*(B + C))*Cos[2*(c + d*x)])*Sin[c + d*x])/(2*Cos[c + d*x]^(5/2))))/(15*d)","A",1
1268,1,99,140,0.7054135,"\int (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{a \sec ^{\frac{3}{2}}(c+d x) \left(2 (A+3 (B+C)) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 (A+B-C) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) (3 (A+B) \cos (c+d x)+A)\right)}{3 d}","\frac{2 a (A+3 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A+B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a (A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}",1,"(a*Sec[c + d*x]^(3/2)*(-6*(A + B - C)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 2*(A + 3*(B + C))*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 2*(A + 3*(A + B)*Cos[c + d*x])*Sin[c + d*x]))/(3*d)","A",1
1269,1,101,141,0.4675641,"\int (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{a \sqrt{\sec (c+d x)} \left(2 (3 A+3 B+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 (A-B-C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) (3 A+C \cos (c+d x))\right)}{3 d}","\frac{2 a (3 A+3 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A-B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a C \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(a*Sqrt[Sec[c + d*x]]*(-6*(A - B - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*(3*A + 3*B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(3*A + C*Cos[c + d*x])*Sin[c + d*x]))/(3*d)","A",1
1270,1,105,147,0.62909,"\int (a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{a \sqrt{\sec (c+d x)} \left(10 (3 A+B+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 (5 A+5 B+3 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (2 (c+d x)) (5 (B+C)+3 C \cos (c+d x))\right)}{15 d}","\frac{2 a (3 A+B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (5 A+5 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (B+C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a C \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a*Sqrt[Sec[c + d*x]]*(6*(5*A + 5*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 10*(3*A + B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (5*(B + C) + 3*C*Cos[c + d*x])*Sin[2*(c + d*x)]))/(15*d)","A",1
1271,1,125,184,0.9771408,"\int \frac{(a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{a \sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) (70 A+42 (B+C) \cos (c+d x)+70 B+15 C \cos (2 (c+d x))+65 C)+20 (7 A+7 B+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+84 (5 A+3 (B+C)) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 a (7 A+7 B+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (7 A+7 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (5 A+3 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (B+C) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a C \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a*Sqrt[Sec[c + d*x]]*(84*(5*A + 3*(B + C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 20*(7*A + 7*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (70*A + 70*B + 65*C + 42*(B + C)*Cos[c + d*x] + 15*C*Cos[2*(c + d*x)])*Sin[2*(c + d*x)]))/(210*d)","A",1
1272,1,149,217,0.8875676,"\int \frac{(a+a \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{a \sqrt{\sec (c+d x)} \left(2 \sin (2 (c+d x)) (7 (36 A+36 B+43 C) \cos (c+d x)+5 (84 A+18 (B+C) \cos (2 (c+d x))+78 B+7 C \cos (3 (c+d x))+78 C))+240 (7 A+5 (B+C)) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+336 (9 A+9 B+7 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{2520 d}","\frac{2 a (9 A+9 B+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (7 A+5 (B+C)) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (7 A+5 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (9 A+9 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a (B+C) \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a C \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(a*Sqrt[Sec[c + d*x]]*(336*(9*A + 9*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 240*(7*A + 5*(B + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(7*(36*A + 36*B + 43*C)*Cos[c + d*x] + 5*(84*A + 78*B + 78*C + 18*(B + C)*Cos[2*(c + d*x)] + 7*C*Cos[3*(c + d*x)]))*Sin[2*(c + d*x)]))/(2520*d)","A",1
1273,1,209,291,2.2968039,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{a^2 \sec ^{\frac{9}{2}}(c+d x) \left(240 (5 A+6 B+7 C) \cos ^{\frac{9}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-336 (8 A+9 B+12 C) \cos ^{\frac{9}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) (90 (9 A+8 B+7 C) \cos (c+d x)+14 (64 A+72 B+81 C) \cos (2 (c+d x))+150 A \cos (3 (c+d x))+168 A \cos (4 (c+d x))+868 A+180 B \cos (3 (c+d x))+189 B \cos (4 (c+d x))+819 B+210 C \cos (3 (c+d x))+252 C \cos (4 (c+d x))+882 C)\right)}{1260 d}","\frac{2 a^2 (19 A+27 B+21 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d}+\frac{4 a^2 (5 A+6 B+7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{4 a^2 (8 A+9 B+12 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^2 (5 A+6 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (8 A+9 B+12 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (4 A+9 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}",1,"(a^2*Sec[c + d*x]^(9/2)*(-336*(8*A + 9*B + 12*C)*Cos[c + d*x]^(9/2)*EllipticE[(c + d*x)/2, 2] + 240*(5*A + 6*B + 7*C)*Cos[c + d*x]^(9/2)*EllipticF[(c + d*x)/2, 2] + 2*(868*A + 819*B + 882*C + 90*(9*A + 8*B + 7*C)*Cos[c + d*x] + 14*(64*A + 72*B + 81*C)*Cos[2*(c + d*x)] + 150*A*Cos[3*(c + d*x)] + 180*B*Cos[3*(c + d*x)] + 210*C*Cos[3*(c + d*x)] + 168*A*Cos[4*(c + d*x)] + 189*B*Cos[4*(c + d*x)] + 252*C*Cos[4*(c + d*x)])*Sin[c + d*x]))/(1260*d)","A",1
1274,1,177,255,2.5781002,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{a^2 \sec ^{\frac{7}{2}}(c+d x) \left(40 (6 A+7 (B+2 C)) \cos ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-168 (3 A+4 B+5 C) \cos ^{\frac{7}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) (21 (13 A+14 B+15 C) \cos (c+d x)+5 (12 A+14 B+7 C) \cos (2 (c+d x))+63 A \cos (3 (c+d x))+90 A+84 B \cos (3 (c+d x))+70 B+105 C \cos (3 (c+d x))+35 C)\right)}{210 d}","\frac{2 a^2 (33 A+49 B+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^2 (3 A+4 B+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^2 (6 A+7 B+14 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (3 A+4 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (4 A+7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2}{7 d}",1,"(a^2*Sec[c + d*x]^(7/2)*(-168*(3*A + 4*B + 5*C)*Cos[c + d*x]^(7/2)*EllipticE[(c + d*x)/2, 2] + 40*(6*A + 7*(B + 2*C))*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] + 2*(90*A + 70*B + 35*C + 21*(13*A + 14*B + 15*C)*Cos[c + d*x] + 5*(12*A + 14*B + 7*C)*Cos[2*(c + d*x)] + 63*A*Cos[3*(c + d*x)] + 84*B*Cos[3*(c + d*x)] + 105*C*Cos[3*(c + d*x)])*Sin[c + d*x]))/(210*d)","A",1
1275,1,135,214,1.3631171,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{a^2 \sec ^{\frac{5}{2}}(c+d x) \left(40 (A+2 B+3 C) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) (3 (8 A+5 (2 B+C)) \cos (2 (c+d x))+10 (2 A+B) \cos (c+d x)+15 (2 A+2 B+C))-24 (4 A+5 B) \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{30 d}","\frac{2 a^2 (17 A+25 B+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^2 (A+2 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (4 A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{15 d}-\frac{4 a^2 (4 A+5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{5 d}",1,"(a^2*Sec[c + d*x]^(5/2)*(-24*(4*A + 5*B)*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 40*(A + 2*B + 3*C)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 2*(15*(2*A + 2*B + C) + 10*(2*A + B)*Cos[c + d*x] + 3*(8*A + 5*(2*B + C))*Cos[2*(c + d*x)])*Sin[c + d*x]))/(30*d)","A",1
1276,1,118,212,1.4632353,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{a^2 \sec ^{\frac{3}{2}}(c+d x) \left(8 (2 A+3 B+2 C) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) (6 (2 A+B) \cos (c+d x)+2 A+C \cos (2 (c+d x))+C)-24 (A-C) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{6 d}","-\frac{2 a^2 (5 A+3 B-C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (2 A+3 B+2 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (4 A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)}{3 d}-\frac{4 a^2 (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{3 d}",1,"(a^2*Sec[c + d*x]^(3/2)*(-24*(A - C)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 8*(2*A + 3*B + 2*C)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 2*(2*A + C + 6*(2*A + B)*Cos[c + d*x] + C*Cos[2*(c + d*x)])*Sin[c + d*x]))/(6*d)","A",1
1277,1,121,212,0.7304208,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{a^2 \sqrt{\sec (c+d x)} \left(2 \sin (c+d x) (3 (10 A+C \cos (2 (c+d x))+C)+10 (B+2 C) \cos (c+d x))+40 (3 A+2 B+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+24 (5 B+4 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{30 d}","-\frac{2 a^2 (15 A-5 B-7 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (3 A+2 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 (5 A-C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{5 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (5 B+4 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2}{d}",1,"(a^2*Sqrt[Sec[c + d*x]]*(24*(5*B + 4*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 40*(3*A + 2*B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(10*(B + 2*C)*Cos[c + d*x] + 3*(10*A + C + C*Cos[2*(c + d*x)]))*Sin[c + d*x]))/(30*d)","A",1
1278,1,133,219,0.9995383,"\int (a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{a^2 \sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) (5 (14 A+28 B+3 C \cos (2 (c+d x))+27 C)+42 (B+2 C) \cos (c+d x))+40 (14 A+7 B+6 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+168 (5 A+4 B+3 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 a^2 (35 A+49 B+33 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (14 A+7 B+6 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (5 A+4 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (7 B+4 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{35 d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \sqrt{\sec (c+d x)}}",1,"(a^2*Sqrt[Sec[c + d*x]]*(168*(5*A + 4*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 40*(14*A + 7*B + 6*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (42*(B + 2*C)*Cos[c + d*x] + 5*(14*A + 28*B + 27*C + 3*C*Cos[2*(c + d*x)]))*Sin[2*(c + d*x)]))/(210*d)","A",1
1279,1,151,255,0.9479051,"\int \frac{(a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{a^2 \sqrt{\sec (c+d x)} \left(2 \sin (2 (c+d x)) (7 (36 A+72 B+79 C) \cos (c+d x)+840 A+90 (B+2 C) \cos (2 (c+d x))+810 B+35 C \cos (3 (c+d x))+780 C)+480 (7 A+6 B+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+672 (12 A+9 B+8 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{2520 d}","\frac{2 a^2 (21 A+27 B+19 C) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (7 A+6 B+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (7 A+6 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (12 A+9 B+8 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (9 B+4 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a^2*Sqrt[Sec[c + d*x]]*(672*(12*A + 9*B + 8*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 480*(7*A + 6*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(840*A + 810*B + 780*C + 7*(36*A + 72*B + 79*C)*Cos[c + d*x] + 90*(B + 2*C)*Cos[2*(c + d*x)] + 35*C*Cos[3*(c + d*x)])*Sin[2*(c + d*x)]))/(2520*d)","A",1
1280,1,174,291,1.4042054,"\int \frac{(a+a \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{a^2 \sqrt{\sec (c+d x)} \left(2 \sin (2 (c+d x)) (154 (72 A+79 B+86 C) \cos (c+d x)+5 (36 (11 A+22 B+27 C) \cos (2 (c+d x))+3564 A+154 (B+2 C) \cos (3 (c+d x))+3432 B+63 C \cos (4 (c+d x))+3309 C))+960 (66 A+55 B+50 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+14784 (9 A+8 B+7 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{55440 d}","\frac{4 a^2 (9 A+8 B+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (99 A+121 B+89 C) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2 (66 A+55 B+50 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (66 A+55 B+50 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^2 (9 A+8 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (11 B+4 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{99 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{11 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a^2*Sqrt[Sec[c + d*x]]*(14784*(9*A + 8*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 960*(66*A + 55*B + 50*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(154*(72*A + 79*B + 86*C)*Cos[c + d*x] + 5*(3564*A + 3432*B + 3309*C + 36*(11*A + 22*B + 27*C)*Cos[2*(c + d*x)] + 154*(B + 2*C)*Cos[3*(c + d*x)] + 63*C*Cos[4*(c + d*x)]))*Sin[2*(c + d*x)]))/(55440*d)","A",1
1281,1,242,343,3.1912894,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{13}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2),x]","\frac{a^3 \sec ^{\frac{11}{2}}(c+d x) \left(480 (105 A+121 B+143 C) \cos ^{\frac{11}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-7392 (15 A+17 B+21 C) \cos ^{\frac{11}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) (154 (375 A+377 B+396 C) \cos (c+d x)+60 (336 A+341 B+319 C) \cos (2 (c+d x))+21945 A \cos (3 (c+d x))+3150 A \cos (4 (c+d x))+3465 A \cos (5 (c+d x))+19530 A+24871 B \cos (3 (c+d x))+3630 B \cos (4 (c+d x))+3927 B \cos (5 (c+d x))+16830 B+28413 C \cos (3 (c+d x))+4290 C \cos (4 (c+d x))+4851 C \cos (5 (c+d x))+14850 C)\right)}{27720 d}","\frac{4 a^3 (210 A+253 B+264 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{1155 d}+\frac{4 a^3 (105 A+121 B+143 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{231 d}+\frac{4 a^3 (15 A+17 B+21 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 (105 A+143 B+99 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{693 d}+\frac{4 a^3 (105 A+121 B+143 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}-\frac{4 a^3 (15 A+17 B+21 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (6 A+11 B) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{99 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}",1,"(a^3*Sec[c + d*x]^(11/2)*(-7392*(15*A + 17*B + 21*C)*Cos[c + d*x]^(11/2)*EllipticE[(c + d*x)/2, 2] + 480*(105*A + 121*B + 143*C)*Cos[c + d*x]^(11/2)*EllipticF[(c + d*x)/2, 2] + 2*(19530*A + 16830*B + 14850*C + 154*(375*A + 377*B + 396*C)*Cos[c + d*x] + 60*(336*A + 341*B + 319*C)*Cos[2*(c + d*x)] + 21945*A*Cos[3*(c + d*x)] + 24871*B*Cos[3*(c + d*x)] + 28413*C*Cos[3*(c + d*x)] + 3150*A*Cos[4*(c + d*x)] + 3630*B*Cos[4*(c + d*x)] + 4290*C*Cos[4*(c + d*x)] + 3465*A*Cos[5*(c + d*x)] + 3927*B*Cos[5*(c + d*x)] + 4851*C*Cos[5*(c + d*x)])*Sin[c + d*x]))/(27720*d)","A",1
1282,1,209,307,2.3601577,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{a^3 \sec ^{\frac{9}{2}}(c+d x) \left(240 (11 A+13 B+21 C) \cos ^{\frac{9}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-336 (17 A+21 B+27 C) \cos ^{\frac{9}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) (45 (34 A+30 B+21 C) \cos (c+d x)+14 (136 A+153 B+171 C) \cos (2 (c+d x))+330 A \cos (3 (c+d x))+357 A \cos (4 (c+d x))+1687 A+390 B \cos (3 (c+d x))+441 B \cos (4 (c+d x))+1701 B+315 C \cos (3 (c+d x))+567 C \cos (4 (c+d x))+1827 C)\right)}{1260 d}","\frac{4 a^3 (32 A+41 B+42 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^3 (17 A+21 B+27 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 (73 A+99 B+63 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{315 d}+\frac{4 a^3 (11 A+13 B+21 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (17 A+21 B+27 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (2 A+3 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^3}{9 d}",1,"(a^3*Sec[c + d*x]^(9/2)*(-336*(17*A + 21*B + 27*C)*Cos[c + d*x]^(9/2)*EllipticE[(c + d*x)/2, 2] + 240*(11*A + 13*B + 21*C)*Cos[c + d*x]^(9/2)*EllipticF[(c + d*x)/2, 2] + 2*(1687*A + 1701*B + 1827*C + 45*(34*A + 30*B + 21*C)*Cos[c + d*x] + 14*(136*A + 153*B + 171*C)*Cos[2*(c + d*x)] + 330*A*Cos[3*(c + d*x)] + 390*B*Cos[3*(c + d*x)] + 315*C*Cos[3*(c + d*x)] + 357*A*Cos[4*(c + d*x)] + 441*B*Cos[4*(c + d*x)] + 567*C*Cos[4*(c + d*x)])*Sin[c + d*x]))/(1260*d)","A",1
1283,1,176,271,3.4851795,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{a^3 \sec ^{\frac{7}{2}}(c+d x) \left(80 (13 A+21 B+35 C) \cos ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-336 (7 A+9 B+5 C) \cos ^{\frac{7}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) (21 (54 A+58 B+45 C) \cos (c+d x)+10 (26 A+21 B+7 C) \cos (2 (c+d x))+294 A \cos (3 (c+d x))+320 A+378 B \cos (3 (c+d x))+210 B+315 C \cos (3 (c+d x))+70 C)\right)}{420 d}","\frac{4 a^3 (106 A+147 B+140 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d}+\frac{2 (7 A+9 B+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d}+\frac{4 a^3 (13 A+21 B+35 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (7 A+9 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (6 A+7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{35 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^3}{7 d}",1,"(a^3*Sec[c + d*x]^(7/2)*(-336*(7*A + 9*B + 5*C)*Cos[c + d*x]^(7/2)*EllipticE[(c + d*x)/2, 2] + 80*(13*A + 21*B + 35*C)*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] + 2*(320*A + 210*B + 70*C + 21*(54*A + 58*B + 45*C)*Cos[c + d*x] + 10*(26*A + 21*B + 7*C)*Cos[2*(c + d*x)] + 294*A*Cos[3*(c + d*x)] + 378*B*Cos[3*(c + d*x)] + 315*C*Cos[3*(c + d*x)])*Sin[c + d*x]))/(420*d)","A",1
1284,1,157,270,2.1199368,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{a^3 \sec ^{\frac{5}{2}}(c+d x) \left(80 (3 A+5 (B+C)) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-48 (9 A+5 B-5 C) \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) (5 (12 A+4 B+3 C) \cos (c+d x)+6 (18 A+5 (3 B+C)) \cos (2 (c+d x))+5 (6 (4 A+3 B+C)+C \cos (3 (c+d x))))\right)}{60 d}","-\frac{4 a^3 (21 A+20 B+5 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 (33 A+35 B+15 C) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{15 d}+\frac{4 a^3 (3 A+5 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (9 A+5 B-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (6 A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{15 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{5 d}",1,"(a^3*Sec[c + d*x]^(5/2)*(-48*(9*A + 5*B - 5*C)*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 80*(3*A + 5*(B + C))*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 2*(5*(12*A + 4*B + 3*C)*Cos[c + d*x] + 6*(18*A + 5*(3*B + C))*Cos[2*(c + d*x)] + 5*(6*(4*A + 3*B + C) + C*Cos[3*(c + d*x)]))*Sin[c + d*x]))/(60*d)","A",1
1285,1,149,267,1.0656139,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{a^3 \sec ^{\frac{3}{2}}(c+d x) \left(80 (5 A+5 B+3 C) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-48 (5 A-5 B-9 C) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) (3 (60 A+20 B+3 C) \cos (c+d x)+20 A+10 (B+3 C) \cos (2 (c+d x))+10 B+3 C \cos (3 (c+d x))+30 C)\right)}{60 d}","-\frac{4 a^3 (20 A+5 B-6 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}-\frac{2 (35 A+15 B-3 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (5 A+5 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (5 A-5 B-9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (2 A+B) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)^2}{a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{3 d}",1,"(a^3*Sec[c + d*x]^(3/2)*(-48*(5*A - 5*B - 9*C)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 80*(5*A + 5*B + 3*C)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 2*(20*A + 10*B + 30*C + 3*(60*A + 20*B + 3*C)*Cos[c + d*x] + 10*(B + 3*C)*Cos[2*(c + d*x)] + 3*C*Cos[3*(c + d*x)])*Sin[c + d*x]))/(60*d)","A",1
1286,1,149,269,1.0142635,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{a^3 \sqrt{\sec (c+d x)} \left(2 \sin (c+d x) (5 (28 A+84 B+113 C) \cos (c+d x)+420 A+42 (B+3 C) \cos (2 (c+d x))+42 B+15 C \cos (3 (c+d x))+126 C)+80 (35 A+21 B+13 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+336 (5 A+9 B+7 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{420 d}","-\frac{4 a^3 (35 A-42 B-41 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}-\frac{2 (35 A-7 B-11 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{35 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (35 A+21 B+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (5 A+9 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{2 (7 A-C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{7 a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^3}{d}",1,"(a^3*Sqrt[Sec[c + d*x]]*(336*(5*A + 9*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 80*(35*A + 21*B + 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(420*A + 42*B + 126*C + 5*(28*A + 84*B + 113*C)*Cos[c + d*x] + 42*(B + 3*C)*Cos[2*(c + d*x)] + 15*C*Cos[3*(c + d*x)])*Sin[c + d*x]))/(420*d)","A",1
1287,1,153,271,1.5666785,"\int (a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{a^3 \sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) (7 (36 A+108 B+151 C) \cos (c+d x)+5 (252 A+18 (B+3 C) \cos (2 (c+d x))+330 B+7 C \cos (3 (c+d x))+318 C))+240 (21 A+13 B+11 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+336 (27 A+21 B+17 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{1260 d}","\frac{4 a^3 (42 A+41 B+32 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{2 (63 A+99 B+73 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{315 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (21 A+13 B+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (27 A+21 B+17 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (3 B+2 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \sqrt{\sec (c+d x)}}",1,"(a^3*Sqrt[Sec[c + d*x]]*(336*(27*A + 21*B + 17*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 240*(21*A + 13*B + 11*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (7*(36*A + 108*B + 151*C)*Cos[c + d*x] + 5*(252*A + 330*B + 318*C + 18*(B + 3*C)*Cos[2*(c + d*x)] + 7*C*Cos[3*(c + d*x)]))*Sin[2*(c + d*x)]))/(1260*d)","A",1
1288,1,174,307,1.554494,"\int \frac{(a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{a^3 \sqrt{\sec (c+d x)} \left(2 \sin (2 (c+d x)) (154 (108 A+151 B+165 C) \cos (c+d x)+5 (36 (11 A+33 B+49 C) \cos (2 (c+d x))+7260 A+154 (B+3 C) \cos (3 (c+d x))+6996 B+63 C \cos (4 (c+d x))+6741 C))+960 (143 A+121 B+105 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+14784 (21 A+17 B+15 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{55440 d}","\frac{4 a^3 (264 A+253 B+210 C) \sin (c+d x)}{1155 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (143 A+121 B+105 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 (99 A+143 B+105 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{693 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (143 A+121 B+105 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (21 A+17 B+15 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (11 B+6 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{99 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a^3*Sqrt[Sec[c + d*x]]*(14784*(21*A + 17*B + 15*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 960*(143*A + 121*B + 105*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(154*(108*A + 151*B + 165*C)*Cos[c + d*x] + 5*(7260*A + 6996*B + 6741*C + 36*(11*A + 33*B + 49*C)*Cos[2*(c + d*x)] + 154*(B + 3*C)*Cos[3*(c + d*x)] + 63*C*Cos[4*(c + d*x)]))*Sin[2*(c + d*x)]))/(55440*d)","A",1
1289,1,197,343,2.1619188,"\int \frac{(a+a \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{a^3 \sqrt{\sec (c+d x)} \left(2 \sin (2 (c+d x)) (154 (3926 A+4290 B+4525 C) \cos (c+d x)+5 (936 (33 A+49 B+59 C) \cos (2 (c+d x))+77 (52 A+156 B+245 C) \cos (3 (c+d x))+3 (60632 A+546 (B+3 C) \cos (4 (c+d x))+58422 B+231 C \cos (5 (c+d x))+56290 C)))+24960 (121 A+105 B+95 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+29568 (221 A+195 B+175 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{1441440 d}","\frac{4 a^3 (221 A+195 B+175 C) \sin (c+d x)}{585 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{20 a^3 (286 A+273 B+236 C) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (121 A+105 B+95 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 (143 A+195 B+145 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{1287 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (121 A+105 B+95 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (221 A+195 B+175 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{195 d}+\frac{2 (13 B+6 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{143 a d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{13 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a^3*Sqrt[Sec[c + d*x]]*(29568*(221*A + 195*B + 175*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 24960*(121*A + 105*B + 95*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(154*(3926*A + 4290*B + 4525*C)*Cos[c + d*x] + 5*(936*(33*A + 49*B + 59*C)*Cos[2*(c + d*x)] + 77*(52*A + 156*B + 245*C)*Cos[3*(c + d*x)] + 3*(60632*A + 58422*B + 56290*C + 546*(B + 3*C)*Cos[4*(c + d*x)] + 231*C*Cos[5*(c + d*x)])))*Sin[2*(c + d*x)]))/(1441440*d)","A",1
1290,1,200,250,3.9162743,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x]),x]","-\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \left(20 (5 A-5 B+3 C) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+36 (7 A-5 B+5 C) \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\tan \left(\frac{1}{2} (c+d x)\right) ((173 A-95 B+135 C) \cos (c+d x)+(76 A-40 B+60 C) \cos (2 (c+d x))+63 A \cos (3 (c+d x))+100 A-45 B \cos (3 (c+d x))-40 B+45 C \cos (3 (c+d x))+60 C)\right)}{30 a d (\cos (c+d x)+1)}","\frac{(7 A-5 B+5 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 a d}-\frac{(5 A-5 B+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}+\frac{3 (7 A-5 B+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 a d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(5 A-5 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 (7 A-5 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}",1,"-1/30*(Cos[(c + d*x)/2]^2*Sec[c + d*x]^(5/2)*(36*(7*A - 5*B + 5*C)*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 20*(5*A - 5*B + 3*C)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] - (100*A - 40*B + 60*C + (173*A - 95*B + 135*C)*Cos[c + d*x] + (76*A - 40*B + 60*C)*Cos[2*(c + d*x)] + 63*A*Cos[3*(c + d*x)] - 45*B*Cos[3*(c + d*x)] + 45*C*Cos[3*(c + d*x)])*Tan[(c + d*x)/2]))/(a*d*(1 + Cos[c + d*x]))","A",1
1291,1,162,205,2.5249969,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x]),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \left(2 (5 A-3 B+3 C) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 (3 A-3 B+C) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\tan \left(\frac{1}{2} (c+d x)\right) (3 (3 A-3 B+C) \cos (2 (c+d x))+4 (2 A-3 B) \cos (c+d x)+5 A-9 B+3 C)\right)}{3 a d (\cos (c+d x)+1)}","\frac{(5 A-3 B+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}-\frac{(3 A-3 B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(5 A-3 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{(3 A-3 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(Cos[(c + d*x)/2]^2*Sec[c + d*x]^(3/2)*(6*(3*A - 3*B + C)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 2*(5*A - 3*B + 3*C)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] - (5*A - 9*B + 3*C + 4*(2*A - 3*B)*Cos[c + d*x] + 3*(3*A - 3*B + C)*Cos[2*(c + d*x)])*Tan[(c + d*x)/2]))/(3*a*d*(1 + Cos[c + d*x]))","A",1
1292,1,132,165,1.1801697,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x]),x]","-\frac{2 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(-\tan \left(\frac{1}{2} (c+d x)\right) ((3 A-B+C) \cos (c+d x)+2 A)+(A-B-C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+(3 A-B+C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a d (\cos (c+d x)+1)}","\frac{(3 A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{(A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \cos (c+d x)+a)}-\frac{(A-B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(3 A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(-2*Cos[(c + d*x)/2]^2*Sqrt[Sec[c + d*x]]*((3*A - B + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + (A - B - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (2*A + (3*A - B + C)*Cos[c + d*x])*Tan[(c + d*x)/2]))/(a*d*(1 + Cos[c + d*x]))","A",1
1293,1,126,130,0.759126,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{a+a \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x]),x]","\frac{2 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(-(A-B+C) \left(\sin (c+d x)-\tan \left(\frac{1}{2} (c+d x)\right)\right)+(A+B-C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+(A-B+3 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a d (\cos (c+d x)+1)}","-\frac{(A-B+C) \sin (c+d x)}{d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)}+\frac{(A+B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A-B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(2*Cos[(c + d*x)/2]^2*Sqrt[Sec[c + d*x]]*((A - B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + (A + B - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (A - B + C)*(Sin[c + d*x] - Tan[(c + d*x)/2])))/(a*d*(1 + Cos[c + d*x]))","A",1
1294,1,163,174,0.78957,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x)) \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]),x]","-\frac{2 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(-(3 A-3 B+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 (A-3 B+3 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{1}{2} \left(\sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) (3 A-3 B+2 C \cos (c+d x)+5 C)\right)}{3 a d (\cos (c+d x)+1)}","\frac{(3 A-3 B+5 C) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{(3 A-3 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{(A-3 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(-2*Cos[(c + d*x)/2]^2*Sqrt[Sec[c + d*x]]*(3*(A - 3*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] - (3*A - 3*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + ((3*A - 3*B + 5*C + 2*C*Cos[c + d*x])*Sec[(c + d*x)/2]*(Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2]))/2))/(3*a*d*(1 + Cos[c + d*x]))","A",1
1295,1,178,214,1.1633388,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(-10 (3 A-5 B+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+18 (5 A-5 B+7 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\left(\sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) (15 A+(4 C-10 B) \cos (c+d x)-25 B-3 C \cos (2 (c+d x))+22 C)\right)}{15 a d (\cos (c+d x)+1)}","\frac{(5 A-5 B+7 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(3 A-5 B+5 C) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)}-\frac{(3 A-5 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (5 A-5 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}",1,"(Cos[(c + d*x)/2]^2*Sqrt[Sec[c + d*x]]*(18*(5*A - 5*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] - 10*(3*A - 5*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (15*A - 25*B + 22*C + (-10*B + 4*C)*Cos[c + d*x] - 3*C*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]*(Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2])))/(15*a*d*(1 + Cos[c + d*x]))","A",1
1296,1,198,250,1.8124191,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)),x]","-\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(-100 (7 A-7 B+9 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+252 (5 A-7 B+7 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\left(\sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) ((140 A-56 B+201 C) \cos (c+d x)+350 A+6 (7 B-2 C) \cos (2 (c+d x))-308 B+15 C \cos (3 (c+d x))+438 C)\right)}{210 a d (\cos (c+d x)+1)}","-\frac{(5 A-7 B+7 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{(7 A-7 B+9 C) \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}+\frac{5 (7 A-7 B+9 C) \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{5 (7 A-7 B+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a d}-\frac{3 (5 A-7 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}",1,"-1/210*(Cos[(c + d*x)/2]^2*Sqrt[Sec[c + d*x]]*(252*(5*A - 7*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] - 100*(7*A - 7*B + 9*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (350*A - 308*B + 438*C + (140*A - 56*B + 201*C)*Cos[c + d*x] + 6*(7*B - 2*C)*Cos[2*(c + d*x)] + 15*C*Cos[3*(c + d*x)])*Sec[(c + d*x)/2]*(Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2])))/(a*d*(1 + Cos[c + d*x]))","A",1
1297,1,212,251,4.2483016,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^2,x]","\frac{2 \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \left(2 (10 A-5 B+2 C) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 (7 A-4 B+C) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{1}{4} \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) ((95 A-60 B+9 C) \cos (c+d x)+(64 A-38 B+8 C) \cos (2 (c+d x))+21 A \cos (3 (c+d x))+56 A-12 B \cos (3 (c+d x))-38 B+3 C \cos (3 (c+d x))+8 C)\right)}{3 a^2 d (\cos (c+d x)+1)^2}","\frac{(10 A-5 B+2 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d}-\frac{(7 A-4 B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}-\frac{(7 A-4 B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{(10 A-5 B+2 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(7 A-4 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(2*Cos[(c + d*x)/2]^4*Sec[c + d*x]^(3/2)*(6*(7*A - 4*B + C)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 2*(10*A - 5*B + 2*C)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] - ((56*A - 38*B + 8*C + (95*A - 60*B + 9*C)*Cos[c + d*x] + (64*A - 38*B + 8*C)*Cos[2*(c + d*x)] + 21*A*Cos[3*(c + d*x)] - 12*B*Cos[3*(c + d*x)] + 3*C*Cos[3*(c + d*x)])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/4))/(3*a^2*d*(1 + Cos[c + d*x])^2)","A",1
1298,1,172,215,3.209591,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^2,x]","-\frac{2 \cos ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(2 (5 A-2 B-C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{1}{2} \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (2 (19 A-4 B+C) \cos (c+d x)+3 (4 A-B) \cos (2 (c+d x))+24 A-3 B)+6 (4 A-B) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 a^2 d (\cos (c+d x)+1)^2}","-\frac{(5 A-2 B-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d (\cos (c+d x)+1)}-\frac{(5 A-2 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(4 A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}-\frac{(4 A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d (a \cos (c+d x)+a)^2}",1,"(-2*Cos[(c + d*x)/2]^4*Sqrt[Sec[c + d*x]]*(6*(4*A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*(5*A - 2*B - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - ((24*A - 3*B + 2*(19*A - 4*B + C)*Cos[c + d*x] + 3*(4*A - B)*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/2))/(3*a^2*d*(1 + Cos[c + d*x])^2)","A",1
1299,1,164,173,2.0541579,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^2,x]","\frac{2 \cos ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(2 (2 A+B+2 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{1}{2} \left(\sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) (3 (A-C) \cos (c+d x)+4 A-B-2 C)+6 (A-C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 a^2 d (\cos (c+d x)+1)^2}","\frac{(2 A+B+2 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(A-C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1) \sqrt{\sec (c+d x)}}+\frac{(A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B+C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2}",1,"(2*Cos[(c + d*x)/2]^4*Sqrt[Sec[c + d*x]]*(6*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*(2*A + B + 2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + ((4*A - B - 2*C + 3*(A - C)*Cos[c + d*x])*Sec[(c + d*x)/2]^3*(Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2]))/2))/(3*a^2*d*(1 + Cos[c + d*x])^2)","A",1
1300,1,162,179,2.1479044,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2 \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]),x]","-\frac{2 \cos ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(-2 (A+2 B-5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{1}{2} \left(\sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) (A+3 (B-2 C) \cos (c+d x)+2 B-5 C)+6 (B-4 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 a^2 d (\cos (c+d x)+1)^2}","\frac{(A+2 B-5 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1) \sqrt{\sec (c+d x)}}+\frac{(A+2 B-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(B-4 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B+C) \sin (c+d x)}{3 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"(-2*Cos[(c + d*x)/2]^4*Sqrt[Sec[c + d*x]]*(6*(B - 4*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] - 2*(A + 2*B - 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + ((A + 2*B - 5*C + 3*(B - 2*C)*Cos[c + d*x])*Sec[(c + d*x)/2]^3*(Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2]))/2))/(3*a^2*d*(1 + Cos[c + d*x])^2)","A",1
1301,1,183,220,2.7627157,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)),x]","-\frac{2 \cos ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(-2 (2 A-5 B+10 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 (A-4 B+7 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{1}{2} \left(\sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) ((3 A-6 B+13 C) \cos (c+d x)+2 A-5 B+C \cos (2 (c+d x))+11 C)\right)}{3 a^2 d (\cos (c+d x)+1)^2}","\frac{(2 A-5 B+10 C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{(A-4 B+7 C) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1) \sec ^{\frac{3}{2}}(c+d x)}+\frac{(2 A-5 B+10 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(A-4 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B+C) \sin (c+d x)}{3 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"(-2*Cos[(c + d*x)/2]^4*Sqrt[Sec[c + d*x]]*(6*(A - 4*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] - 2*(2*A - 5*B + 10*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + ((2*A - 5*B + 11*C + (3*A - 6*B + 13*C)*Cos[c + d*x] + C*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^3*(Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2]))/2))/(3*a^2*d*(1 + Cos[c + d*x])^2)","A",1
1302,1,200,254,3.6290931,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^2 \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)),x]","\frac{2 \cos ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(-50 (A-2 B+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 (20 A-35 B+56 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{1}{4} \left(\sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) ((60 A-130 B+179 C) \cos (c+d x)+50 A+(8 C-10 B) \cos (2 (c+d x))-110 B-3 C \cos (3 (c+d x))+158 C)\right)}{15 a^2 d (\cos (c+d x)+1)^2}","\frac{(20 A-35 B+56 C) \sin (c+d x)}{15 a^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 (A-2 B+3 C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{(A-2 B+3 C) \sin (c+d x)}{a^2 d (\cos (c+d x)+1) \sec ^{\frac{5}{2}}(c+d x)}-\frac{5 (A-2 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(20 A-35 B+56 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{(A-B+C) \sin (c+d x)}{3 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"(2*Cos[(c + d*x)/2]^4*Sqrt[Sec[c + d*x]]*(6*(20*A - 35*B + 56*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] - 50*(A - 2*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + ((50*A - 110*B + 158*C + (60*A - 130*B + 179*C)*Cos[c + d*x] + (-10*B + 8*C)*Cos[2*(c + d*x)] - 3*C*Cos[3*(c + d*x)])*Sec[(c + d*x)/2]^3*(Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2]))/4))/(15*a^2*d*(1 + Cos[c + d*x])^2)","A",1
1303,1,249,310,5.7857479,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^3,x]","\frac{2 \cos ^6\left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \left(10 (33 A-13 B+3 C) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 (119 A-49 B+9 C) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{1}{16} \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) (12 (533 A-228 B+33 C) \cos (c+d x)+8 (526 A-221 B+36 C) \cos (2 (c+d x))+1812 A \cos (3 (c+d x))+357 A \cos (4 (c+d x))+3691 A-752 B \cos (3 (c+d x))-147 B \cos (4 (c+d x))-1621 B+132 C \cos (3 (c+d x))+27 C \cos (4 (c+d x))+261 C)\right)}{15 a^3 d (\cos (c+d x)+1)^3}","\frac{(33 A-13 B+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 a^3 d}-\frac{(119 A-49 B+9 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{(119 A-49 B+9 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{30 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(33 A-13 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(119 A-49 B+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(2 A-B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d (a \cos (c+d x)+a)^2}",1,"(2*Cos[(c + d*x)/2]^6*Sec[c + d*x]^(3/2)*(6*(119*A - 49*B + 9*C)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 10*(33*A - 13*B + 3*C)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] - ((3691*A - 1621*B + 261*C + 12*(533*A - 228*B + 33*C)*Cos[c + d*x] + 8*(526*A - 221*B + 36*C)*Cos[2*(c + d*x)] + 1812*A*Cos[3*(c + d*x)] - 752*B*Cos[3*(c + d*x)] + 132*C*Cos[3*(c + d*x)] + 357*A*Cos[4*(c + d*x)] - 147*B*Cos[4*(c + d*x)] + 27*C*Cos[4*(c + d*x)])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/16))/(15*a^3*d*(1 + Cos[c + d*x])^3)","A",1
1304,1,215,277,2.0682423,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^3,x]","-\frac{2 \cos ^6\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(10 (13 A-3 B-C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 (49 A-9 B-C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{1}{8} \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) ((1621 A-261 B+11 C) \cos (c+d x)+4 (188 A-33 B-2 C) \cos (2 (c+d x))+147 A \cos (3 (c+d x))+992 A-27 B \cos (3 (c+d x))-132 B-3 C \cos (3 (c+d x))-8 C)\right)}{15 a^3 d (\cos (c+d x)+1)^3}","\frac{(49 A-9 B-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{(13 A-3 B-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(13 A-3 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(49 A-9 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(8 A-3 B-2 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d (a \cos (c+d x)+a)^3}",1,"(-2*Cos[(c + d*x)/2]^6*Sqrt[Sec[c + d*x]]*(6*(49*A - 9*B - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 10*(13*A - 3*B - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - ((992*A - 132*B - 8*C + (1621*A - 261*B + 11*C)*Cos[c + d*x] + 4*(188*A - 33*B - 2*C)*Cos[2*(c + d*x)] + 147*A*Cos[3*(c + d*x)] - 27*B*Cos[3*(c + d*x)] - 3*C*Cos[3*(c + d*x)])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/8))/(15*a^3*d*(1 + Cos[c + d*x])^3)","A",1
1305,1,188,233,1.5033511,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^3} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^3,x]","\frac{2 \cos ^6\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(10 (3 A+B+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 (9 A+B-C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{1}{8} \left(\sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) (4 (33 A+2 B-7 C) \cos (c+d x)+3 (9 A+B-C) \cos (2 (c+d x))+117 A-7 B-13 C)\right)}{15 a^3 d (\cos (c+d x)+1)^3}","-\frac{(9 A+B-C) \sin (c+d x)}{10 d \sqrt{\sec (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(3 A+B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(9 A+B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(6 A-B-4 C) \sin (c+d x)}{15 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^3}",1,"(2*Cos[(c + d*x)/2]^6*Sqrt[Sec[c + d*x]]*(6*(9*A + B - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 10*(3*A + B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + ((117*A - 7*B - 13*C + 4*(33*A + 2*B - 7*C)*Cos[c + d*x] + 3*(9*A + B - C)*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^5*(Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2]))/8))/(15*a^3*d*(1 + Cos[c + d*x])^3)","A",1
1306,1,188,231,1.5557151,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]),x]","\frac{2 \cos ^6\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(10 (A+B+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 (A-B-9 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{1}{8} \left(\sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) (4 (2 A-7 B-18 C) \cos (c+d x)+3 (A-B-9 C) \cos (2 (c+d x))-7 A-13 B-57 C)\right)}{15 a^3 d (\cos (c+d x)+1)^3}","-\frac{(A-B-9 C) \sin (c+d x)}{10 d \sqrt{\sec (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A+B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(A-B-9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B+C) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}+\frac{(4 A+B-6 C) \sin (c+d x)}{15 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^2}",1,"(2*Cos[(c + d*x)/2]^6*Sqrt[Sec[c + d*x]]*(6*(A - B - 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 10*(A + B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + ((-7*A - 13*B - 57*C + 4*(2*A - 7*B - 18*C)*Cos[c + d*x] + 3*(A - B - 9*C)*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^5*(Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2]))/8))/(15*a^3*d*(1 + Cos[c + d*x])^3)","A",1
1307,1,190,235,2.1917412,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)),x]","-\frac{2 \cos ^6\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(-10 (A+3 B-13 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 (A+9 B-49 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{1}{8} \left(\sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) (4 (7 A+18 B-73 C) \cos (c+d x)+3 (A+9 B-29 C) \cos (2 (c+d x))+13 A+57 B-217 C)\right)}{15 a^3 d (\cos (c+d x)+1)^3}","\frac{(A+3 B-13 C) \sin (c+d x)}{6 d \sqrt{\sec (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A+3 B-13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(A+9 B-49 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(2 A+3 B-8 C) \sin (c+d x)}{15 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x)}{5 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^3}",1,"(-2*Cos[(c + d*x)/2]^6*Sqrt[Sec[c + d*x]]*(6*(A + 9*B - 49*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] - 10*(A + 3*B - 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + ((13*A + 57*B - 217*C + 4*(7*A + 18*B - 73*C)*Cos[c + d*x] + 3*(A + 9*B - 29*C)*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^5*(Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2]))/8))/(15*a^3*d*(1 + Cos[c + d*x])^3)","A",1
1308,1,206,272,2.9002877,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)),x]","-\frac{2 \cos ^6\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(-10 (3 A-13 B+33 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 (9 A-49 B+119 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{1}{8} \left(\sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) ((72 A-292 B+782 C) \cos (c+d x)+3 (9 A-29 B+79 C) \cos (2 (c+d x))+57 A-217 B+10 C \cos (3 (c+d x))+567 C)\right)}{15 a^3 d (\cos (c+d x)+1)^3}","\frac{(3 A-13 B+33 C) \sin (c+d x)}{6 a^3 d \sqrt{\sec (c+d x)}}-\frac{(9 A-49 B+119 C) \sin (c+d x)}{30 d \sec ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(3 A-13 B+33 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(9 A-49 B+119 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B+C) \sin (c+d x)}{5 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^3}+\frac{(B-2 C) \sin (c+d x)}{3 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"(-2*Cos[(c + d*x)/2]^6*Sqrt[Sec[c + d*x]]*(6*(9*A - 49*B + 119*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] - 10*(3*A - 13*B + 33*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + ((57*A - 217*B + 567*C + (72*A - 292*B + 782*C)*Cos[c + d*x] + 3*(9*A - 29*B + 79*C)*Cos[2*(c + d*x)] + 10*C*Cos[3*(c + d*x)])*Sec[(c + d*x)/2]^5*(Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2]))/8))/(15*a^3*d*(1 + Cos[c + d*x])^3)","A",1
1309,1,229,313,3.0509941,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^3 \sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)),x]","\frac{2 \cos ^6\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(-10 (13 A-33 B+63 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+42 (7 A-17 B+33 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{1}{8} \left(\sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) (2 (146 A-391 B+732 C) \cos (c+d x)+3 (29 A-79 B+143 C) \cos (2 (c+d x))+217 A-10 B \cos (3 (c+d x))-567 B+12 C \cos (3 (c+d x))-3 C \cos (4 (c+d x))+1062 C)\right)}{15 a^3 d (\cos (c+d x)+1)^3}","\frac{7 (7 A-17 B+33 C) \sin (c+d x)}{30 a^3 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(13 A-33 B+63 C) \sin (c+d x)}{6 a^3 d \sqrt{\sec (c+d x)}}-\frac{(13 A-33 B+63 C) \sin (c+d x)}{10 d \sec ^{\frac{5}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(13 A-33 B+63 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{7 (7 A-17 B+33 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(2 A-7 B+12 C) \sin (c+d x)}{15 a d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x)}{5 d \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^3}",1,"(2*Cos[(c + d*x)/2]^6*Sqrt[Sec[c + d*x]]*(42*(7*A - 17*B + 33*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] - 10*(13*A - 33*B + 63*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + ((217*A - 567*B + 1062*C + 2*(146*A - 391*B + 732*C)*Cos[c + d*x] + 3*(29*A - 79*B + 143*C)*Cos[2*(c + d*x)] - 10*B*Cos[3*(c + d*x)] + 12*C*Cos[3*(c + d*x)] - 3*C*Cos[4*(c + d*x)])*Sec[(c + d*x)/2]^5*(Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2]))/8))/(15*a^3*d*(1 + Cos[c + d*x])^3)","A",1
1310,1,155,226,0.9724426,"\int \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} (2 (88 A+99 B+63 C) \cos (c+d x)+11 (16 A+18 B+21 C) \cos (2 (c+d x))+32 A \cos (3 (c+d x))+32 A \cos (4 (c+d x))+214 A+36 B \cos (3 (c+d x))+36 B \cos (4 (c+d x))+162 B+42 C \cos (3 (c+d x))+42 C \cos (4 (c+d x))+189 C)}{315 d}","\frac{2 a (16 A+18 B+21 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a (16 A+18 B+21 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a (16 A+18 B+21 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+9 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{9 d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(214*A + 162*B + 189*C + 2*(88*A + 99*B + 63*C)*Cos[c + d*x] + 11*(16*A + 18*B + 21*C)*Cos[2*(c + d*x)] + 32*A*Cos[3*(c + d*x)] + 36*B*Cos[3*(c + d*x)] + 42*C*Cos[3*(c + d*x)] + 32*A*Cos[4*(c + d*x)] + 36*B*Cos[4*(c + d*x)] + 42*C*Cos[4*(c + d*x)])*Sec[c + d*x]^(9/2)*Tan[(c + d*x)/2])/(315*d)","A",1
1311,1,121,178,0.7120962,"\int \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} (3 (36 A+42 B+35 C) \cos (c+d x)+(24 A+28 B+35 C) \cos (2 (c+d x))+24 A \cos (3 (c+d x))+54 A+28 B \cos (3 (c+d x))+28 B+35 C \cos (3 (c+d x))+35 C)}{105 d}","\frac{2 a (24 A+28 B+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a (24 A+28 B+35 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{7 d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(54*A + 28*B + 35*C + 3*(36*A + 42*B + 35*C)*Cos[c + d*x] + (24*A + 28*B + 35*C)*Cos[2*(c + d*x)] + 24*A*Cos[3*(c + d*x)] + 28*B*Cos[3*(c + d*x)] + 35*C*Cos[3*(c + d*x)])*Sec[c + d*x]^(7/2)*Tan[(c + d*x)/2])/(105*d)","A",1
1312,1,85,130,0.3460315,"\int \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} ((8 A+10 B+15 C) \cos (2 (c+d x))+2 (4 A+5 B) \cos (c+d x)+14 A+10 B+15 C)}{15 d}","\frac{2 a (8 A+10 B+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{5 d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(14*A + 10*B + 15*C + 2*(4*A + 5*B)*Cos[c + d*x] + (8*A + 10*B + 15*C)*Cos[2*(c + d*x)])*Sec[c + d*x]^(5/2)*Tan[(c + d*x)/2])/(15*d)","A",1
1313,1,105,140,0.3879723,"\int \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) ((2 A+3 B) \cos (c+d x)+A)+3 \sqrt{2} C \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{3}{2}}(c+d x)\right)}{3 d}","\frac{2 a (A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{2 \sqrt{a} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^(3/2)*(3*Sqrt[2]*C*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(3/2) + 2*(A + (2*A + 3*B)*Cos[c + d*x])*Sin[(c + d*x)/2]))/(3*d)","A",1
1314,1,104,141,0.3120647,"\int \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) (2 A+C \cos (c+d x))+\sqrt{2} (2 B+C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}\right)}{2 d}","-\frac{a (2 A-C) \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{d}+\frac{\sqrt{a} (2 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(Sqrt[2]*(2*B + C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*(2*A + C*Cos[c + d*x])*Sin[(c + d*x)/2]))/(2*d)","A",1
1315,1,123,151,0.4342968,"\int \sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(\sqrt{2} (8 A+4 B+3 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} (4 B+2 C \cos (c+d x)+3 C)\right)}{8 d}","\frac{\sqrt{a} (8 A+4 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a (4 B+C) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(Sqrt[2]*(8*A + 4*B + 3*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(4*B + 3*C + 2*C*Cos[c + d*x])*Sin[(c + d*x)/2]))/(8*d)","A",1
1316,1,144,199,0.8023057,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} (8 A+6 B+5 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} (24 A+2 (6 B+5 C) \cos (c+d x)+18 B+4 C \cos (2 (c+d x))+19 C)\right)}{48 d}","\frac{\sqrt{a} (8 A+6 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a (8 A+6 B+5 C) \sin (c+d x)}{8 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a (6 B+C) \sin (c+d x)}{12 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(3*Sqrt[2]*(8*A + 6*B + 5*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(24*A + 18*B + 19*C + 2*(6*B + 5*C)*Cos[c + d*x] + 4*C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
1317,1,164,247,1.013927,"\int \frac{\sqrt{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} (48 A+40 B+35 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} (2 (48 A+40 B+53 C) \cos (c+d x)+144 A+4 (8 B+7 C) \cos (2 (c+d x))+152 B+12 C \cos (3 (c+d x))+133 C)\right)}{384 d}","\frac{a (48 A+40 B+35 C) \sin (c+d x)}{96 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (48 A+40 B+35 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a (48 A+40 B+35 C) \sin (c+d x)}{64 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a (8 B+C) \sin (c+d x)}{24 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(3*Sqrt[2]*(48*A + 40*B + 35*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(144*A + 152*B + 133*C + 2*(48*A + 40*B + 53*C)*Cos[c + d*x] + 4*(8*B + 7*C)*Cos[2*(c + d*x)] + 12*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(384*d)","A",1
1318,1,187,284,1.1593149,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{13}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} ((12684 A+12386 B+12441 C) \cos (c+d x)+(4368 A+4862 B+4422 C) \cos (2 (c+d x))+4368 A \cos (3 (c+d x))+672 A \cos (4 (c+d x))+672 A \cos (5 (c+d x))+4956 A+4862 B \cos (3 (c+d x))+748 B \cos (4 (c+d x))+748 B \cos (5 (c+d x))+4114 B+5577 C \cos (3 (c+d x))+858 C \cos (4 (c+d x))+858 C \cos (5 (c+d x))+3564 C)}{6930 d}","\frac{2 a^2 (84 A+110 B+99 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (336 A+374 B+429 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{1155 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a^2 (336 A+374 B+429 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (336 A+374 B+429 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (3 A+11 B) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{99 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(4956*A + 4114*B + 3564*C + (12684*A + 12386*B + 12441*C)*Cos[c + d*x] + (4368*A + 4862*B + 4422*C)*Cos[2*(c + d*x)] + 4368*A*Cos[3*(c + d*x)] + 4862*B*Cos[3*(c + d*x)] + 5577*C*Cos[3*(c + d*x)] + 672*A*Cos[4*(c + d*x)] + 748*B*Cos[4*(c + d*x)] + 858*C*Cos[4*(c + d*x)] + 672*A*Cos[5*(c + d*x)] + 748*B*Cos[5*(c + d*x)] + 858*C*Cos[5*(c + d*x)])*Sec[c + d*x]^(11/2)*Tan[(c + d*x)/2])/(6930*d)","A",1
1319,1,157,232,1.0492057,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} ((748 A+81 (8 B+7 C)) \cos (c+d x)+(748 A+858 B+882 C) \cos (2 (c+d x))+136 A \cos (3 (c+d x))+136 A \cos (4 (c+d x))+752 A+156 B \cos (3 (c+d x))+156 B \cos (4 (c+d x))+702 B+189 C \cos (3 (c+d x))+189 C \cos (4 (c+d x))+693 C)}{630 d}","\frac{2 a^2 (52 A+72 B+63 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (136 A+156 B+189 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (136 A+156 B+189 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+3 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{21 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(752*A + 702*B + 693*C + (748*A + 81*(8*B + 7*C))*Cos[c + d*x] + (748*A + 858*B + 882*C)*Cos[2*(c + d*x)] + 136*A*Cos[3*(c + d*x)] + 156*B*Cos[3*(c + d*x)] + 189*C*Cos[3*(c + d*x)] + 136*A*Cos[4*(c + d*x)] + 156*B*Cos[4*(c + d*x)] + 189*C*Cos[4*(c + d*x)])*Sec[c + d*x]^(9/2)*Tan[(c + d*x)/2])/(630*d)","A",1
1320,1,122,184,0.7705014,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} ((468 A+462 B+525 C) \cos (c+d x)+2 (52 A+63 B+35 C) \cos (2 (c+d x))+104 A \cos (3 (c+d x))+164 A+126 B \cos (3 (c+d x))+126 B+175 C \cos (3 (c+d x))+70 C)}{210 d}","\frac{2 a^2 (4 A+6 B+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (104 A+126 B+175 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (3 A+7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(164*A + 126*B + 70*C + (468*A + 462*B + 525*C)*Cos[c + d*x] + 2*(52*A + 63*B + 35*C)*Cos[2*(c + d*x)] + 104*A*Cos[3*(c + d*x)] + 126*B*Cos[3*(c + d*x)] + 175*C*Cos[3*(c + d*x)])*Sec[c + d*x]^(7/2)*Tan[(c + d*x)/2])/(210*d)","A",1
1321,1,134,192,0.8811053,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) ((18 A+25 B+15 C) \cos (2 (c+d x))+2 (9 A+5 B) \cos (c+d x)+24 A+25 B+15 C)+30 \sqrt{2} C \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{5}{2}}(c+d x)\right)}{30 d}","\frac{2 a^{3/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a^2 (12 A+20 B+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (3 A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^(5/2)*(30*Sqrt[2]*C*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(5/2) + 2*(24*A + 25*B + 15*C + 2*(9*A + 5*B)*Cos[c + d*x] + (18*A + 25*B + 15*C)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(30*d)","A",1
1322,1,128,191,0.7198198,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (4 (5 A+3 B) \cos (c+d x)+4 A+3 C \cos (2 (c+d x))+3 C)+3 \sqrt{2} (2 B+3 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{3}{2}}(c+d x)\right)}{6 d}","\frac{a^{3/2} (2 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a^2 (8 A+6 B-3 C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a (A+B) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^(3/2)*(3*Sqrt[2]*(2*B + 3*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(3/2) + (4*A + 3*C + 4*(5*A + 3*B)*Cos[c + d*x] + 3*C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(6*d)","A",1
1323,1,127,201,0.5746513,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(\sqrt{2} (8 A+12 B+7 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+2 \sin \left(\frac{1}{2} (c+d x)\right) (8 A+(4 B+7 C) \cos (c+d x)+C \cos (2 (c+d x))+C)\right)}{8 d}","\frac{a^{3/2} (8 A+12 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^2 (8 A-4 B-5 C) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a (4 A-C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}{d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(Sqrt[2]*(8*A + 12*B + 7*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*(8*A + C + (4*B + 7*C)*Cos[c + d*x] + C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(8*d)","A",1
1324,1,145,201,0.9058964,"\int (a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{a \sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} (24 A+14 B+11 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} (24 A+2 (6 B+11 C) \cos (c+d x)+42 B+4 C \cos (2 (c+d x))+37 C)\right)}{48 d}","\frac{a^{3/2} (24 A+14 B+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 (24 A+30 B+19 C) \sin (c+d x)}{24 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a (2 B+C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}}",1,"(a*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(3*Sqrt[2]*(24*A + 14*B + 11*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(24*A + 42*B + 37*C + 2*(6*B + 11*C)*Cos[c + d*x] + 4*C*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
1325,1,167,253,0.9288254,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} (112 A+88 B+75 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+\left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (2 (48 A+88 B+93 C) \cos (c+d x)+336 A+4 (8 B+15 C) \cos (2 (c+d x))+296 B+12 C \cos (3 (c+d x))+285 C)\right)}{384 d}","\frac{a^{3/2} (112 A+88 B+75 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (48 A+56 B+39 C) \sin (c+d x)}{96 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (112 A+88 B+75 C) \sin (c+d x)}{64 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a (8 B+3 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{24 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(3*Sqrt[2]*(112*A + 88*B + 75*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + (336*A + 296*B + 285*C + 2*(48*A + 88*B + 93*C)*Cos[c + d*x] + 4*(8*B + 15*C)*Cos[2*(c + d*x)] + 12*C*Cos[3*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(384*d)","A",1
1326,1,190,303,1.9710788,"\int \frac{(a+a \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{a \sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(15 \sqrt{2} (176 A+150 B+133 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} (2 (880 A+930 B+1007 C) \cos (c+d x)+4 (80 A+150 B+181 C) \cos (2 (c+d x))+2960 A+120 B \cos (3 (c+d x))+2850 B+228 C \cos (3 (c+d x))+48 C \cos (4 (c+d x))+2671 C)\right)}{3840 d}","\frac{a^{3/2} (176 A+150 B+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{192 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (80 A+90 B+67 C) \sin (c+d x)}{240 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{128 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a (10 B+3 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{40 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(15*Sqrt[2]*(176*A + 150*B + 133*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(2960*A + 2850*B + 2671*C + 2*(880*A + 930*B + 1007*C)*Cos[c + d*x] + 4*(80*A + 150*B + 181*C)*Cos[2*(c + d*x)] + 120*B*Cos[3*(c + d*x)] + 228*C*Cos[3*(c + d*x)] + 48*C*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]))/(3840*d)","A",1
1327,1,224,334,1.2809356,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{15}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(15/2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{13}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} (70 (5552 A+5083 B+4576 C) \cos (c+d x)+14 (30334 A+31850 B+32747 C) \cos (2 (c+d x))+125520 A \cos (3 (c+d x))+125520 A \cos (4 (c+d x))+16736 A \cos (5 (c+d x))+16736 A \cos (6 (c+d x))+343612 A+138450 B \cos (3 (c+d x))+138450 B \cos (4 (c+d x))+18460 B \cos (5 (c+d x))+18460 B \cos (6 (c+d x))+325910 B+141570 C \cos (3 (c+d x))+156585 C \cos (4 (c+d x))+20878 C \cos (5 (c+d x))+20878 C \cos (6 (c+d x))+322751 C)}{180180 d}","\frac{2 a^3 (2224 A+2522 B+2717 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{9009 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (8368 A+9230 B+10439 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{15015 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a^3 (8368 A+9230 B+10439 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{45045 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^3 (8368 A+9230 B+10439 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{45045 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (136 A+182 B+143 C) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{1287 d}+\frac{2 a (5 A+13 B) \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{143 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{13}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{13 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(343612*A + 325910*B + 322751*C + 70*(5552*A + 5083*B + 4576*C)*Cos[c + d*x] + 14*(30334*A + 31850*B + 32747*C)*Cos[2*(c + d*x)] + 125520*A*Cos[3*(c + d*x)] + 138450*B*Cos[3*(c + d*x)] + 141570*C*Cos[3*(c + d*x)] + 125520*A*Cos[4*(c + d*x)] + 138450*B*Cos[4*(c + d*x)] + 156585*C*Cos[4*(c + d*x)] + 16736*A*Cos[5*(c + d*x)] + 18460*B*Cos[5*(c + d*x)] + 20878*C*Cos[5*(c + d*x)] + 16736*A*Cos[6*(c + d*x)] + 18460*B*Cos[6*(c + d*x)] + 20878*C*Cos[6*(c + d*x)])*Sec[c + d*x]^(13/2)*Tan[(c + d*x)/2])/(180180*d)","A",1
1328,1,190,284,0.9880614,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{13}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} ((50140 A+49654 B+49830 C) \cos (c+d x)+4 (4615 A+4642 B+4290 C) \cos (2 (c+d x))+18460 A \cos (3 (c+d x))+2840 A \cos (4 (c+d x))+2840 A \cos (5 (c+d x))+18140 A+20878 B \cos (3 (c+d x))+3212 B \cos (4 (c+d x))+3212 B \cos (5 (c+d x))+15356 B+22935 C \cos (3 (c+d x))+3795 C \cos (4 (c+d x))+3795 C \cos (5 (c+d x))+13365 C)}{13860 d}","\frac{2 a^3 (1160 A+1364 B+1485 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (2840 A+3212 B+3795 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^3 (2840 A+3212 B+3795 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (32 A+44 B+33 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{231 d}+\frac{2 a (5 A+11 B) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{99 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{11 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(18140*A + 15356*B + 13365*C + (50140*A + 49654*B + 49830*C)*Cos[c + d*x] + 4*(4615*A + 4642*B + 4290*C)*Cos[2*(c + d*x)] + 18460*A*Cos[3*(c + d*x)] + 20878*B*Cos[3*(c + d*x)] + 22935*C*Cos[3*(c + d*x)] + 2840*A*Cos[4*(c + d*x)] + 3212*B*Cos[4*(c + d*x)] + 3795*C*Cos[4*(c + d*x)] + 2840*A*Cos[5*(c + d*x)] + 3212*B*Cos[5*(c + d*x)] + 3795*C*Cos[5*(c + d*x)])*Sec[c + d*x]^(11/2)*Tan[(c + d*x)/2])/(13860*d)","A",1
1329,1,158,234,1.3372255,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} (2 (1396 A+1215 B+882 C) \cos (c+d x)+4 (803 A+870 B+966 C) \cos (2 (c+d x))+584 A \cos (3 (c+d x))+584 A \cos (4 (c+d x))+2908 A+690 B \cos (3 (c+d x))+690 B \cos (4 (c+d x))+2790 B+588 C \cos (3 (c+d x))+903 C \cos (4 (c+d x))+2961 C)}{1260 d}","\frac{2 a^3 (8 A+10 B+11 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (584 A+690 B+903 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (64 A+90 B+63 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 a (5 A+9 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{63 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{9 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(2908*A + 2790*B + 2961*C + 2*(1396*A + 1215*B + 882*C)*Cos[c + d*x] + 4*(803*A + 870*B + 966*C)*Cos[2*(c + d*x)] + 584*A*Cos[3*(c + d*x)] + 690*B*Cos[3*(c + d*x)] + 588*C*Cos[3*(c + d*x)] + 584*A*Cos[4*(c + d*x)] + 690*B*Cos[4*(c + d*x)] + 903*C*Cos[4*(c + d*x)])*Sec[c + d*x]^(9/2)*Tan[(c + d*x)/2])/(1260*d)","A",1
1330,1,172,242,1.636032,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) ((930 A+987 B+840 C) \cos (c+d x)+2 (115 A+98 B+35 C) \cos (2 (c+d x))+230 A \cos (3 (c+d x))+290 A+301 B \cos (3 (c+d x))+196 B+280 C \cos (3 (c+d x))+70 C)+420 \sqrt{2} C \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{7}{2}}(c+d x)\right)}{420 d}","\frac{2 a^{5/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a^3 (160 A+224 B+245 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (40 A+56 B+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 a (5 A+7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^(7/2)*(420*Sqrt[2]*C*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(7/2) + 2*(290*A + 196*B + 70*C + (930*A + 987*B + 840*C)*Cos[c + d*x] + 2*(115*A + 98*B + 35*C)*Cos[2*(c + d*x)] + 230*A*Cos[3*(c + d*x)] + 301*B*Cos[3*(c + d*x)] + 280*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(420*d)","A",1
1331,1,156,243,1.4584404,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) ((112 A+40 B+45 C) \cos (c+d x)+4 (43 A+40 B+15 C) \cos (2 (c+d x))+196 A+160 B+15 C \cos (3 (c+d x))+60 C)+60 \sqrt{2} (2 B+5 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{5}{2}}(c+d x)\right)}{120 d}","\frac{a^{5/2} (2 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a^3 (64 A+70 B+15 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (8 A+10 B+5 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{5 d}+\frac{2 a (A+B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^(5/2)*(60*Sqrt[2]*(2*B + 5*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(5/2) + 2*(196*A + 160*B + 60*C + (112*A + 40*B + 45*C)*Cos[c + d*x] + 4*(43*A + 40*B + 15*C)*Cos[2*(c + d*x)] + 15*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(120*d)","A",1
1332,1,156,253,1.1685062,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(6 \sqrt{2} (8 A+20 B+19 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{3}{2}}(c+d x)+2 \sin \left(\frac{1}{2} (c+d x)\right) ((128 A+48 B+9 C) \cos (c+d x)+16 A+3 (4 B+11 C) \cos (2 (c+d x))+12 B+3 C \cos (3 (c+d x))+33 C)\right)}{48 d}","\frac{a^{5/2} (8 A+20 B+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^3 (56 A+12 B-27 C) \sin (c+d x)}{12 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A+4 B-C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 a (5 A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}{3 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^(3/2)*(6*Sqrt[2]*(8*A + 20*B + 19*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(3/2) + 2*(16*A + 12*B + 33*C + (128*A + 48*B + 9*C)*Cos[c + d*x] + 3*(4*B + 11*C)*Cos[2*(c + d*x)] + 3*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
1333,1,156,251,0.9998507,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} (40 A+38 B+25 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+2 \sin \left(\frac{1}{2} (c+d x)\right) (3 (8 A+22 B+27 C) \cos (c+d x)+48 A+(6 B+17 C) \cos (2 (c+d x))+6 B+2 C \cos (3 (c+d x))+17 C)\right)}{48 d}","\frac{a^{5/2} (40 A+38 B+25 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}-\frac{a^3 (24 A-54 B-49 C) \sin (c+d x)}{24 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A-2 B-3 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sqrt{\sec (c+d x)}}-\frac{a (6 A-C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}{d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(3*Sqrt[2]*(40*A + 38*B + 25*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*(48*A + 6*B + 17*C + 3*(8*A + 22*B + 27*C)*Cos[c + d*x] + (6*B + 17*C)*Cos[2*(c + d*x)] + 2*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
1334,1,166,253,1.4595606,"\int (a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{a^2 \sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} (304 A+200 B+163 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} ((96 A+272 B+362 C) \cos (c+d x)+528 A+4 (8 B+23 C) \cos (2 (c+d x))+632 B+12 C \cos (3 (c+d x))+581 C)\right)}{384 d}","\frac{a^{5/2} (304 A+200 B+163 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^3 (432 A+392 B+299 C) \sin (c+d x)}{192 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (16 A+24 B+17 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{32 d \sqrt{\sec (c+d x)}}+\frac{a (8 B+5 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{24 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{4 d \sqrt{\sec (c+d x)}}",1,"(a^2*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(3*Sqrt[2]*(304*A + 200*B + 163*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(528*A + 632*B + 581*C + (96*A + 272*B + 362*C)*Cos[c + d*x] + 4*(8*B + 23*C)*Cos[2*(c + d*x)] + 12*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(384*d)","A",1
1335,1,193,301,1.3357949,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(15 \sqrt{2} (400 A+326 B+283 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+\left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) ((2720 A+3620 B+3874 C) \cos (c+d x)+4 (80 A+230 B+331 C) \cos (2 (c+d x))+6320 A+120 B \cos (3 (c+d x))+5810 B+348 C \cos (3 (c+d x))+48 C \cos (4 (c+d x))+5521 C)\right)}{3840 d}","\frac{a^{5/2} (400 A+326 B+283 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^3 (1040 A+950 B+787 C) \sin (c+d x)}{960 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (400 A+326 B+283 C) \sin (c+d x)}{128 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (80 A+110 B+79 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{240 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{a (2 B+C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{8 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(15*Sqrt[2]*(400*A + 326*B + 283*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + (6320*A + 5810*B + 5521*C + (2720*A + 3620*B + 3874*C)*Cos[c + d*x] + 4*(80*A + 230*B + 331*C)*Cos[2*(c + d*x)] + 120*B*Cos[3*(c + d*x)] + 348*C*Cos[3*(c + d*x)] + 48*C*Cos[4*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(3840*d)","A",1
1336,1,227,353,1.8064773,"\int \frac{(a+a \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(15 \sqrt{2} (1304 A+1132 B+1015 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+\left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (2 (7240 A+7748 B+8085 C) \cos (c+d x)+4 (920 A+1324 B+1575 C) \cos (2 (c+d x))+480 A \cos (3 (c+d x))+23240 A+1392 B \cos (3 (c+d x))+192 B \cos (4 (c+d x))+22084 B+2140 C \cos (3 (c+d x))+560 C \cos (4 (c+d x))+80 C \cos (5 (c+d x))+20965 C)\right)}{15360 d}","\frac{a^{5/2} (1304 A+1132 B+1015 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{512 d}+\frac{a^3 (1304 A+1132 B+1015 C) \sin (c+d x)}{768 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (680 A+628 B+545 C) \sin (c+d x)}{960 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (1304 A+1132 B+1015 C) \sin (c+d x)}{512 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (120 A+156 B+115 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{480 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a (12 B+5 C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{60 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{6 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(15*Sqrt[2]*(1304*A + 1132*B + 1015*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + (23240*A + 22084*B + 20965*C + 2*(7240*A + 7748*B + 8085*C)*Cos[c + d*x] + 4*(920*A + 1324*B + 1575*C)*Cos[2*(c + d*x)] + 480*A*Cos[3*(c + d*x)] + 1392*B*Cos[3*(c + d*x)] + 2140*C*Cos[3*(c + d*x)] + 192*B*Cos[4*(c + d*x)] + 560*C*Cos[4*(c + d*x)] + 80*C*Cos[5*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(15360*d)","A",1
1337,1,7123,305,35.0280912,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2))/Sqrt[a + a*Cos[c + d*x]],x]","\text{Result too large to show}","\frac{2 (19 A-3 B+21 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (29 A-93 B+21 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (257 A-129 B+273 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-9 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}}",1,"Result too large to show","C",0
1338,1,2646,257,9.9800984,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2))/Sqrt[a + a*Cos[c + d*x]],x]","\text{Result too large to show}","\frac{2 (31 A-7 B+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (43 A-91 B+35 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}",1,"(2*Cos[c/2 + (d*x)/2]*Sqrt[(1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]*((2*B*Sin[c/2 + (d*x)/2])/(7*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2)) - (C*Sin[c/2 + (d*x)/2])/(3*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2)) + ((A - B + C)*Csc[c/2 + (d*x)/2]^9*(363825*Sin[c/2 + (d*x)/2]^2 - 4729725*Sin[c/2 + (d*x)/2]^4 + 26785605*Sin[c/2 + (d*x)/2]^6 - 86790165*Sin[c/2 + (d*x)/2]^8 + 177677808*Sin[c/2 + (d*x)/2]^10 - 239283044*Sin[c/2 + (d*x)/2]^12 + 52080*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12 + 560*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12 + 213120160*Sin[c/2 + (d*x)/2]^14 - 168280*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14 - 2240*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14 - 121497024*Sin[c/2 + (d*x)/2]^16 + 212520*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^16 + 3360*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^16 + 40125184*Sin[c/2 + (d*x)/2]^18 - 124320*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^18 - 2240*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^18 - 5840384*Sin[c/2 + (d*x)/2]^20 + 28000*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^20 + 560*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^20 + 363825*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 5336100*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^2*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 34636140*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^4*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 131060160*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^6*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 320535600*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^8*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 530671680*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^10*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 604296000*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^12*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 468948480*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^14*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 237726720*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^16*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 70963200*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^18*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 9461760*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^20*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 1120*Cos[(c + d*x)/2]^6*HypergeometricPFQ[{2, 2, 2, 11/2}, {1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12*(-6 + 5*Sin[c/2 + (d*x)/2]^2) + 280*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 11/2}, {1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12*(103 - 164*Sin[c/2 + (d*x)/2]^2 + 70*Sin[c/2 + (d*x)/2]^4)))/(40425*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(9/2)*(-1 + 2*Sin[c/2 + (d*x)/2]^2)) + (4*B*((3*Sin[c/2 + (d*x)/2])/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2) + 4*(Sin[c/2 + (d*x)/2]/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2) + (2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])))/35 + (C*((5*Sin[c/2 + (d*x)/2])/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2) + 2*((3*Sin[c/2 + (d*x)/2])/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2) + 4*(Sin[c/2 + (d*x)/2]/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2) + (2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]))))/105))/(d*Sqrt[a*(1 + Cos[c + d*x])])","C",0
1339,1,1882,211,7.7432508,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\frac{1}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}} \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)} \left(-\frac{(A-B+C) \left(440 \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right) \sin ^{16}\left(\frac{c}{2}+\frac{d x}{2}\right)+69120 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^{16}\left(\frac{c}{2}+\frac{d x}{2}\right)-42048 \sin ^{16}\left(\frac{c}{2}+\frac{d x}{2}\right)-1500 \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right) \sin ^{14}\left(\frac{c}{2}+\frac{d x}{2}\right)-414720 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^{14}\left(\frac{c}{2}+\frac{d x}{2}\right)+226656 \sin ^{14}\left(\frac{c}{2}+\frac{d x}{2}\right)+1770 \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right) \sin ^{12}\left(\frac{c}{2}+\frac{d x}{2}\right)+1080000 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^{12}\left(\frac{c}{2}+\frac{d x}{2}\right)-518760 \sin ^{12}\left(\frac{c}{2}+\frac{d x}{2}\right)-710 \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right) \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)-40 \cos ^6\left(\frac{1}{2} (c+d x)\right) \, _4F_3\left(2,2,2,\frac{9}{2};1,1,\frac{11}{2};\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right) \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)+60 \cos ^4\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(2,2,\frac{9}{2};1,\frac{11}{2};\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right) \left(4 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-5\right) \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)-1598400 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)+655812 \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)+1458000 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)-486630 \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)-833760 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)+210105 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)+291060 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-48825 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-56700 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}} \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+4725 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+4725 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right) \csc ^7\left(\frac{c}{2}+\frac{d x}{2}\right)}{675 \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{7/2} \left(2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1\right)}+\frac{8}{15} B \left(\frac{2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}+\frac{\sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{3/2}}\right)+\frac{1}{30} C \left(\frac{3 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{5/2}}+4 \left(\frac{2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}+\frac{\sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{3/2}}\right)\right)+\frac{2 B \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{5/2}}-\frac{C \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{5/2}}\right)}{d \sqrt{a (\cos (c+d x)+1)}}","\frac{2 (13 A-5 B+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}",1,"(2*Cos[c/2 + (d*x)/2]*Sqrt[(1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]*((2*B*Sin[c/2 + (d*x)/2])/(5*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2)) - (C*Sin[c/2 + (d*x)/2])/(2*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2)) + (8*B*(Sin[c/2 + (d*x)/2]/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2) + (2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]))/15 - ((A - B + C)*Csc[c/2 + (d*x)/2]^7*(4725*Sin[c/2 + (d*x)/2]^2 - 48825*Sin[c/2 + (d*x)/2]^4 + 210105*Sin[c/2 + (d*x)/2]^6 - 486630*Sin[c/2 + (d*x)/2]^8 + 655812*Sin[c/2 + (d*x)/2]^10 - 710*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^10 - 40*Cos[(c + d*x)/2]^6*HypergeometricPFQ[{2, 2, 2, 9/2}, {1, 1, 11/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^10 - 518760*Sin[c/2 + (d*x)/2]^12 + 1770*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12 + 226656*Sin[c/2 + (d*x)/2]^14 - 1500*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14 - 42048*Sin[c/2 + (d*x)/2]^16 + 440*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^16 + 4725*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 56700*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^2*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 291060*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^4*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 833760*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^6*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 1458000*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^8*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 1598400*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^10*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 1080000*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^12*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 414720*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^14*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 69120*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^16*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 60*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 9/2}, {1, 11/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^10*(-5 + 4*Sin[c/2 + (d*x)/2]^2)))/(675*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2)*(-1 + 2*Sin[c/2 + (d*x)/2]^2)) + (C*((3*Sin[c/2 + (d*x)/2])/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2) + 4*(Sin[c/2 + (d*x)/2]/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2) + (2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])))/30))/(d*Sqrt[a*(1 + Cos[c + d*x])])","C",0
1340,-1,0,163,0,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/Sqrt[a + a*Cos[c + d*x]],x]","\text{\$Aborted}","\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}",1,"$Aborted","F",-1
1341,1,277,178,4.0261115,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{(A-B+C) \csc ^3\left(\frac{1}{2} (c+d x)\right) \left(5 \cos ^2(c+d x) (\cos (c+d x)+2) \left(-\cos (c+d x)+\cos (c+d x) \sqrt{2-2 \sec (c+d x)} \tanh ^{-1}\left(\sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right) (-\sec (c+d x))}\right)+1\right)-\sin ^4\left(\frac{1}{2} (c+d x)\right) \sin ^2(c+d x) \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};-\sec (c+d x) \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}{10 \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 B \sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}+\sqrt{2} C \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)-\frac{2 C \sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)}{d \sqrt{a (\cos (c+d x)+1)}}","-\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(2*Cos[(c + d*x)/2]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(Sqrt[2]*C*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + (2*B*Sin[(c + d*x)/2])/Sqrt[Cos[c + d*x]] - (2*C*Sin[(c + d*x)/2])/Sqrt[Cos[c + d*x]] + ((A - B + C)*Csc[(c + d*x)/2]^3*(5*Cos[c + d*x]^2*(2 + Cos[c + d*x])*(1 - Cos[c + d*x] + ArcTanh[Sqrt[-(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]]*Cos[c + d*x]*Sqrt[2 - 2*Sec[c + d*x]]) - Hypergeometric2F1[2, 5/2, 7/2, -(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^4*Sin[c + d*x]^2))/(10*Cos[c + d*x]^(5/2))))/(d*Sqrt[a*(1 + Cos[c + d*x])])","C",0
1342,1,132,181,0.3891166,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(2 (A-B+C) \tan ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)+\sqrt{2} (2 B-C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 C \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)}\right)}{d \sqrt{a (\cos (c+d x)+1)}}","\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(2 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{C \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(Sqrt[2]*(2*B - C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*(A - B + C)*ArcTan[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]] + 2*C*Sqrt[Cos[c + d*x]]*Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
1343,1,16855,235,27.6102784,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","\text{Result too large to show}","\frac{(8 A-4 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(4 B-C) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"Result too large to show","C",0
1344,1,16904,281,27.6709501,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)),x]","\text{Result too large to show}","-\frac{(8 A-14 B+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{(8 A-2 B+7 C) \sin (c+d x)}{8 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(6 B-C) \sin (c+d x)}{12 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{C \sin (c+d x)}{3 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"Result too large to show","C",0
1345,1,143,192,0.455218,"\int \frac{\left(a A+(A b+a B) \cos (c+d x)+b B \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((a*A + (A*b + a*B)*Cos[c + d*x] + b*B*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(\sqrt{2} (2 a B+2 A b-b B) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 (a-b) (A-B) \tan ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)+2 b B \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)}\right)}{d \sqrt{a (\cos (c+d x)+1)}}","\frac{(2 a B+2 A b-b B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{\sqrt{2} (a-b) (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{b B \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(Sqrt[2]*(2*A*b + 2*a*B - b*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*(a - b)*(A - B)*ArcTan[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]] + 2*b*B*Sqrt[Cos[c + d*x]]*Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
1346,1,3136,333,9.9598812,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2))/(a + a*Cos[c + d*x])^(3/2),x]","\text{Result too large to show}","\frac{(19 A-15 B+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(11 A-7 B+7 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{14 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(67 A-63 B+35 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{70 a d \sqrt{a \cos (c+d x)+a}}+\frac{(397 A-273 B+245 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{210 a d \sqrt{a \cos (c+d x)+a}}-\frac{(1201 A-1029 B+665 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{210 a d \sqrt{a \cos (c+d x)+a}}",1,"(2*Cos[c/2 + (d*x)/2]^3*Sqrt[(1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]*((4*C*Sin[c/2 + (d*x)/2])/(7*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2)) - ((A - B + C)*(1 - 2*Sin[c/2 + (d*x)/2]))/(28*(1 + Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2)) + ((A - B + C)*(1 + 2*Sin[c/2 + (d*x)/2]))/(28*(1 - Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2)) - ((A - B + C)*(315*ArcTan[(1 - 2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]] + (5 + 3*Sin[c/2 + (d*x)/2])/((1 - Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2)) - (11 + 17*Sin[c/2 + (d*x)/2])/((1 - Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) + (61 + 71*Sin[c/2 + (d*x)/2])/((1 - Sin[c/2 + (d*x)/2])*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) + (193*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])/(1 - Sin[c/2 + (d*x)/2])))/70 + ((A - B + C)*(315*ArcTan[(1 + 2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]] + (5 - 3*Sin[c/2 + (d*x)/2])/((1 + Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2)) - (11 - 17*Sin[c/2 + (d*x)/2])/((1 + Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) + (61 - 71*Sin[c/2 + (d*x)/2])/((1 + Sin[c/2 + (d*x)/2])*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) + (193*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])/(1 + Sin[c/2 + (d*x)/2])))/70 - ((-A - 3*B + 7*C)*Csc[c/2 + (d*x)/2]^9*(363825*Sin[c/2 + (d*x)/2]^2 - 4729725*Sin[c/2 + (d*x)/2]^4 + 26785605*Sin[c/2 + (d*x)/2]^6 - 86790165*Sin[c/2 + (d*x)/2]^8 + 177677808*Sin[c/2 + (d*x)/2]^10 - 239283044*Sin[c/2 + (d*x)/2]^12 + 52080*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12 + 560*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12 + 213120160*Sin[c/2 + (d*x)/2]^14 - 168280*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14 - 2240*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14 - 121497024*Sin[c/2 + (d*x)/2]^16 + 212520*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^16 + 3360*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^16 + 40125184*Sin[c/2 + (d*x)/2]^18 - 124320*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^18 - 2240*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^18 - 5840384*Sin[c/2 + (d*x)/2]^20 + 28000*Hypergeometric2F1[2, 11/2, 13/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^20 + 560*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^20 + 363825*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 5336100*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^2*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 34636140*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^4*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 131060160*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^6*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 320535600*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^8*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 530671680*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^10*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 604296000*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^12*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 468948480*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^14*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 237726720*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^16*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 70963200*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^18*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 9461760*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^20*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 1120*Cos[(c + d*x)/2]^6*HypergeometricPFQ[{2, 2, 2, 11/2}, {1, 1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12*(-6 + 5*Sin[c/2 + (d*x)/2]^2) + 280*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 11/2}, {1, 13/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12*(103 - 164*Sin[c/2 + (d*x)/2]^2 + 70*Sin[c/2 + (d*x)/2]^4)))/(80850*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(9/2)*(-1 + 2*Sin[c/2 + (d*x)/2]^2)) + (8*C*((3*Sin[c/2 + (d*x)/2])/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2) + 4*(Sin[c/2 + (d*x)/2]/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2) + (2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])))/35))/(d*(a*(1 + Cos[c + d*x]))^(3/2))","C",0
1347,1,2295,283,7.842323,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x])^(3/2),x]","\text{Result too large to show}","-\frac{(15 A-11 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(9 A-5 B+5 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(39 A-35 B+15 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{30 a d \sqrt{a \cos (c+d x)+a}}+\frac{(147 A-95 B+75 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{30 a d \sqrt{a \cos (c+d x)+a}}",1,"(2*Cos[c/2 + (d*x)/2]^3*Sqrt[(1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]*((4*C*Sin[c/2 + (d*x)/2])/(5*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2)) - ((A - B + C)*(1 - 2*Sin[c/2 + (d*x)/2]))/(20*(1 + Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2)) + ((A - B + C)*(1 + 2*Sin[c/2 + (d*x)/2]))/(20*(1 - Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(5/2)) + (16*C*(Sin[c/2 + (d*x)/2]/(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2) + (2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]))/15 - ((A - B + C)*(-105*ArcTan[(1 - 2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]] + (4 + 3*Sin[c/2 + (d*x)/2])/((1 - Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) - (19 + 29*Sin[c/2 + (d*x)/2])/((1 - Sin[c/2 + (d*x)/2])*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) - (67*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])/(1 - Sin[c/2 + (d*x)/2])))/30 + ((A - B + C)*(-105*ArcTan[(1 + 2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]] + (4 - 3*Sin[c/2 + (d*x)/2])/((1 + Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) - (19 - 29*Sin[c/2 + (d*x)/2])/((1 + Sin[c/2 + (d*x)/2])*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) - (67*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])/(1 + Sin[c/2 + (d*x)/2])))/30 + ((-A - 3*B + 7*C)*Csc[c/2 + (d*x)/2]^7*(4725*Sin[c/2 + (d*x)/2]^2 - 48825*Sin[c/2 + (d*x)/2]^4 + 210105*Sin[c/2 + (d*x)/2]^6 - 486630*Sin[c/2 + (d*x)/2]^8 + 655812*Sin[c/2 + (d*x)/2]^10 - 710*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^10 - 40*Cos[(c + d*x)/2]^6*HypergeometricPFQ[{2, 2, 2, 9/2}, {1, 1, 11/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^10 - 518760*Sin[c/2 + (d*x)/2]^12 + 1770*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^12 + 226656*Sin[c/2 + (d*x)/2]^14 - 1500*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14 - 42048*Sin[c/2 + (d*x)/2]^16 + 440*Hypergeometric2F1[2, 9/2, 11/2, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^16 + 4725*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 56700*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^2*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 291060*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^4*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 833760*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^6*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 1458000*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^8*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 1598400*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^10*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 1080000*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^12*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] - 414720*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^14*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 69120*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Sin[c/2 + (d*x)/2]^16*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)] + 60*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 9/2}, {1, 11/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^10*(-5 + 4*Sin[c/2 + (d*x)/2]^2)))/(1350*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2)*(-1 + 2*Sin[c/2 + (d*x)/2]^2))))/(d*(a*(1 + Cos[c + d*x]))^(3/2))","C",0
1348,1,1070,233,6.8651496,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{2 \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\frac{1}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}} \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)} \left(\frac{(A+3 B-7 C) \left(-12 \cos ^4\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(2,2,\frac{7}{2};1,\frac{9}{2};-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}\right) \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)-12 \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}\right) \left(3 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-7 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+4\right) \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)+7 \sqrt{-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}} \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3 \left(8 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-20 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+15\right) \left(\left(3-7 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sqrt{-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}-3 \tanh ^{-1}\left(\sqrt{-\frac{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}\right) \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)\right)\right) \csc ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{126 \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{7/2}}-\frac{1}{2} (A-B+C) \left(5 \tan ^{-1}\left(\frac{1-2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}\right)+\frac{3 \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}{1-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)}+\frac{\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1}{\left(1-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}\right)+\frac{1}{2} (A-B+C) \left(5 \tan ^{-1}\left(\frac{2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1}{\sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}\right)+\frac{3 \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}{\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1}+\frac{1-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1\right) \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}\right)+\frac{8 C \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}+\frac{4 C \sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{3/2}}+\frac{(A-B+C) \left(2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1\right)}{12 \left(1-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{3/2}}-\frac{(A-B+C) \left(1-2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{12 \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+1\right) \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{3/2}}\right)}{d (a (\cos (c+d x)+1))^{3/2}}","\frac{(11 A-7 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(7 A-3 B+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(19 A-15 B+3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 a d \sqrt{a \cos (c+d x)+a}}",1,"(2*Cos[c/2 + (d*x)/2]^3*Sqrt[(1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]*((4*C*Sin[c/2 + (d*x)/2])/(3*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) - ((A - B + C)*(1 - 2*Sin[c/2 + (d*x)/2]))/(12*(1 + Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) + ((A - B + C)*(1 + 2*Sin[c/2 + (d*x)/2]))/(12*(1 - Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) + (8*C*Sin[c/2 + (d*x)/2])/(3*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) - ((A - B + C)*(5*ArcTan[(1 - 2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]] + (1 + Sin[c/2 + (d*x)/2])/((1 - Sin[c/2 + (d*x)/2])*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) + (3*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])/(1 - Sin[c/2 + (d*x)/2])))/2 + ((A - B + C)*(5*ArcTan[(1 + 2*Sin[c/2 + (d*x)/2])/Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]] + (1 - Sin[c/2 + (d*x)/2])/((1 + Sin[c/2 + (d*x)/2])*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]) + (3*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2])/(1 + Sin[c/2 + (d*x)/2])))/2 + ((A + 3*B - 7*C)*Csc[c/2 + (d*x)/2]^5*(-12*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 7/2}, {1, 9/2}, -(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]*Sin[c/2 + (d*x)/2]^8 - 12*Hypergeometric2F1[2, 7/2, 9/2, -(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]*Sin[c/2 + (d*x)/2]^8*(4 - 7*Sin[c/2 + (d*x)/2]^2 + 3*Sin[c/2 + (d*x)/2]^4) + 7*Sqrt[-(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]*(1 - 2*Sin[c/2 + (d*x)/2]^2)^3*(15 - 20*Sin[c/2 + (d*x)/2]^2 + 8*Sin[c/2 + (d*x)/2]^4)*((3 - 7*Sin[c/2 + (d*x)/2]^2)*Sqrt[-(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))] - 3*ArcTanh[Sqrt[-(Sin[c/2 + (d*x)/2]^2/(1 - 2*Sin[c/2 + (d*x)/2]^2))]]*(1 - 2*Sin[c/2 + (d*x)/2]^2))))/(126*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2))))/(d*(a*(1 + Cos[c + d*x]))^(3/2))","C",0
1349,1,481,181,5.8045963,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{2 \cos ^3\left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{(A+3 B-7 C) \csc ^3\left(\frac{1}{2} (c+d x)\right) \left(5 (4 \cos (c+d x)+\cos (2 (c+d x))+1) \left(-\cos (c+d x)+\cos (c+d x) \sqrt{2-2 \sec (c+d x)} \tanh ^{-1}\left(\sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right) (-\sec (c+d x))}\right)+1\right)-2 \sin ^4\left(\frac{1}{2} (c+d x)\right) \sin (c+d x) \tan (c+d x) \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};-\sec (c+d x) \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}{40 \cos ^{\frac{3}{2}}(c+d x)}+\frac{(A-B+C) \left(2 \sin \left(\frac{1}{2} (c+d x)\right)-1\right)}{4 \sqrt{\cos (c+d x)} \left(\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)\right)^2}-\frac{(A-B+C) \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+1\right)}{4 \left(\sin \left(\frac{1}{2} (c+d x)\right)-1\right) \sqrt{\cos (c+d x)}}-\frac{(A-B+C) \sqrt{\cos (c+d x)}}{\sin \left(\frac{1}{2} (c+d x)\right)-1}-\frac{(A-B+C) \sqrt{\cos (c+d x)}}{\sin \left(\frac{1}{2} (c+d x)\right)+1}+\frac{3}{2} (A-B+C) \tan ^{-1}\left(\frac{1-2 \sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)-\frac{3}{2} (A-B+C) \tan ^{-1}\left(\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)+1}{\sqrt{\cos (c+d x)}}\right)+\frac{4 C \sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)}{d (a (\cos (c+d x)+1))^{3/2}}","-\frac{(7 A-3 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(2*Cos[(c + d*x)/2]^3*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((3*(A - B + C)*ArcTan[(1 - 2*Sin[(c + d*x)/2])/Sqrt[Cos[c + d*x]]])/2 - (3*(A - B + C)*ArcTan[(1 + 2*Sin[(c + d*x)/2])/Sqrt[Cos[c + d*x]]])/2 - ((A - B + C)*Sqrt[Cos[c + d*x]])/(-1 + Sin[(c + d*x)/2]) + (4*C*Sin[(c + d*x)/2])/Sqrt[Cos[c + d*x]] - ((A - B + C)*Sqrt[Cos[c + d*x]])/(1 + Sin[(c + d*x)/2]) + ((A - B + C)*(-1 + 2*Sin[(c + d*x)/2]))/(4*Sqrt[Cos[c + d*x]]*(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^2) - ((A - B + C)*(1 + 2*Sin[(c + d*x)/2]))/(4*Sqrt[Cos[c + d*x]]*(-1 + Sin[(c + d*x)/2])) + ((A + 3*B - 7*C)*Csc[(c + d*x)/2]^3*(5*(1 + 4*Cos[c + d*x] + Cos[2*(c + d*x)])*(1 - Cos[c + d*x] + ArcTanh[Sqrt[-(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]]*Cos[c + d*x]*Sqrt[2 - 2*Sec[c + d*x]]) - 2*Hypergeometric2F1[2, 5/2, 7/2, -(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^4*Sin[c + d*x]*Tan[c + d*x]))/(40*Cos[c + d*x]^(3/2))))/(d*(a*(1 + Cos[c + d*x]))^(3/2))","C",0
1350,1,16018,189,27.7798549,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^(3/2),x]","\text{Result too large to show}","\frac{(3 A+B-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A-B+C) \sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"Result too large to show","C",0
1351,1,16833,242,27.904782,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]),x]","\text{Result too large to show}","\frac{(A-5 B+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(2 B-3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A-B+C) \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B+3 C) \sin (c+d x)}{2 a d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"Result too large to show","C",0
1352,1,17654,300,28.2086065,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)),x]","\text{Result too large to show}","\frac{(8 A-12 B+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{(5 A-9 B+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(A-B+2 C) \sin (c+d x)}{2 a d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x)}{2 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(2 A-6 B+7 C) \sin (c+d x)}{4 a d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"Result too large to show","C",0
1353,1,7162,333,27.8013127,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x])^(5/2),x]","\text{Result too large to show}","-\frac{(283 A-163 B+75 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(157 A-85 B+45 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{80 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(787 A-475 B+195 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{240 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(2671 A-1495 B+735 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{240 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(21 A-13 B+5 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"Result too large to show","C",0
1354,1,7114,281,26.0308118,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{(163 A-75 B+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(95 A-39 B+15 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{48 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(299 A-147 B+27 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{48 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(17 A-9 B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"Result too large to show","C",0
1355,1,7100,231,25.5846946,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^(5/2),x]","\text{Result too large to show}","-\frac{(75 A-19 B-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(49 A-9 B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(13 A-5 B-3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"Result too large to show","C",0
1356,1,7093,183,24.8230315,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{(19 A+5 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(9 A-B-7 C) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}",1,"Result too large to show","C",0
1357,1,16090,241,27.9615033,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]),x]","\text{Result too large to show}","\frac{(5 A+3 B-43 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}+\frac{(5 A+3 B-11 C) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"Result too large to show","C",0
1358,1,16906,294,28.2031061,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)),x]","\text{Result too large to show}","\frac{(3 A-43 B+115 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(2 B-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}+\frac{(3 A-11 B+35 C) \sin (c+d x)}{16 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{(A+7 B-15 C) \sin (c+d x)}{16 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}",1,"Result too large to show","C",0
1359,1,17727,352,28.4883368,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)),x]","\text{Result too large to show}","\frac{(8 A-20 B+39 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 a^{5/2} d}-\frac{(43 A-115 B+219 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(7 A-15 B+31 C) \sin (c+d x)}{16 a^2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(11 A-35 B+63 C) \sin (c+d x)}{16 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(3 A-11 B+19 C) \sin (c+d x)}{16 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}",1,"Result too large to show","C",0
1360,1,155,205,2.5460956,"\int (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{\sec ^{\frac{7}{2}}(c+d x) \left(2 \sin (c+d x) (10 a (5 A+7 C) \cos (2 (c+d x))+110 a A+70 a C+21 b (13 A+15 C) \cos (c+d x)+63 A b \cos (3 (c+d x))+105 b C \cos (3 (c+d x)))+40 a (5 A+7 C) \cos ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-168 b (3 A+5 C) \cos ^{\frac{7}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{420 d}","\frac{2 a (5 A+7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 a (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{2 b (3 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{2 b (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}",1,"(Sec[c + d*x]^(7/2)*(-168*b*(3*A + 5*C)*Cos[c + d*x]^(7/2)*EllipticE[(c + d*x)/2, 2] + 40*a*(5*A + 7*C)*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] + 2*(110*a*A + 70*a*C + 21*b*(13*A + 15*C)*Cos[c + d*x] + 10*a*(5*A + 7*C)*Cos[2*(c + d*x)] + 63*A*b*Cos[3*(c + d*x)] + 105*b*C*Cos[3*(c + d*x)])*Sin[c + d*x]))/(420*d)","A",1
1361,1,122,172,1.6413378,"\int (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{\sec ^{\frac{5}{2}}(c+d x) \left(2 \sin (c+d x) (3 a ((3 A+5 C) \cos (2 (c+d x))+5 (A+C))+10 A b \cos (c+d x))-12 a (3 A+5 C) \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+20 b (A+3 C) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{30 d}","\frac{2 a (3 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{2 a (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 b (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}",1,"(Sec[c + d*x]^(5/2)*(-12*a*(3*A + 5*C)*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 20*b*(A + 3*C)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 2*(10*A*b*Cos[c + d*x] + 3*a*(5*(A + C) + (3*A + 5*C)*Cos[2*(c + d*x)]))*Sin[c + d*x]))/(30*d)","A",1
1362,1,96,135,0.4587933,"\int (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{\sec ^{\frac{3}{2}}(c+d x) \left(2 A \sin (c+d x) (a+3 b \cos (c+d x))+2 a (A+3 C) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 b (A-C) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d}","\frac{2 a (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{2 b (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A b \sin (c+d x) \sqrt{\sec (c+d x)}}{d}",1,"(Sec[c + d*x]^(3/2)*(-6*b*(A - C)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 2*a*(A + 3*C)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 2*A*(a + 3*b*Cos[c + d*x])*Sin[c + d*x]))/(3*d)","A",1
1363,1,98,135,0.4058787,"\int (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \left(2 \sin (c+d x) (3 a A+b C \cos (c+d x))-6 a (A-C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 b (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d}","-\frac{2 a (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 b (3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 b C \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Sec[c + d*x]]*(-6*a*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*b*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(3*a*A + b*C*Cos[c + d*x])*Sin[c + d*x]))/(3*d)","A",1
1364,1,101,141,0.4825197,"\int (a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(10 a (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (2 (c+d x)) (5 a+3 b \cos (c+d x))+6 b (5 A+3 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d}","\frac{2 a (3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a C \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 b (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b C \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(6*b*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 10*a*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + C*(5*a + 3*b*Cos[c + d*x])*Sin[2*(c + d*x)]))/(15*d)","A",1
1365,1,120,174,0.837193,"\int \frac{(a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) (42 a C \cos (c+d x)+70 A b+15 b C \cos (2 (c+d x))+65 b C)+84 a (5 A+3 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+20 b (7 A+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 a (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a C \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b (7 A+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 b (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 b C \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(84*a*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 20*b*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (70*A*b + 65*b*C + 42*a*C*Cos[c + d*x] + 15*b*C*Cos[2*(c + d*x)])*Sin[2*(c + d*x)]))/(210*d)","A",1
1366,1,141,205,1.2649057,"\int \frac{(a+b \cos (c+d x)) \left(A+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) (5 (84 a A+18 a C \cos (2 (c+d x))+78 a C+7 b C \cos (3 (c+d x)))+7 b (36 A+43 C) \cos (c+d x))+120 a (7 A+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+168 b (9 A+7 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{1260 d}","\frac{2 a (7 A+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a C \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 b (9 A+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b (9 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b C \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(168*b*(9*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 120*a*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (7*b*(36*A + 43*C)*Cos[c + d*x] + 5*(84*a*A + 78*a*C + 18*a*C*Cos[2*(c + d*x)] + 7*b*C*Cos[3*(c + d*x)]))*Sin[2*(c + d*x)]))/(1260*d)","A",1
1367,1,286,292,6.4318619,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{2}{15} \left(7 a^2 A+9 a^2 C+9 A b^2+15 b^2 C\right) \sin (c+d x)+\frac{2}{45} \sec ^2(c+d x) \left(7 a^2 A \sin (c+d x)+9 a^2 C \sin (c+d x)+9 A b^2 \sin (c+d x)\right)+\frac{2}{9} a^2 A \tan (c+d x) \sec ^3(c+d x)+\frac{4}{21} \sec (c+d x) (5 a A b \sin (c+d x)+7 a b C \sin (c+d x))+\frac{4}{7} a A b \tan (c+d x) \sec ^2(c+d x)\right)}{d}+\frac{\frac{2 \left(-49 a^2 A-63 a^2 C-63 A b^2-105 b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}+2 (50 a A b+70 a b C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{105 d}","\frac{2 \left(a^2 (7 A+9 C)+4 A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{45 d}+\frac{2 \left(a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}-\frac{2 \left(a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a b (5 A+7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{4 a b (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{8 a A b \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}",1,"((2*(-49*a^2*A - 63*A*b^2 - 63*a^2*C - 105*b^2*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(50*a*A*b + 70*a*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(105*d) + (Sqrt[Sec[c + d*x]]*((2*(7*a^2*A + 9*A*b^2 + 9*a^2*C + 15*b^2*C)*Sin[c + d*x])/15 + (2*Sec[c + d*x]^2*(7*a^2*A*Sin[c + d*x] + 9*A*b^2*Sin[c + d*x] + 9*a^2*C*Sin[c + d*x]))/45 + (4*Sec[c + d*x]*(5*a*A*b*Sin[c + d*x] + 7*a*b*C*Sin[c + d*x]))/21 + (4*a*A*b*Sec[c + d*x]^2*Tan[c + d*x])/7 + (2*a^2*A*Sec[c + d*x]^3*Tan[c + d*x])/9))/d","A",1
1368,1,218,243,1.3770836,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 \sec ^{\frac{7}{2}}(c+d x) \left(5 \left(a^2 (5 A+7 C)+7 b^2 (A+3 C)\right) \cos ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+15 a^2 A \sin (c+d x)+25 a^2 A \sin (c+d x) \cos ^2(c+d x)+35 a^2 C \sin (c+d x) \cos ^2(c+d x)-42 a b (3 A+5 C) \cos ^{\frac{7}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+21 a A b \sin (2 (c+d x))+126 a A b \sin (c+d x) \cos ^3(c+d x)+210 a b C \sin (c+d x) \cos ^3(c+d x)+35 A b^2 \sin (c+d x) \cos ^2(c+d x)\right)}{105 d}","\frac{2 \left(a^2 (5 A+7 C)+4 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 \left(a^2 (5 A+7 C)+7 b^2 (A+3 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a b (3 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{4 a b (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 a A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}{7 d}",1,"(2*Sec[c + d*x]^(7/2)*(-42*a*b*(3*A + 5*C)*Cos[c + d*x]^(7/2)*EllipticE[(c + d*x)/2, 2] + 5*(7*b^2*(A + 3*C) + a^2*(5*A + 7*C))*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] + 15*a^2*A*Sin[c + d*x] + 25*a^2*A*Cos[c + d*x]^2*Sin[c + d*x] + 35*A*b^2*Cos[c + d*x]^2*Sin[c + d*x] + 35*a^2*C*Cos[c + d*x]^2*Sin[c + d*x] + 126*a*A*b*Cos[c + d*x]^3*Sin[c + d*x] + 210*a*b*C*Cos[c + d*x]^3*Sin[c + d*x] + 21*a*A*b*Sin[2*(c + d*x)]))/(105*d)","A",1
1369,1,147,209,2.1840029,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{2 \sec ^{\frac{5}{2}}(c+d x) \left(-3 \left(a^2 (3 A+5 C)+5 b^2 (A-C)\right) \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 \sin (c+d x) \left(\left(a^2 (3 A+5 C)+5 A b^2\right) \cos ^2(c+d x)+a^2 A\right)+10 a b (A+3 C) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+5 a A b \sin (2 (c+d x))\right)}{15 d}","\frac{2 \left(a^2 (3 A+5 C)+4 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{2 \left(a^2 (3 A+5 C)+5 b^2 (A-C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a b (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{8 a A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{5 d}",1,"(2*Sec[c + d*x]^(5/2)*(-3*(5*b^2*(A - C) + a^2*(3*A + 5*C))*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 10*a*b*(A + 3*C)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 3*(a^2*A + (5*A*b^2 + a^2*(3*A + 5*C))*Cos[c + d*x]^2)*Sin[c + d*x] + 5*a*A*b*Sin[2*(c + d*x)]))/(15*d)","A",1
1370,1,133,194,1.611804,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(2 \left(a^2 (A+3 C)+b^2 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{\sin (c+d x) \left(2 a^2 A+12 a A b \cos (c+d x)+b^2 C \cos (2 (c+d x))+b^2 C\right)}{\cos ^{\frac{3}{2}}(c+d x)}-12 a b (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d}","\frac{2 \left(a^2 (A+3 C)+b^2 (3 A+C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a b (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{8 a A b \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{3 d}-\frac{2 b^2 (A-C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-12*a*b*(A - C)*EllipticE[(c + d*x)/2, 2] + 2*(b^2*(3*A + C) + a^2*(A + 3*C))*EllipticF[(c + d*x)/2, 2] + ((2*a^2*A + b^2*C + 12*a*A*b*Cos[c + d*x] + b^2*C*Cos[2*(c + d*x)])*Sin[c + d*x])/Cos[c + d*x]^(3/2)))/(3*d)","A",1
1371,1,139,206,1.1087302,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(12 \left(b^2 (5 A+3 C)-5 a^2 (A-C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{2 \sin (c+d x) \left(30 a^2 A+20 a b C \cos (c+d x)+3 b^2 C \cos (2 (c+d x))+3 b^2 C\right)}{\sqrt{\cos (c+d x)}}+40 a b (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{30 d}","-\frac{2 \left(5 a^2 (A-C)-b^2 (5 A+3 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{4 a b (3 A-C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{4 a b (3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}{d}-\frac{2 b^2 (5 A-C) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(12*(-5*a^2*(A - C) + b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2] + 40*a*b*(3*A + C)*EllipticF[(c + d*x)/2, 2] + (2*(30*a^2*A + 3*b^2*C + 20*a*b*C*Cos[c + d*x] + 3*b^2*C*Cos[2*(c + d*x)])*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(30*d)","A",1
1372,1,148,211,1.1115291,"\int (a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) \left(70 a^2 C+84 a b C \cos (c+d x)+70 A b^2+15 b^2 C \cos (2 (c+d x))+65 b^2 C\right)+20 \left(7 a^2 (3 A+C)+b^2 (7 A+5 C)\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+168 a b (5 A+3 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 \left(4 a^2 C+b^2 (7 A+5 C)\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left(7 a^2 (3 A+C)+b^2 (7 A+5 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a b (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 a b C \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Sec[c + d*x]]*(168*a*b*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 20*(7*a^2*(3*A + C) + b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (70*A*b^2 + 70*a^2*C + 65*b^2*C + 84*a*b*C*Cos[c + d*x] + 15*b^2*C*Cos[2*(c + d*x)])*Sin[2*(c + d*x)]))/(210*d)","A",1
1373,1,170,245,1.57435,"\int \frac{(a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) \left(7 \left(36 a^2 C+36 A b^2+43 b^2 C\right) \cos (c+d x)+5 b (168 a A+36 a C \cos (2 (c+d x))+156 a C+7 b C \cos (3 (c+d x)))\right)+168 \left(3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+240 a b (7 A+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{1260 d}","\frac{2 \left(4 a^2 C+b^2 (9 A+7 C)\right) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a b (7 A+5 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a b (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{8 a b C \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(168*(3*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 240*a*b*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (7*(36*A*b^2 + 36*a^2*C + 43*b^2*C)*Cos[c + d*x] + 5*b*(168*a*A + 156*a*C + 36*a*C*Cos[2*(c + d*x)] + 7*b*C*Cos[3*(c + d*x)]))*Sin[2*(c + d*x)]))/(1260*d)","A",1
1374,1,209,294,2.4157881,"\int \frac{(a+b \cos (c+d x))^2 \left(A+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \left(2 \sin (2 (c+d x)) \left(5 \left(36 \left(11 a^2 C+11 A b^2+16 b^2 C\right) \cos (2 (c+d x))+132 a^2 (14 A+13 C)+308 a b C \cos (3 (c+d x))+3 b^2 (572 A+531 C)+63 b^2 C \cos (4 (c+d x))\right)+308 a b (36 A+43 C) \cos (c+d x)\right)+480 \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+14784 a b (9 A+7 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{55440 d}","\frac{2 \left(4 a^2 C+b^2 (11 A+9 C)\right) \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a b (9 A+7 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a b (9 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{8 a b C \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^2}{11 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(14784*a*b*(9*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 480*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(308*a*b*(36*A + 43*C)*Cos[c + d*x] + 5*(132*a^2*(14*A + 13*C) + 3*b^2*(572*A + 531*C) + 36*(11*A*b^2 + 11*a^2*C + 16*b^2*C)*Cos[2*(c + d*x)] + 308*a*b*C*Cos[3*(c + d*x)] + 63*b^2*C*Cos[4*(c + d*x)]))*Sin[2*(c + d*x)]))/(55440*d)","A",1
1375,1,324,333,6.5700834,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{2}{45} \sec ^2(c+d x) \left(7 a^3 A \sin (c+d x)+9 a^3 C \sin (c+d x)+27 a A b^2 \sin (c+d x)\right)+\frac{2}{9} a^3 A \tan (c+d x) \sec ^3(c+d x)+\frac{2}{21} \sec (c+d x) \left(15 a^2 A b \sin (c+d x)+21 a^2 b C \sin (c+d x)+7 A b^3 \sin (c+d x)\right)+\frac{2}{15} a \left(7 a^2 A+9 a^2 C+27 A b^2+45 b^2 C\right) \sin (c+d x)+\frac{6}{7} a^2 A b \tan (c+d x) \sec ^2(c+d x)\right)}{d}+\frac{\frac{2 \left(-49 a^3 A-63 a^3 C-189 a A b^2-315 a b^2 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}+2 \left(75 a^2 A b+105 a^2 b C+35 A b^3+105 b^3 C\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{105 d}","\frac{2 a \left(7 a^2 (7 A+9 C)+24 A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{315 d}+\frac{2 b \left(9 a^2 (5 A+7 C)+8 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{63 d}+\frac{2 a \left(a^2 (7 A+9 C)+9 b^2 (3 A+5 C)\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 b \left(3 a^2 (5 A+7 C)+7 b^2 (A+3 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a \left(a^2 (7 A+9 C)+9 b^2 (3 A+5 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^3}{9 d}+\frac{4 A b \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}{21 d}",1,"((2*(-49*a^3*A - 189*a*A*b^2 - 63*a^3*C - 315*a*b^2*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(75*a^2*A*b + 35*A*b^3 + 105*a^2*b*C + 105*b^3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(105*d) + (Sqrt[Sec[c + d*x]]*((2*a*(7*a^2*A + 27*A*b^2 + 9*a^2*C + 45*b^2*C)*Sin[c + d*x])/15 + (2*Sec[c + d*x]^2*(7*a^3*A*Sin[c + d*x] + 27*a*A*b^2*Sin[c + d*x] + 9*a^3*C*Sin[c + d*x]))/45 + (2*Sec[c + d*x]*(15*a^2*A*b*Sin[c + d*x] + 7*A*b^3*Sin[c + d*x] + 21*a^2*b*C*Sin[c + d*x]))/21 + (6*a^2*A*b*Sec[c + d*x]^2*Tan[c + d*x])/7 + (2*a^3*A*Sec[c + d*x]^3*Tan[c + d*x])/9))/d","A",1
1376,1,261,283,2.249075,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{\sec ^{\frac{7}{2}}(c+d x) \left(30 a^3 A \sin (c+d x)+50 a^3 A \sin (c+d x) \cos ^2(c+d x)+70 a^3 C \sin (c+d x) \cos ^2(c+d x)+10 a \left(a^2 (5 A+7 C)+21 b^2 (A+3 C)\right) \cos ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-42 b \left(3 a^2 (3 A+5 C)+5 b^2 (A-C)\right) \cos ^{\frac{7}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+63 a^2 A b \sin (2 (c+d x))+378 a^2 A b \sin (c+d x) \cos ^3(c+d x)+630 a^2 b C \sin (c+d x) \cos ^3(c+d x)+210 a A b^2 \sin (c+d x) \cos ^2(c+d x)+210 A b^3 \sin (c+d x) \cos ^3(c+d x)\right)}{105 d}","\frac{2 a \left(5 a^2 (5 A+7 C)+24 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{6 b \left(7 a^2 (3 A+5 C)+8 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{35 d}+\frac{2 a \left(a^2 (5 A+7 C)+21 b^2 (A+3 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 b \left(3 a^2 (3 A+5 C)+5 b^2 (A-C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^3}{7 d}+\frac{12 A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{35 d}",1,"(Sec[c + d*x]^(7/2)*(-42*b*(5*b^2*(A - C) + 3*a^2*(3*A + 5*C))*Cos[c + d*x]^(7/2)*EllipticE[(c + d*x)/2, 2] + 10*a*(21*b^2*(A + 3*C) + a^2*(5*A + 7*C))*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] + 30*a^3*A*Sin[c + d*x] + 50*a^3*A*Cos[c + d*x]^2*Sin[c + d*x] + 210*a*A*b^2*Cos[c + d*x]^2*Sin[c + d*x] + 70*a^3*C*Cos[c + d*x]^2*Sin[c + d*x] + 378*a^2*A*b*Cos[c + d*x]^3*Sin[c + d*x] + 210*A*b^3*Cos[c + d*x]^3*Sin[c + d*x] + 630*a^2*b*C*Cos[c + d*x]^3*Sin[c + d*x] + 63*a^2*A*b*Sin[2*(c + d*x)]))/(105*d)","A",1
1377,1,216,269,2.1191779,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{\sec ^{\frac{5}{2}}(c+d x) \left(6 a^3 A \sin (c+d x)+18 a^3 A \sin (c+d x) \cos ^2(c+d x)+30 a^3 C \sin (c+d x) \cos ^2(c+d x)+10 b \left(3 a^2 (A+3 C)+b^2 (3 A+C)\right) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 a \left(a^2 (3 A+5 C)+15 b^2 (A-C)\right) \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+15 a^2 A b \sin (2 (c+d x))+90 a A b^2 \sin (c+d x) \cos ^2(c+d x)+10 b^3 C \sin (c+d x) \cos ^3(c+d x)\right)}{15 d}","\frac{2 a \left(a^2 (3 A+5 C)+8 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 b \left(3 a^2 (A+3 C)+b^2 (3 A+C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a \left(a^2 (3 A+5 C)+15 b^2 (A-C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{5 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3}{5 d}-\frac{2 b^3 (9 A-5 C) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}",1,"(Sec[c + d*x]^(5/2)*(-6*a*(15*b^2*(A - C) + a^2*(3*A + 5*C))*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 10*b*(b^2*(3*A + C) + 3*a^2*(A + 3*C))*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 6*a^3*A*Sin[c + d*x] + 18*a^3*A*Cos[c + d*x]^2*Sin[c + d*x] + 90*a*A*b^2*Cos[c + d*x]^2*Sin[c + d*x] + 30*a^3*C*Cos[c + d*x]^2*Sin[c + d*x] + 10*b^3*C*Cos[c + d*x]^3*Sin[c + d*x] + 15*a^2*A*b*Sin[2*(c + d*x)]))/(15*d)","A",1
1378,1,179,258,2.6628123,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(20 a \left(a^2 (A+3 C)+3 b^2 (3 A+C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+12 b \left(b^2 (5 A+3 C)-15 a^2 (A-C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{\sin (c+d x) \left(20 a^3 A+9 b \left(20 a^2 A+b^2 C\right) \cos (c+d x)+30 a b^2 C \cos (2 (c+d x))+30 a b^2 C+3 b^3 C \cos (3 (c+d x))\right)}{\cos ^{\frac{3}{2}}(c+d x)}\right)}{30 d}","\frac{2 a \left(a^2 (A+3 C)+3 b^2 (3 A+C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 b \left(15 a^2 (A-C)-b^2 (5 A+3 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{2 a b^2 (5 A-C) \sin (c+d x)}{d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{3 d}+\frac{4 A b \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}{d}-\frac{2 b^3 (35 A-3 C) \sin (c+d x)}{15 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(12*b*(-15*a^2*(A - C) + b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2] + 20*a*(3*b^2*(3*A + C) + a^2*(A + 3*C))*EllipticF[(c + d*x)/2, 2] + ((20*a^3*A + 30*a*b^2*C + 9*b*(20*a^2*A + b^2*C)*Cos[c + d*x] + 30*a*b^2*C*Cos[2*(c + d*x)] + 3*b^3*C*Cos[3*(c + d*x)])*Sin[c + d*x])/Cos[c + d*x]^(3/2)))/(30*d)","A",1
1379,1,193,284,2.0487472,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \left(40 b \left(21 a^2 (3 A+C)+b^2 (7 A+5 C)\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-168 a \left(5 a^2 (A-C)-3 b^2 (5 A+3 C)\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) \left(3 \left(140 a^3 A+42 a b^2 C \cos (2 (c+d x))+42 a b^2 C+5 b^3 C \cos (3 (c+d x))\right)+5 b \left(84 a^2 C+28 A b^2+29 b^2 C\right) \cos (c+d x)\right)\right)}{420 d}","-\frac{2 b \left(6 a^2 (7 A-3 C)-b^2 (7 A+5 C)\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 b \left(21 a^2 (3 A+C)+b^2 (7 A+5 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a \left(5 a^2 (A-C)-3 b^2 (5 A+3 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{2 a b^2 (35 A-11 C) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b (7 A-C) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^3}{d}",1,"(Sqrt[Sec[c + d*x]]*(-168*a*(5*a^2*(A - C) - 3*b^2*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 40*b*(21*a^2*(3*A + C) + b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(5*b*(28*A*b^2 + 84*a^2*C + 29*b^2*C)*Cos[c + d*x] + 3*(140*a^3*A + 42*a*b^2*C + 42*a*b^2*C*Cos[2*(c + d*x)] + 5*b^3*C*Cos[3*(c + d*x)]))*Sin[c + d*x]))/(420*d)","A",1
1380,1,203,285,1.7100836,"\int (a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(120 a \left(7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+168 b \left(9 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (2 (c+d x)) \left(5 \left(84 a^3 C+252 a A b^2+54 a b^2 C \cos (2 (c+d x))+234 a b^2 C+7 b^3 C \cos (3 (c+d x))\right)+7 b \left(108 a^2 C+36 A b^2+43 b^2 C\right) \cos (c+d x)\right)\right)}{1260 d}","\frac{2 b \left(24 a^2 C+7 b^2 (9 A+7 C)\right) \sin (c+d x)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left(8 a^2 C+63 A b^2+45 b^2 C\right) \sin (c+d x)}{63 d \sqrt{\sec (c+d x)}}+\frac{2 a \left(7 a^2 (3 A+C)+3 b^2 (7 A+5 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 b \left(9 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \sqrt{\sec (c+d x)}}+\frac{4 a C \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Sec[c + d*x]]*(168*b*(9*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 120*a*(7*a^2*(3*A + C) + 3*b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (7*b*(36*A*b^2 + 108*a^2*C + 43*b^2*C)*Cos[c + d*x] + 5*(252*a*A*b^2 + 84*a^3*C + 234*a*b^2*C + 54*a*b^2*C*Cos[2*(c + d*x)] + 7*b^3*C*Cos[3*(c + d*x)]))*Sin[2*(c + d*x)]))/(1260*d)","A",1
1381,1,236,335,2.4189533,"\int \frac{(a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) \left(154 a \left(12 a^2 C+36 A b^2+43 b^2 C\right) \cos (c+d x)+5 b \left(12 \left(33 a^2 C+11 A b^2+16 b^2 C\right) \cos (2 (c+d x))+1848 a^2 A+1716 a^2 C+154 a b C \cos (3 (c+d x))+572 A b^2+21 b^2 C \cos (4 (c+d x))+531 b^2 C\right)\right)+80 b \left(33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3696 a \left(a^2 (5 A+3 C)+b^2 (9 A+7 C)\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{9240 d}","\frac{2 a \left(8 a^2 C+99 A b^2+77 b^2 C\right) \sin (c+d x)}{165 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left(8 a^2 C+3 b^2 (11 A+9 C)\right) \sin (c+d x)}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 b \left(33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 b \left(33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 a \left(a^2 (5 A+3 C)+b^2 (9 A+7 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^3}{11 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a C \sin (c+d x) (a+b \cos (c+d x))^2}{33 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(3696*a*(a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 80*b*(33*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (154*a*(36*A*b^2 + 12*a^2*C + 43*b^2*C)*Cos[c + d*x] + 5*b*(1848*a^2*A + 572*A*b^2 + 1716*a^2*C + 531*b^2*C + 12*(11*A*b^2 + 33*a^2*C + 16*b^2*C)*Cos[2*(c + d*x)] + 154*a*b*C*Cos[3*(c + d*x)] + 21*b^2*C*Cos[4*(c + d*x)]))*Sin[2*(c + d*x)]))/(9240*d)","A",1
1382,1,276,386,2.6392987,"\int \frac{(a+b \cos (c+d x))^3 \left(A+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \left(6240 a \left(11 a^2 (7 A+5 C)+15 b^2 (11 A+9 C)\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+7392 b \left(39 a^2 (9 A+7 C)+7 b^2 (13 A+11 C)\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (2 (c+d x)) \left(154 b \left(78 a^2 (36 A+43 C)+b^2 (1118 A+1171 C)\right) \cos (c+d x)+5 \left(3432 a^3 (14 A+13 C)+77 \left(156 a^2 b C+52 A b^3+89 b^3 C\right) \cos (3 (c+d x))+936 a \left(11 a^2 C+33 A b^2+48 b^2 C\right) \cos (2 (c+d x))+234 a b^2 (572 A+531 C)+4914 a b^2 C \cos (4 (c+d x))+693 b^3 C \cos (5 (c+d x))\right)\right)\right)}{720720 d}","\frac{2 b \left(39 a^2 (9 A+7 C)+7 b^2 (13 A+11 C)\right) \sin (c+d x)}{585 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{6 a \left(8 a^2 C+143 A b^2+117 b^2 C\right) \sin (c+d x)}{1001 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 b \left(24 a^2 C+11 b^2 (13 A+11 C)\right) \sin (c+d x)}{1287 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a \left(11 a^2 (7 A+5 C)+15 b^2 (11 A+9 C)\right) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 a \left(11 a^2 (7 A+5 C)+15 b^2 (11 A+9 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 b \left(39 a^2 (9 A+7 C)+7 b^2 (13 A+11 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{195 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^3}{13 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{12 a C \sin (c+d x) (a+b \cos (c+d x))^2}{143 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(7392*b*(39*a^2*(9*A + 7*C) + 7*b^2*(13*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 6240*a*(11*a^2*(7*A + 5*C) + 15*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (154*b*(78*a^2*(36*A + 43*C) + b^2*(1118*A + 1171*C))*Cos[c + d*x] + 5*(3432*a^3*(14*A + 13*C) + 234*a*b^2*(572*A + 531*C) + 936*a*(33*A*b^2 + 11*a^2*C + 48*b^2*C)*Cos[2*(c + d*x)] + 77*(52*A*b^3 + 156*a^2*b*C + 89*b^3*C)*Cos[3*(c + d*x)] + 4914*a*b^2*C*Cos[4*(c + d*x)] + 693*b^3*C*Cos[5*(c + d*x)]))*Sin[2*(c + d*x)]))/(720720*d)","A",1
1383,1,425,417,6.8847033,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{13}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{2}{11} a^4 A \tan (c+d x) \sec ^4(c+d x)+\frac{8}{45} \sec ^2(c+d x) \left(7 a^3 A b \sin (c+d x)+9 a^3 b C \sin (c+d x)+9 a A b^3 \sin (c+d x)\right)+\frac{8}{9} a^3 A b \tan (c+d x) \sec ^3(c+d x)+\frac{8}{15} a b \left(7 a^2 A+9 a^2 C+9 A b^2+15 b^2 C\right) \sin (c+d x)+\frac{2}{77} \sec ^3(c+d x) \left(9 a^4 A \sin (c+d x)+11 a^4 C \sin (c+d x)+66 a^2 A b^2 \sin (c+d x)\right)+\frac{2}{231} \sec (c+d x) \left(45 a^4 A \sin (c+d x)+55 a^4 C \sin (c+d x)+330 a^2 A b^2 \sin (c+d x)+462 a^2 b^2 C \sin (c+d x)+77 A b^4 \sin (c+d x)\right)\right)}{d}+\frac{\frac{2 \left(-2156 a^3 A b-2772 a^3 b C-2772 a A b^3-4620 a b^3 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}+2 \left(225 a^4 A+275 a^4 C+1650 a^2 A b^2+2310 a^2 b^2 C+385 A b^4+1155 b^4 C\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{1155 d}","\frac{4 a b \left(a^2 (673 A+891 C)+96 A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{3465 d}+\frac{8 a b \left(a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 \left(3 a^2 (9 A+11 C)+16 A b^2\right) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}{231 d}-\frac{8 a b \left(a^2 (7 A+9 C)+3 b^2 (3 A+5 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)+64 A b^4\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{693 d}+\frac{2 \left(5 a^4 (9 A+11 C)+66 a^2 b^2 (5 A+7 C)+77 b^4 (A+3 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a+b \cos (c+d x))^4}{11 d}+\frac{16 A b \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^3}{99 d}",1,"((2*(-2156*a^3*A*b - 2772*a*A*b^3 - 2772*a^3*b*C - 4620*a*b^3*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(225*a^4*A + 1650*a^2*A*b^2 + 385*A*b^4 + 275*a^4*C + 2310*a^2*b^2*C + 1155*b^4*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(1155*d) + (Sqrt[Sec[c + d*x]]*((8*a*b*(7*a^2*A + 9*A*b^2 + 9*a^2*C + 15*b^2*C)*Sin[c + d*x])/15 + (2*Sec[c + d*x]^3*(9*a^4*A*Sin[c + d*x] + 66*a^2*A*b^2*Sin[c + d*x] + 11*a^4*C*Sin[c + d*x]))/77 + (8*Sec[c + d*x]^2*(7*a^3*A*b*Sin[c + d*x] + 9*a*A*b^3*Sin[c + d*x] + 9*a^3*b*C*Sin[c + d*x]))/45 + (2*Sec[c + d*x]*(45*a^4*A*Sin[c + d*x] + 330*a^2*A*b^2*Sin[c + d*x] + 77*A*b^4*Sin[c + d*x] + 55*a^4*C*Sin[c + d*x] + 462*a^2*b^2*C*Sin[c + d*x]))/231 + (8*a^3*A*b*Sec[c + d*x]^3*Tan[c + d*x])/9 + (2*a^4*A*Sec[c + d*x]^4*Tan[c + d*x])/11))/d","A",1
1384,1,356,365,6.7531757,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{2}{9} a^4 A \tan (c+d x) \sec ^3(c+d x)+\frac{8}{21} \sec (c+d x) \left(5 a^3 A b \sin (c+d x)+7 a^3 b C \sin (c+d x)+7 a A b^3 \sin (c+d x)\right)+\frac{8}{7} a^3 A b \tan (c+d x) \sec ^2(c+d x)+\frac{2}{45} \sec ^2(c+d x) \left(7 a^4 A \sin (c+d x)+9 a^4 C \sin (c+d x)+54 a^2 A b^2 \sin (c+d x)\right)+\frac{2}{15} \left(7 a^4 A+9 a^4 C+54 a^2 A b^2+90 a^2 b^2 C+15 A b^4\right) \sin (c+d x)\right)}{d}+\frac{2 \left(100 a^3 A b+140 a^3 b C+140 a A b^3+420 a b^3 C\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{2 \left(-49 a^4 A-63 a^4 C-378 a^2 A b^2-630 a^2 b^2 C-105 A b^4+105 b^4 C\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}}{105 d}","\frac{4 a b \left(a^2 (101 A+147 C)+32 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d}+\frac{2 \left(7 a^2 (7 A+9 C)+48 A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{315 d}+\frac{8 a b \left(a^2 (5 A+7 C)+7 b^2 (A+3 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(21 a^4 (7 A+9 C)+7 a^2 b^2 (155 A+261 C)+192 A b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d}-\frac{2 \left(a^4 (7 A+9 C)+18 a^2 b^2 (3 A+5 C)+15 b^4 (A-C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^4}{9 d}+\frac{16 A b \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^3}{63 d}",1,"((2*(-49*a^4*A - 378*a^2*A*b^2 - 105*A*b^4 - 63*a^4*C - 630*a^2*b^2*C + 105*b^4*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(100*a^3*A*b + 140*a*A*b^3 + 140*a^3*b*C + 420*a*b^3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(105*d) + (Sqrt[Sec[c + d*x]]*((2*(7*a^4*A + 54*a^2*A*b^2 + 15*A*b^4 + 9*a^4*C + 90*a^2*b^2*C)*Sin[c + d*x])/15 + (2*Sec[c + d*x]^2*(7*a^4*A*Sin[c + d*x] + 54*a^2*A*b^2*Sin[c + d*x] + 9*a^4*C*Sin[c + d*x]))/45 + (8*Sec[c + d*x]*(5*a^3*A*b*Sin[c + d*x] + 7*a*A*b^3*Sin[c + d*x] + 7*a^3*b*C*Sin[c + d*x]))/21 + (8*a^3*A*b*Sec[c + d*x]^2*Tan[c + d*x])/7 + (2*a^4*A*Sec[c + d*x]^3*Tan[c + d*x])/9))/d","A",1
1385,1,296,356,3.3015546,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 \sec ^{\frac{7}{2}}(c+d x) \left(15 a^4 A \sin (c+d x)+25 a^4 A \sin (c+d x) \cos ^2(c+d x)+35 a^4 C \sin (c+d x) \cos ^2(c+d x)+42 a^3 A b \sin (2 (c+d x))+252 a^3 A b \sin (c+d x) \cos ^3(c+d x)+420 a^3 b C \sin (c+d x) \cos ^3(c+d x)-84 a b \left(a^2 (3 A+5 C)+5 b^2 (A-C)\right) \cos ^{\frac{7}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+210 a^2 A b^2 \sin (c+d x) \cos ^2(c+d x)+5 \left(a^4 (5 A+7 C)+42 a^2 b^2 (A+3 C)+7 b^4 (3 A+C)\right) \cos ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+420 a A b^3 \sin (c+d x) \cos ^3(c+d x)+35 b^4 C \sin (c+d x) \cos ^4(c+d x)\right)}{105 d}","\frac{4 a b \left(a^2 (101 A+175 C)+96 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d}-\frac{2 b^2 \left(5 a^2 (5 A+7 C)+b^2 (87 A-35 C)\right) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{2 \left(5 a^2 (5 A+7 C)+48 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{105 d}-\frac{8 a b \left(a^2 (3 A+5 C)+5 b^2 (A-C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(a^4 (5 A+7 C)+42 a^2 b^2 (A+3 C)+7 b^4 (3 A+C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^4}{7 d}+\frac{16 A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3}{35 d}",1,"(2*Sec[c + d*x]^(7/2)*(-84*a*b*(5*b^2*(A - C) + a^2*(3*A + 5*C))*Cos[c + d*x]^(7/2)*EllipticE[(c + d*x)/2, 2] + 5*(7*b^4*(3*A + C) + 42*a^2*b^2*(A + 3*C) + a^4*(5*A + 7*C))*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] + 15*a^4*A*Sin[c + d*x] + 25*a^4*A*Cos[c + d*x]^2*Sin[c + d*x] + 210*a^2*A*b^2*Cos[c + d*x]^2*Sin[c + d*x] + 35*a^4*C*Cos[c + d*x]^2*Sin[c + d*x] + 252*a^3*A*b*Cos[c + d*x]^3*Sin[c + d*x] + 420*a*A*b^3*Cos[c + d*x]^3*Sin[c + d*x] + 420*a^3*b*C*Cos[c + d*x]^3*Sin[c + d*x] + 35*b^4*C*Cos[c + d*x]^4*Sin[c + d*x] + 42*a^3*A*b*Sin[2*(c + d*x)]))/(105*d)","A",1
1386,1,233,361,3.2740745,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{\sqrt{\sec (c+d x)} \left(36 a^4 A \sin (c+d x)+12 a^4 A \tan (c+d x) \sec (c+d x)+60 a^4 C \sin (c+d x)+80 a^3 A b \tan (c+d x)+80 a b \left(a^2 (A+3 C)+b^2 (3 A+C)\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+360 a^2 A b^2 \sin (c+d x)-12 \left(a^4 (3 A+5 C)+30 a^2 b^2 (A-C)-b^4 (5 A+3 C)\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+40 a b^3 C \sin (2 (c+d x))+3 b^4 C \sin (c+d x)+3 b^4 C \sin (3 (c+d x))\right)}{30 d}","-\frac{2 b^2 \left(3 a^2 (3 A+5 C)+b^2 (59 A-3 C)\right) \sin (c+d x)}{15 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{4 a b \left(3 a^2 (3 A+5 C)+2 b^2 (33 A-5 C)\right) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 \left(a^2 (3 A+5 C)+16 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}{5 d}+\frac{8 a b \left(a^2 (A+3 C)+b^2 (3 A+C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(a^4 (3 A+5 C)+30 a^2 b^2 (A-C)-b^4 (5 A+3 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^4}{5 d}+\frac{16 A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{15 d}",1,"(Sqrt[Sec[c + d*x]]*(-12*(30*a^2*b^2*(A - C) - b^4*(5*A + 3*C) + a^4*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 80*a*b*(b^2*(3*A + C) + a^2*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 36*a^4*A*Sin[c + d*x] + 360*a^2*A*b^2*Sin[c + d*x] + 60*a^4*C*Sin[c + d*x] + 3*b^4*C*Sin[c + d*x] + 40*a*b^3*C*Sin[2*(c + d*x)] + 3*b^4*C*Sin[3*(c + d*x)] + 80*a^3*A*b*Tan[c + d*x] + 12*a^4*A*Sec[c + d*x]*Tan[c + d*x]))/(30*d)","A",1
1387,1,243,340,2.0173355,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-672 a b \left(5 a^2 (A-C)-b^2 (5 A+3 C)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+40 \left(7 a^4 (A+3 C)+42 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{\sin (c+d x) \left(280 a^4 A+168 a b \left(20 a^2 A+3 b^2 C\right) \cos (c+d x)+20 \left(42 a^2 b^2 C+7 A b^4+8 b^4 C\right) \cos (2 (c+d x))+840 a^2 b^2 C+168 a b^3 C \cos (3 (c+d x))+140 A b^4+15 b^4 C \cos (4 (c+d x))+145 b^4 C\right)}{\cos ^{\frac{3}{2}}(c+d x)}\right)}{420 d}","-\frac{2 b^2 \left(3 a^2 (49 A-13 C)-b^2 (7 A+5 C)\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}-\frac{8 a b \left(5 a^2 (A-C)-b^2 (5 A+3 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(7 a^4 (A+3 C)+42 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a b^3 (175 A-27 C) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b^2 (21 A-C) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4}{3 d}+\frac{16 A b \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^3}{3 d}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-672*a*b*(5*a^2*(A - C) - b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2] + 40*(42*a^2*b^2*(3*A + C) + 7*a^4*(A + 3*C) + b^4*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2] + ((280*a^4*A + 140*A*b^4 + 840*a^2*b^2*C + 145*b^4*C + 168*a*b*(20*a^2*A + 3*b^2*C)*Cos[c + d*x] + 20*(7*A*b^4 + 42*a^2*b^2*C + 8*b^4*C)*Cos[2*(c + d*x)] + 168*a*b^3*C*Cos[3*(c + d*x)] + 15*b^4*C*Cos[4*(c + d*x)])*Sin[c + d*x])/Cos[c + d*x]^(3/2)))/(420*d)","A",1
1388,1,252,360,1.7650031,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \left(960 a b \left(7 a^2 (3 A+C)+b^2 (7 A+5 C)\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-336 \left(15 a^4 (A-C)-18 a^2 b^2 (5 A+3 C)-b^4 (9 A+7 C)\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) \left(2520 a^4 A+120 a b \left(28 a^2 C+28 A b^2+29 b^2 C\right) \cos (c+d x)+84 \left(18 a^2 b^2 C+3 A b^4+4 b^4 C\right) \cos (2 (c+d x))+1512 a^2 b^2 C+360 a b^3 C \cos (3 (c+d x))+252 A b^4+35 b^4 C \cos (4 (c+d x))+301 b^4 C\right)\right)}{2520 d}","-\frac{2 b^2 \left(3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right) \sin (c+d x)}{315 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{4 a b \left(a^2 (63 A-31 C)-6 b^2 (7 A+5 C)\right) \sin (c+d x)}{63 d \sqrt{\sec (c+d x)}}+\frac{8 a b \left(7 a^2 (3 A+C)+b^2 (7 A+5 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 \left(15 a^4 (A-C)-18 a^2 b^2 (5 A+3 C)-b^4 (9 A+7 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}-\frac{2 b (9 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \sqrt{\sec (c+d x)}}-\frac{2 a b (21 A-5 C) \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^4}{d}",1,"(Sqrt[Sec[c + d*x]]*(-336*(15*a^4*(A - C) - 18*a^2*b^2*(5*A + 3*C) - b^4*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 960*a*b*(7*a^2*(3*A + C) + b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(2520*a^4*A + 252*A*b^4 + 1512*a^2*b^2*C + 301*b^4*C + 120*a*b*(28*A*b^2 + 28*a^2*C + 29*b^2*C)*Cos[c + d*x] + 84*(3*A*b^4 + 18*a^2*b^2*C + 4*b^4*C)*Cos[2*(c + d*x)] + 360*a*b^3*C*Cos[3*(c + d*x)] + 35*b^4*C*Cos[4*(c + d*x)])*Sin[c + d*x]))/(2520*d)","A",1
1389,1,265,369,1.764722,"\int (a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(29568 a b \left(3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+480 \left(77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (2 (c+d x)) \left(616 a b \left(36 a^2 C+36 A b^2+43 b^2 C\right) \cos (c+d x)+5 \left(1848 a^4 C+792 a^2 b^2 (14 A+13 C)+36 \left(66 a^2 b^2 C+11 A b^4+16 b^4 C\right) \cos (2 (c+d x))+616 a b^3 C \cos (3 (c+d x))+3 b^4 (572 A+531 C)+63 b^4 C \cos (4 (c+d x))\right)\right)\right)}{55440 d}","\frac{4 a b \left(96 a^2 C+891 A b^2+673 b^2 C\right) \sin (c+d x)}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(16 a^2 C+3 b^2 (11 A+9 C)\right) \sin (c+d x) (a+b \cos (c+d x))^2}{231 d \sqrt{\sec (c+d x)}}+\frac{8 a b \left(3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(64 a^4 C+9 a^2 b^2 (143 A+101 C)+15 b^4 (11 A+9 C)\right) \sin (c+d x)}{693 d \sqrt{\sec (c+d x)}}+\frac{2 \left(77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \sqrt{\sec (c+d x)}}+\frac{16 a C \sin (c+d x) (a+b \cos (c+d x))^3}{99 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Sec[c + d*x]]*(29568*a*b*(3*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 480*(77*a^4*(3*A + C) + 66*a^2*b^2*(7*A + 5*C) + 5*b^4*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(616*a*b*(36*A*b^2 + 36*a^2*C + 43*b^2*C)*Cos[c + d*x] + 5*(1848*a^4*C + 792*a^2*b^2*(14*A + 13*C) + 3*b^4*(572*A + 531*C) + 36*(11*A*b^4 + 66*a^2*b^2*C + 16*b^4*C)*Cos[2*(c + d*x)] + 616*a*b^3*C*Cos[3*(c + d*x)] + 63*b^4*C*Cos[4*(c + d*x)]))*Sin[2*(c + d*x)]))/(55440*d)","A",1
1390,1,303,422,2.7784753,"\int \frac{(a+b \cos (c+d x))^4 \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(49920 a b \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+14784 \left(39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (2 (c+d x)) \left(5 b \left(77 \left(312 a^2 b C+52 A b^3+89 b^3 C\right) \cos (3 (c+d x))+3744 a \left(11 a^2 C+11 A b^2+16 b^2 C\right) \cos (2 (c+d x))+312 a \left(44 a^2 (14 A+13 C)+b^2 (572 A+531 C)\right)+6552 a b^2 C \cos (4 (c+d x))+693 b^3 C \cos (5 (c+d x))\right)+154 \left(936 a^4 C+156 a^2 b^2 (36 A+43 C)+b^4 (1118 A+1171 C)\right) \cos (c+d x)\right)\right)}{1441440 d}","\frac{4 a b \left(96 a^2 C+1573 A b^2+1259 b^2 C\right) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a b \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \left(48 a^2 C+11 b^2 (13 A+11 C)\right) \sin (c+d x) (a+b \cos (c+d x))^2}{1287 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 a b \left(11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 \left(192 a^4 C+11 a^2 b^2 (637 A+491 C)+77 b^4 (13 A+11 C)\right) \sin (c+d x)}{6435 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{195 d}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^4}{13 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{16 a C \sin (c+d x) (a+b \cos (c+d x))^3}{143 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(14784*(39*a^4*(5*A + 3*C) + 78*a^2*b^2*(9*A + 7*C) + 7*b^4*(13*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 49920*a*b*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(154*(936*a^4*C + 156*a^2*b^2*(36*A + 43*C) + b^4*(1118*A + 1171*C))*Cos[c + d*x] + 5*b*(312*a*(44*a^2*(14*A + 13*C) + b^2*(572*A + 531*C)) + 3744*a*(11*A*b^2 + 11*a^2*C + 16*b^2*C)*Cos[2*(c + d*x)] + 77*(52*A*b^3 + 312*a^2*b*C + 89*b^3*C)*Cos[3*(c + d*x)] + 6552*a*b^2*C*Cos[4*(c + d*x)] + 693*b^3*C*Cos[5*(c + d*x)]))*Sin[2*(c + d*x)]))/(1441440*d)","A",1
1391,1,642,266,6.8866759,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + b*Cos[c + d*x]),x]","\frac{\sqrt{\sec (c+d x)} \left(-\frac{2 A b \tan (c+d x)}{3 a^2}+\frac{2 \left(3 a^2 A+5 a^2 C+5 A b^2\right) \sin (c+d x)}{5 a^3}+\frac{2 A \tan (c+d x) \sec (c+d x)}{5 a}\right)}{d}-\frac{\frac{2 \left(18 a^3 A+30 a^3 C+40 a A b^2\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{b \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{2 \left(19 a^2 A b+45 a^2 b C+45 A b^3\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{\left(9 a^2 A b+15 a^2 b C+15 A b^3\right) \sin (c+d x) \cos (2 (c+d x)) (a \sec (c+d x)+b) \left(-4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a b^2 \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a+b \cos (c+d x))}}{30 a^3 d}","-\frac{2 A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d}-\frac{2 A b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{2 \left(a^2 (3 A+5 C)+5 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 a^3 d}-\frac{2 \left(a^2 (3 A+5 C)+5 A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^3 d}-\frac{2 b \left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a+b)}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 a d}",1,"-1/30*((2*(19*a^2*A*b + 45*A*b^3 + 45*a^2*b*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(18*a^3*A + 40*a*A*b^2 + 30*a^3*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((9*a^2*A*b + 15*A*b^3 + 15*a^2*b*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(a^3*d) + (Sqrt[Sec[c + d*x]]*((2*(3*a^2*A + 5*A*b^2 + 5*a^2*C)*Sin[c + d*x])/(5*a^3) - (2*A*b*Tan[c + d*x])/(3*a^2) + (2*A*Sec[c + d*x]*Tan[c + d*x])/(5*a)))/d","B",0
1392,1,216,200,2.8889517,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x]),x]","-\frac{\cot (c+d x) \left(-2 \left(a^2 (A+3 C)+3 a A b+3 A b^2\right) \sqrt{-\tan ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-a^2 A \sec ^{\frac{5}{2}}(c+d x)+a^2 A \cos (2 (c+d x)) \sec ^{\frac{5}{2}}(c+d x)+6 a^2 C \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+6 A b^2 \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+6 a A b \sqrt{-\tan ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{3 a^3 d}","\frac{2 \left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a+b)}-\frac{2 A b \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}+\frac{2 A b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}+\frac{2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}",1,"-1/3*(Cot[c + d*x]*(-(a^2*A*Sec[c + d*x]^(5/2)) + a^2*A*Cos[2*(c + d*x)]*Sec[c + d*x]^(5/2) + 6*a*A*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] - 2*(3*a*A*b + 3*A*b^2 + a^2*(A + 3*C))*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] + 6*A*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] + 6*a^2*C*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2]))/(a^3*d)","A",1
1393,1,126,172,1.1967489,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x]),x]","-\frac{2 \cos (2 (c+d x)) \sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sec (c+d x) \left(\left(a^2 C+A b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-A b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+a A b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a^2 b d \left(\sec ^2(c+d x)-2\right)}","-\frac{2 \left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d (a+b)}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(-2*Cos[2*(c + d*x)]*Csc[c + d*x]*(a*A*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1] - A*b*(a + b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] + (A*b^2 + a^2*C)*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*Sec[c + d*x]*Sqrt[-Tan[c + d*x]^2])/(a^2*b*d*(-2 + Sec[c + d*x]^2))","A",1
1394,1,238,145,2.0503668,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{a+b \cos (c+d x)} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x]),x]","\frac{\cot (c+d x) \left(-2 a^2 C \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-2 A b^2 \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b (a C+A b) \sqrt{-\tan ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+a b C \sec ^{\frac{7}{2}}(c+d x)-a b C \sec ^{\frac{3}{2}}(c+d x)+a b C \cos (2 (c+d x)) \sec ^{\frac{7}{2}}(c+d x)-a b C \cos (2 (c+d x)) \sec ^{\frac{3}{2}}(c+d x)-2 a b C \sqrt{-\tan ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a b^2 d}","\frac{2 \left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}-\frac{2 a C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(Cot[c + d*x]*(-(a*b*C*Sec[c + d*x]^(3/2)) - a*b*C*Cos[2*(c + d*x)]*Sec[c + d*x]^(3/2) + a*b*C*Sec[c + d*x]^(7/2) + a*b*C*Cos[2*(c + d*x)]*Sec[c + d*x]^(7/2) - 2*a*b*C*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] + 2*b*(A*b + a*C)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] - 2*A*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] - 2*a^2*C*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2]))/(a*b^2*d)","A",1
1395,1,533,190,6.6818595,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x)) \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]]),x]","\frac{-\frac{3 C \sin (c+d x) \cos (2 (c+d x)) (a \sec (c+d x)+b) \left(-4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{b^2 \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{2 (6 A b+2 b C) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{b \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}-\frac{2 C \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{\left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}}{6 b d}+\frac{C \sin (2 (c+d x)) \sqrt{\sec (c+d x)}}{3 b d}","\frac{2 \left(3 a^2 C+b^2 (3 A+C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^3 d}-\frac{2 a \left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a+b)}-\frac{2 a C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C \sin (c+d x)}{3 b d \sqrt{\sec (c+d x)}}",1,"((-2*C*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/((a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(6*A*b + 2*b*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) - (3*C*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(6*b*d) + (C*Sqrt[Sec[c + d*x]]*Sin[2*(c + d*x)])/(3*b*d)","B",0
1396,1,597,241,6.8284536,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])*Sec[c + d*x]^(3/2)),x]","\frac{\frac{2 \left(5 a^2 C+15 A b^2+9 b^2 C\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{\left(15 a^2 C+15 A b^2+9 b^2 C\right) \sin (c+d x) \cos (2 (c+d x)) (a \sec (c+d x)+b) \left(-4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a b^2 \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{16 a C \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{\left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}}{30 b^2 d}+\frac{\sqrt{\sec (c+d x)} \left(-\frac{a C \sin (2 (c+d x))}{3 b^2}+\frac{C \sin (c+d x)}{10 b}+\frac{C \sin (3 (c+d x))}{10 b}\right)}{d}","-\frac{2 a \left(C \left(3 a^2+b^2\right)+3 A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d}+\frac{2 a^2 \left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a+b)}+\frac{2 \left(5 a^2 C+b^2 (5 A+3 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^3 d}-\frac{2 a C \sin (c+d x)}{3 b^2 d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x)}{5 b d \sec ^{\frac{3}{2}}(c+d x)}",1,"((2*(15*A*b^2 + 5*a^2*C + 9*b^2*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (16*a*C*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/((a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((15*A*b^2 + 15*a^2*C + 9*b^2*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(30*b^2*d) + (Sqrt[Sec[c + d*x]]*((C*Sin[c + d*x])/(10*b) - (a*C*Sin[2*(c + d*x)])/(3*b^2) + (C*Sin[3*(c + d*x)])/(10*b)))/d","B",0
1397,1,657,299,6.9556829,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])*Sec[c + d*x]^(5/2)),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{\left(14 a^2 C+14 A b^2+13 b^2 C\right) \sin (2 (c+d x))}{42 b^3}-\frac{a C \sin (c+d x)}{10 b^2}-\frac{a C \sin (3 (c+d x))}{10 b^2}+\frac{C \sin (4 (c+d x))}{28 b}\right)}{d}-\frac{\frac{2 \left(35 a^3 C+35 a A b^2+13 a b^2 C\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{2 \left(56 a^2 b C-70 A b^3-50 b^3 C\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{b \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{\left(105 a^3 C+105 a A b^2+63 a b^2 C\right) \sin (c+d x) \cos (2 (c+d x)) (a \sec (c+d x)+b) \left(-4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a b^2 \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a+b \cos (c+d x))}}{210 b^3 d}","-\frac{2 a \left(5 a^2 C+5 A b^2+3 b^2 C\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^4 d}+\frac{2 \left(7 a^2 C+b^2 (7 A+5 C)\right) \sin (c+d x)}{21 b^3 d \sqrt{\sec (c+d x)}}+\frac{2 \left(21 a^4 C+7 a^2 b^2 (3 A+C)+b^4 (7 A+5 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b^5 d}-\frac{2 a^3 \left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a+b)}-\frac{2 a C \sin (c+d x)}{5 b^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x)}{7 b d \sec ^{\frac{5}{2}}(c+d x)}",1,"-1/210*((2*(35*a*A*b^2 + 35*a^3*C + 13*a*b^2*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-70*A*b^3 + 56*a^2*b*C - 50*b^3*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((105*a*A*b^2 + 105*a^3*C + 63*a*b^2*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(b^3*d) + (Sqrt[Sec[c + d*x]]*(-1/10*(a*C*Sin[c + d*x])/b^2 + ((14*A*b^2 + 14*a^2*C + 13*b^2*C)*Sin[2*(c + d*x)])/(42*b^3) - (a*C*Sin[3*(c + d*x)])/(10*b^2) + (C*Sin[4*(c + d*x)])/(28*b)))/d","B",0
1398,1,718,396,7.14587,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^2,x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{-a^2 b C \sin (c+d x)-A b^3 \sin (c+d x)}{a^2 \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{2 A \tan (c+d x)}{3 a^2}-\frac{b \left(4 a^2 A-a^2 C-5 A b^2\right) \sin (c+d x)}{a^3 \left(a^2-b^2\right)}\right)}{d}+\frac{\frac{2 \left(-28 a^3 A b+12 a^3 b C+40 a A b^3\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{b \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{\left(-12 a^2 A b^2+3 a^2 b^2 C+15 A b^4\right) \sin (c+d x) \cos (2 (c+d x)) (a \sec (c+d x)+b) \left(-4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a b^2 \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{2 \left(-4 a^4 A-12 a^4 C-44 a^2 A b^2+9 a^2 b^2 C+45 A b^4\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}}{12 a^3 d (b-a) (a+b)}","-\frac{\left(5 A b^2-a^2 (2 A-3 C)\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d \left(a^2-b^2\right)}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\left(5 A b^2-a^2 (2 A-3 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d \left(a^2-b^2\right)}+\frac{b \left(5 A b^2-a^2 (4 A-C)\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^3 d \left(a^2-b^2\right)}-\frac{b \left(5 A b^2-a^2 (4 A-C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \left(a^2-b^2\right)}-\frac{\left(-3 a^4 C-a^2 b^2 (7 A-C)+5 A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}",1,"((2*(-4*a^4*A - 44*a^2*A*b^2 + 45*A*b^4 - 12*a^4*C + 9*a^2*b^2*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-28*a^3*A*b + 40*a*A*b^3 + 12*a^3*b*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-12*a^2*A*b^2 + 15*A*b^4 + 3*a^2*b^2*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(12*a^3*(-a + b)*(a + b)*d) + (Sqrt[Sec[c + d*x]]*(-((b*(4*a^2*A - 5*A*b^2 - a^2*C)*Sin[c + d*x])/(a^3*(a^2 - b^2))) + (-(A*b^3*Sin[c + d*x]) - a^2*b*C*Sin[c + d*x])/(a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])) + (2*A*Tan[c + d*x])/(3*a^2)))/d","A",0
1399,1,676,330,7.0195146,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^2,x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{\left(2 a^2 A-a^2 C-3 A b^2\right) \sin (c+d x)}{a^2 \left(a^2-b^2\right)}+\frac{a^2 C \sin (c+d x)+A b^2 \sin (c+d x)}{a \left(a^2-b^2\right) (a+b \cos (c+d x))}\right)}{d}-\frac{\frac{2 \left(4 a^3 A-4 a^3 C-8 a A b^2\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{b \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{2 \left(10 a^2 A b+a^2 b C-9 A b^3\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{\left(2 a^2 A b-a^2 b C-3 A b^3\right) \sin (c+d x) \cos (2 (c+d x)) (a \sec (c+d x)+b) \left(-4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a b^2 \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a+b \cos (c+d x))}}{4 a^2 d (a-b) (a+b)}","-\frac{\left(3 A b^2-a^2 (2 A-C)\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d \left(a^2-b^2\right)}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d \left(a^2-b^2\right)}+\frac{\left(3 A b^2-a^2 (2 A-C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{\left(a^4 (-C)-a^2 b^2 (5 A+C)+3 A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 b d (a-b) (a+b)^2}",1,"-1/4*((2*(10*a^2*A*b - 9*A*b^3 + a^2*b*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(4*a^3*A - 8*a*A*b^2 - 4*a^3*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((2*a^2*A*b - 3*A*b^3 - a^2*b*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(a^2*(a - b)*(a + b)*d) + (Sqrt[Sec[c + d*x]]*(((2*a^2*A - 3*A*b^2 - a^2*C)*Sin[c + d*x])/(a^2*(a^2 - b^2)) + (A*b^2*Sin[c + d*x] + a^2*C*Sin[c + d*x])/(a*(a^2 - b^2)*(a + b*Cos[c + d*x]))))/d","B",0
1400,1,651,274,6.8562744,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^2} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^2,x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{a b \left(a^2-b^2\right)}+\frac{a^2 C \sin (c+d x)+A b^2 \sin (c+d x)}{b \left(b^2-a^2\right) (a+b \cos (c+d x))}\right)}{d}+\frac{\frac{2 \left(-4 a^2 A-a^2 C+3 A b^2\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos (2 (c+d x)) (a \sec (c+d x)+b) \left(-4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a b^2 \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{2 (4 a A b+4 a b C) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{b \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}}{4 a d (b-a) (a+b)}","\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}-\frac{\left(a^2 (-C)+A b^2+2 b^2 C\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}-\frac{\left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d \left(a^2-b^2\right)}-\frac{\left(a^4 C-3 a^2 b^2 (A+C)+A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b^2 d (a-b) (a+b)^2}",1,"((2*(-4*a^2*A + 3*A*b^2 - a^2*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(4*a*A*b + 4*a*b*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((A*b^2 + a^2*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(4*a*(-a + b)*(a + b)*d) + (Sqrt[Sec[c + d*x]]*(((A*b^2 + a^2*C)*Sin[c + d*x])/(a*b*(a^2 - b^2)) + (A*b^2*Sin[c + d*x] + a^2*C*Sin[c + d*x])/(b*(-a^2 + b^2)*(a + b*Cos[c + d*x]))))/d","B",0
1401,1,657,277,6.8761054,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^2 \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]),x]","\frac{\frac{2 \left(a^2 C-A b^2-2 b^2 C\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{\left(3 a^2 C+A b^2-2 b^2 C\right) \sin (c+d x) \cos (2 (c+d x)) (a \sec (c+d x)+b) \left(-4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a b^2 \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{2 (4 a A b+4 a b C) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{b \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}}{4 b d (a-b) (a+b)}+\frac{\sqrt{\sec (c+d x)} \left(\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{b^2 \left(b^2-a^2\right)}+\frac{a^3 (-C) \sin (c+d x)-a A b^2 \sin (c+d x)}{b^2 \left(b^2-a^2\right) (a+b \cos (c+d x))}\right)}{d}","-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}+\frac{\left(3 a^2 C+A b^2-2 b^2 C\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}+\frac{a \left(-3 a^2 C+A b^2+4 b^2 C\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}-\frac{\left(-3 a^4 C+a^2 b^2 (A+5 C)+A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}",1,"((2*(-(A*b^2) + a^2*C - 2*b^2*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(4*a*A*b + 4*a*b*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((A*b^2 + 3*a^2*C - 2*b^2*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(4*(a - b)*b*(a + b)*d) + (Sqrt[Sec[c + d*x]]*(((A*b^2 + a^2*C)*Sin[c + d*x])/(b^2*(-a^2 + b^2)) + (-(a*A*b^2*Sin[c + d*x]) - a^3*C*Sin[c + d*x])/(b^2*(-a^2 + b^2)*(a + b*Cos[c + d*x]))))/d","B",0
1402,1,692,352,7.0279779,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{a \left(a^2 C+A b^2\right) \sin (c+d x)}{b^3 \left(a^2-b^2\right)}-\frac{a^4 (-C) \sin (c+d x)-a^2 A b^2 \sin (c+d x)}{b^3 \left(b^2-a^2\right) (a+b \cos (c+d x))}+\frac{C \sin (2 (c+d x))}{3 b^2}\right)}{d}+\frac{\frac{2 \left(5 a^3 C-3 a A b^2-8 a b^2 C\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{2 \left(8 a^2 b C+12 A b^3+4 b^3 C\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{b \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{\left(15 a^3 C+3 a A b^2-12 a b^2 C\right) \sin (c+d x) \cos (2 (c+d x)) (a \sec (c+d x)+b) \left(-4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a b^2 \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a+b \cos (c+d x))}}{12 b^2 d (b-a) (a+b)}","\frac{\left(5 a^2 C+3 A b^2-2 b^2 C\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{b d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{a \left(5 a^2 C+A b^2-4 b^2 C\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}+\frac{\left(15 a^4 C+a^2 b^2 (3 A-16 C)-2 b^4 (3 A+C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d \left(a^2-b^2\right)}+\frac{a \left(-5 a^4 C-a^2 b^2 (A-7 C)+3 A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a-b) (a+b)^2}",1,"((2*(-3*a*A*b^2 + 5*a^3*C - 8*a*b^2*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(12*A*b^3 + 8*a^2*b*C + 4*b^3*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((3*a*A*b^2 + 15*a^3*C - 12*a*b^2*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(12*b^2*(-a + b)*(a + b)*d) + (Sqrt[Sec[c + d*x]]*((a*(A*b^2 + a^2*C)*Sin[c + d*x])/(b^3*(a^2 - b^2)) - (-(a^2*A*b^2*Sin[c + d*x]) - a^4*C*Sin[c + d*x])/(b^3*(-a^2 + b^2)*(a + b*Cos[c + d*x])) + (C*Sin[2*(c + d*x)])/(3*b^2)))/d","A",0
1403,1,762,430,7.1708862,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^2 \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)),x]","\frac{\frac{2 \left(56 a^3 b C+60 a A b^3+4 a b^3 C\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{b \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{2 \left(35 a^4 C+15 a^2 A b^2-32 a^2 b^2 C-30 A b^4-18 b^4 C\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{\left(105 a^4 C+45 a^2 A b^2-72 a^2 b^2 C-30 A b^4-18 b^4 C\right) \sin (c+d x) \cos (2 (c+d x)) (a \sec (c+d x)+b) \left(-4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a b^2 \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a+b \cos (c+d x))}}{60 b^3 d (a-b) (a+b)}+\frac{\sqrt{\sec (c+d x)} \left(-\frac{\left(10 a^4 C+10 a^2 A b^2-a^2 b^2 C+b^4 C\right) \sin (c+d x)}{10 b^4 \left(a^2-b^2\right)}-\frac{a^5 C \sin (c+d x)+a^3 A b^2 \sin (c+d x)}{b^4 \left(b^2-a^2\right) (a+b \cos (c+d x))}-\frac{2 a C \sin (2 (c+d x))}{3 b^3}+\frac{C \sin (3 (c+d x))}{10 b^2}\right)}{d}","\frac{\left(7 a^2 C+5 A b^2-2 b^2 C\right) \sin (c+d x)}{5 b^2 d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x)}-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{b d \left(a^2-b^2\right) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{a \left(7 a^2 C+3 A b^2-4 b^2 C\right) \sin (c+d x)}{3 b^3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}+\frac{\left(35 a^4 C+3 a^2 b^2 (5 A-8 C)-2 b^4 (5 A+3 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^4 d \left(a^2-b^2\right)}-\frac{a \left(21 a^4 C+a^2 b^2 (9 A-20 C)-4 b^4 (3 A+C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^5 d \left(a^2-b^2\right)}-\frac{a^2 \left(-7 a^4 C-3 a^2 b^2 (A-3 C)+5 A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a-b) (a+b)^2}",1,"((2*(15*a^2*A*b^2 - 30*A*b^4 + 35*a^4*C - 32*a^2*b^2*C - 18*b^4*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(60*a*A*b^3 + 56*a^3*b*C + 4*a*b^3*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((45*a^2*A*b^2 - 30*A*b^4 + 105*a^4*C - 72*a^2*b^2*C - 18*b^4*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(60*(a - b)*b^3*(a + b)*d) + (Sqrt[Sec[c + d*x]]*(-1/10*((10*a^2*A*b^2 + 10*a^4*C - a^2*b^2*C + b^4*C)*Sin[c + d*x])/(b^4*(a^2 - b^2)) - (a^3*A*b^2*Sin[c + d*x] + a^5*C*Sin[c + d*x])/(b^4*(-a^2 + b^2)*(a + b*Cos[c + d*x])) - (2*a*C*Sin[2*(c + d*x)])/(3*b^3) + (C*Sin[3*(c + d*x)])/(10*b^2)))/d","A",0
1404,1,880,554,7.3797583,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{2 \left(16 A a^6+48 C a^6+328 A b^2 a^4-57 b^2 C a^4-641 A b^4 a^2+27 b^4 C a^2+315 A b^6\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(160 A b a^5-96 b C a^5-512 A b^3 a^3+24 b^3 C a^3+280 A b^5 a\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(105 A b^6-195 a^2 A b^4+9 a^2 C b^4+72 a^4 A b^2-27 a^4 C b^2\right) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left(-4 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a+2 (2 a-b) b F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}+2 b^2 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}}{48 a^4 (a-b)^2 (a+b)^2 d}+\frac{\sqrt{\sec (c+d x)} \left(-\frac{b \left(24 A a^4-9 C a^4-65 A b^2 a^2+3 b^2 C a^2+35 A b^4\right) \sin (c+d x)}{4 a^4 \left(a^2-b^2\right)^2}+\frac{-A \sin (c+d x) b^3-a^2 C \sin (c+d x) b}{2 a^2 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{9 A \sin (c+d x) b^5-15 a^2 A \sin (c+d x) b^3+a^2 C \sin (c+d x) b^3-7 a^4 C \sin (c+d x) b}{4 a^3 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{2 A \tan (c+d x)}{3 a^3}\right)}{d}","\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{b \left(3 a^4 (8 A-3 C)-a^2 b^2 (65 A-3 C)+35 A b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^4 d \left(a^2-b^2\right)^2}-\frac{\left(-5 a^4 C-a^2 b^2 (13 A+C)+7 A b^4\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{b \left(3 a^4 (8 A-3 C)-a^2 b^2 (65 A-3 C)+35 A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{\left(15 a^6 C+3 a^4 b^2 (21 A-2 C)-a^2 b^4 (86 A-3 C)+35 A b^6\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}+\frac{\left(a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)+35 A b^4\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{12 a^3 d \left(a^2-b^2\right)^2}+\frac{\left(a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)+35 A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 a^3 d \left(a^2-b^2\right)^2}",1,"((2*(16*a^6*A + 328*a^4*A*b^2 - 641*a^2*A*b^4 + 315*A*b^6 + 48*a^6*C - 57*a^4*b^2*C + 27*a^2*b^4*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(160*a^5*A*b - 512*a^3*A*b^3 + 280*a*A*b^5 - 96*a^5*b*C + 24*a^3*b^3*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((72*a^4*A*b^2 - 195*a^2*A*b^4 + 105*A*b^6 - 27*a^4*b^2*C + 9*a^2*b^4*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(48*a^4*(a - b)^2*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(-1/4*(b*(24*a^4*A - 65*a^2*A*b^2 + 35*A*b^4 - 9*a^4*C + 3*a^2*b^2*C)*Sin[c + d*x])/(a^4*(a^2 - b^2)^2) + (-(A*b^3*Sin[c + d*x]) - a^2*b*C*Sin[c + d*x])/(2*a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) + (-15*a^2*A*b^3*Sin[c + d*x] + 9*A*b^5*Sin[c + d*x] - 7*a^4*b*C*Sin[c + d*x] + a^2*b^3*C*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) + (2*A*Tan[c + d*x])/(3*a^3)))/d","A",0
1405,1,840,477,7.2480955,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^3,x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{\left(8 A a^4-5 C a^4-29 A b^2 a^2-b^2 C a^2+15 A b^4\right) \sin (c+d x)}{4 a^3 \left(a^2-b^2\right)^2}+\frac{C \sin (c+d x) a^2+A b^2 \sin (c+d x)}{2 a \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{3 C \sin (c+d x) a^4+11 A b^2 \sin (c+d x) a^2+3 b^2 C \sin (c+d x) a^2-5 A b^4 \sin (c+d x)}{4 a^2 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}\right)}{d}-\frac{\frac{2 \left(45 A b^5-95 a^2 A b^3-3 a^2 C b^3+56 a^4 A b+9 a^4 C b\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(16 A a^5-16 C a^5-80 A b^2 a^3-8 b^2 C a^3+40 A b^4 a\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(15 A b^5-29 a^2 A b^3-a^2 C b^3+8 a^4 A b-5 a^4 C b\right) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left(-4 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a+2 (2 a-b) b F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}+2 b^2 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}}{16 a^3 (a-b)^2 (a+b)^2 d}","\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{\left(-3 a^4 C-a^2 b^2 (11 A+3 C)+5 A b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\left(-3 a^4 C-a^2 b^2 (11 A+3 C)+5 A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b d \left(a^2-b^2\right)^2}+\frac{\left(a^4 (8 A-5 C)-a^2 b^2 (29 A+C)+15 A b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(a^4 (8 A-5 C)-a^2 b^2 (29 A+C)+15 A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(3 a^6 C+5 a^4 b^2 (7 A+2 C)-a^2 b^4 (38 A+C)+15 A b^6\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 b d (a-b)^2 (a+b)^3}",1,"-1/16*((2*(56*a^4*A*b - 95*a^2*A*b^3 + 45*A*b^5 + 9*a^4*b*C - 3*a^2*b^3*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(16*a^5*A - 80*a^3*A*b^2 + 40*a*A*b^4 - 16*a^5*C - 8*a^3*b^2*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((8*a^4*A*b - 29*a^2*A*b^3 + 15*A*b^5 - 5*a^4*b*C - a^2*b^3*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(a^3*(a - b)^2*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 - 5*a^4*C - a^2*b^2*C)*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2) + (A*b^2*Sin[c + d*x] + a^2*C*Sin[c + d*x])/(2*a*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) + (11*a^2*A*b^2*Sin[c + d*x] - 5*A*b^4*Sin[c + d*x] + 3*a^4*C*Sin[c + d*x] + 3*a^2*b^2*C*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/d","A",0
1406,1,810,405,7.1212208,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^3} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{2 \left(16 A a^4+5 C a^4-19 A b^2 a^2+b^2 C a^2+9 A b^4\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(-32 A b a^3-24 b C a^3+8 A b^3 a\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(-C a^4-9 A b^2 a^2-5 b^2 C a^2+3 A b^4\right) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left(-4 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a+2 (2 a-b) b F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}+2 b^2 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}}{16 a^2 (a-b)^2 (a+b)^2 d}+\frac{\sqrt{\sec (c+d x)} \left(\frac{\left(C a^4+9 A b^2 a^2+5 b^2 C a^2-3 A b^4\right) \sin (c+d x)}{4 a^2 b \left(a^2-b^2\right)^2}+\frac{C \sin (c+d x) a^2+A b^2 \sin (c+d x)}{2 b \left(b^2-a^2\right) (a+b \cos (c+d x))^2}+\frac{C \sin (c+d x) a^4-7 A b^2 \sin (c+d x) a^2-7 b^2 C \sin (c+d x) a^2+A b^4 \sin (c+d x)}{4 a b \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}\right)}{d}","\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}-\frac{\left(a^4 (-C)-a^2 b^2 (9 A+5 C)+3 A b^4\right) \sin (c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}+\frac{\left(a^4 C-7 a^2 b^2 (A+C)+A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^2 d \left(a^2-b^2\right)^2}+\frac{\left(a^4 (-C)-a^2 b^2 (9 A+5 C)+3 A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b d \left(a^2-b^2\right)^2}+\frac{\left(a^6 (-C)+5 a^4 b^2 (3 A+2 C)-3 a^2 b^4 (2 A-C)+3 A b^6\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b^2 d (a-b)^2 (a+b)^3}",1,"((2*(16*a^4*A - 19*a^2*A*b^2 + 9*A*b^4 + 5*a^4*C + a^2*b^2*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-32*a^3*A*b + 8*a*A*b^3 - 24*a^3*b*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-9*a^2*A*b^2 + 3*A*b^4 - a^4*C - 5*a^2*b^2*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(16*a^2*(a - b)^2*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(((9*a^2*A*b^2 - 3*A*b^4 + a^4*C + 5*a^2*b^2*C)*Sin[c + d*x])/(4*a^2*b*(a^2 - b^2)^2) + (A*b^2*Sin[c + d*x] + a^2*C*Sin[c + d*x])/(2*b*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (-7*a^2*A*b^2*Sin[c + d*x] + A*b^4*Sin[c + d*x] + a^4*C*Sin[c + d*x] - 7*a^2*b^2*C*Sin[c + d*x])/(4*a*b*(a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/d","A",0
1407,1,815,408,7.0779317,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{\left(3 C a^4-5 A b^2 a^2-9 b^2 C a^2-A b^4\right) \sin (c+d x)}{4 a b^2 \left(a^2-b^2\right)^2}-\frac{C \sin (c+d x) a^3+A b^2 \sin (c+d x) a}{2 b^2 \left(b^2-a^2\right) (a+b \cos (c+d x))^2}+\frac{-5 C \sin (c+d x) a^4+3 A b^2 \sin (c+d x) a^2+11 b^2 C \sin (c+d x) a^2+3 A b^4 \sin (c+d x)}{4 b^2 \left(b^2-a^2\right)^2 (a+b \cos (c+d x))}\right)}{d}-\frac{\frac{2 \left(C a^4+9 A b^2 a^2+5 b^2 C a^2-3 A b^4\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(-16 A b a^3-8 b C a^3-8 A b^3 a-16 b^3 C a\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(3 C a^4-5 A b^2 a^2-9 b^2 C a^2-A b^4\right) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left(-4 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a+2 (2 a-b) b F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}+2 b^2 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}}{16 a (a-b)^2 b (a+b)^2 d}","-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}-\frac{\left(-3 a^4 C+a^2 b^2 (5 A+9 C)+A b^4\right) \sin (c+d x)}{4 a b d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}+\frac{\left(-3 a^4 C+a^2 b^2 (5 A+9 C)+A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^2 d \left(a^2-b^2\right)^2}+\frac{\left(3 a^4 C+a^2 b^2 (3 A-5 C)+b^4 (3 A+8 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d \left(a^2-b^2\right)^2}+\frac{\left(-3 a^6 C-3 a^4 b^2 (A-2 C)-5 a^2 b^4 (2 A+3 C)+A b^6\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^3 d (a-b)^2 (a+b)^3}",1,"-1/16*((2*(9*a^2*A*b^2 - 3*A*b^4 + a^4*C + 5*a^2*b^2*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-16*a^3*A*b - 8*a*A*b^3 - 8*a^3*b*C - 16*a*b^3*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-5*a^2*A*b^2 - A*b^4 + 3*a^4*C - 9*a^2*b^2*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(a*(a - b)^2*b*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(((-5*a^2*A*b^2 - A*b^4 + 3*a^4*C - 9*a^2*b^2*C)*Sin[c + d*x])/(4*a*b^2*(a^2 - b^2)^2) - (a*A*b^2*Sin[c + d*x] + a^3*C*Sin[c + d*x])/(2*b^2*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (3*a^2*A*b^2*Sin[c + d*x] + 3*A*b^4*Sin[c + d*x] - 5*a^4*C*Sin[c + d*x] + 11*a^2*b^2*C*Sin[c + d*x])/(4*b^2*(-a^2 + b^2)^2*(a + b*Cos[c + d*x]))))/d","A",0
1408,1,818,405,7.0803757,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)),x]","\frac{\frac{2 \left(5 C a^4+5 A b^2 a^2-7 b^2 C a^2+A b^4+8 b^4 C\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(8 b C a^3-24 A b^3 a-32 b^3 C a\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(15 C a^4-A b^2 a^2-29 b^2 C a^2-5 A b^4+8 b^4 C\right) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left(-4 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a+2 (2 a-b) b F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}+2 b^2 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}}{16 (a-b)^2 b^2 (a+b)^2 d}+\frac{\sqrt{\sec (c+d x)} \left(-\frac{\left(7 C a^4-A b^2 a^2-13 b^2 C a^2-5 A b^4\right) \sin (c+d x)}{4 b^3 \left(a^2-b^2\right)^2}-\frac{-C \sin (c+d x) a^4-A b^2 \sin (c+d x) a^2}{2 b^3 \left(b^2-a^2\right) (a+b \cos (c+d x))^2}+\frac{9 C \sin (c+d x) a^5+A b^2 \sin (c+d x) a^3-15 b^2 C \sin (c+d x) a^3-7 A b^4 \sin (c+d x) a}{4 b^3 \left(b^2-a^2\right)^2 (a+b \cos (c+d x))}\right)}{d}","-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac{\left(-5 a^4 C+a^2 b^2 (3 A+11 C)+3 A b^4\right) \sin (c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}-\frac{a \left(15 a^4 C-a^2 b^2 (A+33 C)+b^4 (7 A+24 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d \left(a^2-b^2\right)^2}-\frac{\left(-15 a^4 C+a^2 b^2 (A+29 C)+b^4 (5 A-8 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d \left(a^2-b^2\right)^2}+\frac{\left(15 a^6 C-a^4 b^2 (A+38 C)+5 a^2 b^4 (2 A+7 C)+3 A b^6\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d (a-b)^2 (a+b)^3}",1,"((2*(5*a^2*A*b^2 + A*b^4 + 5*a^4*C - 7*a^2*b^2*C + 8*b^4*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-24*a*A*b^3 + 8*a^3*b*C - 32*a*b^3*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-(a^2*A*b^2) - 5*A*b^4 + 15*a^4*C - 29*a^2*b^2*C + 8*b^4*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(16*(a - b)^2*b^2*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(-1/4*((-(a^2*A*b^2) - 5*A*b^4 + 7*a^4*C - 13*a^2*b^2*C)*Sin[c + d*x])/(b^3*(a^2 - b^2)^2) - (-(a^2*A*b^2*Sin[c + d*x]) - a^4*C*Sin[c + d*x])/(2*b^3*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (a^3*A*b^2*Sin[c + d*x] - 7*a*A*b^4*Sin[c + d*x] + 9*a^5*C*Sin[c + d*x] - 15*a^3*b^2*C*Sin[c + d*x])/(4*b^3*(-a^2 + b^2)^2*(a + b*Cos[c + d*x]))))/d","B",0
1409,1,857,493,7.2971291,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{a \left(11 C a^4+3 A b^2 a^2-17 b^2 C a^2-9 A b^4\right) \sin (c+d x)}{4 b^4 \left(a^2-b^2\right)^2}-\frac{C \sin (c+d x) a^5+A b^2 \sin (c+d x) a^3}{2 b^4 \left(b^2-a^2\right) (a+b \cos (c+d x))^2}+\frac{-13 C \sin (c+d x) a^6-5 A b^2 \sin (c+d x) a^4+19 b^2 C \sin (c+d x) a^4+11 A b^4 \sin (c+d x) a^2}{4 b^4 \left(b^2-a^2\right)^2 (a+b \cos (c+d x))}+\frac{C \sin (2 (c+d x))}{3 b^3}\right)}{d}-\frac{\frac{2 \left(35 C a^5+3 A b^2 a^3-73 b^2 C a^3+15 A b^4 a+56 b^4 C a\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(-48 A b^5-16 C b^5-24 a^2 A b^3-112 a^2 C b^3+56 a^4 C b\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(105 C a^5+9 A b^2 a^3-195 b^2 C a^3-27 A b^4 a+72 b^4 C a\right) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left(-4 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a+2 (2 a-b) b F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}+2 b^2 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}}{48 (a-b)^2 b^3 (a+b)^2 d}","-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac{\left(-7 a^4 C+a^2 b^2 (A+13 C)+5 A b^4\right) \sin (c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{a \left(35 a^4 C+a^2 b^2 (3 A-65 C)-3 b^4 (3 A-8 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d \left(a^2-b^2\right)^2}+\frac{\left(35 a^4 C+a^2 b^2 (3 A-61 C)-b^4 (21 A-8 C)\right) \sin (c+d x)}{12 b^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}+\frac{\left(105 a^6 C+a^4 b^2 (9 A-223 C)-a^2 b^4 (15 A-128 C)+8 b^6 (3 A+C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 b^5 d \left(a^2-b^2\right)^2}-\frac{a \left(35 a^6 C+a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+15 A b^6\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^5 d (a-b)^2 (a+b)^3}",1,"-1/48*((2*(3*a^3*A*b^2 + 15*a*A*b^4 + 35*a^5*C - 73*a^3*b^2*C + 56*a*b^4*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-24*a^2*A*b^3 - 48*A*b^5 + 56*a^4*b*C - 112*a^2*b^3*C - 16*b^5*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((9*a^3*A*b^2 - 27*a*A*b^4 + 105*a^5*C - 195*a^3*b^2*C + 72*a*b^4*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/((a - b)^2*b^3*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*((a*(3*a^2*A*b^2 - 9*A*b^4 + 11*a^4*C - 17*a^2*b^2*C)*Sin[c + d*x])/(4*b^4*(a^2 - b^2)^2) - (a^3*A*b^2*Sin[c + d*x] + a^5*C*Sin[c + d*x])/(2*b^4*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (-5*a^4*A*b^2*Sin[c + d*x] + 11*a^2*A*b^4*Sin[c + d*x] - 13*a^6*C*Sin[c + d*x] + 19*a^4*b^2*C*Sin[c + d*x])/(4*b^4*(-a^2 + b^2)^2*(a + b*Cos[c + d*x])) + (C*Sin[2*(c + d*x)])/(3*b^3)))/d","A",0
1410,1,925,579,7.5098166,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)),x]","\frac{\frac{2 \left(105 C a^6+25 A b^2 a^4-211 b^2 C a^4-35 A b^4 a^2+112 b^4 C a^2+40 A b^6+24 b^6 C\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(168 b C a^5+40 A b^3 a^3-256 b^3 C a^3-160 A b^5 a-32 b^5 C a\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(315 C a^6+75 A b^2 a^4-561 b^2 C a^4-145 A b^4 a^2+192 b^4 C a^2+40 A b^6+24 b^6 C\right) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left(-4 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a+2 (2 a-b) b F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}+2 b^2 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}}{80 (a-b)^2 b^4 (a+b)^2 d}+\frac{\sqrt{\sec (c+d x)} \left(-\frac{\left(75 C a^6+35 A b^2 a^4-107 b^2 C a^4-65 A b^4 a^2+4 b^4 C a^2-2 b^6 C\right) \sin (c+d x)}{20 b^5 \left(a^2-b^2\right)^2}-\frac{-C \sin (c+d x) a^6-A b^2 \sin (c+d x) a^4}{2 b^5 \left(b^2-a^2\right) (a+b \cos (c+d x))^2}+\frac{17 C \sin (c+d x) a^7+9 A b^2 \sin (c+d x) a^5-23 b^2 C \sin (c+d x) a^5-15 A b^4 \sin (c+d x) a^3}{4 b^5 \left(b^2-a^2\right)^2 (a+b \cos (c+d x))}-\frac{a C \sin (2 (c+d x))}{b^4}+\frac{C \sin (3 (c+d x))}{10 b^3}\right)}{d}","-\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac{a \left(-21 a^4 C-5 a^2 b^2 (A-7 C)+b^4 (11 A-8 C)\right) \sin (c+d x)}{4 b^4 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}+\frac{\left(-9 a^4 C-a^2 b^2 (A-15 C)+7 A b^4\right) \sin (c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}+\frac{\left(63 a^4 C+a^2 b^2 (15 A-101 C)-b^4 (45 A-8 C)\right) \sin (c+d x)}{20 b^3 d \left(a^2-b^2\right)^2 \sec ^{\frac{3}{2}}(c+d x)}+\frac{a \left(-63 a^6 C-3 a^4 b^2 (5 A-43 C)+a^2 b^4 (33 A-64 C)-8 b^6 (3 A+C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^6 d \left(a^2-b^2\right)^2}+\frac{a^2 \left(63 a^6 C+15 a^4 b^2 (A-10 C)-a^2 b^4 (38 A-99 C)+35 A b^6\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^6 d (a-b)^2 (a+b)^3}-\frac{\left(-315 a^6 C-3 a^4 b^2 (25 A-187 C)+a^2 b^4 (145 A-192 C)-8 b^6 (5 A+3 C)\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{20 b^5 d \left(a^2-b^2\right)^2}",1,"((2*(25*a^4*A*b^2 - 35*a^2*A*b^4 + 40*A*b^6 + 105*a^6*C - 211*a^4*b^2*C + 112*a^2*b^4*C + 24*b^6*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(40*a^3*A*b^3 - 160*a*A*b^5 + 168*a^5*b*C - 256*a^3*b^3*C - 32*a*b^5*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((75*a^4*A*b^2 - 145*a^2*A*b^4 + 40*A*b^6 + 315*a^6*C - 561*a^4*b^2*C + 192*a^2*b^4*C + 24*b^6*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(80*(a - b)^2*b^4*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(-1/20*((35*a^4*A*b^2 - 65*a^2*A*b^4 + 75*a^6*C - 107*a^4*b^2*C + 4*a^2*b^4*C - 2*b^6*C)*Sin[c + d*x])/(b^5*(a^2 - b^2)^2) - (-(a^4*A*b^2*Sin[c + d*x]) - a^6*C*Sin[c + d*x])/(2*b^5*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (9*a^5*A*b^2*Sin[c + d*x] - 15*a^3*A*b^4*Sin[c + d*x] + 17*a^7*C*Sin[c + d*x] - 23*a^5*b^2*C*Sin[c + d*x])/(4*b^5*(-a^2 + b^2)^2*(a + b*Cos[c + d*x])) - (a*C*Sin[2*(c + d*x)])/b^4 + (C*Sin[3*(c + d*x)])/(10*b^3)))/d","A",0
1411,1,3619,544,25.640739,"\int \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\text{Result too large to show}","-\frac{2 \left(6 A b^2-7 a^2 (7 A+9 C)\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 a^2 d}+\frac{2 b \left(a^2 (13 A+21 C)+8 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 a^3 d}-\frac{2 (a-b) \sqrt{a+b} \left(-21 a^4 (7 A+9 C)+6 a^2 b^2 (4 A+7 C)+16 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^5 d \sqrt{\sec (c+d x)}}-\frac{2 (a-b) \sqrt{a+b} \left(21 a^3 (7 A+9 C)+6 a^2 b (6 A+7 C)+12 a A b^2+16 A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{9 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{63 a d}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(147*a^4*A - 24*a^2*A*b^2 - 16*A*b^4 + 189*a^4*C - 42*a^2*b^2*C)*Sin[c + d*x])/(315*a^4) + (2*Sec[c + d*x]^2*(49*a^2*A*Sin[c + d*x] - 6*A*b^2*Sin[c + d*x] + 63*a^2*C*Sin[c + d*x]))/(315*a^2) + (2*Sec[c + d*x]*(13*a^2*A*b*Sin[c + d*x] + 8*A*b^3*Sin[c + d*x] + 21*a^2*b*C*Sin[c + d*x]))/(315*a^3) + (2*A*b*Sec[c + d*x]^2*Tan[c + d*x])/(63*a) + (2*A*Sec[c + d*x]^3*Tan[c + d*x])/9))/d + (2*((-7*a*A)/(15*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (8*A*b^2)/(105*a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*A*b^4)/(315*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (3*a*C)/(5*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b^2*C)/(15*a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*A*b*Sqrt[Sec[c + d*x]])/(35*Sqrt[a + b*Cos[c + d*x]]) + (4*A*b^3*Sqrt[Sec[c + d*x]])/(63*a^2*Sqrt[a + b*Cos[c + d*x]]) + (16*A*b^5*Sqrt[Sec[c + d*x]])/(315*a^4*Sqrt[a + b*Cos[c + d*x]]) - (2*b*C*Sqrt[Sec[c + d*x]])/(15*Sqrt[a + b*Cos[c + d*x]]) + (2*b^3*C*Sqrt[Sec[c + d*x]])/(15*a^2*Sqrt[a + b*Cos[c + d*x]]) - (7*A*b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[a + b*Cos[c + d*x]]) + (8*A*b^3*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*a^2*Sqrt[a + b*Cos[c + d*x]]) + (16*A*b^5*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(315*a^4*Sqrt[a + b*Cos[c + d*x]]) - (3*b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*Sqrt[a + b*Cos[c + d*x]]) + (2*b^3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*a^2*Sqrt[a + b*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(-16*A*b^4 - 6*a^2*b^2*(4*A + 7*C) + 21*a^4*(7*A + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(12*a*A*b^2 - 16*A*b^3 - 6*a^2*b*(6*A + 7*C) + 21*a^3*(7*A + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (16*A*b^4 + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*a^4*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*((b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(-2*(a + b)*(-16*A*b^4 - 6*a^2*b^2*(4*A + 7*C) + 21*a^4*(7*A + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(12*a*A*b^2 - 16*A*b^3 - 6*a^2*b*(6*A + 7*C) + 21*a^3*(7*A + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (16*A*b^4 + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*a^4*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(-2*(a + b)*(-16*A*b^4 - 6*a^2*b^2*(4*A + 7*C) + 21*a^4*(7*A + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(12*a*A*b^2 - 16*A*b^3 - 6*a^2*b*(6*A + 7*C) + 21*a^3*(7*A + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (16*A*b^4 + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*a^4*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((16*A*b^4 + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 - ((a + b)*(-16*A*b^4 - 6*a^2*b^2*(4*A + 7*C) + 21*a^4*(7*A + 9*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (a*(a + b)*(12*a*A*b^2 - 16*A*b^3 - 6*a^2*b*(6*A + 7*C) + 21*a^3*(7*A + 9*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - ((a + b)*(-16*A*b^4 - 6*a^2*b^2*(4*A + 7*C) + 21*a^4*(7*A + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (a*(a + b)*(12*a*A*b^2 - 16*A*b^3 - 6*a^2*b*(6*A + 7*C) + 21*a^3*(7*A + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - b*(16*A*b^4 + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (16*A*b^4 + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (16*A*b^4 + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (a*(a + b)*(12*a*A*b^2 - 16*A*b^3 - 6*a^2*b*(6*A + 7*C) + 21*a^3*(7*A + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a + b)*(-16*A*b^4 - 6*a^2*b^2*(4*A + 7*C) + 21*a^4*(7*A + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(315*a^4*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((-2*(a + b)*(-16*A*b^4 - 6*a^2*b^2*(4*A + 7*C) + 21*a^4*(7*A + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(12*a*A*b^2 - 16*A*b^3 - 6*a^2*b*(6*A + 7*C) + 21*a^3*(7*A + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (16*A*b^4 + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(315*a^4*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
1412,1,478,455,18.33402,"\int \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{\sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \sec (c+d x) \left(25 a^2 A \sin (c+d x)+35 a^2 C \sin (c+d x)-4 A b^2 \sin (c+d x)\right)}{105 a^2}+\frac{2 b \left(19 a^2 A+35 a^2 C+8 A b^2\right) \sin (c+d x)}{105 a^3}+\frac{2 A b \tan (c+d x) \sec (c+d x)}{35 a}+\frac{2}{7} A \tan (c+d x) \sec ^2(c+d x)\right)}{d}+\frac{2 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(-b \left(a^2 (19 A+35 C)+8 A b^2\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))+2 a (a+b) \left(5 a^2 (5 A+7 C)-6 a A b+8 A b^2\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-2 b (a+b) \left(a^2 (19 A+35 C)+8 A b^2\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{105 a^3 d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(4 A b^2-5 a^2 (5 A+7 C)\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{105 a^2 d}+\frac{2 b (a-b) \sqrt{a+b} \left(a^2 (19 A+35 C)+8 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \left(5 a^2 (5 A+7 C)+6 a A b+8 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{7 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{35 a d}",1,"(2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*b*(a + b)*(8*A*b^2 + a^2*(19*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(-6*a*A*b + 8*A*b^2 + 5*a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - b*(8*A*b^2 + a^2*(19*A + 35*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*a^3*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*b*(19*a^2*A + 8*A*b^2 + 35*a^2*C)*Sin[c + d*x])/(105*a^3) + (2*Sec[c + d*x]*(25*a^2*A*Sin[c + d*x] - 4*A*b^2*Sin[c + d*x] + 35*a^2*C*Sin[c + d*x]))/(105*a^2) + (2*A*b*Sec[c + d*x]*Tan[c + d*x])/(35*a) + (2*A*Sec[c + d*x]^2*Tan[c + d*x])/7))/d","A",0
1413,1,429,385,17.5605008,"\int \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{\sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \left(9 a^2 A+15 a^2 C-2 A b^2\right) \sin (c+d x)}{15 a^2}+\frac{2 A b \tan (c+d x)}{15 a}+\frac{2}{5} A \tan (c+d x) \sec (c+d x)\right)}{d}+\frac{2 \sqrt{2} \sqrt{\frac{\cos (c+d x)}{(\cos (c+d x)+1)^2}} \sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \left(\left(2 A b^2-3 a^2 (3 A+5 C)\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))-(a+b) \sec (c+d x) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2} \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))}{a+b}} \left(\left(3 a^2 (3 A+5 C)-2 A b^2\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+a (2 A b-3 a (3 A+5 C)) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)\right)}{15 a^2 d \sqrt{\frac{1}{\cos (c+d x)+1}} \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} \sqrt{a+b \cos (c+d x)}}","-\frac{2 (a-b) \sqrt{a+b} (9 a A+15 a C+2 A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^2 d \sqrt{\sec (c+d x)}}-\frac{2 (a-b) \sqrt{a+b} \left(2 A b^2-3 a^2 (3 A+5 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{15 a d}",1,"(2*Sqrt[2]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])^2]*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(-((a + b)*((-2*A*b^2 + 3*a^2*(3*A + 5*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*(2*A*b - 3*a*(3*A + 5*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]) + (2*A*b^2 - 3*a^2*(3*A + 5*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(15*a^2*d*Sqrt[(1 + Cos[c + d*x])^(-1)]*Sqrt[a + b*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(9*a^2*A - 2*A*b^2 + 15*a^2*C)*Sin[c + d*x])/(15*a^2) + (2*A*b*Tan[c + d*x])/(15*a) + (2*A*Sec[c + d*x]*Tan[c + d*x])/5))/d","A",0
1414,1,398,454,12.6787193,"\int \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{\sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 A b \sin (c+d x)}{3 a}+\frac{2}{3} A \tan (c+d x)\right)}{d}-\frac{2 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(-2 a (a (A+3 C)+b (A-3 C)) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+b \left(A \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))-12 a C \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)+2 A b (a+b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{3 a d \sqrt{a+b \cos (c+d x)}}","\frac{2 A b (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{a+b} (A b-a (A+3 C)) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}-\frac{2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}",1,"(-2*Cos[(c + d*x)/2]^2*Sqrt[Sec[c + d*x]]*(2*A*b*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 2*a*(b*(A - 3*C) + a*(A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + b*(-12*a*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + A*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])))/(3*a*d*Sqrt[a + b*Cos[c + d*x]]) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*A*b*Sin[c + d*x])/(3*a) + (2*A*Tan[c + d*x])/3))/d","A",1
1415,1,699,499,18.1671521,"\int \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(2 (a (A-C)+A b) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-(a+b) (2 A-C) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+2 a A \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a A \tan \left(\frac{1}{2} (c+d x)\right)+2 a C \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+2 a C \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-a C \tan ^5\left(\frac{1}{2} (c+d x)\right)+a C \tan \left(\frac{1}{2} (c+d x)\right)-2 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)+4 A b \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 A b \tan \left(\frac{1}{2} (c+d x)\right)+b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 b C \tan ^3\left(\frac{1}{2} (c+d x)\right)+b C \tan \left(\frac{1}{2} (c+d x)\right)\right)}{d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}","-\frac{(2 A-C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{\sqrt{a+b} (2 a A-a C-2 A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} (2 A-C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{a C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d \sqrt{\sec (c+d x)}}",1,"(2*A*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-2*a*A*Tan[(c + d*x)/2] - 2*A*b*Tan[(c + d*x)/2] + a*C*Tan[(c + d*x)/2] + b*C*Tan[(c + d*x)/2] + 4*A*b*Tan[(c + d*x)/2]^3 - 2*b*C*Tan[(c + d*x)/2]^3 + 2*a*A*Tan[(c + d*x)/2]^5 - 2*A*b*Tan[(c + d*x)/2]^5 - a*C*Tan[(c + d*x)/2]^5 + b*C*Tan[(c + d*x)/2]^5 + 2*a*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*a*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (a + b)*(2*A - C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*(A*b + a*(A - C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(d*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","A",0
1416,1,1391,515,18.7993513,"\int \sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{C \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (2 (c+d x))}{4 d}+\frac{a^2 \sqrt{\frac{a-b}{a+b}} C \tan ^5\left(\frac{1}{2} (c+d x)\right)-a b \sqrt{\frac{a-b}{a+b}} C \tan ^5\left(\frac{1}{2} (c+d x)\right)+2 a b \sqrt{\frac{a-b}{a+b}} C \tan ^3\left(\frac{1}{2} (c+d x)\right)+16 i A b^2 \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-2 i a^2 C \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+8 i b^2 C \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-a^2 \sqrt{\frac{a-b}{a+b}} C \tan \left(\frac{1}{2} (c+d x)\right)-a b \sqrt{\frac{a-b}{a+b}} C \tan \left(\frac{1}{2} (c+d x)\right)-i a (a-b) C E\left(i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 i (a-b) (4 A b+(a+2 b) C) F\left(i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+16 i A b^2 \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 i a^2 C \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+8 i b^2 C \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}}{4 b \sqrt{\frac{a-b}{a+b}} d \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\sqrt{a+b} \left(a^2 C-4 b^2 (2 A+C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} (C (a+2 b)+8 A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d \sqrt{\sec (c+d x)}}+\frac{a C \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}-\frac{C (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d \sqrt{\sec (c+d x)}}",1,"(C*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[2*(c + d*x)])/(4*d) + (-(a^2*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2]) - a*b*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2] + 2*a*b*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2]^3 + a^2*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2]^5 - a*b*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2]^5 + (16*I)*A*b^2*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*a^2*C*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*I)*b^2*C*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (16*I)*A*b^2*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*a^2*C*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*I)*b^2*C*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - I*a*(a - b)*C*EllipticE[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*(a - b)*(4*A*b + (a + 2*b)*C)*EllipticF[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)])/(4*b*Sqrt[(a - b)/(a + b)]*d*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","C",0
1417,1,1306,613,19.1736542,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{1}{12} C \sin (c+d x)+\frac{a C \sin (2 (c+d x))}{24 b}+\frac{1}{12} C \sin (3 (c+d x))\right)}{d}-\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(24 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-24 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 a^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+16 b^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 a b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 a^2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-48 A b^3 \tan ^3\left(\frac{1}{2} (c+d x)\right)-32 b^3 C \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 a^2 b C \tan ^3\left(\frac{1}{2} (c+d x)\right)+48 a A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+6 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+24 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+24 A b^3 \tan \left(\frac{1}{2} (c+d x)\right)+24 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)-3 a^3 C \tan \left(\frac{1}{2} (c+d x)\right)+16 b^3 C \tan \left(\frac{1}{2} (c+d x)\right)+16 a b^2 C \tan \left(\frac{1}{2} (c+d x)\right)-3 a^2 b C \tan \left(\frac{1}{2} (c+d x)\right)-(a+b) \left(3 C a^2-24 A b^2-16 b^2 C\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 a b ((a-14 b) C-24 A b) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+48 a A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+6 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+24 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{24 b^2 d \sqrt{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1} \left(b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","-\frac{\left(3 a^2 C-8 b^2 (3 A+2 C)\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{24 b^2 d}-\frac{\sqrt{a+b} \left(3 a^2 C-2 a b C-8 b^2 (3 A+2 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^2 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \left(3 a^2 C-8 b^2 (3 A+2 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^2 d \sqrt{\sec (c+d x)}}-\frac{a \sqrt{a+b} \left(C \left(a^2+4 b^2\right)+8 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^3 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 b d \sqrt{\sec (c+d x)}}-\frac{a C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b d \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((C*Sin[c + d*x])/12 + (a*C*Sin[2*(c + d*x)])/(24*b) + (C*Sin[3*(c + d*x)])/12))/d - (Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(24*a*A*b^2*Tan[(c + d*x)/2] + 24*A*b^3*Tan[(c + d*x)/2] - 3*a^3*C*Tan[(c + d*x)/2] - 3*a^2*b*C*Tan[(c + d*x)/2] + 16*a*b^2*C*Tan[(c + d*x)/2] + 16*b^3*C*Tan[(c + d*x)/2] - 48*A*b^3*Tan[(c + d*x)/2]^3 + 6*a^2*b*C*Tan[(c + d*x)/2]^3 - 32*b^3*C*Tan[(c + d*x)/2]^3 - 24*a*A*b^2*Tan[(c + d*x)/2]^5 + 24*A*b^3*Tan[(c + d*x)/2]^5 + 3*a^3*C*Tan[(c + d*x)/2]^5 - 3*a^2*b*C*Tan[(c + d*x)/2]^5 - 16*a*b^2*C*Tan[(c + d*x)/2]^5 + 16*b^3*C*Tan[(c + d*x)/2]^5 + 48*a*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 24*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 24*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (a + b)*(-24*A*b^2 + 3*a^2*C - 16*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*a*b*(-24*A*b + (a - 14*b)*C)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(24*b^2*d*Sqrt[1 + Tan[(c + d*x)/2]^2]*(b*(-1 + Tan[(c + d*x)/2]^2) - a*(1 + Tan[(c + d*x)/2]^2)))","B",0
1418,1,1798,698,15.6916381,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{a C \sin (c+d x)}{96 b}+\frac{\left(-5 C a^2+48 A b^2+48 b^2 C\right) \sin (2 (c+d x))}{192 b^2}+\frac{a C \sin (3 (c+d x))}{96 b}+\frac{1}{32} C \sin (4 (c+d x))\right)}{d}-\frac{\sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(-48 a A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)+48 a^2 A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+15 a^4 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-28 a b^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+28 a^2 b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-15 a^3 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)+96 a A b^3 \tan ^3\left(\frac{1}{2} (c+d x)\right)+56 a b^3 C \tan ^3\left(\frac{1}{2} (c+d x)\right)+30 a^3 b C \tan ^3\left(\frac{1}{2} (c+d x)\right)-384 A b^4 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+96 a^2 A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+30 a^4 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-288 b^4 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+48 a^2 b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-48 a A b^3 \tan \left(\frac{1}{2} (c+d x)\right)-48 a^2 A b^2 \tan \left(\frac{1}{2} (c+d x)\right)-15 a^4 C \tan \left(\frac{1}{2} (c+d x)\right)-28 a b^3 C \tan \left(\frac{1}{2} (c+d x)\right)-28 a^2 b^2 C \tan \left(\frac{1}{2} (c+d x)\right)-15 a^3 b C \tan \left(\frac{1}{2} (c+d x)\right)-a (a+b) \left(15 C a^2+48 A b^2+28 b^2 C\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 b \left(5 C a^3+2 b C a^2-12 b^2 (4 A+3 C) a+24 b^3 (4 A+3 C)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-384 A b^4 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+96 a^2 A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+30 a^4 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-288 b^4 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+48 a^2 b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{192 b^3 d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \sqrt{\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","\frac{\left(5 a^2 C+4 b^2 (4 A+3 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{32 b^2 d \sqrt{\sec (c+d x)}}+\frac{a \left(15 a^2 C+48 A b^2+28 b^2 C\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{192 b^3 d}-\frac{(a-b) \sqrt{a+b} \left(15 a^2 C+48 A b^2+28 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b^3 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left(5 a^4 C+8 a^2 b^2 (2 A+C)-16 b^4 (4 A+3 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^4 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left(15 a^3 C-10 a^2 b C+4 a b^2 (12 A+7 C)+24 b^3 (4 A+3 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b^3 d \sqrt{\sec (c+d x)}}-\frac{5 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((a*C*Sin[c + d*x])/(96*b) + ((48*A*b^2 - 5*a^2*C + 48*b^2*C)*Sin[2*(c + d*x)])/(192*b^2) + (a*C*Sin[3*(c + d*x)])/(96*b) + (C*Sin[4*(c + d*x)])/32))/d - (Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(-48*a^2*A*b^2*Tan[(c + d*x)/2] - 48*a*A*b^3*Tan[(c + d*x)/2] - 15*a^4*C*Tan[(c + d*x)/2] - 15*a^3*b*C*Tan[(c + d*x)/2] - 28*a^2*b^2*C*Tan[(c + d*x)/2] - 28*a*b^3*C*Tan[(c + d*x)/2] + 96*a*A*b^3*Tan[(c + d*x)/2]^3 + 30*a^3*b*C*Tan[(c + d*x)/2]^3 + 56*a*b^3*C*Tan[(c + d*x)/2]^3 + 48*a^2*A*b^2*Tan[(c + d*x)/2]^5 - 48*a*A*b^3*Tan[(c + d*x)/2]^5 + 15*a^4*C*Tan[(c + d*x)/2]^5 - 15*a^3*b*C*Tan[(c + d*x)/2]^5 + 28*a^2*b^2*C*Tan[(c + d*x)/2]^5 - 28*a*b^3*C*Tan[(c + d*x)/2]^5 + 96*a^2*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 384*A*b^4*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*a^4*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^2*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 288*b^4*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 96*a^2*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 384*A*b^4*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*a^4*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^2*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 288*b^4*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - a*(a + b)*(48*A*b^2 + 15*a^2*C + 28*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*b*(5*a^3*C + 2*a^2*b*C - 12*a*b^2*(4*A + 3*C) + 24*b^3*(4*A + 3*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(192*b^3*d*(-1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(b*(-1 + Tan[(c + d*x)/2]^2) - a*(1 + Tan[(c + d*x)/2]^2)))","B",0
1419,1,3622,542,25.5860642,"\int (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\text{Result too large to show}","\frac{2 \left(7 a^2 (7 A+9 C)+3 A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d}-\frac{4 b \left(2 A b^2-a^2 (44 A+63 C)\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \left(21 a^4 (7 A+9 C)+3 a^2 b^2 (11 A+21 C)+8 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \left(-21 a^3 (7 A+9 C)+a^2 (39 A b+63 b C)+6 a A b^2+8 A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{9 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{21 d}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 189*a^4*C + 63*a^2*b^2*C)*Sin[c + d*x])/(315*a^3) + (2*Sec[c + d*x]^2*(49*a^2*A*Sin[c + d*x] + 3*A*b^2*Sin[c + d*x] + 63*a^2*C*Sin[c + d*x]))/(315*a) + (4*Sec[c + d*x]*(44*a^2*A*b*Sin[c + d*x] - 2*A*b^3*Sin[c + d*x] + 63*a^2*b*C*Sin[c + d*x]))/(315*a^2) + (20*A*b*Sec[c + d*x]^2*Tan[c + d*x])/63 + (2*a*A*Sec[c + d*x]^3*Tan[c + d*x])/9))/d + (2*((-7*a^2*A)/(15*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (11*A*b^2)/(105*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*A*b^4)/(315*a^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (3*a^2*C)/(5*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (b^2*C)/(5*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (13*a*A*b*Sqrt[Sec[c + d*x]])/(105*Sqrt[a + b*Cos[c + d*x]]) - (31*A*b^3*Sqrt[Sec[c + d*x]])/(315*a*Sqrt[a + b*Cos[c + d*x]]) - (8*A*b^5*Sqrt[Sec[c + d*x]])/(315*a^3*Sqrt[a + b*Cos[c + d*x]]) + (a*b*C*Sqrt[Sec[c + d*x]])/(5*Sqrt[a + b*Cos[c + d*x]]) - (b^3*C*Sqrt[Sec[c + d*x]])/(5*a*Sqrt[a + b*Cos[c + d*x]]) - (7*a*A*b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[a + b*Cos[c + d*x]]) - (11*A*b^3*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*a*Sqrt[a + b*Cos[c + d*x]]) - (8*A*b^5*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(315*a^3*Sqrt[a + b*Cos[c + d*x]]) - (3*a*b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*Sqrt[a + b*Cos[c + d*x]]) - (b^3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*a*Sqrt[a + b*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(8*A*b^4 + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(-6*a*A*b^2 + 8*A*b^3 + 21*a^3*(7*A + 9*C) + a^2*(39*A*b + 63*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (8*A*b^4 + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*a^3*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*((b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(-2*(a + b)*(8*A*b^4 + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(-6*a*A*b^2 + 8*A*b^3 + 21*a^3*(7*A + 9*C) + a^2*(39*A*b + 63*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (8*A*b^4 + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*a^3*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(-2*(a + b)*(8*A*b^4 + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(-6*a*A*b^2 + 8*A*b^3 + 21*a^3*(7*A + 9*C) + a^2*(39*A*b + 63*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (8*A*b^4 + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1/2*((8*A*b^4 + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4) - ((a + b)*(8*A*b^4 + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (a*(a + b)*(-6*a*A*b^2 + 8*A*b^3 + 21*a^3*(7*A + 9*C) + a^2*(39*A*b + 63*b*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - ((a + b)*(8*A*b^4 + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (a*(a + b)*(-6*a*A*b^2 + 8*A*b^3 + 21*a^3*(7*A + 9*C) + a^2*(39*A*b + 63*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + b*(8*A*b^4 + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (8*A*b^4 + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (8*A*b^4 + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (a*(a + b)*(-6*a*A*b^2 + 8*A*b^3 + 21*a^3*(7*A + 9*C) + a^2*(39*A*b + 63*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a + b)*(8*A*b^4 + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(315*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((-2*(a + b)*(8*A*b^4 + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(-6*a*A*b^2 + 8*A*b^3 + 21*a^3*(7*A + 9*C) + a^2*(39*A*b + 63*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (8*A*b^4 + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(315*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
1420,1,482,458,18.9259697,"\int (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{\sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)} \left(-\frac{4 b \left(-41 a^2 A-70 a^2 C+3 A b^2\right) \sin (c+d x)}{105 a^2}+\frac{2 \sec (c+d x) \left(25 a^2 A \sin (c+d x)+35 a^2 C \sin (c+d x)+3 A b^2 \sin (c+d x)\right)}{105 a}+\frac{2}{7} a A \tan (c+d x) \sec ^2(c+d x)+\frac{16}{35} A b \tan (c+d x) \sec (c+d x)\right)}{d}+\frac{4 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(b \left(3 A b^2-a^2 (41 A+70 C)\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))+a (a+b) \left(5 a^2 (5 A+7 C)+3 a b (19 A+35 C)-6 A b^2\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+2 b (a+b) \left(3 A b^2-a^2 (41 A+70 C)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{105 a^2 d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(5 a^2 (5 A+7 C)+3 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{105 a d}+\frac{2 (a-b) \sqrt{a+b} \left(25 a^2 A+35 a^2 C-57 a A b-105 a b C-6 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d \sqrt{\sec (c+d x)}}-\frac{4 b (a-b) \sqrt{a+b} \left(3 A b^2-a^2 (41 A+70 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{7 d}+\frac{6 A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{35 d}",1,"(4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*b*(a + b)*(3*A*b^2 - a^2*(41*A + 70*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*(a + b)*(-6*A*b^2 + 5*a^2*(5*A + 7*C) + 3*a*b*(19*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + b*(3*A*b^2 - a^2*(41*A + 70*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*a^2*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((-4*b*(-41*a^2*A + 3*A*b^2 - 70*a^2*C)*Sin[c + d*x])/(105*a^2) + (2*Sec[c + d*x]*(25*a^2*A*Sin[c + d*x] + 3*A*b^2*Sin[c + d*x] + 35*a^2*C*Sin[c + d*x]))/(105*a) + (16*A*b*Sec[c + d*x]*Tan[c + d*x])/35 + (2*a*A*Sec[c + d*x]^2*Tan[c + d*x])/7))/d","A",0
1421,1,6023,525,24.9654033,"\int (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\text{Result too large to show}","-\frac{2 \sqrt{a+b} \left(a^2 (3 A+5 C)-2 a b (2 A+5 C)+A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \left(a^2 (3 A+5 C)+A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}-\frac{2 b C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}",1,"Result too large to show","B",0
1422,1,3930,560,23.6289977,"\int (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\text{Result too large to show}","\frac{\sqrt{a+b} \left(2 a^2 (A+3 C)-a (8 A b-3 b C)+6 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d \sqrt{\sec (c+d x)}}-\frac{b (8 A-3 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{b (a-b) \sqrt{a+b} (8 A-3 C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}+\frac{2 A b \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{3 a C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((8*A*b*Sin[c + d*x])/3 + (2*a*A*Tan[c + d*x])/3))/d + (((-4*a*A*b)/(3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a*b*C)/(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^2*A*Sqrt[Sec[c + d*x]])/(3*Sqrt[a + b*Cos[c + d*x]]) - (A*b^2*Sqrt[Sec[c + d*x]])/(3*Sqrt[a + b*Cos[c + d*x]]) + (a^2*C*Sqrt[Sec[c + d*x]])/Sqrt[a + b*Cos[c + d*x]] + (b^2*C*Sqrt[Sec[c + d*x]])/(2*Sqrt[a + b*Cos[c + d*x]]) - (4*A*b^2*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*Sqrt[a + b*Cos[c + d*x]]) + (b^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(2*Sqrt[a + b*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*b*(a + b)*(8*A - 3*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2 + 4*(3*A*b^2 + a^2*(A + 3*C) + a*(4*A*b - 6*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2 + b*(36*a*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2 - (8*A - 3*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])))/(3*d*Sqrt[a + b*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)*((b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(-2*b*(a + b)*(8*A - 3*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2 + 4*(3*A*b^2 + a^2*(A + 3*C) + a*(4*A*b - 6*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2 + b*(36*a*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2 - (8*A - 3*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])))/(6*(a + b*Cos[c + d*x])^(3/2)*(Sec[(c + d*x)/2]^2)^(3/2)) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(-2*b*(a + b)*(8*A - 3*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2 + 4*(3*A*b^2 + a^2*(A + 3*C) + a*(4*A*b - 6*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2 + b*(36*a*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2 - (8*A - 3*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])))/(2*Sqrt[a + b*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) + (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-((b*(a + b)*(8*A - 3*C)*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]) + (2*(3*A*b^2 + a^2*(A + 3*C) + a*(4*A*b - 6*b*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (b*(a + b)*(8*A - 3*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (2*(3*A*b^2 + a^2*(A + 3*C) + a*(4*A*b - 6*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - 2*b*(a + b)*(8*A - 3*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] + 4*(3*A*b^2 + a^2*(A + 3*C) + a*(4*A*b - 6*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] + (2*(3*A*b^2 + a^2*(A + 3*C) + a*(4*A*b - 6*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^4)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - (b*(a + b)*(8*A - 3*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^4*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2] + b*(-1/2*((8*A - 3*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^6) + (18*a*C*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (18*a*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + 36*a*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] + b*(8*A - 3*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^4*Sin[c + d*x]*Tan[(c + d*x)/2] + (8*A - 3*C)*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Sin[c + d*x]*Tan[(c + d*x)/2] - 2*(8*A - 3*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]^2 + (18*a*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^4)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]))))/(3*Sqrt[a + b*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) + ((-2*b*(a + b)*(8*A - 3*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2 + 4*(3*A*b^2 + a^2*(A + 3*C) + a*(4*A*b - 6*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2 + b*(36*a*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2 - (8*A - 3*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(6*Sqrt[a + b*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
1423,1,1166,569,18.5949986,"\int (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(2 a A \sin (c+d x)+\frac{1}{4} b C \sin (2 (c+d x))\right)}{d}+\frac{-8 a^2 A \tan ^5\left(\frac{1}{2} (c+d x)\right)+8 a A b \tan ^5\left(\frac{1}{2} (c+d x)\right)+5 a^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-5 a b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 a A b \tan ^3\left(\frac{1}{2} (c+d x)\right)+10 a b C \tan ^3\left(\frac{1}{2} (c+d x)\right)-16 A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-6 a^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-8 b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+8 a^2 A \tan \left(\frac{1}{2} (c+d x)\right)+8 a A b \tan \left(\frac{1}{2} (c+d x)\right)-5 a^2 C \tan \left(\frac{1}{2} (c+d x)\right)-5 a b C \tan \left(\frac{1}{2} (c+d x)\right)+a (a+b) (8 A-5 C) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 \left(4 (A-C) a^2+b (8 A+C) a-2 b^2 (2 A+C)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-16 A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-6 a^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-8 b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}}{4 d \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","-\frac{\sqrt{a+b} \left(3 a^2 C+8 A b^2+4 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d \sqrt{\sec (c+d x)}}-\frac{a (8 A-5 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{b (4 A-C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} (8 a A-5 a C-16 A b-2 b C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} (8 A-5 C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}{d}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(2*a*A*Sin[c + d*x] + (b*C*Sin[2*(c + d*x)])/4))/d + (8*a^2*A*Tan[(c + d*x)/2] + 8*a*A*b*Tan[(c + d*x)/2] - 5*a^2*C*Tan[(c + d*x)/2] - 5*a*b*C*Tan[(c + d*x)/2] - 16*a*A*b*Tan[(c + d*x)/2]^3 + 10*a*b*C*Tan[(c + d*x)/2]^3 - 8*a^2*A*Tan[(c + d*x)/2]^5 + 8*a*A*b*Tan[(c + d*x)/2]^5 + 5*a^2*C*Tan[(c + d*x)/2]^5 - 5*a*b*C*Tan[(c + d*x)/2]^5 - 16*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 6*a^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 8*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 16*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 6*a^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 8*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + a*(a + b)*(8*A - 5*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*(4*a^2*(A - C) - 2*b^2*(2*A + C) + a*b*(8*A + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)])/(4*d*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",1
1424,1,1273,613,17.003673,"\int (a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{1}{12} b C \sin (c+d x)+\frac{7}{24} a C \sin (2 (c+d x))+\frac{1}{12} b C \sin (3 (c+d x))\right)}{d}+\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(24 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-24 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 a^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+16 b^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 a b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 a^2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-48 A b^3 \tan ^3\left(\frac{1}{2} (c+d x)\right)-32 b^3 C \tan ^3\left(\frac{1}{2} (c+d x)\right)-6 a^2 b C \tan ^3\left(\frac{1}{2} (c+d x)\right)+144 a A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-6 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+72 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+24 A b^3 \tan \left(\frac{1}{2} (c+d x)\right)+24 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+3 a^3 C \tan \left(\frac{1}{2} (c+d x)\right)+16 b^3 C \tan \left(\frac{1}{2} (c+d x)\right)+16 a b^2 C \tan \left(\frac{1}{2} (c+d x)\right)+3 a^2 b C \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(3 C a^2+24 A b^2+16 b^2 C\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 a b (24 a A-48 b A+7 a C-26 b C) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+144 a A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-6 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+72 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{24 b d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\left(3 a^2 C+8 b^2 (3 A+2 C)\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{24 b d}+\frac{\sqrt{a+b} \left(3 a^2 C+48 a A b+14 a b C+24 A b^2+16 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(3 a^2 C+8 b^2 (3 A+2 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b d \sqrt{\sec (c+d x)}}-\frac{a \sqrt{a+b} \left(a^2 (-C)+24 A b^2+12 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \sqrt{\sec (c+d x)}}+\frac{a C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((b*C*Sin[c + d*x])/12 + (7*a*C*Sin[2*(c + d*x)])/24 + (b*C*Sin[3*(c + d*x)])/12))/d + (Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(24*a*A*b^2*Tan[(c + d*x)/2] + 24*A*b^3*Tan[(c + d*x)/2] + 3*a^3*C*Tan[(c + d*x)/2] + 3*a^2*b*C*Tan[(c + d*x)/2] + 16*a*b^2*C*Tan[(c + d*x)/2] + 16*b^3*C*Tan[(c + d*x)/2] - 48*A*b^3*Tan[(c + d*x)/2]^3 - 6*a^2*b*C*Tan[(c + d*x)/2]^3 - 32*b^3*C*Tan[(c + d*x)/2]^3 - 24*a*A*b^2*Tan[(c + d*x)/2]^5 + 24*A*b^3*Tan[(c + d*x)/2]^5 - 3*a^3*C*Tan[(c + d*x)/2]^5 + 3*a^2*b*C*Tan[(c + d*x)/2]^5 - 16*a*b^2*C*Tan[(c + d*x)/2]^5 + 16*b^3*C*Tan[(c + d*x)/2]^5 + 144*a*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 6*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 72*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 144*a*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 6*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 72*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(24*A*b^2 + 3*a^2*C + 16*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*a*b*(24*a*A - 48*A*b + 7*a*C - 26*b*C)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(24*b*d*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",1
1425,1,1797,698,15.3432711,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{3}{32} a C \sin (c+d x)+\frac{\left(C a^2+16 A b^2+16 b^2 C\right) \sin (2 (c+d x))}{64 b}+\frac{3}{32} a C \sin (3 (c+d x))+\frac{1}{32} b C \sin (4 (c+d x))\right)}{d}+\frac{\sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(80 a A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-80 a^2 A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 a^4 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+52 a b^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-52 a^2 b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 a^3 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-160 a A b^3 \tan ^3\left(\frac{1}{2} (c+d x)\right)-104 a b^3 C \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 a^3 b C \tan ^3\left(\frac{1}{2} (c+d x)\right)+128 A b^4 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+96 a^2 A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+6 a^4 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+96 b^4 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+48 a^2 b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+80 a A b^3 \tan \left(\frac{1}{2} (c+d x)\right)+80 a^2 A b^2 \tan \left(\frac{1}{2} (c+d x)\right)-3 a^4 C \tan \left(\frac{1}{2} (c+d x)\right)+52 a b^3 C \tan \left(\frac{1}{2} (c+d x)\right)+52 a^2 b^2 C \tan \left(\frac{1}{2} (c+d x)\right)-3 a^3 b C \tan \left(\frac{1}{2} (c+d x)\right)-a (a+b) \left(3 C a^2-80 A b^2-52 b^2 C\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 b \left(C a^3-2 b (32 A+19 C) a^2+4 b^2 (4 A+3 C) a-8 b^3 (4 A+3 C)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+128 A b^4 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+96 a^2 A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+6 a^4 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+96 b^4 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+48 a^2 b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{64 b^2 d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \sqrt{\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","\frac{a \left(-3 a^2 C+80 A b^2+52 b^2 C\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{64 b^2 d}-\frac{\left(3 a^2 C-4 b^2 (4 A+3 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{32 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(-3 a^2 C+80 A b^2+52 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left(3 a^4 C+24 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^3 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left(3 a^3 C-2 a^2 b C-4 a b^2 (20 A+13 C)-8 b^3 (4 A+3 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{4 b d \sqrt{\sec (c+d x)}}-\frac{a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{8 b d \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((3*a*C*Sin[c + d*x])/32 + ((16*A*b^2 + a^2*C + 16*b^2*C)*Sin[2*(c + d*x)])/(64*b) + (3*a*C*Sin[3*(c + d*x)])/32 + (b*C*Sin[4*(c + d*x)])/32))/d + (Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(80*a^2*A*b^2*Tan[(c + d*x)/2] + 80*a*A*b^3*Tan[(c + d*x)/2] - 3*a^4*C*Tan[(c + d*x)/2] - 3*a^3*b*C*Tan[(c + d*x)/2] + 52*a^2*b^2*C*Tan[(c + d*x)/2] + 52*a*b^3*C*Tan[(c + d*x)/2] - 160*a*A*b^3*Tan[(c + d*x)/2]^3 + 6*a^3*b*C*Tan[(c + d*x)/2]^3 - 104*a*b^3*C*Tan[(c + d*x)/2]^3 - 80*a^2*A*b^2*Tan[(c + d*x)/2]^5 + 80*a*A*b^3*Tan[(c + d*x)/2]^5 + 3*a^4*C*Tan[(c + d*x)/2]^5 - 3*a^3*b*C*Tan[(c + d*x)/2]^5 - 52*a^2*b^2*C*Tan[(c + d*x)/2]^5 + 52*a*b^3*C*Tan[(c + d*x)/2]^5 + 96*a^2*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 128*A*b^4*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*a^4*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^2*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 96*b^4*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 96*a^2*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 128*A*b^4*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*a^4*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^2*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 96*b^4*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - a*(a + b)*(-80*A*b^2 + 3*a^2*C - 52*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*b*(a^3*C + 4*a*b^2*(4*A + 3*C) - 8*b^3*(4*A + 3*C) - 2*a^2*b*(32*A + 19*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(64*b^2*d*(-1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(b*(-1 + Tan[(c + d*x)/2]^2) - a*(1 + Tan[(c + d*x)/2]^2)))","B",1
1426,1,3885,627,26.4466798,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{13}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2),x]","\text{Result too large to show}","\frac{2 \left(3 a^2 (9 A+11 C)+5 A b^2\right) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{231 d}+\frac{2 b \left(a^2 (229 A+297 C)+3 A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{693 a d}-\frac{2 \left(-15 a^4 (9 A+11 C)-a^2 b^2 (205 A+297 C)+4 A b^4\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{693 a^2 d}+\frac{2 b (a-b) \sqrt{a+b} \left(a^4 (741 A+957 C)+3 a^2 b^2 (17 A+33 C)+8 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{693 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \left(15 a^4 (9 A+11 C)-6 a^3 b (101 A+132 C)+3 a^2 b^2 (19 A+33 C)+6 a A b^3+8 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{693 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{11 d}+\frac{10 A b \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{99 d}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*b*(741*a^4*A + 51*a^2*A*b^2 + 8*A*b^4 + 957*a^4*C + 99*a^2*b^2*C)*Sin[c + d*x])/(693*a^3) + (2*Sec[c + d*x]^3*(81*a^2*A*Sin[c + d*x] + 113*A*b^2*Sin[c + d*x] + 99*a^2*C*Sin[c + d*x]))/693 + (2*Sec[c + d*x]^2*(229*a^2*A*b*Sin[c + d*x] + 3*A*b^3*Sin[c + d*x] + 297*a^2*b*C*Sin[c + d*x]))/(693*a) + (2*Sec[c + d*x]*(135*a^4*A*Sin[c + d*x] + 205*a^2*A*b^2*Sin[c + d*x] - 4*A*b^4*Sin[c + d*x] + 165*a^4*C*Sin[c + d*x] + 297*a^2*b^2*C*Sin[c + d*x]))/(693*a^2) + (46*a*A*b*Sec[c + d*x]^3*Tan[c + d*x])/99 + (2*a^2*A*Sec[c + d*x]^4*Tan[c + d*x])/11))/d + (2*((-247*a^2*A*b)/(231*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (17*A*b^3)/(231*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*A*b^5)/(693*a^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (29*a^2*b*C)/(21*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (b^3*C)/(7*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (15*a^3*A*Sqrt[Sec[c + d*x]])/(77*Sqrt[a + b*Cos[c + d*x]]) - (26*a*A*b^2*Sqrt[Sec[c + d*x]])/(231*Sqrt[a + b*Cos[c + d*x]]) - (7*A*b^4*Sqrt[Sec[c + d*x]])/(99*a*Sqrt[a + b*Cos[c + d*x]]) - (8*A*b^6*Sqrt[Sec[c + d*x]])/(693*a^3*Sqrt[a + b*Cos[c + d*x]]) + (5*a^3*C*Sqrt[Sec[c + d*x]])/(21*Sqrt[a + b*Cos[c + d*x]]) - (2*a*b^2*C*Sqrt[Sec[c + d*x]])/(21*Sqrt[a + b*Cos[c + d*x]]) - (b^4*C*Sqrt[Sec[c + d*x]])/(7*a*Sqrt[a + b*Cos[c + d*x]]) - (247*a*A*b^2*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(231*Sqrt[a + b*Cos[c + d*x]]) - (17*A*b^4*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(231*a*Sqrt[a + b*Cos[c + d*x]]) - (8*A*b^6*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(693*a^3*Sqrt[a + b*Cos[c + d*x]]) - (29*a*b^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(21*Sqrt[a + b*Cos[c + d*x]]) - (b^4*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(7*a*Sqrt[a + b*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*b*(a + b)*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(-6*a*A*b^3 + 8*A*b^4 + 15*a^4*(9*A + 11*C) + 3*a^2*b^2*(19*A + 33*C) + a^3*(606*A*b + 792*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - b*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(693*a^3*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*((b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(-2*b*(a + b)*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(-6*a*A*b^3 + 8*A*b^4 + 15*a^4*(9*A + 11*C) + 3*a^2*b^2*(19*A + 33*C) + a^3*(606*A*b + 792*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - b*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(693*a^3*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(-2*b*(a + b)*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(-6*a*A*b^3 + 8*A*b^4 + 15*a^4*(9*A + 11*C) + 3*a^2*b^2*(19*A + 33*C) + a^3*(606*A*b + 792*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - b*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(693*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1/2*(b*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4) - (b*(a + b)*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (a*(a + b)*(-6*a*A*b^3 + 8*A*b^4 + 15*a^4*(9*A + 11*C) + 3*a^2*b^2*(19*A + 33*C) + a^3*(606*A*b + 792*b*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (b*(a + b)*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (a*(a + b)*(-6*a*A*b^3 + 8*A*b^4 + 15*a^4*(9*A + 11*C) + 3*a^2*b^2*(19*A + 33*C) + a^3*(606*A*b + 792*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + b^2*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + b*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - b*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (a*(a + b)*(-6*a*A*b^3 + 8*A*b^4 + 15*a^4*(9*A + 11*C) + 3*a^2*b^2*(19*A + 33*C) + a^3*(606*A*b + 792*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - (b*(a + b)*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(693*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((-2*b*(a + b)*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(-6*a*A*b^3 + 8*A*b^4 + 15*a^4*(9*A + 11*C) + 3*a^2*b^2*(19*A + 33*C) + a^3*(606*A*b + 792*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - b*(8*A*b^4 + 3*a^2*b^2*(17*A + 33*C) + a^4*(741*A + 957*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(693*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
1427,1,621,544,21.5528952,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{\sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \sec (c+d x) \left(163 a^2 A b \sin (c+d x)+231 a^2 b C \sin (c+d x)+5 A b^3 \sin (c+d x)\right)}{315 a}+\frac{2}{315} \sec ^2(c+d x) \left(49 a^2 A \sin (c+d x)+63 a^2 C \sin (c+d x)+75 A b^2 \sin (c+d x)\right)+\frac{2}{9} a^2 A \tan (c+d x) \sec ^3(c+d x)-\frac{2 \left(-147 a^4 A-189 a^4 C-279 a^2 A b^2-483 a^2 b^2 C+10 A b^4\right) \sin (c+d x)}{315 a^2}+\frac{38}{63} a A b \tan (c+d x) \sec ^2(c+d x)\right)}{d}+\frac{2 \sqrt{2} \sqrt{\frac{\cos (c+d x)}{(\cos (c+d x)+1)^2}} \sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \left(-\left(\left(21 a^4 (7 A+9 C)+3 a^2 b^2 (93 A+161 C)-10 A b^4\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))\right)-(a+b) \sec (c+d x) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2} \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))}{a+b}} \left(\left(21 a^4 (7 A+9 C)+3 a^2 b^2 (93 A+161 C)-10 A b^4\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-a \left(21 a^3 (7 A+9 C)+6 a^2 b (19 A+28 C)+15 a b^2 (11 A+21 C)-10 A b^3\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)\right)}{315 a^2 d \sqrt{\frac{1}{\cos (c+d x)+1}} \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(7 a^2 (7 A+9 C)+15 A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 d}+\frac{2 b \left(a^2 (163 A+231 C)+5 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d}-\frac{2 (a-b) \sqrt{a+b} \left(21 a^3 (7 A+9 C)-6 a^2 b (19 A+28 C)+15 a b^2 (11 A+21 C)+10 A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^2 d \sqrt{\sec (c+d x)}}-\frac{2 (a-b) \sqrt{a+b} \left(-21 a^4 (7 A+9 C)-3 a^2 b^2 (93 A+161 C)+10 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{9 d}+\frac{10 A b \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{63 d}",1,"(2*Sqrt[2]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])^2]*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(-((a + b)*((-10*A*b^4 + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(93*A + 161*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - a*(-10*A*b^3 + 21*a^3*(7*A + 9*C) + 15*a*b^2*(11*A + 21*C) + 6*a^2*b*(19*A + 28*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]) - (-10*A*b^4 + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(93*A + 161*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(315*a^2*d*Sqrt[(1 + Cos[c + d*x])^(-1)]*Sqrt[a + b*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((-2*(-147*a^4*A - 279*a^2*A*b^2 + 10*A*b^4 - 189*a^4*C - 483*a^2*b^2*C)*Sin[c + d*x])/(315*a^2) + (2*Sec[c + d*x]^2*(49*a^2*A*Sin[c + d*x] + 75*A*b^2*Sin[c + d*x] + 63*a^2*C*Sin[c + d*x]))/315 + (2*Sec[c + d*x]*(163*a^2*A*b*Sin[c + d*x] + 5*A*b^3*Sin[c + d*x] + 231*a^2*b*C*Sin[c + d*x]))/(315*a) + (38*a*A*b*Sec[c + d*x]^2*Tan[c + d*x])/63 + (2*a^2*A*Sec[c + d*x]^3*Tan[c + d*x])/9))/d","A",0
1428,1,3967,600,25.5376761,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\text{Result too large to show}","\frac{2 \left(a^2 (5 A+7 C)+3 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{21 d}+\frac{2 b (a-b) \sqrt{a+b} \left(a^2 (29 A+49 C)+3 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{21 a^2 d \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{a+b} \left(-\left(a^3 (5 A+7 C)\right)+a^2 b (29 A+49 C)-9 a b^2 (3 A+7 C)+3 A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{21 a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{7 d}-\frac{2 b^2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*b*(29*a^2*A + 3*A*b^2 + 49*a^2*C)*Sin[c + d*x])/(21*a) + (2*Sec[c + d*x]*(5*a^2*A*Sin[c + d*x] + 9*A*b^2*Sin[c + d*x] + 7*a^2*C*Sin[c + d*x]))/21 + (6*a*A*b*Sec[c + d*x]*Tan[c + d*x])/7 + (2*a^2*A*Sec[c + d*x]^2*Tan[c + d*x])/7))/d + (2*((-29*a^2*A*b)/(21*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (A*b^3)/(7*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (7*a^2*b*C)/(3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (b^3*C)/(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (5*a^3*A*Sqrt[Sec[c + d*x]])/(21*Sqrt[a + b*Cos[c + d*x]]) - (2*a*A*b^2*Sqrt[Sec[c + d*x]])/(21*Sqrt[a + b*Cos[c + d*x]]) - (A*b^4*Sqrt[Sec[c + d*x]])/(7*a*Sqrt[a + b*Cos[c + d*x]]) + (a^3*C*Sqrt[Sec[c + d*x]])/(3*Sqrt[a + b*Cos[c + d*x]]) + (2*a*b^2*C*Sqrt[Sec[c + d*x]])/(3*Sqrt[a + b*Cos[c + d*x]]) - (29*a*A*b^2*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(21*Sqrt[a + b*Cos[c + d*x]]) - (A*b^4*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(7*a*Sqrt[a + b*Cos[c + d*x]]) - (7*a*b^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*Sqrt[a + b*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*b*(a + b)*(3*A*b^2 + a^2*(29*A + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(3*b^3*(A - 7*C) + 9*a*b^2*(3*A + 7*C) + a^3*(5*A + 7*C) + a^2*b*(29*A + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 84*a*b^3*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - b*(3*A*b^2 + a^2*(29*A + 49*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(21*a*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*((b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(-2*b*(a + b)*(3*A*b^2 + a^2*(29*A + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(3*b^3*(A - 7*C) + 9*a*b^2*(3*A + 7*C) + a^3*(5*A + 7*C) + a^2*b*(29*A + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 84*a*b^3*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - b*(3*A*b^2 + a^2*(29*A + 49*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(21*a*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(-2*b*(a + b)*(3*A*b^2 + a^2*(29*A + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(3*b^3*(A - 7*C) + 9*a*b^2*(3*A + 7*C) + a^3*(5*A + 7*C) + a^2*b*(29*A + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 84*a*b^3*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - b*(3*A*b^2 + a^2*(29*A + 49*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(21*a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1/2*(b*(3*A*b^2 + a^2*(29*A + 49*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4) - (b*(a + b)*(3*A*b^2 + a^2*(29*A + 49*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (a*(3*b^3*(A - 7*C) + 9*a*b^2*(3*A + 7*C) + a^3*(5*A + 7*C) + a^2*b*(29*A + 49*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (42*a*b^3*C*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (b*(a + b)*(3*A*b^2 + a^2*(29*A + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (a*(3*b^3*(A - 7*C) + 9*a*b^2*(3*A + 7*C) + a^3*(5*A + 7*C) + a^2*b*(29*A + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (42*a*b^3*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + b^2*(3*A*b^2 + a^2*(29*A + 49*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + b*(3*A*b^2 + a^2*(29*A + 49*C))*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - b*(3*A*b^2 + a^2*(29*A + 49*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (a*(3*b^3*(A - 7*C) + 9*a*b^2*(3*A + 7*C) + a^3*(5*A + 7*C) + a^2*b*(29*A + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) + (42*a*b^3*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - (b*(a + b)*(3*A*b^2 + a^2*(29*A + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(21*a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((-2*b*(a + b)*(3*A*b^2 + a^2*(29*A + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(3*b^3*(A - 7*C) + 9*a*b^2*(3*A + 7*C) + a^3*(5*A + 7*C) + a^2*b*(29*A + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 84*a*b^3*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - b*(3*A*b^2 + a^2*(29*A + 49*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(21*a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
1429,1,6694,666,25.8790151,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\text{Result too large to show}","\frac{2 \left(a^2 (3 A+5 C)+5 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{5 d}-\frac{\left(6 a^2 (3 A+5 C)+b^2 (46 A-15 C)\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{(a-b) \sqrt{a+b} \left(6 a^2 (3 A+5 C)+b^2 (46 A-15 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left(-6 a^3 (3 A+5 C)+a^2 (34 A b+90 b C)-a b^2 (46 A-15 C)+30 A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}+\frac{2 A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}-\frac{5 a b C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}",1,"Result too large to show","B",0
1430,1,4240,627,25.7748338,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\text{Result too large to show}","\frac{\sqrt{a+b} \left(8 a^2 (A+3 C)-a (56 A b-27 b C)+6 b^2 (12 A+C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{12 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left(15 a^2 C+8 A b^2+4 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d \sqrt{\sec (c+d x)}}-\frac{b^2 (8 A-C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}-\frac{a b (56 A-27 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{12 d}+\frac{b (a-b) \sqrt{a+b} (56 A-27 C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{12 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}+\frac{10 A b \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((14*a*A*b*Sin[c + d*x])/3 + (b^2*C*Sin[2*(c + d*x)])/4 + (2*a^2*A*Tan[c + d*x])/3))/d + (((-7*a^2*A*b)/(3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (A*b^3)/(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (3*a^2*b*C)/(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (b^3*C)/(2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^3*A*Sqrt[Sec[c + d*x]])/(3*Sqrt[a + b*Cos[c + d*x]]) + (2*a*A*b^2*Sqrt[Sec[c + d*x]])/(3*Sqrt[a + b*Cos[c + d*x]]) + (a^3*C*Sqrt[Sec[c + d*x]])/Sqrt[a + b*Cos[c + d*x]] + (11*a*b^2*C*Sqrt[Sec[c + d*x]])/(8*Sqrt[a + b*Cos[c + d*x]]) - (7*a*A*b^2*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*Sqrt[a + b*Cos[c + d*x]]) + (9*a*b^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(8*Sqrt[a + b*Cos[c + d*x]]))*(2*a*b*(a + b)*(56*A - 27*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 4*(4*a^2*b*(7*A - 9*C) - 6*b^3*(2*A + C) + 3*a*b^2*(12*A + C) + 4*a^3*(A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 12*b*(8*A*b^2 + 15*a^2*C + 4*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*b*(56*A - 27*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(12*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)*(-1/12*(Sqrt[Sec[(c + d*x)/2]^2]*Tan[(c + d*x)/2]*(2*a*b*(a + b)*(56*A - 27*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 4*(4*a^2*b*(7*A - 9*C) - 6*b^3*(2*A + C) + 3*a*b^2*(12*A + C) + 4*a^3*(A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 12*b*(8*A*b^2 + 15*a^2*C + 4*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*b*(56*A - 27*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)^2) + (b*Sin[c + d*x]*(2*a*b*(a + b)*(56*A - 27*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 4*(4*a^2*b*(7*A - 9*C) - 6*b^3*(2*A + C) + 3*a*b^2*(12*A + C) + 4*a^3*(A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 12*b*(8*A*b^2 + 15*a^2*C + 4*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*b*(56*A - 27*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(24*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)) - (Tan[(c + d*x)/2]*(2*a*b*(a + b)*(56*A - 27*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 4*(4*a^2*b*(7*A - 9*C) - 6*b^3*(2*A + C) + 3*a*b^2*(12*A + C) + 4*a^3*(A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 12*b*(8*A*b^2 + 15*a^2*C + 4*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*b*(56*A - 27*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(24*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)) + ((a*b*(56*A - 27*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + (a*b*(a + b)*(56*A - 27*C)*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (2*(4*a^2*b*(7*A - 9*C) - 6*b^3*(2*A + C) + 3*a*b^2*(12*A + C) + 4*a^3*(A + 3*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (6*b*(8*A*b^2 + 15*a^2*C + 4*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (a*b*(a + b)*(56*A - 27*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (2*(4*a^2*b*(7*A - 9*C) - 6*b^3*(2*A + C) + 3*a*b^2*(12*A + C) + 4*a^3*(A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (6*b*(8*A*b^2 + 15*a^2*C + 4*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*b^2*(56*A - 27*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - a*b*(56*A - 27*C)*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + a*b*(56*A - 27*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 - (2*(4*a^2*b*(7*A - 9*C) - 6*b^3*(2*A + C) + 3*a*b^2*(12*A + C) + 4*a^3*(A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - (6*b*(8*A*b^2 + 15*a^2*C + 4*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) + (a*b*(a + b)*(56*A - 27*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2])/(12*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)) - ((2*a*b*(a + b)*(56*A - 27*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 4*(4*a^2*b*(7*A - 9*C) - 6*b^3*(2*A + C) + 3*a*b^2*(12*A + C) + 4*a^3*(A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 12*b*(8*A*b^2 + 15*a^2*C + 4*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*b*(56*A - 27*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(24*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(-1 + Tan[(c + d*x)/2]^2))))","B",0
1431,1,1393,669,19.5419354,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{1}{12} C \sin (3 (c+d x)) b^2+\frac{13}{24} a C \sin (2 (c+d x)) b+\frac{1}{12} \left(24 A a^2+b^2 C\right) \sin (c+d x)\right)}{d}+\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(24 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-24 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+48 a^3 A \tan ^5\left(\frac{1}{2} (c+d x)\right)-48 a^2 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)-33 a^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+16 b^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 a b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+33 a^2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-48 A b^3 \tan ^3\left(\frac{1}{2} (c+d x)\right)+96 a^2 A b \tan ^3\left(\frac{1}{2} (c+d x)\right)-32 b^3 C \tan ^3\left(\frac{1}{2} (c+d x)\right)-66 a^2 b C \tan ^3\left(\frac{1}{2} (c+d x)\right)+240 a A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+30 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+120 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+24 A b^3 \tan \left(\frac{1}{2} (c+d x)\right)+24 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)-48 a^3 A \tan \left(\frac{1}{2} (c+d x)\right)-48 a^2 A b \tan \left(\frac{1}{2} (c+d x)\right)+33 a^3 C \tan \left(\frac{1}{2} (c+d x)\right)+16 b^3 C \tan \left(\frac{1}{2} (c+d x)\right)+16 a b^2 C \tan \left(\frac{1}{2} (c+d x)\right)+33 a^2 b C \tan \left(\frac{1}{2} (c+d x)\right)-(a+b) \left(a^2 (48 A-33 C)-8 b^2 (3 A+2 C)\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 a \left(24 (A-C) a^2+b (72 A+13 C) a-2 b^2 (36 A+19 C)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+240 a A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+30 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+120 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{24 d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","-\frac{\left(a^2 (48 A-33 C)-8 b^2 (3 A+2 C)\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{24 d}-\frac{\sqrt{a+b} \left(a^2 (48 A-33 C)-2 a b (72 A+13 C)-8 b^2 (3 A+2 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \left(a^2 (48 A-33 C)-8 b^2 (3 A+2 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a d \sqrt{\sec (c+d x)}}-\frac{5 a \sqrt{a+b} \left(C \left(a^2+4 b^2\right)+8 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b d \sqrt{\sec (c+d x)}}-\frac{b (6 A-C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \sqrt{\sec (c+d x)}}-\frac{a b (8 A-3 C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{5/2}}{d}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(((24*a^2*A + b^2*C)*Sin[c + d*x])/12 + (13*a*b*C*Sin[2*(c + d*x)])/24 + (b^2*C*Sin[3*(c + d*x)])/12))/d + (Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-48*a^3*A*Tan[(c + d*x)/2] - 48*a^2*A*b*Tan[(c + d*x)/2] + 24*a*A*b^2*Tan[(c + d*x)/2] + 24*A*b^3*Tan[(c + d*x)/2] + 33*a^3*C*Tan[(c + d*x)/2] + 33*a^2*b*C*Tan[(c + d*x)/2] + 16*a*b^2*C*Tan[(c + d*x)/2] + 16*b^3*C*Tan[(c + d*x)/2] + 96*a^2*A*b*Tan[(c + d*x)/2]^3 - 48*A*b^3*Tan[(c + d*x)/2]^3 - 66*a^2*b*C*Tan[(c + d*x)/2]^3 - 32*b^3*C*Tan[(c + d*x)/2]^3 + 48*a^3*A*Tan[(c + d*x)/2]^5 - 48*a^2*A*b*Tan[(c + d*x)/2]^5 - 24*a*A*b^2*Tan[(c + d*x)/2]^5 + 24*A*b^3*Tan[(c + d*x)/2]^5 - 33*a^3*C*Tan[(c + d*x)/2]^5 + 33*a^2*b*C*Tan[(c + d*x)/2]^5 - 16*a*b^2*C*Tan[(c + d*x)/2]^5 + 16*b^3*C*Tan[(c + d*x)/2]^5 + 240*a*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 120*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 240*a*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 120*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (a + b)*(a^2*(48*A - 33*C) - 8*b^2*(3*A + 2*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*a*(24*a^2*(A - C) + a*b*(72*A + 13*C) - 2*b^2*(36*A + 19*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(24*d*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",1
1432,1,601,695,19.9342302,"\int (a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{1}{192} \left(59 a^2 C+48 A b^2+48 b^2 C\right) \sin (2 (c+d x))+\frac{17}{96} a b C \sin (c+d x)+\frac{17}{96} a b C \sin (3 (c+d x))+\frac{1}{32} b^2 C \sin (4 (c+d x))\right)}{d}+\frac{\sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(a b \left(15 a^2 C+432 A b^2+284 b^2 C\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2} (a+b \cos (c+d x))+a b (a+b) \left(15 a^2 C+432 A b^2+284 b^2 C\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-3 \left(5 a^4 C-120 a^2 b^2 (2 A+C)-16 b^4 (4 A+3 C)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))}{a+b}} \left((a-b) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+2 b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)+a (a+b) \left(15 a^3 C-30 a^2 b C-4 a b^2 (84 A+53 C)-24 b^3 (4 A+3 C)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))}{a+b}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{192 b^2 d \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} \sqrt{a+b \cos (c+d x)}}","\frac{a \left(15 a^2 C+432 A b^2+284 b^2 C\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{192 b d}+\frac{\left(5 a^2 C+4 b^2 (4 A+3 C)\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{32 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(15 a^2 C+432 A b^2+284 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left(5 a^4 C-120 a^2 b^2 (2 A+C)-16 b^4 (4 A+3 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left(15 a^3 C+2 a^2 b (192 A+59 C)+4 a b^2 (108 A+71 C)+24 b^3 (4 A+3 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{4 d \sqrt{\sec (c+d x)}}+\frac{5 a C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((17*a*b*C*Sin[c + d*x])/96 + ((48*A*b^2 + 59*a^2*C + 48*b^2*C)*Sin[2*(c + d*x)])/192 + (17*a*b*C*Sin[3*(c + d*x)])/96 + (b^2*C*Sin[4*(c + d*x)])/32))/d + (Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a*b*(a + b)*(432*A*b^2 + 15*a^2*C + 284*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + a*(a + b)*(15*a^3*C - 30*a^2*b*C - 24*b^3*(4*A + 3*C) - 4*a*b^2*(84*A + 53*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - 3*(5*a^4*C - 120*a^2*b^2*(2*A + C) - 16*b^4*(4*A + 3*C))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + a*b*(432*A*b^2 + 15*a^2*C + 284*b^2*C)*(a + b*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(192*b^2*d*Sqrt[a + b*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2))","A",0
1433,1,2045,806,21.7512148,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\text{Result too large to show}","\frac{C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d \sqrt{\sec (c+d x)}}-\frac{3 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{40 b d \sqrt{\sec (c+d x)}}-\frac{\left(15 a^2 C-16 b^2 (5 A+4 C)\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{240 b d \sqrt{\sec (c+d x)}}-\frac{\left(45 C a^4-12 b^2 (220 A+141 C) a^2-256 b^4 (5 A+4 C)\right) \sqrt{\sec (c+d x)} \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d}+\frac{a \left(-15 C a^2+240 A b^2+172 b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{320 b d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \left(45 C a^4-12 b^2 (220 A+141 C) a^2-256 b^4 (5 A+4 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 a b^2 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left(45 C a^4-30 b C a^3-12 b^2 (220 A+141 C) a^2-8 b^3 (260 A+193 C) a-256 b^4 (5 A+4 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 b^2 d \sqrt{\sec (c+d x)}}-\frac{a \sqrt{a+b} \left(3 C a^4+40 b^2 (2 A+C) a^2+80 b^4 (4 A+3 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{128 b^3 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(((80*A*b^2 + 93*a^2*C + 88*b^2*C)*Sin[c + d*x])/960 + (a*(1040*A*b^2 + 15*a^2*C + 1024*b^2*C)*Sin[2*(c + d*x)])/(1920*b) + ((80*A*b^2 + 93*a^2*C + 100*b^2*C)*Sin[3*(c + d*x)])/960 + (21*a*b*C*Sin[4*(c + d*x)])/320 + (b^2*C*Sin[5*(c + d*x)])/80))/d - (Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(2640*a^3*A*b^2*Tan[(c + d*x)/2] + 2640*a^2*A*b^3*Tan[(c + d*x)/2] + 1280*a*A*b^4*Tan[(c + d*x)/2] + 1280*A*b^5*Tan[(c + d*x)/2] - 45*a^5*C*Tan[(c + d*x)/2] - 45*a^4*b*C*Tan[(c + d*x)/2] + 1692*a^3*b^2*C*Tan[(c + d*x)/2] + 1692*a^2*b^3*C*Tan[(c + d*x)/2] + 1024*a*b^4*C*Tan[(c + d*x)/2] + 1024*b^5*C*Tan[(c + d*x)/2] - 5280*a^2*A*b^3*Tan[(c + d*x)/2]^3 - 2560*A*b^5*Tan[(c + d*x)/2]^3 + 90*a^4*b*C*Tan[(c + d*x)/2]^3 - 3384*a^2*b^3*C*Tan[(c + d*x)/2]^3 - 2048*b^5*C*Tan[(c + d*x)/2]^3 - 2640*a^3*A*b^2*Tan[(c + d*x)/2]^5 + 2640*a^2*A*b^3*Tan[(c + d*x)/2]^5 - 1280*a*A*b^4*Tan[(c + d*x)/2]^5 + 1280*A*b^5*Tan[(c + d*x)/2]^5 + 45*a^5*C*Tan[(c + d*x)/2]^5 - 45*a^4*b*C*Tan[(c + d*x)/2]^5 - 1692*a^3*b^2*C*Tan[(c + d*x)/2]^5 + 1692*a^2*b^3*C*Tan[(c + d*x)/2]^5 - 1024*a*b^4*C*Tan[(c + d*x)/2]^5 + 1024*b^5*C*Tan[(c + d*x)/2]^5 + 2400*a^3*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 9600*a*A*b^4*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 90*a^5*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 1200*a^3*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 7200*a*b^4*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 2400*a^3*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 9600*a*A*b^4*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 90*a^5*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 1200*a^3*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 7200*a*b^4*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (a + b)*(45*a^4*C - 256*b^4*(5*A + 4*C) - 12*a^2*b^2*(220*A + 141*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*a*b*(15*a^3*C - 6*a^2*b*(320*A + 191*C) + 4*a*b^2*(260*A + 193*C) - 8*b^3*(380*A + 289*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(1920*b^2*d*Sqrt[1 + Tan[(c + d*x)/2]^2]*(b*(-1 + Tan[(c + d*x)/2]^2) - a*(1 + Tan[(c + d*x)/2]^2)))","B",0
1434,1,3164,469,22.794175,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2))/Sqrt[a + b*Cos[c + d*x]],x]","\text{Result too large to show}","-\frac{12 A b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{35 a^2 d}-\frac{4 b (a-b) \sqrt{a+b} \left(a^2 (22 A+35 C)+24 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^5 d \sqrt{\sec (c+d x)}}+\frac{2 \left(5 a^2 (5 A+7 C)+24 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{105 a^3 d}-\frac{2 \sqrt{a+b} \left(-5 a^3 (5 A+7 C)-a^2 (44 A b+70 b C)+12 a A b^2-48 A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{7 a d}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((-4*b*(22*a^2*A + 24*A*b^2 + 35*a^2*C)*Sin[c + d*x])/(105*a^4) + (2*Sec[c + d*x]*(25*a^2*A*Sin[c + d*x] + 24*A*b^2*Sin[c + d*x] + 35*a^2*C*Sin[c + d*x]))/(105*a^3) - (12*A*b*Sec[c + d*x]*Tan[c + d*x])/(35*a^2) + (2*A*Sec[c + d*x]^2*Tan[c + d*x])/(7*a)))/d + (4*((44*A*b)/(105*a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*A*b^3)/(35*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b*C)/(3*a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (5*A*Sqrt[Sec[c + d*x]])/(21*Sqrt[a + b*Cos[c + d*x]]) + (32*A*b^2*Sqrt[Sec[c + d*x]])/(105*a^2*Sqrt[a + b*Cos[c + d*x]]) + (16*A*b^4*Sqrt[Sec[c + d*x]])/(35*a^4*Sqrt[a + b*Cos[c + d*x]]) + (C*Sqrt[Sec[c + d*x]])/(3*Sqrt[a + b*Cos[c + d*x]]) + (2*b^2*C*Sqrt[Sec[c + d*x]])/(3*a^2*Sqrt[a + b*Cos[c + d*x]]) + (44*A*b^2*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*a^2*Sqrt[a + b*Cos[c + d*x]]) + (16*A*b^4*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(35*a^4*Sqrt[a + b*Cos[c + d*x]]) + (2*b^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*a^2*Sqrt[a + b*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*b*(a + b)*(24*A*b^2 + a^2*(22*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*(-12*a*A*b^2 - 48*A*b^3 + 5*a^3*(5*A + 7*C) - 2*a^2*b*(22*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + b*(24*A*b^2 + a^2*(22*A + 35*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*a^4*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*((2*b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*b*(a + b)*(24*A*b^2 + a^2*(22*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*(-12*a*A*b^2 - 48*A*b^3 + 5*a^3*(5*A + 7*C) - 2*a^2*b*(22*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + b*(24*A*b^2 + a^2*(22*A + 35*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*a^4*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*b*(a + b)*(24*A*b^2 + a^2*(22*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*(-12*a*A*b^2 - 48*A*b^3 + 5*a^3*(5*A + 7*C) - 2*a^2*b*(22*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + b*(24*A*b^2 + a^2*(22*A + 35*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*a^4*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((b*(24*A*b^2 + a^2*(22*A + 35*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + (b*(a + b)*(24*A*b^2 + a^2*(22*A + 35*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (a*(-12*a*A*b^2 - 48*A*b^3 + 5*a^3*(5*A + 7*C) - 2*a^2*b*(22*A + 35*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/(2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]) + (b*(a + b)*(24*A*b^2 + a^2*(22*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (a*(-12*a*A*b^2 - 48*A*b^3 + 5*a^3*(5*A + 7*C) - 2*a^2*b*(22*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/(2*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - b^2*(24*A*b^2 + a^2*(22*A + 35*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - b*(24*A*b^2 + a^2*(22*A + 35*C))*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + b*(24*A*b^2 + a^2*(22*A + 35*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (a*(-12*a*A*b^2 - 48*A*b^3 + 5*a^3*(5*A + 7*C) - 2*a^2*b*(22*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) + (b*(a + b)*(24*A*b^2 + a^2*(22*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(105*a^4*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*(2*b*(a + b)*(24*A*b^2 + a^2*(22*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*(-12*a*A*b^2 - 48*A*b^3 + 5*a^3*(5*A + 7*C) - 2*a^2*b*(22*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + b*(24*A*b^2 + a^2*(22*A + 35*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(105*a^4*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
1435,1,2920,394,21.3629135,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/Sqrt[a + b*Cos[c + d*x]],x]","\text{Result too large to show}","-\frac{8 A b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{15 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \left(3 a^2 (3 A+5 C)+8 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{a+b} \left(-3 a^2 (3 A+5 C)+2 a A b-8 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 a d}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(9*a^2*A + 8*A*b^2 + 15*a^2*C)*Sin[c + d*x])/(15*a^3) - (8*A*b*Tan[c + d*x])/(15*a^2) + (2*A*Sec[c + d*x]*Tan[c + d*x])/(5*a)))/d + (2*((-3*A)/(5*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*A*b^2)/(15*a^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - C/(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (7*A*b*Sqrt[Sec[c + d*x]])/(15*a*Sqrt[a + b*Cos[c + d*x]]) - (8*A*b^3*Sqrt[Sec[c + d*x]])/(15*a^3*Sqrt[a + b*Cos[c + d*x]]) - (b*C*Sqrt[Sec[c + d*x]])/(a*Sqrt[a + b*Cos[c + d*x]]) - (3*A*b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*a*Sqrt[a + b*Cos[c + d*x]]) - (8*A*b^3*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*a^3*Sqrt[a + b*Cos[c + d*x]]) - (b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(a*Sqrt[a + b*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(8*A*b^2 + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(2*a*A*b + 8*A*b^2 + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (8*A*b^2 + 3*a^2*(3*A + 5*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*a^3*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*((b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(-2*(a + b)*(8*A*b^2 + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(2*a*A*b + 8*A*b^2 + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (8*A*b^2 + 3*a^2*(3*A + 5*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*a^3*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(-2*(a + b)*(8*A*b^2 + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(2*a*A*b + 8*A*b^2 + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (8*A*b^2 + 3*a^2*(3*A + 5*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1/2*((8*A*b^2 + 3*a^2*(3*A + 5*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4) - ((a + b)*(8*A*b^2 + 3*a^2*(3*A + 5*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (a*(2*a*A*b + 8*A*b^2 + 3*a^2*(3*A + 5*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - ((a + b)*(8*A*b^2 + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (a*(2*a*A*b + 8*A*b^2 + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + b*(8*A*b^2 + 3*a^2*(3*A + 5*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (8*A*b^2 + 3*a^2*(3*A + 5*C))*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (8*A*b^2 + 3*a^2*(3*A + 5*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (a*(2*a*A*b + 8*A*b^2 + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a + b)*(8*A*b^2 + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(15*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((-2*(a + b)*(8*A*b^2 + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(2*a*A*b + 8*A*b^2 + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (8*A*b^2 + 3*a^2*(3*A + 5*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(15*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
1436,1,303,323,13.4178032,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \left(\frac{2 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(a (a (A+3 C)-2 A b) \sqrt{\frac{1}{\sec (c+d x)+1}} \sqrt{\frac{a \sec (c+d x)+b}{(a+b) (\sec (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+A b \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))+2 A b (a+b) \sqrt{\frac{1}{\sec (c+d x)+1}} \sqrt{\frac{a \sec (c+d x)+b}{(a+b) (\sec (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{\sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)}}+A \sqrt{\sec (c+d x)} (a+b \cos (c+d x)) (a \tan (c+d x)-2 b \sin (c+d x))\right)}{3 a^2 d \sqrt{a+b \cos (c+d x)}}","-\frac{4 A b (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{a+b} (a (A+3 C)+2 A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 a d}",1,"(2*((2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*A*b*(a + b)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[(b + a*Sec[c + d*x])/((a + b)*(1 + Sec[c + d*x]))] + a*(-2*A*b + a*(A + 3*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[(b + a*Sec[c + d*x])/((a + b)*(1 + Sec[c + d*x]))] + A*b*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/Sqrt[Sec[(c + d*x)/2]^2] + A*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*(-2*b*Sin[c + d*x] + a*Tan[c + d*x])))/(3*a^2*d*Sqrt[a + b*Cos[c + d*x]])","A",0
1437,1,620,403,15.6641322,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(a (A-C) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-A (a+b) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+a A \tan ^5\left(\frac{1}{2} (c+d x)\right)-a A \tan \left(\frac{1}{2} (c+d x)\right)+2 a C \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+2 a C \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-A b \tan ^5\left(\frac{1}{2} (c+d x)\right)+2 A b \tan ^3\left(\frac{1}{2} (c+d x)\right)-A b \tan \left(\frac{1}{2} (c+d x)\right)\right)}{a d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{a d}","\frac{2 A (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d \sqrt{\sec (c+d x)}}-\frac{2 A \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{\sec (c+d x)}}-\frac{2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d \sqrt{\sec (c+d x)}}",1,"(2*A*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) + (2*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-(a*A*Tan[(c + d*x)/2]) - A*b*Tan[(c + d*x)/2] + 2*A*b*Tan[(c + d*x)/2]^3 + a*A*Tan[(c + d*x)/2]^5 - A*b*Tan[(c + d*x)/2]^5 + 2*a*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*a*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - A*(a + b)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + a*(A - C)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(a*d*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","A",0
1438,1,338,453,10.8622125,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(4 A b \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+C \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))+2 C (a+b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-4 a C \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{b d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \cos (c+d x)}}","\frac{\sqrt{a+b} (a C+2 A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{\sec (c+d x)}}+\frac{a C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{b d}-\frac{C (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a + b)*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 4*A*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 4*a*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + C*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(b*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2])","A",0
1439,1,1399,515,14.7836525,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","\frac{C \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (2 (c+d x))}{4 b d}+\frac{-3 a^2 \sqrt{\frac{a-b}{a+b}} C \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 a b \sqrt{\frac{a-b}{a+b}} C \tan ^5\left(\frac{1}{2} (c+d x)\right)-6 a b \sqrt{\frac{a-b}{a+b}} C \tan ^3\left(\frac{1}{2} (c+d x)\right)+16 i A b^2 \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+6 i a^2 C \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+8 i b^2 C \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+3 a^2 \sqrt{\frac{a-b}{a+b}} C \tan \left(\frac{1}{2} (c+d x)\right)+3 a b \sqrt{\frac{a-b}{a+b}} C \tan \left(\frac{1}{2} (c+d x)\right)+3 i a (a-b) C E\left(i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 i \left(3 C a^2-b C a+4 A b^2+2 b^2 C\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+16 i A b^2 \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+6 i a^2 C \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+8 i b^2 C \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}}{4 b^2 \sqrt{\frac{a-b}{a+b}} d \sqrt{\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(\tan ^4\left(\frac{1}{2} (c+d x)\right)-1\right)}","-\frac{\sqrt{a+b} \left(3 a^2 C+4 b^2 (2 A+C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{\sec (c+d x)}}-\frac{3 a C \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b^2 d}-\frac{C (3 a-2 b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d \sqrt{\sec (c+d x)}}+\frac{3 C (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 b d \sqrt{\sec (c+d x)}}",1,"(C*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[2*(c + d*x)])/(4*b*d) + (3*a^2*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2] + 3*a*b*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2] - 6*a*b*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2]^3 - 3*a^2*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2]^5 + 3*a*b*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2]^5 + (16*I)*A*b^2*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (6*I)*a^2*C*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*I)*b^2*C*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (16*I)*A*b^2*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (6*I)*a^2*C*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*I)*b^2*C*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (3*I)*a*(a - b)*C*EllipticE[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*(4*A*b^2 + 3*a^2*C - a*b*C + 2*b^2*C)*EllipticF[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)])/(4*b^2*Sqrt[(a - b)/(a + b)]*d*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(-1 + Tan[(c + d*x)/2]^4))","C",0
1440,1,3767,534,25.759608,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + b*Cos[c + d*x])^(3/2),x]","\text{Result too large to show}","-\frac{2 \left(6 A b^2-a^2 (A-5 C)\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 a^2 d \left(a^2-b^2\right)}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 b \left(8 A b^2-a^2 (3 A-5 C)\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 a^3 d \left(a^2-b^2\right)}-\frac{2 \left(-\left(a^4 (3 A+5 C)\right)-2 a^2 b^2 (4 A-5 C)+16 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^5 d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{2 \left(a^3 (3 A+5 C)+2 a^2 b (2 A+5 C)+12 a A b^2+16 A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^4 d \sqrt{a+b} \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(3*a^4*A + 8*a^2*A*b^2 - 16*A*b^4 + 5*a^4*C - 10*a^2*b^2*C)*Sin[c + d*x])/(5*a^4*(a^2 - b^2)) + (2*(A*b^4*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(a^3*(a^2 - b^2)*(a + b*Cos[c + d*x])) - (6*A*b*Tan[c + d*x])/(5*a^3) + (2*A*Sec[c + d*x]*Tan[c + d*x])/(5*a^2)))/d + (2*((-3*a*A)/(5*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*A*b^2)/(5*a*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*A*b^4)/(5*a^3*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a*C)/((a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b^2*C)/(a*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*A*b*Sqrt[Sec[c + d*x]])/(5*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]) - (12*A*b^3*Sqrt[Sec[c + d*x]])/(5*a^2*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]) + (16*A*b^5*Sqrt[Sec[c + d*x]])/(5*a^4*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]) - (2*b*C*Sqrt[Sec[c + d*x]])/((a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]) + (2*b^3*C*Sqrt[Sec[c + d*x]])/(a^2*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]) - (3*A*b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]) - (8*A*b^3*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*a^2*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]) + (16*A*b^5*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*a^4*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]) - (b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]) + (2*b^3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(a^2*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(-16*A*b^4 + 2*a^2*b^2*(4*A - 5*C) + a^4*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(12*a*A*b^2 - 16*A*b^3 - 2*a^2*b*(2*A + 5*C) + a^3*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (16*A*b^4 + 2*a^2*b^2*(-4*A + 5*C) - a^4*(3*A + 5*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(5*a^4*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*((b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(-2*(a + b)*(-16*A*b^4 + 2*a^2*b^2*(4*A - 5*C) + a^4*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(12*a*A*b^2 - 16*A*b^3 - 2*a^2*b*(2*A + 5*C) + a^3*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (16*A*b^4 + 2*a^2*b^2*(-4*A + 5*C) - a^4*(3*A + 5*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(5*a^4*(a^2 - b^2)*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(-2*(a + b)*(-16*A*b^4 + 2*a^2*b^2*(4*A - 5*C) + a^4*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(12*a*A*b^2 - 16*A*b^3 - 2*a^2*b*(2*A + 5*C) + a^3*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (16*A*b^4 + 2*a^2*b^2*(-4*A + 5*C) - a^4*(3*A + 5*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(5*a^4*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((16*A*b^4 + 2*a^2*b^2*(-4*A + 5*C) - a^4*(3*A + 5*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 - ((a + b)*(-16*A*b^4 + 2*a^2*b^2*(4*A - 5*C) + a^4*(3*A + 5*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (a*(a + b)*(12*a*A*b^2 - 16*A*b^3 - 2*a^2*b*(2*A + 5*C) + a^3*(3*A + 5*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - ((a + b)*(-16*A*b^4 + 2*a^2*b^2*(4*A - 5*C) + a^4*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (a*(a + b)*(12*a*A*b^2 - 16*A*b^3 - 2*a^2*b*(2*A + 5*C) + a^3*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - b*(16*A*b^4 + 2*a^2*b^2*(-4*A + 5*C) - a^4*(3*A + 5*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (16*A*b^4 + 2*a^2*b^2*(-4*A + 5*C) - a^4*(3*A + 5*C))*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (16*A*b^4 + 2*a^2*b^2*(-4*A + 5*C) - a^4*(3*A + 5*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (a*(a + b)*(12*a*A*b^2 - 16*A*b^3 - 2*a^2*b*(2*A + 5*C) + a^3*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a + b)*(-16*A*b^4 + 2*a^2*b^2*(4*A - 5*C) + a^4*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(5*a^4*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((-2*(a + b)*(-16*A*b^4 + 2*a^2*b^2*(4*A - 5*C) + a^4*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(12*a*A*b^2 - 16*A*b^3 - 2*a^2*b*(2*A + 5*C) + a^3*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (16*A*b^4 + 2*a^2*b^2*(-4*A + 5*C) - a^4*(3*A + 5*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(5*a^4*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
1441,1,472,432,19.0629008,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)} \left(-\frac{2 \left(a^2 b C \sin (c+d x)+A b^3 \sin (c+d x)\right)}{a^2 \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{2 A \tan (c+d x)}{3 a^2}-\frac{2 b \left(5 a^2 A-3 a^2 C-8 A b^2\right) \sin (c+d x)}{3 a^3 \left(a^2-b^2\right)}\right)}{d}-\frac{2 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(b \left(a^2 (3 C-5 A)+8 A b^2\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))-2 a (a+b) \left(a^2 (A+3 C)-6 a A b+8 A b^2\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+2 b (a+b) \left(a^2 (3 C-5 A)+8 A b^2\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{3 a^3 d \left(a^2-b^2\right) \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(4 A b^2-a^2 (A-3 C)\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 a^2 d \left(a^2-b^2\right)}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 b \left(8 A b^2-a^2 (5 A-3 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{2 \left(a^2 (A+3 C)+6 a A b+8 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}",1,"(-2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*b*(a + b)*(8*A*b^2 + a^2*(-5*A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 2*a*(a + b)*(-6*a*A*b + 8*A*b^2 + a^2*(A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + b*(8*A*b^2 + a^2*(-5*A + 3*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*a^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((-2*b*(5*a^2*A - 8*A*b^2 - 3*a^2*C)*Sin[c + d*x])/(3*a^3*(a^2 - b^2)) - (2*(A*b^3*Sin[c + d*x] + a^2*b*C*Sin[c + d*x]))/(a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])) + (2*A*Tan[c + d*x])/(3*a^2)))/d","A",0
1442,1,456,348,17.7319168,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \left(a^2 A-a^2 C-2 A b^2\right) \sin (c+d x)}{a^2 \left(a^2-b^2\right)}+\frac{2 \left(a^2 C \sin (c+d x)+A b^2 \sin (c+d x)\right)}{a \left(a^2-b^2\right) (a+b \cos (c+d x))}\right)}{d}+\frac{2 \sqrt{2} \sqrt{\frac{\cos (c+d x)}{(\cos (c+d x)+1)^2}} \sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \left(\left(a^2 (C-A)+2 A b^2\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))-(a+b) \sec (c+d x) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2} \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))}{a+b}} \left(\left(a^2 (A-C)-2 A b^2\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+a (a (C-A)+2 A b) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\frac{1}{\cos (c+d x)+1}} \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 (a (A-C)+2 A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{2 \left(2 A b^2-a^2 (A-C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(a^2*A - 2*A*b^2 - a^2*C)*Sin[c + d*x])/(a^2*(a^2 - b^2)) + (2*(A*b^2*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(a*(a^2 - b^2)*(a + b*Cos[c + d*x]))))/d + (2*Sqrt[2]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])^2]*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(-((a + b)*((-2*A*b^2 + a^2*(A - C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*(2*A*b + a*(-A + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]) + (2*A*b^2 + a^2*(-A + C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(a^2*(a^2 - b^2)*d*Sqrt[(1 + Cos[c + d*x])^(-1)]*Sqrt[a + b*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2))","A",0
1443,1,1019,481,17.9675799,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{2 \left(C a^2+A b^2\right) \sin (c+d x)}{a b \left(a^2-b^2\right)}+\frac{2 \left(C \sin (c+d x) a^2+A b^2 \sin (c+d x)\right)}{b \left(b^2-a^2\right) (a+b \cos (c+d x))}\right)}{d}-\frac{2 \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)-a^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+a^2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 A b^3 \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 a^2 b C \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+2 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+A b^3 \tan \left(\frac{1}{2} (c+d x)\right)+a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+a^3 C \tan \left(\frac{1}{2} (c+d x)\right)+a^2 b C \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(C a^2+A b^2\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-a b (a+b) (A+C) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{b \left(a^3-a b^2\right) d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 b d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{2 (A b-a C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(A*b^2 + a^2*C)*Sin[c + d*x])/(a*b*(a^2 - b^2)) + (2*(A*b^2*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(b*(-a^2 + b^2)*(a + b*Cos[c + d*x]))))/d - (2*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(a*A*b^2*Tan[(c + d*x)/2] + A*b^3*Tan[(c + d*x)/2] + a^3*C*Tan[(c + d*x)/2] + a^2*b*C*Tan[(c + d*x)/2] - 2*A*b^3*Tan[(c + d*x)/2]^3 - 2*a^2*b*C*Tan[(c + d*x)/2]^3 - a*A*b^2*Tan[(c + d*x)/2]^5 + A*b^3*Tan[(c + d*x)/2]^5 - a^3*C*Tan[(c + d*x)/2]^5 + a^2*b*C*Tan[(c + d*x)/2]^5 - 2*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(A*b^2 + a^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - a*b*(a + b)*(A + C)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(b*(a^3 - a*b^2)*d*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",0
1444,1,1155,563,18.8255995,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]),x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{2 \left(C a^2+A b^2\right) \sin (c+d x)}{b^2 \left(b^2-a^2\right)}-\frac{2 \left(C \sin (c+d x) a^3+A b^2 \sin (c+d x) a\right)}{b^2 \left(b^2-a^2\right) (a+b \cos (c+d x))}\right)}{d}+\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(2 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 a^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-b^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+a b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 a^2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-4 A b^3 \tan ^3\left(\frac{1}{2} (c+d x)\right)+2 b^3 C \tan ^3\left(\frac{1}{2} (c+d x)\right)-6 a^2 b C \tan ^3\left(\frac{1}{2} (c+d x)\right)-6 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+6 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+2 A b^3 \tan \left(\frac{1}{2} (c+d x)\right)+2 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+3 a^3 C \tan \left(\frac{1}{2} (c+d x)\right)-b^3 C \tan \left(\frac{1}{2} (c+d x)\right)-a b^2 C \tan \left(\frac{1}{2} (c+d x)\right)+3 a^2 b C \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(3 C a^2+2 A b^2-b^2 C\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 b (a+b) (A b+a C) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-6 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+6 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{b^2 \left(b^2-a^2\right) d \sqrt{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1} \left(b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","\frac{\left(3 a^2 C+2 A b^2-b^2 C\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{b^2 d \left(a^2-b^2\right)}-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\left(3 a^2 C+2 A b^2-b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{\left(a C (3 a+b)+2 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{3 a C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(A*b^2 + a^2*C)*Sin[c + d*x])/(b^2*(-a^2 + b^2)) - (2*(a*A*b^2*Sin[c + d*x] + a^3*C*Sin[c + d*x]))/(b^2*(-a^2 + b^2)*(a + b*Cos[c + d*x]))))/d + (Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(2*a*A*b^2*Tan[(c + d*x)/2] + 2*A*b^3*Tan[(c + d*x)/2] + 3*a^3*C*Tan[(c + d*x)/2] + 3*a^2*b*C*Tan[(c + d*x)/2] - a*b^2*C*Tan[(c + d*x)/2] - b^3*C*Tan[(c + d*x)/2] - 4*A*b^3*Tan[(c + d*x)/2]^3 - 6*a^2*b*C*Tan[(c + d*x)/2]^3 + 2*b^3*C*Tan[(c + d*x)/2]^3 - 2*a*A*b^2*Tan[(c + d*x)/2]^5 + 2*A*b^3*Tan[(c + d*x)/2]^5 - 3*a^3*C*Tan[(c + d*x)/2]^5 + 3*a^2*b*C*Tan[(c + d*x)/2]^5 + a*b^2*C*Tan[(c + d*x)/2]^5 - b^3*C*Tan[(c + d*x)/2]^5 - 6*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 6*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(2*A*b^2 + 3*a^2*C - b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*b*(a + b)*(A*b + a*C)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(b^2*(-a^2 + b^2)*d*Sqrt[1 + Tan[(c + d*x)/2]^2]*(b*(-1 + Tan[(c + d*x)/2]^2) - a*(1 + Tan[(c + d*x)/2]^2)))","B",0
1445,1,2415,664,16.7809082,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)),x]","\text{Result too large to show}","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{b d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}+\frac{\left(5 a^2 C+4 A b^2-b^2 C\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 b^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left(15 a^2 C+8 A b^2+4 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^4 d \sqrt{\sec (c+d x)}}-\frac{a \left(15 a^2 C+8 A b^2-7 b^2 C\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b^3 d \left(a^2-b^2\right)}-\frac{\left(C \left(15 a^2+5 a b-2 b^2\right)+8 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{\left(15 a^2 C+8 A b^2-7 b^2 C\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*a*(A*b^2 + a^2*C)*Sin[c + d*x])/(b^3*(a^2 - b^2)) + (2*(a^2*A*b^2*Sin[c + d*x] + a^4*C*Sin[c + d*x]))/(b^3*(-a^2 + b^2)*(a + b*Cos[c + d*x])) + (C*Sin[2*(c + d*x)])/(4*b^2)))/d + (8*a^2*A*b^2*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2] + 8*a*A*b^3*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2] + 15*a^4*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2] + 15*a^3*b*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2] - 7*a^2*b^2*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2] - 7*a*b^3*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2] - 16*a*A*b^3*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^3 - 30*a^3*b*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2]^3 + 14*a*b^3*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2]^3 - 8*a^2*A*b^2*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^5 + 8*a*A*b^3*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^5 - 15*a^4*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2]^5 + 15*a^3*b*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2]^5 + 7*a^2*b^2*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2]^5 - 7*a*b^3*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2]^5 + (16*I)*a^2*A*b^2*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (16*I)*A*b^4*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (30*I)*a^4*C*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (22*I)*a^2*b^2*C*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (8*I)*b^4*C*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (16*I)*a^2*A*b^2*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (16*I)*A*b^4*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (30*I)*a^4*C*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (22*I)*a^2*b^2*C*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (8*I)*b^4*C*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + I*a*(a - b)*(8*A*b^2 + 15*a^2*C - 7*b^2*C)*EllipticE[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*(a - b)*(15*a^3*C + 10*a^2*b*C + 2*b^3*(2*A + C) + a*b^2*(8*A + C))*EllipticF[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)])/(4*b^3*Sqrt[(a - b)/(a + b)]*(a^2 - b^2)*d*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(-1 + Tan[(c + d*x)/2]^4))","C",0
1446,1,3973,589,26.2822126,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{4 \left(2 a^4 C+5 a^2 A b^2-3 A b^4\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{4 b \left(a^4 (4 A-3 C)-a^2 b^2 (14 A-C)+8 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^5 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}+\frac{2 \left(a^4 (A-5 C)-a^2 b^2 (13 A-C)+8 A b^4\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 a^3 d \left(a^2-b^2\right)^2}-\frac{2 \left(-\left(a^4 (A+3 C)\right)-a^3 (9 A b-3 b C)-2 a^2 b^2 (8 A-C)+12 a A b^3+16 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((-4*b*(4*a^4*A - 14*a^2*A*b^2 + 8*A*b^4 - 3*a^4*C + a^2*b^2*C)*Sin[c + d*x])/(3*a^4*(a^2 - b^2)^2) - (2*(A*b^3*Sin[c + d*x] + a^2*b*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) - (2*(11*a^2*A*b^3*Sin[c + d*x] - 7*A*b^5*Sin[c + d*x] + 5*a^4*b*C*Sin[c + d*x] - a^2*b^3*C*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) + (2*A*Tan[c + d*x])/(3*a^3)))/d + (4*((8*a*A*b)/(3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (28*A*b^3)/(3*a*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*A*b^5)/(3*a^3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*a*b*C)/((a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b^3*C)/(3*a*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^2*A*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) + (5*A*b^2*Sqrt[Sec[c + d*x]])/((a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) - (32*A*b^4*Sqrt[Sec[c + d*x]])/(3*a^2*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) + (16*A*b^6*Sqrt[Sec[c + d*x]])/(3*a^4*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) + (a^2*C*Sqrt[Sec[c + d*x]])/((a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) - (5*b^2*C*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) + (2*b^4*C*Sqrt[Sec[c + d*x]])/(3*a^2*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) + (8*A*b^2*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) - (28*A*b^4*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*a^2*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) + (16*A*b^6*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*a^4*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) - (2*b^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) + (2*b^4*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*a^2*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*b*(a + b)*(8*A*b^4 + a^4*(4*A - 3*C) + a^2*b^2*(-14*A + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*(a + b)*(12*a*A*b^3 - 16*A*b^4 + 2*a^2*b^2*(8*A - C) + 3*a^3*b*(-3*A + C) + a^4*(A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + b*(8*A*b^4 + a^4*(4*A - 3*C) + a^2*b^2*(-14*A + C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*a^4*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*((2*b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*b*(a + b)*(8*A*b^4 + a^4*(4*A - 3*C) + a^2*b^2*(-14*A + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*(a + b)*(12*a*A*b^3 - 16*A*b^4 + 2*a^2*b^2*(8*A - C) + 3*a^3*b*(-3*A + C) + a^4*(A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + b*(8*A*b^4 + a^4*(4*A - 3*C) + a^2*b^2*(-14*A + C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*a^4*(a^2 - b^2)^2*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*b*(a + b)*(8*A*b^4 + a^4*(4*A - 3*C) + a^2*b^2*(-14*A + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*(a + b)*(12*a*A*b^3 - 16*A*b^4 + 2*a^2*b^2*(8*A - C) + 3*a^3*b*(-3*A + C) + a^4*(A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + b*(8*A*b^4 + a^4*(4*A - 3*C) + a^2*b^2*(-14*A + C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*a^4*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((b*(8*A*b^4 + a^4*(4*A - 3*C) + a^2*b^2*(-14*A + C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + (b*(a + b)*(8*A*b^4 + a^4*(4*A - 3*C) + a^2*b^2*(-14*A + C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (a*(a + b)*(12*a*A*b^3 - 16*A*b^4 + 2*a^2*b^2*(8*A - C) + 3*a^3*b*(-3*A + C) + a^4*(A + 3*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/(2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]) + (b*(a + b)*(8*A*b^4 + a^4*(4*A - 3*C) + a^2*b^2*(-14*A + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (a*(a + b)*(12*a*A*b^3 - 16*A*b^4 + 2*a^2*b^2*(8*A - C) + 3*a^3*b*(-3*A + C) + a^4*(A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/(2*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - b^2*(8*A*b^4 + a^4*(4*A - 3*C) + a^2*b^2*(-14*A + C))*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - b*(8*A*b^4 + a^4*(4*A - 3*C) + a^2*b^2*(-14*A + C))*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + b*(8*A*b^4 + a^4*(4*A - 3*C) + a^2*b^2*(-14*A + C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (a*(a + b)*(12*a*A*b^3 - 16*A*b^4 + 2*a^2*b^2*(8*A - C) + 3*a^3*b*(-3*A + C) + a^4*(A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) + (b*(a + b)*(8*A*b^4 + a^4*(4*A - 3*C) + a^2*b^2*(-14*A + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(3*a^4*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*(2*b*(a + b)*(8*A*b^4 + a^4*(4*A - 3*C) + a^2*b^2*(-14*A + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*(a + b)*(12*a*A*b^3 - 16*A*b^4 + 2*a^2*b^2*(8*A - C) + 3*a^3*b*(-3*A + C) + a^4*(A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + b*(8*A*b^4 + a^4*(4*A - 3*C) + a^2*b^2*(-14*A + C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(3*a^4*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
1447,1,3741,489,26.1293273,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{4 \left(a^4 (-C)-a^2 b^2 (4 A+C)+2 A b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(3 a^4 (A-C)-a^2 b^2 (15 A+C)+8 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}+\frac{2 \left(-3 a^3 (A-C)-a^2 b (9 A+C)+6 a A b^2+8 A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 - 3*a^4*C - a^2*b^2*C)*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2) + (2*(A*b^2*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(3*a*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) + (4*(4*a^2*A*b^2*Sin[c + d*x] - 2*A*b^4*Sin[c + d*x] + a^4*C*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/d + (2*(-((a^2*A)/((a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])) + (5*A*b^2)/((a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*A*b^4)/(3*a^2*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^2*C)/((a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (b^2*C)/(3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (3*a*A*b*Sqrt[Sec[c + d*x]])/((a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) + (17*A*b^3*Sqrt[Sec[c + d*x]])/(3*a*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) - (8*A*b^5*Sqrt[Sec[c + d*x]])/(3*a^3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) - (a*b*C*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) + (b^3*C*Sqrt[Sec[c + d*x]])/(3*a*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) - (a*A*b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) + (5*A*b^3*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(a*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) - (8*A*b^5*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*a^3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) + (a*b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) + (b^3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*a*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(8*A*b^4 + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(-6*a*A*b^2 + 8*A*b^3 + 3*a^3*(A - C) - a^2*b*(9*A + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (8*A*b^4 + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*a^3*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*((b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(-2*(a + b)*(8*A*b^4 + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(-6*a*A*b^2 + 8*A*b^3 + 3*a^3*(A - C) - a^2*b*(9*A + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (8*A*b^4 + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*a^3*(a^2 - b^2)^2*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(-2*(a + b)*(8*A*b^4 + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(-6*a*A*b^2 + 8*A*b^3 + 3*a^3*(A - C) - a^2*b*(9*A + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (8*A*b^4 + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*a^3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1/2*((8*A*b^4 + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4) - ((a + b)*(8*A*b^4 + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (a*(a + b)*(-6*a*A*b^2 + 8*A*b^3 + 3*a^3*(A - C) - a^2*b*(9*A + C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - ((a + b)*(8*A*b^4 + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (a*(a + b)*(-6*a*A*b^2 + 8*A*b^3 + 3*a^3*(A - C) - a^2*b*(9*A + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + b*(8*A*b^4 + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (8*A*b^4 + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (8*A*b^4 + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (a*(a + b)*(-6*a*A*b^2 + 8*A*b^3 + 3*a^3*(A - C) - a^2*b*(9*A + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a + b)*(8*A*b^4 + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(3*a^3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((-2*(a + b)*(8*A*b^4 + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(-6*a*A*b^2 + 8*A*b^3 + 3*a^3*(A - C) - a^2*b*(9*A + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (8*A*b^4 + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(3*a^3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
1448,1,3279,456,22.3309244,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{4 b \left(A b^2-a^2 (3 A+2 C)\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(-\left(a^2 (3 A+C)\right)+3 a b (A+C)+2 A b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{4 b \left(A b^2-a^2 (3 A+2 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((4*b*(3*a^2*A - A*b^2 + 2*a^2*C)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2) + (2*(A*b^2*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(3*b*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (2*(-5*a^2*A*b^2*Sin[c + d*x] + A*b^4*Sin[c + d*x] + a^4*C*Sin[c + d*x] - 5*a^2*b^2*C*Sin[c + d*x]))/(3*a*b*(a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/d + (4*((-2*a*A*b)/((a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*A*b^3)/(3*a*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*a*b*C)/(3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^2*A*Sqrt[Sec[c + d*x]])/((a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) - (5*A*b^2*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) + (2*A*b^4*Sqrt[Sec[c + d*x]])/(3*a^2*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) + (a^2*C*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) - (b^2*C*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) - (2*A*b^2*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) + (2*A*b^4*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*a^2*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) - (4*b^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*b*(a + b)*(A*b^2 - a^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*(a + b)*(-2*A*b^2 + 3*a*b*(A + C) + a^2*(3*A + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + b*(A*b^2 - a^2*(3*A + 2*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(a^3 - a*b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*((2*b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*b*(a + b)*(A*b^2 - a^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*(a + b)*(-2*A*b^2 + 3*a*b*(A + C) + a^2*(3*A + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + b*(A*b^2 - a^2*(3*A + 2*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(a^3 - a*b^2)^2*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*b*(a + b)*(A*b^2 - a^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*(a + b)*(-2*A*b^2 + 3*a*b*(A + C) + a^2*(3*A + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + b*(A*b^2 - a^2*(3*A + 2*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(a^3 - a*b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((b*(A*b^2 - a^2*(3*A + 2*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + (b*(a + b)*(A*b^2 - a^2*(3*A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (a*(a + b)*(-2*A*b^2 + 3*a*b*(A + C) + a^2*(3*A + C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/(2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]) + (b*(a + b)*(A*b^2 - a^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (a*(a + b)*(-2*A*b^2 + 3*a*b*(A + C) + a^2*(3*A + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/(2*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - b^2*(A*b^2 - a^2*(3*A + 2*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - b*(A*b^2 - a^2*(3*A + 2*C))*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + b*(A*b^2 - a^2*(3*A + 2*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (a*(a + b)*(-2*A*b^2 + 3*a*b*(A + C) + a^2*(3*A + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) + (b*(a + b)*(A*b^2 - a^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(3*(a^3 - a*b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*(2*b*(a + b)*(A*b^2 - a^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*(a + b)*(-2*A*b^2 + 3*a*b*(A + C) + a^2*(3*A + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + b*(A*b^2 - a^2*(3*A + 2*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(3*(a^3 - a*b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
1449,1,1576,618,18.1381332,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]),x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{2 \left(3 C a^4-3 A b^2 a^2-7 b^2 C a^2-A b^4\right) \sin (c+d x)}{3 a b^2 \left(a^2-b^2\right)^2}-\frac{2 \left(C \sin (c+d x) a^3+A b^2 \sin (c+d x) a\right)}{3 b^2 \left(b^2-a^2\right) (a+b \cos (c+d x))^2}+\frac{4 \left(-2 C \sin (c+d x) a^4+A b^2 \sin (c+d x) a^2+4 b^2 C \sin (c+d x) a^2+A b^4 \sin (c+d x)\right)}{3 b^2 \left(b^2-a^2\right)^2 (a+b \cos (c+d x))}\right)}{d}+\frac{2 \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(3 C \tan ^5\left(\frac{1}{2} (c+d x)\right) a^5-3 C \tan \left(\frac{1}{2} (c+d x)\right) a^5+6 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a^5+6 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a^5-3 b C \tan ^5\left(\frac{1}{2} (c+d x)\right) a^4+6 b C \tan ^3\left(\frac{1}{2} (c+d x)\right) a^4-3 b C \tan \left(\frac{1}{2} (c+d x)\right) a^4-3 A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right) a^3-7 b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right) a^3+3 A b^2 \tan \left(\frac{1}{2} (c+d x)\right) a^3+7 b^2 C \tan \left(\frac{1}{2} (c+d x)\right) a^3-12 b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a^3-12 b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a^3+3 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right) a^2+7 b^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right) a^2-6 A b^3 \tan ^3\left(\frac{1}{2} (c+d x)\right) a^2-14 b^3 C \tan ^3\left(\frac{1}{2} (c+d x)\right) a^2+3 A b^3 \tan \left(\frac{1}{2} (c+d x)\right) a^2+7 b^3 C \tan \left(\frac{1}{2} (c+d x)\right) a^2-A b^4 \tan ^5\left(\frac{1}{2} (c+d x)\right) a+A b^4 \tan \left(\frac{1}{2} (c+d x)\right) a+b (a+b) \left(2 C a^2-3 b (A+C) a-b^2 (A+3 C)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a+6 b^4 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a+6 b^4 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a+A b^5 \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 A b^5 \tan ^3\left(\frac{1}{2} (c+d x)\right)+A b^5 \tan \left(\frac{1}{2} (c+d x)\right)-(a+b) \left(3 C a^4-b^2 (3 A+7 C) a^2-A b^4\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{3 a b^2 \left(a^2-b^2\right)^2 d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{3 b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(-3 a^4 C+a^2 b^2 (3 A+7 C)+A b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-3 a^4 C+a^2 b^2 (3 A+7 C)+A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 b^2 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{2 \left(3 a^3 C+a^2 b C-3 a b^2 (A+2 C)+A b^3\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^2 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}-\frac{2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(-3*a^2*A*b^2 - A*b^4 + 3*a^4*C - 7*a^2*b^2*C)*Sin[c + d*x])/(3*a*b^2*(a^2 - b^2)^2) - (2*(a*A*b^2*Sin[c + d*x] + a^3*C*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (4*(a^2*A*b^2*Sin[c + d*x] + A*b^4*Sin[c + d*x] - 2*a^4*C*Sin[c + d*x] + 4*a^2*b^2*C*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)^2*(a + b*Cos[c + d*x]))))/d + (2*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(3*a^3*A*b^2*Tan[(c + d*x)/2] + 3*a^2*A*b^3*Tan[(c + d*x)/2] + a*A*b^4*Tan[(c + d*x)/2] + A*b^5*Tan[(c + d*x)/2] - 3*a^5*C*Tan[(c + d*x)/2] - 3*a^4*b*C*Tan[(c + d*x)/2] + 7*a^3*b^2*C*Tan[(c + d*x)/2] + 7*a^2*b^3*C*Tan[(c + d*x)/2] - 6*a^2*A*b^3*Tan[(c + d*x)/2]^3 - 2*A*b^5*Tan[(c + d*x)/2]^3 + 6*a^4*b*C*Tan[(c + d*x)/2]^3 - 14*a^2*b^3*C*Tan[(c + d*x)/2]^3 - 3*a^3*A*b^2*Tan[(c + d*x)/2]^5 + 3*a^2*A*b^3*Tan[(c + d*x)/2]^5 - a*A*b^4*Tan[(c + d*x)/2]^5 + A*b^5*Tan[(c + d*x)/2]^5 + 3*a^5*C*Tan[(c + d*x)/2]^5 - 3*a^4*b*C*Tan[(c + d*x)/2]^5 - 7*a^3*b^2*C*Tan[(c + d*x)/2]^5 + 7*a^2*b^3*C*Tan[(c + d*x)/2]^5 + 6*a^5*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 12*a^3*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*a*b^4*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*a^5*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 12*a^3*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*a*b^4*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (a + b)*(-(A*b^4) + 3*a^4*C - a^2*b^2*(3*A + 7*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + a*b*(a + b)*(2*a^2*C - 3*a*b*(A + C) - b^2*(A + 3*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(3*a*b^2*(a^2 - b^2)^2*d*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",0
1450,1,1597,710,20.3383721,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{5/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)),x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(-\frac{4 \left(3 C a^4-5 b^2 C a^2-2 A b^4\right) \sin (c+d x)}{3 b^3 \left(a^2-b^2\right)^2}+\frac{2 \left(C \sin (c+d x) a^4+A b^2 \sin (c+d x) a^2\right)}{3 b^3 \left(b^2-a^2\right) (a+b \cos (c+d x))^2}+\frac{2 \left(7 C \sin (c+d x) a^5+A b^2 \sin (c+d x) a^3-11 b^2 C \sin (c+d x) a^3-5 A b^4 \sin (c+d x) a\right)}{3 b^3 \left(b^2-a^2\right)^2 (a+b \cos (c+d x))}\right)}{d}+\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(15 C \tan ^5\left(\frac{1}{2} (c+d x)\right) a^5-15 C \tan \left(\frac{1}{2} (c+d x)\right) a^5+30 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a^5+30 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a^5-15 b C \tan ^5\left(\frac{1}{2} (c+d x)\right) a^4+30 b C \tan ^3\left(\frac{1}{2} (c+d x)\right) a^4-15 b C \tan \left(\frac{1}{2} (c+d x)\right) a^4-26 b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right) a^3+26 b^2 C \tan \left(\frac{1}{2} (c+d x)\right) a^3-60 b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a^3-60 b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a^3+26 b^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right) a^2-52 b^3 C \tan ^3\left(\frac{1}{2} (c+d x)\right) a^2+26 b^3 C \tan \left(\frac{1}{2} (c+d x)\right) a^2-8 A b^4 \tan ^5\left(\frac{1}{2} (c+d x)\right) a+3 b^4 C \tan ^5\left(\frac{1}{2} (c+d x)\right) a+8 A b^4 \tan \left(\frac{1}{2} (c+d x)\right) a-3 b^4 C \tan \left(\frac{1}{2} (c+d x)\right) a+30 b^4 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a+30 b^4 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a+8 A b^5 \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 b^5 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 A b^5 \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 b^5 C \tan ^3\left(\frac{1}{2} (c+d x)\right)+8 A b^5 \tan \left(\frac{1}{2} (c+d x)\right)-3 b^5 C \tan \left(\frac{1}{2} (c+d x)\right)-(a+b) \left(\left(15 a^4-26 b^2 a^2+3 b^4\right) C-8 A b^4\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 b (a+b) \left(-5 C a^3+3 b C a^2+b^2 (A+6 C) a+3 A b^3\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{3 b^3 \left(a^2-b^2\right)^2 d \sqrt{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1} \left(b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","-\frac{2 \left(a^2 C+A b^2\right) \sin (c+d x)}{3 b d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left(-5 a^4 C+a^2 b^2 (A+9 C)+3 A b^4\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\left(8 A b^4-C \left(15 a^4-26 a^2 b^2+3 b^4\right)\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)^2}+\frac{\left(8 A b^4-C \left(15 a^4-26 a^2 b^2+3 b^4\right)\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^3 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}-\frac{\left(-15 a^4 C-5 a^3 b C+21 a^2 b^2 C-a b^3 (2 A-3 C)+6 A b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a b^3 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}+\frac{5 a C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^4 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((-4*(-2*A*b^4 + 3*a^4*C - 5*a^2*b^2*C)*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2) + (2*(a^2*A*b^2*Sin[c + d*x] + a^4*C*Sin[c + d*x]))/(3*b^3*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (2*(a^3*A*b^2*Sin[c + d*x] - 5*a*A*b^4*Sin[c + d*x] + 7*a^5*C*Sin[c + d*x] - 11*a^3*b^2*C*Sin[c + d*x]))/(3*b^3*(-a^2 + b^2)^2*(a + b*Cos[c + d*x]))))/d + (Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(8*a*A*b^4*Tan[(c + d*x)/2] + 8*A*b^5*Tan[(c + d*x)/2] - 15*a^5*C*Tan[(c + d*x)/2] - 15*a^4*b*C*Tan[(c + d*x)/2] + 26*a^3*b^2*C*Tan[(c + d*x)/2] + 26*a^2*b^3*C*Tan[(c + d*x)/2] - 3*a*b^4*C*Tan[(c + d*x)/2] - 3*b^5*C*Tan[(c + d*x)/2] - 16*A*b^5*Tan[(c + d*x)/2]^3 + 30*a^4*b*C*Tan[(c + d*x)/2]^3 - 52*a^2*b^3*C*Tan[(c + d*x)/2]^3 + 6*b^5*C*Tan[(c + d*x)/2]^3 - 8*a*A*b^4*Tan[(c + d*x)/2]^5 + 8*A*b^5*Tan[(c + d*x)/2]^5 + 15*a^5*C*Tan[(c + d*x)/2]^5 - 15*a^4*b*C*Tan[(c + d*x)/2]^5 - 26*a^3*b^2*C*Tan[(c + d*x)/2]^5 + 26*a^2*b^3*C*Tan[(c + d*x)/2]^5 + 3*a*b^4*C*Tan[(c + d*x)/2]^5 - 3*b^5*C*Tan[(c + d*x)/2]^5 + 30*a^5*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 60*a^3*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*a*b^4*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*a^5*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 60*a^3*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*a*b^4*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (a + b)*(-8*A*b^4 + (15*a^4 - 26*a^2*b^2 + 3*b^4)*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*b*(a + b)*(3*A*b^3 - 5*a^3*C + 3*a^2*b*C + a*b^2*(A + 6*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(3*b^3*(a^2 - b^2)^2*d*Sqrt[1 + Tan[(c + d*x)/2]^2]*(b*(-1 + Tan[(c + d*x)/2]^2) - a*(1 + Tan[(c + d*x)/2]^2)))","B",0
1451,1,191,230,3.1614293,"\int (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{\sec ^{\frac{7}{2}}(c+d x) \left(40 \cos ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a A+7 a C+7 b B)-168 \cos ^{\frac{7}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a B+3 A b+5 b C)+2 \sin (c+d x) (21 \cos (c+d x) (13 a B+13 A b+15 b C)+10 \cos (2 (c+d x)) (5 a A+7 a C+7 b B)+110 a A+63 a B \cos (3 (c+d x))+70 a C+63 A b \cos (3 (c+d x))+70 b B+105 b C \cos (3 (c+d x)))\right)}{420 d}","\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (5 a A+7 a C+7 b B)}{21 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} (3 a B+3 A b+5 b C)}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a A+7 a C+7 b B)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a B+3 A b+5 b C)}{5 d}+\frac{2 (a B+A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}",1,"(Sec[c + d*x]^(7/2)*(-168*(3*A*b + 3*a*B + 5*b*C)*Cos[c + d*x]^(7/2)*EllipticE[(c + d*x)/2, 2] + 40*(5*a*A + 7*b*B + 7*a*C)*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] + 2*(110*a*A + 70*b*B + 70*a*C + 21*(13*A*b + 13*a*B + 15*b*C)*Cos[c + d*x] + 10*(5*a*A + 7*b*B + 7*a*C)*Cos[2*(c + d*x)] + 63*A*b*Cos[3*(c + d*x)] + 63*a*B*Cos[3*(c + d*x)] + 105*b*C*Cos[3*(c + d*x)])*Sin[c + d*x]))/(420*d)","A",1
1452,1,149,192,1.2121823,"\int (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a B+A b+3 b C)-3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a A+5 a C+5 b B)+\frac{\sin (c+d x) (3 (\cos (2 (c+d x)) (3 a A+5 a C+5 b B)+5 a (A+C)+5 b B)+10 (a B+A b) \cos (c+d x))}{2 \cos ^{\frac{5}{2}}(c+d x)}\right)}{15 d}","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} (3 a A+5 a C+5 b B)}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a B+A b+3 b C)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a A+5 a C+5 b B)}{5 d}+\frac{2 (a B+A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}",1,"(2*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-3*(3*a*A + 5*b*B + 5*a*C)*EllipticE[(c + d*x)/2, 2] + 5*(A*b + a*B + 3*b*C)*EllipticF[(c + d*x)/2, 2] + ((10*(A*b + a*B)*Cos[c + d*x] + 3*(5*b*B + 5*a*(A + C) + (3*a*A + 5*b*B + 5*a*C)*Cos[2*(c + d*x)]))*Sin[c + d*x])/(2*Cos[c + d*x]^(5/2))))/(15*d)","A",1
1453,1,112,151,1.076649,"\int (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a (A+3 C)+3 b B)-3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a B+A b-b C)+\frac{\sin (c+d x) (3 (a B+A b) \cos (c+d x)+a A)}{\cos ^{\frac{3}{2}}(c+d x)}\right)}{3 d}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a (A+3 C)+3 b B)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a B+A b-b C)}{d}+\frac{2 (a B+A b) \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}",1,"(2*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-3*(A*b + a*B - b*C)*EllipticE[(c + d*x)/2, 2] + (3*b*B + a*(A + 3*C))*EllipticF[(c + d*x)/2, 2] + ((a*A + 3*(A*b + a*B)*Cos[c + d*x])*Sin[c + d*x])/Cos[c + d*x]^(3/2)))/(3*d)","A",1
1454,1,109,147,0.7203995,"\int (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \left(2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a B+3 A b+b C)+6 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a (C-A)+b B)+2 \sin (c+d x) (3 a A+b C \cos (c+d x))\right)}{3 d}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a B+3 A b+b C)}{3 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (b B-a (A-C))}{d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 b C \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Sec[c + d*x]]*(6*(b*B + a*(-A + C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*(3*A*b + 3*a*B + b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(3*a*A + b*C*Cos[c + d*x])*Sin[c + d*x]))/(3*d)","A",1
1455,1,116,156,0.9238873,"\int (a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(10 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a A+a C+b B)+6 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a B+5 A b+3 b C)+\sin (2 (c+d x)) (5 a C+5 b B+3 b C \cos (c+d x))\right)}{15 d}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a (3 A+C)+b B)}{3 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a B+5 A b+3 b C)}{5 d}+\frac{2 (a C+b B) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 b C \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(6*(5*A*b + 5*a*B + 3*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 10*(3*a*A + b*B + a*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (5*b*B + 5*a*C + 3*b*C*Cos[c + d*x])*Sin[2*(c + d*x)]))/(15*d)","A",1
1456,1,139,194,1.1121809,"\int \frac{(a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) (42 (a C+b B) \cos (c+d x)+70 a B+70 A b+15 b C \cos (2 (c+d x))+65 b C)+20 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (7 a B+7 A b+5 b C)+84 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a A+3 a C+3 b B)\right)}{210 d}","\frac{2 \sin (c+d x) (7 a B+7 A b+5 b C)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (7 a B+7 A b+5 b C)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a A+3 a C+3 b B)}{5 d}+\frac{2 (a C+b B) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b C \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(84*(5*a*A + 3*b*B + 3*a*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 20*(7*A*b + 7*a*B + 5*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (70*A*b + 70*a*B + 65*b*C + 42*(b*B + a*C)*Cos[c + d*x] + 15*b*C*Cos[2*(c + d*x)])*Sin[2*(c + d*x)]))/(210*d)","A",1
1457,1,165,230,1.4989985,"\int \frac{(a+b \cos (c+d x)) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) (7 \cos (c+d x) (36 a B+36 A b+43 b C)+5 (84 a A+18 (a C+b B) \cos (2 (c+d x))+78 a C+78 b B+7 b C \cos (3 (c+d x))))+120 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (7 a A+5 a C+5 b B)+168 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (9 a B+9 A b+7 b C)\right)}{1260 d}","\frac{2 \sin (c+d x) (9 a B+9 A b+7 b C)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) (7 a A+5 a C+5 b B)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (7 a A+5 a C+5 b B)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (9 a B+9 A b+7 b C)}{15 d}+\frac{2 (a C+b B) \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 b C \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(168*(9*A*b + 9*a*B + 7*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 120*(7*a*A + 5*b*B + 5*a*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (7*(36*A*b + 36*a*B + 43*b*C)*Cos[c + d*x] + 5*(84*a*A + 78*b*B + 78*a*C + 18*(b*B + a*C)*Cos[2*(c + d*x)] + 7*b*C*Cos[3*(c + d*x)]))*Sin[2*(c + d*x)]))/(1260*d)","A",1
1458,1,357,342,6.6567171,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{2}{15} \sin (c+d x) \left(7 a^2 A+9 a^2 C+18 a b B+9 A b^2+15 b^2 C\right)+\frac{2}{45} \sec ^2(c+d x) \left(7 a^2 A \sin (c+d x)+9 a^2 C \sin (c+d x)+18 a b B \sin (c+d x)+9 A b^2 \sin (c+d x)\right)+\frac{2}{21} \sec (c+d x) \left(5 a^2 B \sin (c+d x)+10 a A b \sin (c+d x)+14 a b C \sin (c+d x)+7 b^2 B \sin (c+d x)\right)+\frac{2}{7} \sec ^3(c+d x) \left(a^2 B \sin (c+d x)+2 a A b \sin (c+d x)\right)+\frac{2}{9} a^2 A \tan (c+d x) \sec ^3(c+d x)\right)}{d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(25 a^2 B+50 a A b+70 a b C+35 b^2 B\right)+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-49 a^2 A-63 a^2 C-126 a b B-63 A b^2-105 b^2 C\right)}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}}{105 d}","\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^2 (7 A+9 C)+18 a b B+4 A b^2\right)}{45 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(5 a^2 B+10 a A b+14 a b C+7 b^2 B\right)}{21 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right)}{15 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 B+10 a A b+14 a b C+7 b^2 B\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right)}{15 d}+\frac{2 a (9 a B+4 A b) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}",1,"((2*(-49*a^2*A - 63*A*b^2 - 126*a*b*B - 63*a^2*C - 105*b^2*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(50*a*A*b + 25*a^2*B + 35*b^2*B + 70*a*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(105*d) + (Sqrt[Sec[c + d*x]]*((2*(7*a^2*A + 9*A*b^2 + 18*a*b*B + 9*a^2*C + 15*b^2*C)*Sin[c + d*x])/15 + (2*Sec[c + d*x]^3*(2*a*A*b*Sin[c + d*x] + a^2*B*Sin[c + d*x]))/7 + (2*Sec[c + d*x]^2*(7*a^2*A*Sin[c + d*x] + 9*A*b^2*Sin[c + d*x] + 18*a*b*B*Sin[c + d*x] + 9*a^2*C*Sin[c + d*x]))/45 + (2*Sec[c + d*x]*(10*a*A*b*Sin[c + d*x] + 5*a^2*B*Sin[c + d*x] + 7*b^2*B*Sin[c + d*x] + 14*a*b*C*Sin[c + d*x]))/21 + (2*a^2*A*Sec[c + d*x]^3*Tan[c + d*x])/9))/d","A",1
1459,1,221,288,4.3654396,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 \sqrt{\sec (c+d x)} \left(21 \sin (c+d x) \left(3 a^2 B+2 a b (3 A+5 C)+5 b^2 B\right)+5 \tan (c+d x) \left(a^2 (5 A+7 C)+14 a b B+7 A b^2\right)+5 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (5 A+7 C)+14 a b B+7 b^2 (A+3 C)\right)-21 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 B+2 a b (3 A+5 C)+5 b^2 B\right)+15 a^2 A \tan (c+d x) \sec ^2(c+d x)+21 a (a B+2 A b) \tan (c+d x) \sec (c+d x)\right)}{105 d}","\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 (5 A+7 C)+14 a b B+4 A b^2\right)}{21 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(3 a^2 B+6 a A b+10 a b C+5 b^2 B\right)}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (5 A+7 C)+14 a b B+7 b^2 (A+3 C)\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 B+6 a A b+10 a b C+5 b^2 B\right)}{5 d}+\frac{2 a (7 a B+4 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}{7 d}",1,"(2*Sqrt[Sec[c + d*x]]*(-21*(3*a^2*B + 5*b^2*B + 2*a*b*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 5*(14*a*b*B + 7*b^2*(A + 3*C) + a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 21*(3*a^2*B + 5*b^2*B + 2*a*b*(3*A + 5*C))*Sin[c + d*x] + 5*(7*A*b^2 + 14*a*b*B + a^2*(5*A + 7*C))*Tan[c + d*x] + 21*a*(2*A*b + a*B)*Sec[c + d*x]*Tan[c + d*x] + 15*a^2*A*Sec[c + d*x]^2*Tan[c + d*x]))/(105*d)","A",1
1460,1,193,240,2.2878754,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 B+2 a b (A+3 C)+3 b^2 B\right)-3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (3 A+5 C)+10 a b B+5 b^2 (A-C)\right)+\frac{\sin (c+d x) \left(3 \cos (2 (c+d x)) \left(a^2 (3 A+5 C)+10 a b B+5 A b^2\right)+15 \left(a^2 (A+C)+2 a b B+A b^2\right)+10 a (a B+2 A b) \cos (c+d x)\right)}{2 \cos ^{\frac{5}{2}}(c+d x)}\right)}{15 d}","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 (3 A+5 C)+10 a b B+4 A b^2\right)}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 B+2 a b (A+3 C)+3 b^2 B\right)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (3 A+5 C)+10 a b B+5 b^2 (A-C)\right)}{5 d}+\frac{2 a (5 a B+4 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{5 d}",1,"(2*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-3*(10*a*b*B + 5*b^2*(A - C) + a^2*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2] + 5*(a^2*B + 3*b^2*B + 2*a*b*(A + 3*C))*EllipticF[(c + d*x)/2, 2] + ((15*(A*b^2 + 2*a*b*B + a^2*(A + C)) + 10*a*(2*A*b + a*B)*Cos[c + d*x] + 3*(5*A*b^2 + 10*a*b*B + a^2*(3*A + 5*C))*Cos[2*(c + d*x)])*Sin[c + d*x])/(2*Cos[c + d*x]^(5/2))))/(15*d)","A",1
1461,1,158,220,1.1853601,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (A+3 C)+6 a b B+b^2 (3 A+C)\right)-6 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 B+2 a b (A-C)-b^2 B\right)+\frac{\sin (c+d x) \left(2 a^2 A+6 a (a B+2 A b) \cos (c+d x)+b^2 C \cos (2 (c+d x))+b^2 C\right)}{\cos ^{\frac{3}{2}}(c+d x)}\right)}{3 d}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (A+3 C)+6 a b B+b^2 (3 A+C)\right)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 B+2 a b (A-C)-b^2 B\right)}{d}+\frac{2 a (3 a B+4 A b) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{3 d}-\frac{2 b^2 (A-C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-6*(a^2*B - b^2*B + 2*a*b*(A - C))*EllipticE[(c + d*x)/2, 2] + 2*(6*a*b*B + b^2*(3*A + C) + a^2*(A + 3*C))*EllipticF[(c + d*x)/2, 2] + ((2*a^2*A + b^2*C + 6*a*(2*A*b + a*B)*Cos[c + d*x] + b^2*C*Cos[2*(c + d*x)])*Sin[c + d*x])/Cos[c + d*x]^(3/2)))/(3*d)","A",1
1462,1,165,229,1.2901693,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(20 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 B+2 a b (3 A+C)+b^2 B\right)+12 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^2 (A-C)+10 a b B+b^2 (5 A+3 C)\right)+\frac{2 \sin (c+d x) \left(3 \left(10 a^2 A+b^2 C \cos (2 (c+d x))+b^2 C\right)+10 b (2 a C+b B) \cos (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)}{30 d}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 B+2 a b (3 A+C)+b^2 B\right)}{3 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^2 (A-C)+10 a b B+b^2 (5 A+3 C)\right)}{5 d}-\frac{2 b \sin (c+d x) (6 a A-2 a C-b B)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}{d}-\frac{2 b^2 (5 A-C) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(12*(10*a*b*B - 5*a^2*(A - C) + b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2] + 20*(3*a^2*B + b^2*B + 2*a*b*(3*A + C))*EllipticF[(c + d*x)/2, 2] + (2*(10*b*(b*B + 2*a*C)*Cos[c + d*x] + 3*(10*a^2*A + b^2*C + b^2*C*Cos[2*(c + d*x)]))*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(30*d)","A",1
1463,1,183,243,0.9862374,"\int (a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(2 \sin (2 (c+d x)) \left(5 \left(14 a^2 C+28 a b B+14 A b^2+3 b^2 C \cos (2 (c+d x))+13 b^2 C\right)+42 b (2 a C+b B) \cos (c+d x)\right)+40 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^2 (3 A+C)+14 a b B+b^2 (7 A+5 C)\right)+168 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 B+2 a b (5 A+3 C)+3 b^2 B\right)\right)}{420 d}","\frac{2 \sin (c+d x) \left(4 a^2 C+14 a b B+7 A b^2+5 b^2 C\right)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^2 (3 A+C)+14 a b B+b^2 (7 A+5 C)\right)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 B+10 a A b+6 a b C+3 b^2 B\right)}{5 d}+\frac{2 b (4 a C+7 b B) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Sec[c + d*x]]*(168*(5*a^2*B + 3*b^2*B + 2*a*b*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 40*(14*a*b*B + 7*a^2*(3*A + C) + b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(42*b*(b*B + 2*a*C)*Cos[c + d*x] + 5*(14*A*b^2 + 28*a*b*B + 14*a^2*C + 13*b^2*C + 3*b^2*C*Cos[2*(c + d*x)]))*Sin[2*(c + d*x)]))/(420*d)","A",1
1464,1,218,291,1.3988439,"\int \frac{(a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(2 \sin (2 (c+d x)) \left(7 \cos (c+d x) \left(36 a^2 C+72 a b B+36 A b^2+43 b^2 C\right)+5 \left(84 a^2 B+168 a A b+18 b (2 a C+b B) \cos (2 (c+d x))+156 a b C+78 b^2 B+7 b^2 C \cos (3 (c+d x))\right)\right)+240 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^2 B+2 a b (7 A+5 C)+5 b^2 B\right)+336 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 (5 A+3 C)+18 a b B+b^2 (9 A+7 C)\right)\right)}{2520 d}","\frac{2 \sin (c+d x) \left(4 a^2 C+18 a b B+9 A b^2+7 b^2 C\right)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(7 a^2 B+14 a A b+10 a b C+5 b^2 B\right)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^2 B+14 a A b+10 a b C+5 b^2 B\right)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 (5 A+3 C)+18 a b B+b^2 (9 A+7 C)\right)}{15 d}+\frac{2 b (4 a C+9 b B) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^2}{9 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(336*(18*a*b*B + 3*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 240*(7*a^2*B + 5*b^2*B + 2*a*b*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(7*(36*A*b^2 + 72*a*b*B + 36*a^2*C + 43*b^2*C)*Cos[c + d*x] + 5*(168*a*A*b + 84*a^2*B + 78*b^2*B + 156*a*b*C + 18*b*(b*B + 2*a*C)*Cos[2*(c + d*x)] + 7*b^2*C*Cos[3*(c + d*x)]))*Sin[2*(c + d*x)]))/(2520*d)","A",1
1465,1,259,345,2.0601241,"\int \frac{(a+b \cos (c+d x))^2 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \left(2 \sin (2 (c+d x)) \left(154 \cos (c+d x) \left(36 a^2 B+72 a A b+86 a b C+43 b^2 B\right)+5 \left(36 \cos (2 (c+d x)) \left(11 a^2 C+22 a b B+11 A b^2+16 b^2 C\right)+132 a^2 (14 A+13 C)+154 b (2 a C+b B) \cos (3 (c+d x))+3432 a b B+3 b^2 (572 A+531 C)+63 b^2 C \cos (4 (c+d x))\right)\right)+480 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right)+7392 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(9 a^2 B+2 a b (9 A+7 C)+7 b^2 B\right)\right)}{55440 d}","\frac{2 \sin (c+d x) \left(9 a^2 B+18 a A b+14 a b C+7 b^2 B\right)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(4 a^2 C+22 a b B+11 A b^2+9 b^2 C\right)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(11 a^2 (7 A+5 C)+110 a b B+5 b^2 (11 A+9 C)\right)}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(9 a^2 B+18 a A b+14 a b C+7 b^2 B\right)}{15 d}+\frac{2 b (4 a C+11 b B) \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^2}{11 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(7392*(9*a^2*B + 7*b^2*B + 2*a*b*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 480*(110*a*b*B + 11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(154*(72*a*A*b + 36*a^2*B + 43*b^2*B + 86*a*b*C)*Cos[c + d*x] + 5*(3432*a*b*B + 132*a^2*(14*A + 13*C) + 3*b^2*(572*A + 531*C) + 36*(11*A*b^2 + 22*a*b*B + 11*a^2*C + 16*b^2*C)*Cos[2*(c + d*x)] + 154*b*(b*B + 2*a*C)*Cos[3*(c + d*x)] + 63*b^2*C*Cos[4*(c + d*x)]))*Sin[2*(c + d*x)]))/(55440*d)","A",1
1466,1,416,397,6.9248026,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{2}{9} a^3 A \tan (c+d x) \sec ^3(c+d x)+\frac{2}{45} \sec ^2(c+d x) \left(7 a^3 A \sin (c+d x)+9 a^3 C \sin (c+d x)+27 a^2 b B \sin (c+d x)+27 a A b^2 \sin (c+d x)\right)+\frac{2}{15} \sin (c+d x) \left(7 a^3 A+9 a^3 C+27 a^2 b B+27 a A b^2+45 a b^2 C+15 b^3 B\right)+\frac{2}{21} \sec (c+d x) \left(5 a^3 B \sin (c+d x)+15 a^2 A b \sin (c+d x)+21 a^2 b C \sin (c+d x)+21 a b^2 B \sin (c+d x)+7 A b^3 \sin (c+d x)\right)+\frac{2}{7} \sec ^3(c+d x) \left(a^3 B \sin (c+d x)+3 a^2 A b \sin (c+d x)\right)\right)}{d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(25 a^3 B+75 a^2 A b+105 a^2 b C+105 a b^2 B+35 A b^3+105 b^3 C\right)+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-49 a^3 A-63 a^3 C-189 a^2 b B-189 a A b^2-315 a b^2 C-105 b^3 B\right)}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}}{105 d}","\frac{2 a \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right)}{315 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right)}{63 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right)}{15 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^3 B+3 a^2 b (5 A+7 C)+21 a b^2 B+7 b^3 (A+3 C)\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right)}{15 d}+\frac{2 (3 a B+2 A b) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}{21 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^3}{9 d}",1,"((2*(-49*a^3*A - 189*a*A*b^2 - 189*a^2*b*B - 105*b^3*B - 63*a^3*C - 315*a*b^2*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(75*a^2*A*b + 35*A*b^3 + 25*a^3*B + 105*a*b^2*B + 105*a^2*b*C + 105*b^3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(105*d) + (Sqrt[Sec[c + d*x]]*((2*(7*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 15*b^3*B + 9*a^3*C + 45*a*b^2*C)*Sin[c + d*x])/15 + (2*Sec[c + d*x]^3*(3*a^2*A*b*Sin[c + d*x] + a^3*B*Sin[c + d*x]))/7 + (2*Sec[c + d*x]^2*(7*a^3*A*Sin[c + d*x] + 27*a*A*b^2*Sin[c + d*x] + 27*a^2*b*B*Sin[c + d*x] + 9*a^3*C*Sin[c + d*x]))/45 + (2*Sec[c + d*x]*(15*a^2*A*b*Sin[c + d*x] + 7*A*b^3*Sin[c + d*x] + 5*a^3*B*Sin[c + d*x] + 21*a*b^2*B*Sin[c + d*x] + 21*a^2*b*C*Sin[c + d*x]))/21 + (2*a^3*A*Sec[c + d*x]^3*Tan[c + d*x])/9))/d","A",1
1467,1,255,334,4.5736078,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 \sqrt{\sec (c+d x)} \left(15 a^3 A \tan (c+d x) \sec ^2(c+d x)+5 a \tan (c+d x) \left(a^2 (5 A+7 C)+21 a b B+21 A b^2\right)+21 a^2 (a B+3 A b) \tan (c+d x) \sec (c+d x)+21 \sin (c+d x) \left(3 a^3 B+3 a^2 b (3 A+5 C)+15 a b^2 B+5 A b^3\right)+5 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right)-21 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^3 B+3 a^2 b (3 A+5 C)+15 a b^2 B+5 b^3 (A-C)\right)\right)}{105 d}","\frac{2 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right)}{105 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(21 a^3 B+21 a^2 b (3 A+5 C)+98 a b^2 B+24 A b^3\right)}{35 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^3 B+3 a^2 b (3 A+5 C)+15 a b^2 B+5 b^3 (A-C)\right)}{5 d}+\frac{2 (7 a B+6 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^3}{7 d}",1,"(2*Sqrt[Sec[c + d*x]]*(-21*(3*a^3*B + 15*a*b^2*B + 5*b^3*(A - C) + 3*a^2*b*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 5*(21*a^2*b*B + 21*b^3*B + 21*a*b^2*(A + 3*C) + a^3*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 21*(5*A*b^3 + 3*a^3*B + 15*a*b^2*B + 3*a^2*b*(3*A + 5*C))*Sin[c + d*x] + 5*a*(21*A*b^2 + 21*a*b*B + a^2*(5*A + 7*C))*Tan[c + d*x] + 21*a^2*(3*A*b + a*B)*Sec[c + d*x]*Tan[c + d*x] + 15*a^3*A*Sec[c + d*x]^2*Tan[c + d*x]))/(105*d)","A",1
1468,1,276,313,2.2403742,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{2}{5} a^3 A \tan (c+d x) \sec (c+d x)+\frac{2}{5} a \sin (c+d x) \left(3 a^2 A+5 a^2 C+15 a b B+15 A b^2\right)+\frac{2}{3} \sec (c+d x) \left(a^3 B \sin (c+d x)+3 a^2 A b \sin (c+d x)\right)+\frac{1}{3} b^3 C \sin (2 (c+d x))\right)}{d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^3 B+15 a^2 A b+45 a^2 b C+45 a b^2 B+15 A b^3+5 b^3 C\right)+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-9 a^3 A-15 a^3 C-45 a^2 b B-45 a A b^2+45 a b^2 C+15 b^3 B\right)}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}}{15 d}","\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)} \left(3 a^2 (3 A+5 C)+35 a b B+24 A b^2\right)}{15 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 B+3 a^2 b (A+3 C)+9 a b^2 B+b^3 (3 A+C)\right)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (3 A+5 C)+15 a^2 b B+15 a b^2 (A-C)-5 b^3 B\right)}{5 d}-\frac{2 b^2 \sin (c+d x) (5 a B+9 A b-5 b C)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 (5 a B+6 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}{15 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3}{5 d}",1,"((2*(-9*a^3*A - 45*a*A*b^2 - 45*a^2*b*B + 15*b^3*B - 15*a^3*C + 45*a*b^2*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(15*a^2*A*b + 15*A*b^3 + 5*a^3*B + 45*a*b^2*B + 45*a^2*b*C + 5*b^3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (Sqrt[Sec[c + d*x]]*((2*a*(3*a^2*A + 15*A*b^2 + 15*a*b*B + 5*a^2*C)*Sin[c + d*x])/5 + (2*Sec[c + d*x]*(3*a^2*A*b*Sin[c + d*x] + a^3*B*Sin[c + d*x]))/3 + (b^3*C*Sin[2*(c + d*x)])/3 + (2*a^3*A*Sec[c + d*x]*Tan[c + d*x])/5))/d","A",1
1469,1,224,311,2.0662554,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(20 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (A+3 C)+9 a^2 b B+3 a b^2 (3 A+C)+b^3 B\right)-12 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^3 B+15 a^2 b (A-C)-15 a b^2 B-b^3 (5 A+3 C)\right)+\frac{\sin (c+d x) \left(20 a^3 A+3 \cos (c+d x) \left(20 a^3 B+60 a^2 A b+3 b^3 C\right)+10 b^2 (3 a C+b B) \cos (2 (c+d x))+30 a b^2 C+10 b^3 B+3 b^3 C \cos (3 (c+d x))\right)}{\cos ^{\frac{3}{2}}(c+d x)}\right)}{30 d}","-\frac{2 b \sin (c+d x) \left(6 a^2 B+3 a b (5 A-C)-b^2 B\right)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (A+3 C)+9 a^2 b B+3 a b^2 (3 A+C)+b^3 B\right)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^3 B+15 a^2 b (A-C)-15 a b^2 B-b^3 (5 A+3 C)\right)}{5 d}-\frac{2 b^2 \sin (c+d x) (15 a B+35 A b-3 b C)}{15 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (a B+2 A b) \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}{d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{3 d}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-12*(5*a^3*B - 15*a*b^2*B + 15*a^2*b*(A - C) - b^3*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2] + 20*(9*a^2*b*B + b^3*B + 3*a*b^2*(3*A + C) + a^3*(A + 3*C))*EllipticF[(c + d*x)/2, 2] + ((20*a^3*A + 10*b^3*B + 30*a*b^2*C + 3*(60*a^2*A*b + 20*a^3*B + 3*b^3*C)*Cos[c + d*x] + 10*b^2*(b*B + 3*a*C)*Cos[2*(c + d*x)] + 3*b^3*C*Cos[3*(c + d*x)])*Sin[c + d*x])/Cos[c + d*x]^(3/2)))/(30*d)","A",1
1470,1,233,319,1.9399112,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(40 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(21 a^3 B+21 a^2 b (3 A+C)+21 a b^2 B+b^3 (7 A+5 C)\right)-168 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^3 (A-C)-15 a^2 b B-3 a b^2 (5 A+3 C)-3 b^3 B\right)+\frac{2 \sin (c+d x) \left(420 a^3 A+5 b \cos (c+d x) \left(84 a^2 C+84 a b B+28 A b^2+29 b^2 C\right)+42 b^2 (3 a C+b B) \cos (2 (c+d x))+126 a b^2 C+42 b^3 B+15 b^3 C \cos (3 (c+d x))\right)}{\sqrt{\cos (c+d x)}}\right)}{420 d}","\frac{2 b \sin (c+d x) \left(-6 a^2 (7 A-3 C)+21 a b B+b^2 (7 A+5 C)\right)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(21 a^3 B+21 a^2 b (3 A+C)+21 a b^2 B+b^3 (7 A+5 C)\right)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^3 (A-C)+15 a^2 b B+3 a b^2 (5 A+3 C)+3 b^3 B\right)}{5 d}-\frac{2 b^2 \sin (c+d x) (35 a A-11 a C-7 b B)}{35 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b (7 A-C) \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^3}{d}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-168*(-15*a^2*b*B - 3*b^3*B + 5*a^3*(A - C) - 3*a*b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2] + 40*(21*a^3*B + 21*a*b^2*B + 21*a^2*b*(3*A + C) + b^3*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2] + (2*(420*a^3*A + 42*b^3*B + 126*a*b^2*C + 5*b*(28*A*b^2 + 84*a*b*B + 84*a^2*C + 29*b^2*C)*Cos[c + d*x] + 42*b^2*(b*B + 3*a*C)*Cos[2*(c + d*x)] + 15*b^3*C*Cos[3*(c + d*x)])*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(420*d)","A",1
1471,1,253,336,1.7538312,"\int (a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(2 \sin (2 (c+d x)) \left(7 b \cos (c+d x) \left(108 a^2 C+108 a b B+36 A b^2+43 b^2 C\right)+5 \left(84 a^3 C+252 a^2 b B+18 a b^2 (14 A+13 C)+18 b^2 (3 a C+b B) \cos (2 (c+d x))+78 b^3 B+7 b^3 C \cos (3 (c+d x))\right)\right)+240 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right)+336 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right)\right)}{2520 d}","\frac{2 b \sin (c+d x) \left(24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(8 a^3 C+54 a^2 b B+9 a b^2 (7 A+5 C)+15 b^3 B\right)}{63 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right)}{15 d}+\frac{2 (2 a C+3 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Sec[c + d*x]]*(336*(15*a^3*B + 27*a*b^2*B + 9*a^2*b*(5*A + 3*C) + b^3*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 240*(21*a^2*b*B + 5*b^3*B + 7*a^3*(3*A + C) + 3*a*b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(7*b*(36*A*b^2 + 108*a*b*B + 108*a^2*C + 43*b^2*C)*Cos[c + d*x] + 5*(252*a^2*b*B + 78*b^3*B + 84*a^3*C + 18*a*b^2*(14*A + 13*C) + 18*b^2*(b*B + 3*a*C)*Cos[2*(c + d*x)] + 7*b^3*C*Cos[3*(c + d*x)]))*Sin[2*(c + d*x)]))/(2520*d)","A",1
1472,1,304,401,3.4965954,"\int \frac{(a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(240 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(77 a^3 B+33 a^2 b (7 A+5 C)+165 a b^2 B+5 b^3 (11 A+9 C)\right)+3696 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^3 (5 A+3 C)+27 a^2 b B+3 a b^2 (9 A+7 C)+7 b^3 B\right)+\frac{\sin (2 (c+d x)) \left(154 \cos (c+d x) \left(36 a^3 C+108 a^2 b B+3 a b^2 (36 A+43 C)+43 b^3 B\right)+5 \left(1848 a^3 B+36 b \cos (2 (c+d x)) \left(33 a^2 C+33 a b B+11 A b^2+16 b^2 C\right)+396 a^2 b (14 A+13 C)+154 b^2 (3 a C+b B) \cos (3 (c+d x))+5148 a b^2 B+3 b^3 (572 A+531 C)+63 b^3 C \cos (4 (c+d x))\right)\right)}{\sqrt{\cos (c+d x)}}\right)}{27720 d}","\frac{2 b \sin (c+d x) \left(24 a^2 C+143 a b B+99 A b^2+81 b^2 C\right)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(24 a^3 C+242 a^2 b B+33 a b^2 (9 A+7 C)+77 b^3 B\right)}{495 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(77 a^3 B+33 a^2 b (7 A+5 C)+165 a b^2 B+5 b^3 (11 A+9 C)\right)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(77 a^3 B+33 a^2 b (7 A+5 C)+165 a b^2 B+5 b^3 (11 A+9 C)\right)}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^3 (5 A+3 C)+27 a^2 b B+3 a b^2 (9 A+7 C)+7 b^3 B\right)}{15 d}+\frac{2 (6 a C+11 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{99 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^3}{11 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(3696*(27*a^2*b*B + 7*b^3*B + 3*a^3*(5*A + 3*C) + 3*a*b^2*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2] + 240*(77*a^3*B + 165*a*b^2*B + 33*a^2*b*(7*A + 5*C) + 5*b^3*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2] + ((154*(108*a^2*b*B + 43*b^3*B + 36*a^3*C + 3*a*b^2*(36*A + 43*C))*Cos[c + d*x] + 5*(1848*a^3*B + 5148*a*b^2*B + 396*a^2*b*(14*A + 13*C) + 3*b^3*(572*A + 531*C) + 36*b*(11*A*b^2 + 33*a*b*B + 33*a^2*C + 16*b^2*C)*Cos[2*(c + d*x)] + 154*b^2*(b*B + 3*a*C)*Cos[3*(c + d*x)] + 63*b^3*C*Cos[4*(c + d*x)]))*Sin[2*(c + d*x)])/Sqrt[Cos[c + d*x]]))/(27720*d)","A",1
1473,1,355,463,3.8854699,"\int \frac{(a+b \cos (c+d x))^3 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) \left(154 \cos (c+d x) \left(936 a^3 B+78 a^2 b (36 A+43 C)+3354 a b^2 B+b^3 (1118 A+1171 C)\right)+5 \left(77 b \cos (3 (c+d x)) \left(156 a^2 C+156 a b B+52 A b^2+89 b^2 C\right)+936 \cos (2 (c+d x)) \left(11 a^3 C+33 a^2 b B+3 a b^2 (11 A+16 C)+16 b^3 B\right)+78 \left(44 a^3 (14 A+13 C)+1716 a^2 b B+3 a b^2 (572 A+531 C)+531 b^3 B\right)+1638 b^2 (3 a C+b B) \cos (4 (c+d x))+693 b^3 C \cos (5 (c+d x))\right)\right)+6240 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(11 a^3 (7 A+5 C)+165 a^2 b B+15 a b^2 (11 A+9 C)+45 b^3 B\right)+7392 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(117 a^3 B+39 a^2 b (9 A+7 C)+273 a b^2 B+7 b^3 (13 A+11 C)\right)\right)}{720720 d}","\frac{2 b \sin (c+d x) \left(24 a^2 C+195 a b B+143 A b^2+121 b^2 C\right)}{1287 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(117 a^3 B+39 a^2 b (9 A+7 C)+273 a b^2 B+7 b^3 (13 A+11 C)\right)}{585 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(24 a^3 C+338 a^2 b B+39 a b^2 (11 A+9 C)+117 b^3 B\right)}{1001 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(11 a^3 (7 A+5 C)+165 a^2 b B+15 a b^2 (11 A+9 C)+45 b^3 B\right)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(11 a^3 (7 A+5 C)+165 a^2 b B+15 a b^2 (11 A+9 C)+45 b^3 B\right)}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(117 a^3 B+39 a^2 b (9 A+7 C)+273 a b^2 B+7 b^3 (13 A+11 C)\right)}{195 d}+\frac{2 (6 a C+13 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{143 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^3}{13 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(7392*(117*a^3*B + 273*a*b^2*B + 39*a^2*b*(9*A + 7*C) + 7*b^3*(13*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 6240*(165*a^2*b*B + 45*b^3*B + 11*a^3*(7*A + 5*C) + 15*a*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (154*(936*a^3*B + 3354*a*b^2*B + 78*a^2*b*(36*A + 43*C) + b^3*(1118*A + 1171*C))*Cos[c + d*x] + 5*(78*(1716*a^2*b*B + 531*b^3*B + 44*a^3*(14*A + 13*C) + 3*a*b^2*(572*A + 531*C)) + 936*(33*a^2*b*B + 16*b^3*B + 11*a^3*C + 3*a*b^2*(11*A + 16*C))*Cos[2*(c + d*x)] + 77*b*(52*A*b^2 + 156*a*b*B + 156*a^2*C + 89*b^2*C)*Cos[3*(c + d*x)] + 1638*b^2*(b*B + 3*a*C)*Cos[4*(c + d*x)] + 693*b^3*C*Cos[5*(c + d*x)]))*Sin[2*(c + d*x)]))/(720720*d)","A",1
1474,1,563,515,7.3820002,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{13}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{2}{11} a^4 A \tan (c+d x) \sec ^4(c+d x)+\frac{2}{9} \sec ^4(c+d x) \left(a^4 B \sin (c+d x)+4 a^3 A b \sin (c+d x)\right)+\frac{2}{77} \sec ^3(c+d x) \left(9 a^4 A \sin (c+d x)+11 a^4 C \sin (c+d x)+44 a^3 b B \sin (c+d x)+66 a^2 A b^2 \sin (c+d x)\right)+\frac{2}{45} \sec ^2(c+d x) \left(7 a^4 B \sin (c+d x)+28 a^3 A b \sin (c+d x)+36 a^3 b C \sin (c+d x)+54 a^2 b^2 B \sin (c+d x)+36 a A b^3 \sin (c+d x)\right)+\frac{2}{15} \sin (c+d x) \left(7 a^4 B+28 a^3 A b+36 a^3 b C+54 a^2 b^2 B+36 a A b^3+60 a b^3 C+15 b^4 B\right)+\frac{2}{231} \sec (c+d x) \left(45 a^4 A \sin (c+d x)+55 a^4 C \sin (c+d x)+220 a^3 b B \sin (c+d x)+330 a^2 A b^2 \sin (c+d x)+462 a^2 b^2 C \sin (c+d x)+308 a b^3 B \sin (c+d x)+77 A b^4 \sin (c+d x)\right)\right)}{d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(225 a^4 A+275 a^4 C+1100 a^3 b B+1650 a^2 A b^2+2310 a^2 b^2 C+1540 a b^3 B+385 A b^4+1155 b^4 C\right)+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-539 a^4 B-2156 a^3 A b-2772 a^3 b C-4158 a^2 b^2 B-2772 a A b^3-4620 a b^3 C-1155 b^4 B\right)}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}}{1155 d}","\frac{2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right) (a+b \cos (c+d x))^2}{231 d}+\frac{2 a \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right)}{3465 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right)}{693 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right)}{15 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right)}{231 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right)}{15 d}+\frac{2 (11 a B+8 A b) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^3}{99 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a+b \cos (c+d x))^4}{11 d}",1,"((2*(-2156*a^3*A*b - 2772*a*A*b^3 - 539*a^4*B - 4158*a^2*b^2*B - 1155*b^4*B - 2772*a^3*b*C - 4620*a*b^3*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(225*a^4*A + 1650*a^2*A*b^2 + 385*A*b^4 + 1100*a^3*b*B + 1540*a*b^3*B + 275*a^4*C + 2310*a^2*b^2*C + 1155*b^4*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(1155*d) + (Sqrt[Sec[c + d*x]]*((2*(28*a^3*A*b + 36*a*A*b^3 + 7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 36*a^3*b*C + 60*a*b^3*C)*Sin[c + d*x])/15 + (2*Sec[c + d*x]^4*(4*a^3*A*b*Sin[c + d*x] + a^4*B*Sin[c + d*x]))/9 + (2*Sec[c + d*x]^3*(9*a^4*A*Sin[c + d*x] + 66*a^2*A*b^2*Sin[c + d*x] + 44*a^3*b*B*Sin[c + d*x] + 11*a^4*C*Sin[c + d*x]))/77 + (2*Sec[c + d*x]^2*(28*a^3*A*b*Sin[c + d*x] + 36*a*A*b^3*Sin[c + d*x] + 7*a^4*B*Sin[c + d*x] + 54*a^2*b^2*B*Sin[c + d*x] + 36*a^3*b*C*Sin[c + d*x]))/45 + (2*Sec[c + d*x]*(45*a^4*A*Sin[c + d*x] + 330*a^2*A*b^2*Sin[c + d*x] + 77*A*b^4*Sin[c + d*x] + 220*a^3*b*B*Sin[c + d*x] + 308*a*b^3*B*Sin[c + d*x] + 55*a^4*C*Sin[c + d*x] + 462*a^2*b^2*C*Sin[c + d*x]))/231 + (2*a^4*A*Sec[c + d*x]^4*Tan[c + d*x])/11))/d","A",1
1475,1,459,441,7.2594356,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{2}{9} a^4 A \tan (c+d x) \sec ^3(c+d x)+\frac{2}{7} \sec ^3(c+d x) \left(a^4 B \sin (c+d x)+4 a^3 A b \sin (c+d x)\right)+\frac{2}{45} \sec ^2(c+d x) \left(7 a^4 A \sin (c+d x)+9 a^4 C \sin (c+d x)+36 a^3 b B \sin (c+d x)+54 a^2 A b^2 \sin (c+d x)\right)+\frac{2}{21} \sec (c+d x) \left(5 a^4 B \sin (c+d x)+20 a^3 A b \sin (c+d x)+28 a^3 b C \sin (c+d x)+42 a^2 b^2 B \sin (c+d x)+28 a A b^3 \sin (c+d x)\right)+\frac{2}{15} \sin (c+d x) \left(7 a^4 A+9 a^4 C+36 a^3 b B+54 a^2 A b^2+90 a^2 b^2 C+60 a b^3 B+15 A b^4\right)\right)}{d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(25 a^4 B+100 a^3 A b+140 a^3 b C+210 a^2 b^2 B+140 a A b^3+420 a b^3 C+105 b^4 B\right)+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-49 a^4 A-63 a^4 C-252 a^3 b B-378 a^2 A b^2-630 a^2 b^2 C-420 a b^3 B-105 A b^4+105 b^4 C\right)}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}}{105 d}","\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right) (a+b \cos (c+d x))^2}{315 d}+\frac{2 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right)}{315 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(21 a^4 (7 A+9 C)+756 a^3 b B+7 a^2 b^2 (155 A+261 C)+1098 a b^3 B+192 A b^4\right)}{315 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^4 B+4 a^3 b (5 A+7 C)+42 a^2 b^2 B+28 a b^3 (A+3 C)+21 b^4 B\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (7 A+9 C)+36 a^3 b B+18 a^2 b^2 (3 A+5 C)+60 a b^3 B+15 b^4 (A-C)\right)}{15 d}+\frac{2 (9 a B+8 A b) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^3}{63 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^4}{9 d}",1,"((2*(-49*a^4*A - 378*a^2*A*b^2 - 105*A*b^4 - 252*a^3*b*B - 420*a*b^3*B - 63*a^4*C - 630*a^2*b^2*C + 105*b^4*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(100*a^3*A*b + 140*a*A*b^3 + 25*a^4*B + 210*a^2*b^2*B + 105*b^4*B + 140*a^3*b*C + 420*a*b^3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(105*d) + (Sqrt[Sec[c + d*x]]*((2*(7*a^4*A + 54*a^2*A*b^2 + 15*A*b^4 + 36*a^3*b*B + 60*a*b^3*B + 9*a^4*C + 90*a^2*b^2*C)*Sin[c + d*x])/15 + (2*Sec[c + d*x]^3*(4*a^3*A*b*Sin[c + d*x] + a^4*B*Sin[c + d*x]))/7 + (2*Sec[c + d*x]^2*(7*a^4*A*Sin[c + d*x] + 54*a^2*A*b^2*Sin[c + d*x] + 36*a^3*b*B*Sin[c + d*x] + 9*a^4*C*Sin[c + d*x]))/45 + (2*Sec[c + d*x]*(20*a^3*A*b*Sin[c + d*x] + 28*a*A*b^3*Sin[c + d*x] + 5*a^4*B*Sin[c + d*x] + 42*a^2*b^2*B*Sin[c + d*x] + 28*a^3*b*C*Sin[c + d*x]))/21 + (2*a^4*A*Sec[c + d*x]^3*Tan[c + d*x])/9))/d","A",1
1476,1,294,423,4.4758288,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{\sqrt{\sec (c+d x)} \left(30 a^4 A \tan (c+d x) \sec ^2(c+d x)+42 a^3 (a B+4 A b) \tan (c+d x) \sec (c+d x)+10 a^2 \tan (c+d x) \left(a^2 (5 A+7 C)+28 a b B+42 A b^2\right)+42 a \sin (c+d x) \left(3 a^3 B+4 a^2 b (3 A+5 C)+30 a b^2 B+20 A b^3\right)+10 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right)-42 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^4 B+4 a^3 b (3 A+5 C)+30 a^2 b^2 B+20 a b^3 (A-C)-5 b^4 B\right)+35 b^4 C \sin (2 (c+d x))\right)}{105 d}","-\frac{2 b^2 \sin (c+d x) \left(5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right)}{105 d \sqrt{\sec (c+d x)}}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right) (a+b \cos (c+d x))^2}{105 d}+\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)} \left(63 a^3 B+a^2 (202 A b+350 b C)+413 a b^2 B+192 A b^3\right)}{105 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^4 B+4 a^3 b (3 A+5 C)+30 a^2 b^2 B+20 a b^3 (A-C)-5 b^4 B\right)}{5 d}+\frac{2 (7 a B+8 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^4}{7 d}",1,"(Sqrt[Sec[c + d*x]]*(-42*(3*a^4*B + 30*a^2*b^2*B - 5*b^4*B + 20*a*b^3*(A - C) + 4*a^3*b*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 10*(28*a^3*b*B + 84*a*b^3*B + 7*b^4*(3*A + C) + 42*a^2*b^2*(A + 3*C) + a^4*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 42*a*(20*A*b^3 + 3*a^3*B + 30*a*b^2*B + 4*a^2*b*(3*A + 5*C))*Sin[c + d*x] + 35*b^4*C*Sin[2*(c + d*x)] + 10*a^2*(42*A*b^2 + 28*a*b*B + a^2*(5*A + 7*C))*Tan[c + d*x] + 42*a^3*(4*A*b + a*B)*Sec[c + d*x]*Tan[c + d*x] + 30*a^4*A*Sec[c + d*x]^2*Tan[c + d*x]))/(105*d)","A",1
1477,1,307,426,4.8380055,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{\sqrt{\sec (c+d x)} \left(36 a^4 A \sin (c+d x)+12 a^4 A \tan (c+d x) \sec (c+d x)+20 a^4 B \tan (c+d x)+60 a^4 C \sin (c+d x)+80 a^3 A b \tan (c+d x)+240 a^3 b B \sin (c+d x)+360 a^2 A b^2 \sin (c+d x)+20 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 B+4 a^3 b (A+3 C)+18 a^2 b^2 B+4 a b^3 (3 A+C)+b^4 B\right)-12 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (3 A+5 C)+20 a^3 b B+30 a^2 b^2 (A-C)-20 a b^3 B-b^4 (5 A+3 C)\right)+40 a b^3 C \sin (2 (c+d x))+10 b^4 B \sin (2 (c+d x))+3 b^4 C \sin (c+d x)+3 b^4 C \sin (3 (c+d x))\right)}{30 d}","-\frac{2 b^2 \sin (c+d x) \left(3 a^2 (3 A+5 C)+50 a b B+b^2 (59 A-3 C)\right)}{15 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 (3 A+5 C)+15 a b B+16 A b^2\right) (a+b \cos (c+d x))^2}{5 d}-\frac{2 b \sin (c+d x) \left(6 a^3 (3 A+5 C)+105 a^2 b B+4 a b^2 (33 A-5 C)-5 b^3 B\right)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 B+4 a^3 b (A+3 C)+18 a^2 b^2 B+4 a b^3 (3 A+C)+b^4 B\right)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (3 A+5 C)+20 a^3 b B+30 a^2 b^2 (A-C)-20 a b^3 B-b^4 (5 A+3 C)\right)}{5 d}+\frac{2 (5 a B+8 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3}{15 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^4}{5 d}",1,"(Sqrt[Sec[c + d*x]]*(-12*(20*a^3*b*B - 20*a*b^3*B + 30*a^2*b^2*(A - C) - b^4*(5*A + 3*C) + a^4*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 20*(a^4*B + 18*a^2*b^2*B + b^4*B + 4*a*b^3*(3*A + C) + 4*a^3*b*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 36*a^4*A*Sin[c + d*x] + 360*a^2*A*b^2*Sin[c + d*x] + 240*a^3*b*B*Sin[c + d*x] + 60*a^4*C*Sin[c + d*x] + 3*b^4*C*Sin[c + d*x] + 10*b^4*B*Sin[2*(c + d*x)] + 40*a*b^3*C*Sin[2*(c + d*x)] + 3*b^4*C*Sin[3*(c + d*x)] + 80*a^3*A*b*Tan[c + d*x] + 20*a^4*B*Tan[c + d*x] + 12*a^4*A*Sec[c + d*x]*Tan[c + d*x]))/(30*d)","A",1
1478,1,316,413,3.1073664,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(40 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^4 (A+3 C)+84 a^3 b B+42 a^2 b^2 (3 A+C)+28 a b^3 B+b^4 (7 A+5 C)\right)-168 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^4 B+20 a^3 b (A-C)-30 a^2 b^2 B-4 a b^3 (5 A+3 C)-3 b^4 B\right)+\frac{\sin (c+d x) \left(280 a^4 A+20 b^2 \cos (2 (c+d x)) \left(42 a^2 C+28 a b B+7 A b^2+8 b^2 C\right)+840 a^2 b^2 C+42 \cos (c+d x) \left(20 a^4 B+80 a^3 A b+12 a b^3 C+3 b^4 B\right)+560 a b^3 B+168 a b^3 C \cos (3 (c+d x))+140 A b^4+42 b^4 B \cos (3 (c+d x))+15 b^4 C \cos (4 (c+d x))+145 b^4 C\right)}{\cos ^{\frac{3}{2}}(c+d x)}\right)}{420 d}","-\frac{2 b^2 \sin (c+d x) \left(105 a^2 B+350 a A b-54 a b C-21 b^2 B\right)}{105 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b \sin (c+d x) \left(42 a^3 B+3 a^2 b (49 A-13 C)-28 a b^2 B-b^3 (7 A+5 C)\right)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^4 (A+3 C)+84 a^3 b B+42 a^2 b^2 (3 A+C)+28 a b^3 B+b^4 (7 A+5 C)\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^4 B+20 a^3 b (A-C)-30 a^2 b^2 B-4 a b^3 (5 A+3 C)-3 b^4 B\right)}{5 d}-\frac{2 b \sin (c+d x) (7 a B+21 A b-b C) (a+b \cos (c+d x))^2}{7 d \sqrt{\sec (c+d x)}}+\frac{2 (3 a B+8 A b) \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^3}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^4}{3 d}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-168*(5*a^4*B - 30*a^2*b^2*B - 3*b^4*B + 20*a^3*b*(A - C) - 4*a*b^3*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2] + 40*(84*a^3*b*B + 28*a*b^3*B + 42*a^2*b^2*(3*A + C) + 7*a^4*(A + 3*C) + b^4*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2] + ((280*a^4*A + 140*A*b^4 + 560*a*b^3*B + 840*a^2*b^2*C + 145*b^4*C + 42*(80*a^3*A*b + 20*a^4*B + 3*b^4*B + 12*a*b^3*C)*Cos[c + d*x] + 20*b^2*(7*A*b^2 + 28*a*b*B + 42*a^2*C + 8*b^2*C)*Cos[2*(c + d*x)] + 42*b^4*B*Cos[3*(c + d*x)] + 168*a*b^3*C*Cos[3*(c + d*x)] + 15*b^4*C*Cos[4*(c + d*x)])*Sin[c + d*x])/Cos[c + d*x]^(3/2)))/(420*d)","A",1
1479,1,327,419,2.7564022,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \left(2 \sin (c+d x) \left(2520 a^4 A+84 b^2 \cos (2 (c+d x)) \left(18 a^2 C+12 a b B+3 A b^2+4 b^2 C\right)+1512 a^2 b^2 C+30 b \cos (c+d x) \left(112 a^3 C+168 a^2 b B+4 a b^2 (28 A+29 C)+29 b^3 B\right)+1008 a b^3 B+360 a b^3 C \cos (3 (c+d x))+252 A b^4+90 b^4 B \cos (3 (c+d x))+35 b^4 C \cos (4 (c+d x))+301 b^4 C\right)+240 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right)-336 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(15 a^4 (A-C)-60 a^3 b B-18 a^2 b^2 (5 A+3 C)-36 a b^3 B-b^4 (9 A+7 C)\right)\right)}{2520 d}","\frac{2 b^2 \sin (c+d x) \left(-\left(a^2 (315 A-123 C)\right)+162 a b B+7 b^2 (9 A+7 C)\right)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \sin (c+d x) \left(-\left(a^3 (126 A-62 C)\right)+117 a^2 b B+12 a b^2 (7 A+5 C)+15 b^3 B\right)}{63 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-15 a^4 (A-C)+60 a^3 b B+18 a^2 b^2 (5 A+3 C)+36 a b^3 B+b^4 (9 A+7 C)\right)}{15 d}-\frac{2 b \sin (c+d x) (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{21 d \sqrt{\sec (c+d x)}}-\frac{2 b (9 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^4}{d}",1,"(Sqrt[Sec[c + d*x]]*(-336*(-60*a^3*b*B - 36*a*b^3*B + 15*a^4*(A - C) - 18*a^2*b^2*(5*A + 3*C) - b^4*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 240*(21*a^4*B + 42*a^2*b^2*B + 5*b^4*B + 28*a^3*b*(3*A + C) + 4*a*b^3*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(2520*a^4*A + 252*A*b^4 + 1008*a*b^3*B + 1512*a^2*b^2*C + 301*b^4*C + 30*b*(168*a^2*b*B + 29*b^3*B + 112*a^3*C + 4*a*b^2*(28*A + 29*C))*Cos[c + d*x] + 84*b^2*(3*A*b^2 + 12*a*b*B + 18*a^2*C + 4*b^2*C)*Cos[2*(c + d*x)] + 90*b^4*B*Cos[3*(c + d*x)] + 360*a*b^3*C*Cos[3*(c + d*x)] + 35*b^4*C*Cos[4*(c + d*x)])*Sin[c + d*x]))/(2520*d)","A",1
1480,1,338,444,3.9322163,"\int (a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(240 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(77 a^4 (3 A+C)+308 a^3 b B+66 a^2 b^2 (7 A+5 C)+220 a b^3 B+5 b^4 (11 A+9 C)\right)+3696 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right)+\frac{\sin (2 (c+d x)) \left(154 b \cos (c+d x) \left(144 a^3 C+216 a^2 b B+4 a b^2 (36 A+43 C)+43 b^3 B\right)+5 \left(1848 a^4 C+7392 a^3 b B+36 b^2 \cos (2 (c+d x)) \left(66 a^2 C+44 a b B+11 A b^2+16 b^2 C\right)+792 a^2 b^2 (14 A+13 C)+154 b^3 (4 a C+b B) \cos (3 (c+d x))+6864 a b^3 B+3 b^4 (572 A+531 C)+63 b^4 C \cos (4 (c+d x))\right)\right)}{\sqrt{\cos (c+d x)}}\right)}{27720 d}","\frac{2 \sin (c+d x) \left(16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right) (a+b \cos (c+d x))^2}{231 d \sqrt{\sec (c+d x)}}+\frac{2 b \sin (c+d x) \left(192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right)}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(64 a^4 C+682 a^3 b B+9 a^2 b^2 (143 A+101 C)+660 a b^3 B+15 b^4 (11 A+9 C)\right)}{693 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(77 a^4 (3 A+C)+308 a^3 b B+66 a^2 b^2 (7 A+5 C)+220 a b^3 B+5 b^4 (11 A+9 C)\right)}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right)}{15 d}+\frac{2 (8 a C+11 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{99 d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^4}{11 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(3696*(15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 12*a^3*b*(5*A + 3*C) + 4*a*b^3*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2] + 240*(308*a^3*b*B + 220*a*b^3*B + 77*a^4*(3*A + C) + 66*a^2*b^2*(7*A + 5*C) + 5*b^4*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2] + ((154*b*(216*a^2*b*B + 43*b^3*B + 144*a^3*C + 4*a*b^2*(36*A + 43*C))*Cos[c + d*x] + 5*(7392*a^3*b*B + 6864*a*b^3*B + 1848*a^4*C + 792*a^2*b^2*(14*A + 13*C) + 3*b^4*(572*A + 531*C) + 36*b^2*(11*A*b^2 + 44*a*b*B + 66*a^2*C + 16*b^2*C)*Cos[2*(c + d*x)] + 154*b^3*(b*B + 4*a*C)*Cos[3*(c + d*x)] + 63*b^4*C*Cos[4*(c + d*x)]))*Sin[2*(c + d*x)])/Sqrt[Cos[c + d*x]]))/(27720*d)","A",1
1481,1,400,517,4.3600241,"\int \frac{(a+b \cos (c+d x))^4 \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) \left(154 \cos (c+d x) \left(936 a^4 C+3744 a^3 b B+156 a^2 b^2 (36 A+43 C)+4472 a b^3 B+b^4 (1118 A+1171 C)\right)+5 \left(77 b^2 \cos (3 (c+d x)) \left(312 a^2 C+208 a b B+52 A b^2+89 b^2 C\right)+1872 b \cos (2 (c+d x)) \left(22 a^3 C+33 a^2 b B+2 a b^2 (11 A+16 C)+8 b^3 B\right)+78 \left(616 a^4 B+176 a^3 b (14 A+13 C)+3432 a^2 b^2 B+4 a b^3 (572 A+531 C)+531 b^4 B\right)+1638 b^3 (4 a C+b B) \cos (4 (c+d x))+693 b^4 C \cos (5 (c+d x))\right)\right)+6240 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right)+7392 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(39 a^4 (5 A+3 C)+468 a^3 b B+78 a^2 b^2 (9 A+7 C)+364 a b^3 B+7 b^4 (13 A+11 C)\right)\right)}{720720 d}","\frac{2 \sin (c+d x) \left(48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right) (a+b \cos (c+d x))^2}{1287 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \sin (c+d x) \left(192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(192 a^4 C+3458 a^3 b B+11 a^2 b^2 (637 A+491 C)+4004 a b^3 B+77 b^4 (13 A+11 C)\right)}{6435 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right)}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(39 a^4 (5 A+3 C)+468 a^3 b B+78 a^2 b^2 (9 A+7 C)+364 a b^3 B+7 b^4 (13 A+11 C)\right)}{195 d}+\frac{2 (8 a C+13 b B) \sin (c+d x) (a+b \cos (c+d x))^3}{143 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a+b \cos (c+d x))^4}{13 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(7392*(468*a^3*b*B + 364*a*b^3*B + 39*a^4*(5*A + 3*C) + 78*a^2*b^2*(9*A + 7*C) + 7*b^4*(13*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 6240*(77*a^4*B + 330*a^2*b^2*B + 45*b^4*B + 44*a^3*b*(7*A + 5*C) + 20*a*b^3*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (154*(3744*a^3*b*B + 4472*a*b^3*B + 936*a^4*C + 156*a^2*b^2*(36*A + 43*C) + b^4*(1118*A + 1171*C))*Cos[c + d*x] + 5*(78*(616*a^4*B + 3432*a^2*b^2*B + 531*b^4*B + 176*a^3*b*(14*A + 13*C) + 4*a*b^3*(572*A + 531*C)) + 1872*b*(33*a^2*b*B + 8*b^3*B + 22*a^3*C + 2*a*b^2*(11*A + 16*C))*Cos[2*(c + d*x)] + 77*b^2*(52*A*b^2 + 208*a*b*B + 312*a^2*C + 89*b^2*C)*Cos[3*(c + d*x)] + 1638*b^3*(b*B + 4*a*C)*Cos[4*(c + d*x)] + 693*b^4*C*Cos[5*(c + d*x)]))*Sin[2*(c + d*x)]))/(720720*d)","A",1
1482,1,692,294,7.0481074,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + b*Cos[c + d*x]),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{2 \sec (c+d x) (a B \sin (c+d x)-A b \sin (c+d x))}{3 a^2}+\frac{2 \sin (c+d x) \left(3 a^2 A+5 a^2 C-5 a b B+5 A b^2\right)}{5 a^3}+\frac{2 A \tan (c+d x) \sec (c+d x)}{5 a}\right)}{d}-\frac{\frac{\sin (c+d x) \cos (2 (c+d x)) \left(9 a^2 A b+15 a^2 b C-15 a b^2 B+15 A b^3\right) (a \sec (c+d x)+b) \left(-4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a b^2 \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{2 \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} \left(18 a^3 A+30 a^3 C-40 a^2 b B+40 a A b^2\right) (a \sec (c+d x)+b) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{b \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{2 \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} \left(-10 a^3 B+19 a^2 A b+45 a^2 b C-45 a b^2 B+45 A b^3\right) (a \sec (c+d x)+b) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}}{30 a^3 d}","-\frac{2 b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a+b)}-\frac{2 (A b-a B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d}-\frac{2 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 (3 A+5 C)-5 a b B+5 A b^2\right)}{5 a^3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (3 A+5 C)-5 a b B+5 A b^2\right)}{5 a^3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 a d}",1,"-1/30*((2*(19*a^2*A*b + 45*A*b^3 - 10*a^3*B - 45*a*b^2*B + 45*a^2*b*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(18*a^3*A + 40*a*A*b^2 - 40*a^2*b*B + 30*a^3*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((9*a^2*A*b + 15*A*b^3 - 15*a*b^2*B + 15*a^2*b*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(a^3*d) + (Sqrt[Sec[c + d*x]]*((2*(3*a^2*A + 5*A*b^2 - 5*a*b*B + 5*a^2*C)*Sin[c + d*x])/(5*a^3) + (2*Sec[c + d*x]*(-(A*b*Sin[c + d*x]) + a*B*Sin[c + d*x]))/(3*a^2) + (2*A*Sec[c + d*x]*Tan[c + d*x])/(5*a)))/d","B",0
1483,1,267,218,4.0497269,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x]),x]","-\frac{\cot (c+d x) \left(-2 \sqrt{-\tan ^2(c+d x)} \left(a^2 (A-3 B+3 C)+3 a b (A-B)+3 A b^2\right) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-a^2 A \sec ^{\frac{5}{2}}(c+d x)+a^2 A \cos (2 (c+d x)) \sec ^{\frac{5}{2}}(c+d x)+6 a^2 C \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+6 A b^2 \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-6 a (a B-A b) \sqrt{-\tan ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-6 a b B \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{3 a^3 d}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a+b)}-\frac{2 (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}+\frac{2 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}+\frac{2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}",1,"-1/3*(Cot[c + d*x]*(-(a^2*A*Sec[c + d*x]^(5/2)) + a^2*A*Cos[2*(c + d*x)]*Sec[c + d*x]^(5/2) - 6*a*(-(A*b) + a*B)*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] - 2*(3*A*b^2 + 3*a*b*(A - B) + a^2*(A - 3*B + 3*C))*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] + 6*A*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] - 6*a*b*B*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] + 6*a^2*C*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2]))/(a^3*d)","A",1
1484,1,138,178,1.4572353,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x]),x]","-\frac{2 \cos (2 (c+d x)) \sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sec (c+d x) \left(\left(a^2 C-a b B+A b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-b (a A-a B+A b) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+a A b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a^2 b d \left(\sec ^2(c+d x)-2\right)}","-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d (a+b)}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(-2*Cos[2*(c + d*x)]*Csc[c + d*x]*(a*A*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1] - b*(a*A + A*b - a*B)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] + (A*b^2 - a*b*B + a^2*C)*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*Sec[c + d*x]*Sqrt[-Tan[c + d*x]^2])/(a^2*b*d*(-2 + Sec[c + d*x]^2))","A",1
1485,1,276,157,2.0653989,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{a+b \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x]),x]","\frac{\cot (c+d x) \left(-2 a^2 C \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-2 A b^2 \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b (a C+A b) \sqrt{-\tan ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 a b B \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+a b C \sec ^{\frac{7}{2}}(c+d x)-a b C \sec ^{\frac{3}{2}}(c+d x)+a b C \cos (2 (c+d x)) \sec ^{\frac{7}{2}}(c+d x)-a b C \cos (2 (c+d x)) \sec ^{\frac{3}{2}}(c+d x)-2 a b C \sqrt{-\tan ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a b^2 d}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}+\frac{2 (b B-a C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(Cot[c + d*x]*(-(a*b*C*Sec[c + d*x]^(3/2)) - a*b*C*Cos[2*(c + d*x)]*Sec[c + d*x]^(3/2) + a*b*C*Sec[c + d*x]^(7/2) + a*b*C*Cos[2*(c + d*x)]*Sec[c + d*x]^(7/2) - 2*a*b*C*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] + 2*b*(A*b + a*C)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] - 2*A*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] + 2*a*b*B*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] - 2*a^2*C*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2]))/(a*b^2*d)","A",1
1486,1,554,207,6.8034098,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x)) \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]]),x]","\frac{\frac{(3 b B-3 a C) \sin (c+d x) \cos (2 (c+d x)) (a \sec (c+d x)+b) \left(-4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a b^2 \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{2 (6 A b+2 b C) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{b \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{2 (3 b B-a C) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}}{6 b d}+\frac{C \sin (2 (c+d x)) \sqrt{\sec (c+d x)}}{3 b d}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(b^2 (3 A+C)-3 a (b B-a C)\right)}{3 b^3 d}-\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a+b)}+\frac{2 (b B-a C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C \sin (c+d x)}{3 b d \sqrt{\sec (c+d x)}}",1,"((2*(3*b*B - a*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(6*A*b + 2*b*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((3*b*B - 3*a*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(6*b*d) + (C*Sqrt[Sec[c + d*x]]*Sin[2*(c + d*x)])/(3*b*d)","B",0
1487,1,626,270,7.0119469,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])*Sec[c + d*x]^(3/2)),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{(b B-a C) \sin (2 (c+d x))}{3 b^2}+\frac{C \sin (c+d x)}{10 b}+\frac{C \sin (3 (c+d x))}{10 b}\right)}{d}-\frac{\frac{2 \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} \left(-5 a^2 C+5 a b B-15 A b^2-9 b^2 C\right) (a \sec (c+d x)+b) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \cos (2 (c+d x)) \left(-15 a^2 C+15 a b B-15 A b^2-9 b^2 C\right) (a \sec (c+d x)+b) \left(-4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a b^2 \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{2 \left(-8 a b C-10 b^2 B\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{b \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}}{30 b^2 d}","\frac{2 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a+b)}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 C-5 a b B+5 A b^2+3 b^2 C\right)}{5 b^3 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-3 a^3 C+3 a^2 b B-a b^2 (3 A+C)+b^3 B\right)}{3 b^4 d}+\frac{2 (b B-a C) \sin (c+d x)}{3 b^2 d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x)}{5 b d \sec ^{\frac{3}{2}}(c+d x)}",1,"-1/30*((2*(-15*A*b^2 + 5*a*b*B - 5*a^2*C - 9*b^2*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-10*b^2*B - 8*a*b*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-15*A*b^2 + 15*a*b*B - 15*a^2*C - 9*b^2*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(b^2*d) + (Sqrt[Sec[c + d*x]]*((C*Sin[c + d*x])/(10*b) + ((b*B - a*C)*Sin[2*(c + d*x)])/(3*b^2) + (C*Sin[3*(c + d*x)])/(10*b)))/d","B",0
1488,1,532,345,6.6692529,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])*Sec[c + d*x]^(5/2)),x]","\frac{4 b^2 \sin (c+d x) \left(70 a^2 C+42 b (b B-a C) \cos (c+d x)-70 a b B+70 A b^2+15 b^2 C \cos (2 (c+d x))+65 b^2 C\right)-\frac{2 \cos (c+d x) \cot (c+d x) (a \sec (c+d x)+b) \left(-4 a b^2 \sqrt{-\tan ^2(c+d x)} \sqrt{\sec (c+d x)} \left(-28 a^2 C+28 a b B+35 A b^2+25 b^2 C\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-2 b^2 \sqrt{-\tan ^2(c+d x)} \sqrt{\sec (c+d x)} \left(-35 a^3 C+35 a^2 b B-a b^2 (35 A+13 C)+63 b^3 B\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)-21 \left(5 a^3 C-5 a^2 b B+a b^2 (5 A+3 C)-3 b^3 B\right) \left(4 a^2 \sqrt{-\tan ^2(c+d x)} \sqrt{\sec (c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-2 b^2 \sqrt{-\tan ^2(c+d x)} \sqrt{\sec (c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sec ^2(c+d x)-2 b (2 a-b) \sqrt{-\tan ^2(c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sqrt{-\tan ^2(c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b\right)\right)}{a (a+b \cos (c+d x))}}{420 b^5 d \sqrt{\sec (c+d x)}}","-\frac{2 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a+b)}+\frac{2 \sin (c+d x) \left(7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right)}{21 b^3 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^3 C+5 a^2 b B-a b^2 (5 A+3 C)+3 b^3 B\right)}{5 b^4 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-21 a^4 C+21 a^3 b B-7 a^2 b^2 (3 A+C)+7 a b^3 B-b^4 (7 A+5 C)\right)}{21 b^5 d}+\frac{2 (b B-a C) \sin (c+d x)}{5 b^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x)}{7 b d \sec ^{\frac{5}{2}}(c+d x)}",1,"(4*b^2*(70*A*b^2 - 70*a*b*B + 70*a^2*C + 65*b^2*C + 42*b*(b*B - a*C)*Cos[c + d*x] + 15*b^2*C*Cos[2*(c + d*x)])*Sin[c + d*x] - (2*Cos[c + d*x]*Cot[c + d*x]*(b + a*Sec[c + d*x])*(-2*b^2*(35*a^2*b*B + 63*b^3*B - 35*a^3*C - a*b^2*(35*A + 13*C))*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*Sqrt[Sec[c + d*x]]*Sqrt[-Tan[c + d*x]^2] - 4*a*b^2*(35*A*b^2 + 28*a*b*B - 28*a^2*C + 25*b^2*C)*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[-Tan[c + d*x]^2] - 21*(-5*a^2*b*B - 3*b^3*B + 5*a^3*C + a*b^2*(5*A + 3*C))*(4*a*b - 4*a*b*Sec[c + d*x]^2 + 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[-Tan[c + d*x]^2] - 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[-Tan[c + d*x]^2] + 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[-Tan[c + d*x]^2] - 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[-Tan[c + d*x]^2])))/(a*(a + b*Cos[c + d*x])))/(420*b^5*d*Sqrt[Sec[c + d*x]])","A",1
1489,1,785,452,7.317064,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^2,x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{-a^2 b C \sin (c+d x)+a b^2 B \sin (c+d x)-A b^3 \sin (c+d x)}{a^2 \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{2 A \tan (c+d x)}{3 a^2}+\frac{\sin (c+d x) \left(2 a^3 B-4 a^2 A b+a^2 b C-3 a b^2 B+5 A b^3\right)}{a^3 \left(a^2-b^2\right)}\right)}{d}+\frac{\frac{\sin (c+d x) \cos (2 (c+d x)) \left(6 a^3 b B-12 a^2 A b^2+3 a^2 b^2 C-9 a b^3 B+15 A b^4\right) (a \sec (c+d x)+b) \left(-4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a b^2 \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{2 \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} \left(12 a^4 B-28 a^3 A b+12 a^3 b C-24 a^2 b^2 B+40 a A b^3\right) (a \sec (c+d x)+b) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{b \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{2 \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} \left(-4 a^4 A-12 a^4 C+30 a^3 b B-44 a^2 A b^2+9 a^2 b^2 C-27 a b^3 B+45 A b^4\right) (a \sec (c+d x)+b) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}}{12 a^3 d (b-a) (a+b)}","-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(-\left(a^2 (2 A-3 C)\right)-3 a b B+5 A b^2\right)}{3 a^2 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-\left(a^2 (2 A-3 C)\right)-3 a b B+5 A b^2\right)}{3 a^2 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(2 a^3 B-a^2 b (4 A-C)-3 a b^2 B+5 A b^3\right)}{a^3 d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(2 a^3 B-a^2 b (4 A-C)-3 a b^2 B+5 A b^3\right)}{a^3 d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-3 a^4 C+5 a^3 b B-a^2 b^2 (7 A-C)-3 a b^3 B+5 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}",1,"((2*(-4*a^4*A - 44*a^2*A*b^2 + 45*A*b^4 + 30*a^3*b*B - 27*a*b^3*B - 12*a^4*C + 9*a^2*b^2*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-28*a^3*A*b + 40*a*A*b^3 + 12*a^4*B - 24*a^2*b^2*B + 12*a^3*b*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-12*a^2*A*b^2 + 15*A*b^4 + 6*a^3*b*B - 9*a*b^3*B + 3*a^2*b^2*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(12*a^3*(-a + b)*(a + b)*d) + (Sqrt[Sec[c + d*x]]*(((-4*a^2*A*b + 5*A*b^3 + 2*a^3*B - 3*a*b^2*B + a^2*b*C)*Sin[c + d*x])/(a^3*(a^2 - b^2)) + (-(A*b^3*Sin[c + d*x]) + a*b^2*B*Sin[c + d*x] - a^2*b*C*Sin[c + d*x])/(a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])) + (2*A*Tan[c + d*x])/(3*a^2)))/d","A",0
1490,1,717,366,7.1407338,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^2,x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{\sin (c+d x) \left(2 a^2 A-a^2 C+a b B-3 A b^2\right)}{a^2 \left(a^2-b^2\right)}+\frac{a^2 C \sin (c+d x)-a b B \sin (c+d x)+A b^2 \sin (c+d x)}{a \left(a^2-b^2\right) (a+b \cos (c+d x))}\right)}{d}-\frac{\frac{\sin (c+d x) \cos (2 (c+d x)) \left(2 a^2 A b-a^2 b C+a b^2 B-3 A b^3\right) (a \sec (c+d x)+b) \left(-4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a b^2 \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{2 \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} \left(4 a^3 A-4 a^3 C+4 a^2 b B-8 a A b^2\right) (a \sec (c+d x)+b) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{b \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{2 \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} \left(-4 a^3 B+10 a^2 A b+a^2 b C+3 a b^2 B-9 A b^3\right) (a \sec (c+d x)+b) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}}{4 a^2 d (a-b) (a+b)}","-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-\left(a^2 (2 A-C)\right)-a b B+3 A b^2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(A b^2-a (b B-a C)\right)}{a b d \left(a^2-b^2\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-\left(a^2 (2 A-C)\right)-a b B+3 A b^2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(a^4 (-C)+3 a^3 b B-a^2 b^2 (5 A+C)-a b^3 B+3 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 b d (a-b) (a+b)^2}",1,"-1/4*((2*(10*a^2*A*b - 9*A*b^3 - 4*a^3*B + 3*a*b^2*B + a^2*b*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(4*a^3*A - 8*a*A*b^2 + 4*a^2*b*B - 4*a^3*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((2*a^2*A*b - 3*A*b^3 + a*b^2*B - a^2*b*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(a^2*(a - b)*(a + b)*d) + (Sqrt[Sec[c + d*x]]*(((2*a^2*A - 3*A*b^2 + a*b*B - a^2*C)*Sin[c + d*x])/(a^2*(a^2 - b^2)) + (A*b^2*Sin[c + d*x] - a*b*B*Sin[c + d*x] + a^2*C*Sin[c + d*x])/(a*(a^2 - b^2)*(a + b*Cos[c + d*x]))))/d","A",0
1491,1,682,303,6.8513848,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^2,x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{\sin (c+d x) \left(a^2 C-a b B+A b^2\right)}{a b \left(a^2-b^2\right)}+\frac{a^2 C \sin (c+d x)-a b B \sin (c+d x)+A b^2 \sin (c+d x)}{b \left(b^2-a^2\right) (a+b \cos (c+d x))}\right)}{d}+\frac{\frac{2 \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} \left(-4 a^2 A-a^2 C+a b B+3 A b^2\right) (a \sec (c+d x)+b) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \cos (2 (c+d x)) \left(a^2 C-a b B+A b^2\right) (a \sec (c+d x)+b) \left(-4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a b^2 \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{2 \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} \left(-4 a^2 B+4 a A b+4 a b C\right) (a \sec (c+d x)+b) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{b \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}}{4 a d (b-a) (a+b)}","\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (-C)-a b B+A b^2+2 b^2 C\right)}{b^2 d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(A b^2-a (b B-a C)\right)}{a b d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(a^4 C+a^3 b B-3 a^2 b^2 (A+C)+a b^3 B+A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b^2 d (a-b) (a+b)^2}",1,"((2*(-4*a^2*A + 3*A*b^2 + a*b*B - a^2*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(4*a*A*b - 4*a^2*B + 4*a*b*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((A*b^2 - a*b*B + a^2*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(4*a*(-a + b)*(a + b)*d) + (Sqrt[Sec[c + d*x]]*(((A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(a*b*(a^2 - b^2)) + (A*b^2*Sin[c + d*x] - a*b*B*Sin[c + d*x] + a^2*C*Sin[c + d*x])/(b*(-a^2 + b^2)*(a + b*Cos[c + d*x]))))/d","B",0
1492,1,689,311,7.0494281,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^2 \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]),x]","\frac{\frac{2 \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} \left(a^2 C+a b B-A b^2-2 b^2 C\right) (a \sec (c+d x)+b) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \cos (2 (c+d x)) \left(3 a^2 C-a b B+A b^2-2 b^2 C\right) (a \sec (c+d x)+b) \left(-4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a b^2 \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{2 \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} \left(4 a A b+4 a b C-4 b^2 B\right) (a \sec (c+d x)+b) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{b \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}}{4 b d (a-b) (a+b)}+\frac{\sqrt{\sec (c+d x)} \left(\frac{\sin (c+d x) \left(a^2 C-a b B+A b^2\right)}{b^2 \left(b^2-a^2\right)}+\frac{a^3 (-C) \sin (c+d x)+a^2 b B \sin (c+d x)-a A b^2 \sin (c+d x)}{b^2 \left(b^2-a^2\right) (a+b \cos (c+d x))}\right)}{d}","-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 C-a b B+A b^2-2 b^2 C\right)}{b^2 d \left(a^2-b^2\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-3 a^3 C+a^2 b B+a b^2 (A+4 C)-2 b^3 B\right)}{b^3 d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-3 a^4 C+a^3 b B+a^2 b^2 (A+5 C)-3 a b^3 B+A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}",1,"((2*(-(A*b^2) + a*b*B + a^2*C - 2*b^2*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(4*a*A*b - 4*b^2*B + 4*a*b*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((A*b^2 - a*b*B + 3*a^2*C - 2*b^2*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(4*(a - b)*b*(a + b)*d) + (Sqrt[Sec[c + d*x]]*(((A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(b^2*(-a^2 + b^2)) + (-(a*A*b^2*Sin[c + d*x]) + a^2*b*B*Sin[c + d*x] - a^3*C*Sin[c + d*x])/(b^2*(-a^2 + b^2)*(a + b*Cos[c + d*x]))))/d","B",0
1493,1,742,403,7.0904245,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)),x]","\frac{\frac{2 \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} \left(8 a^2 b C-12 a b^2 B+12 A b^3+4 b^3 C\right) (a \sec (c+d x)+b) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{b \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{2 \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} \left(5 a^3 C-3 a^2 b B-3 a A b^2-8 a b^2 C+6 b^3 B\right) (a \sec (c+d x)+b) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \cos (2 (c+d x)) \left(15 a^3 C-9 a^2 b B+3 a A b^2-12 a b^2 C+6 b^3 B\right) (a \sec (c+d x)+b) \left(-4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a b^2 \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a+b \cos (c+d x))}}{12 b^2 d (b-a) (a+b)}+\frac{\sqrt{\sec (c+d x)} \left(\frac{a \sin (c+d x) \left(a^2 C-a b B+A b^2\right)}{b^3 \left(a^2-b^2\right)}-\frac{a^4 (-C) \sin (c+d x)+a^3 b B \sin (c+d x)-a^2 A b^2 \sin (c+d x)}{b^3 \left(b^2-a^2\right) (a+b \cos (c+d x))}+\frac{C \sin (2 (c+d x))}{3 b^2}\right)}{d}","\frac{\sin (c+d x) \left(5 a^2 C-3 a b B+3 A b^2-2 b^2 C\right)}{3 b^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^3 C+3 a^2 b B-a b^2 (A-4 C)-2 b^3 B\right)}{b^3 d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-15 a^4 C+9 a^3 b B-a^2 b^2 (3 A-16 C)-12 a b^3 B+2 b^4 (3 A+C)\right)}{3 b^4 d \left(a^2-b^2\right)}+\frac{a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-5 a^4 C+3 a^3 b B-a^2 b^2 (A-7 C)-5 a b^3 B+3 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a-b) (a+b)^2}",1,"((2*(-3*a*A*b^2 - 3*a^2*b*B + 6*b^3*B + 5*a^3*C - 8*a*b^2*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(12*A*b^3 - 12*a*b^2*B + 8*a^2*b*C + 4*b^3*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((3*a*A*b^2 - 9*a^2*b*B + 6*b^3*B + 15*a^3*C - 12*a*b^2*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(12*b^2*(-a + b)*(a + b)*d) + (Sqrt[Sec[c + d*x]]*((a*(A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(b^3*(a^2 - b^2)) - (-(a^2*A*b^2*Sin[c + d*x]) + a^3*b*B*Sin[c + d*x] - a^4*C*Sin[c + d*x])/(b^3*(-a^2 + b^2)*(a + b*Cos[c + d*x])) + (C*Sin[2*(c + d*x)])/(3*b^2)))/d","A",0
1494,1,831,505,7.3711941,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^2 \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)),x]","\frac{\frac{2 \left(35 C a^4-25 b B a^3+15 A b^2 a^2-32 b^2 C a^2+40 b^3 B a-30 A b^4-18 b^4 C\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(-20 B b^4+60 a A b^3+4 a C b^3-40 a^2 B b^2+56 a^3 C b\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(105 C a^4-75 b B a^3+45 A b^2 a^2-72 b^2 C a^2+60 b^3 B a-30 A b^4-18 b^4 C\right) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left(-4 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a+2 (2 a-b) b F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}+2 b^2 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}}{60 (a-b) b^3 (a+b) d}+\frac{\sqrt{\sec (c+d x)} \left(-\frac{\left(10 C a^4-10 b B a^3+10 A b^2 a^2-b^2 C a^2+b^4 C\right) \sin (c+d x)}{10 b^4 \left(a^2-b^2\right)}-\frac{C \sin (c+d x) a^5-b B \sin (c+d x) a^4+A b^2 \sin (c+d x) a^3}{b^4 \left(b^2-a^2\right) (a+b \cos (c+d x))}+\frac{(b B-2 a C) \sin (2 (c+d x))}{3 b^3}+\frac{C \sin (3 (c+d x))}{10 b^2}\right)}{d}","\frac{\sin (c+d x) \left(7 a^2 C-5 a b B+5 A b^2-2 b^2 C\right)}{5 b^2 d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x)}-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \left(-7 a^3 C+5 a^2 b B-a b^2 (3 A-4 C)-2 b^3 B\right)}{3 b^3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-35 a^4 C+25 a^3 b B-3 a^2 b^2 (5 A-8 C)-20 a b^3 B+2 b^4 (5 A+3 C)\right)}{5 b^4 d \left(a^2-b^2\right)}-\frac{a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-7 a^4 C+5 a^3 b B-3 a^2 b^2 (A-3 C)-7 a b^3 B+5 A b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d (a-b) (a+b)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-21 a^5 C+15 a^4 b B-a^3 b^2 (9 A-20 C)-16 a^2 b^3 B+4 a b^4 (3 A+C)-2 b^5 B\right)}{3 b^5 d \left(a^2-b^2\right)}",1,"((2*(15*a^2*A*b^2 - 30*A*b^4 - 25*a^3*b*B + 40*a*b^3*B + 35*a^4*C - 32*a^2*b^2*C - 18*b^4*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(60*a*A*b^3 - 40*a^2*b^2*B - 20*b^4*B + 56*a^3*b*C + 4*a*b^3*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((45*a^2*A*b^2 - 30*A*b^4 - 75*a^3*b*B + 60*a*b^3*B + 105*a^4*C - 72*a^2*b^2*C - 18*b^4*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(60*(a - b)*b^3*(a + b)*d) + (Sqrt[Sec[c + d*x]]*(-1/10*((10*a^2*A*b^2 - 10*a^3*b*B + 10*a^4*C - a^2*b^2*C + b^4*C)*Sin[c + d*x])/(b^4*(a^2 - b^2)) - (a^3*A*b^2*Sin[c + d*x] - a^4*b*B*Sin[c + d*x] + a^5*C*Sin[c + d*x])/(b^4*(-a^2 + b^2)*(a + b*Cos[c + d*x])) + ((b*B - 2*a*C)*Sin[2*(c + d*x)])/(3*b^3) + (C*Sin[3*(c + d*x)])/(10*b^2)))/d","A",0
1495,1,1013,669,7.8417972,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{2 \left(16 A a^6+48 C a^6-168 b B a^5+328 A b^2 a^4-57 b^2 C a^4+285 b^3 B a^3-641 A b^4 a^2+27 b^4 C a^2-135 b^5 B a+315 A b^6\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(-48 B a^6+160 A b a^5-96 b C a^5+240 b^2 B a^4-512 A b^3 a^3+24 b^3 C a^3-120 b^4 B a^2+280 A b^5 a\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(105 A b^6-45 a B b^5-195 a^2 A b^4+9 a^2 C b^4+87 a^3 B b^3+72 a^4 A b^2-27 a^4 C b^2-24 a^5 B b\right) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left(-4 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a+2 (2 a-b) b F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}+2 b^2 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}}{48 a^4 (a-b)^2 (a+b)^2 d}+\frac{\sqrt{\sec (c+d x)} \left(\frac{\left(8 B a^5-24 A b a^4+9 b C a^4-29 b^2 B a^3+65 A b^3 a^2-3 b^3 C a^2+15 b^4 B a-35 A b^5\right) \sin (c+d x)}{4 a^4 \left(a^2-b^2\right)^2}+\frac{-A \sin (c+d x) b^3+a B \sin (c+d x) b^2-a^2 C \sin (c+d x) b}{2 a^2 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{9 A \sin (c+d x) b^5-5 a B \sin (c+d x) b^4-15 a^2 A \sin (c+d x) b^3+a^2 C \sin (c+d x) b^3+11 a^3 B \sin (c+d x) b^2-7 a^4 C \sin (c+d x) b}{4 a^3 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{2 A \tan (c+d x)}{3 a^3}\right)}{d}","\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right)}{12 a^3 d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right)}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right)}{12 a^3 d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}",1,"((2*(16*a^6*A + 328*a^4*A*b^2 - 641*a^2*A*b^4 + 315*A*b^6 - 168*a^5*b*B + 285*a^3*b^3*B - 135*a*b^5*B + 48*a^6*C - 57*a^4*b^2*C + 27*a^2*b^4*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(160*a^5*A*b - 512*a^3*A*b^3 + 280*a*A*b^5 - 48*a^6*B + 240*a^4*b^2*B - 120*a^2*b^4*B - 96*a^5*b*C + 24*a^3*b^3*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((72*a^4*A*b^2 - 195*a^2*A*b^4 + 105*A*b^6 - 24*a^5*b*B + 87*a^3*b^3*B - 45*a*b^5*B - 27*a^4*b^2*C + 9*a^2*b^4*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(48*a^4*(a - b)^2*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(((-24*a^4*A*b + 65*a^2*A*b^3 - 35*A*b^5 + 8*a^5*B - 29*a^3*b^2*B + 15*a*b^4*B + 9*a^4*b*C - 3*a^2*b^3*C)*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2) + (-(A*b^3*Sin[c + d*x]) + a*b^2*B*Sin[c + d*x] - a^2*b*C*Sin[c + d*x])/(2*a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) + (-15*a^2*A*b^3*Sin[c + d*x] + 9*A*b^5*Sin[c + d*x] + 11*a^3*b^2*B*Sin[c + d*x] - 5*a*b^4*B*Sin[c + d*x] - 7*a^4*b*C*Sin[c + d*x] + a^2*b^3*C*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) + (2*A*Tan[c + d*x])/(3*a^3)))/d","A",0
1496,1,944,562,7.5256263,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^3,x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{\left(8 A a^4-5 C a^4+9 b B a^3-29 A b^2 a^2-b^2 C a^2-3 b^3 B a+15 A b^4\right) \sin (c+d x)}{4 a^3 \left(a^2-b^2\right)^2}+\frac{C \sin (c+d x) a^2-b B \sin (c+d x) a+A b^2 \sin (c+d x)}{2 a \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{3 C \sin (c+d x) a^4-7 b B \sin (c+d x) a^3+11 A b^2 \sin (c+d x) a^2+3 b^2 C \sin (c+d x) a^2+b^3 B \sin (c+d x) a-5 A b^4 \sin (c+d x)}{4 a^2 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}\right)}{d}-\frac{\frac{2 \left(-16 B a^5+56 A b a^4+9 b C a^4+19 b^2 B a^3-95 A b^3 a^2-3 b^3 C a^2-9 b^4 B a+45 A b^5\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(16 A a^5-16 C a^5+32 b B a^4-80 A b^2 a^3-8 b^2 C a^3-8 b^3 B a^2+40 A b^4 a\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(15 A b^5-3 a B b^4-29 a^2 A b^3-a^2 C b^3+9 a^3 B b^2+8 a^4 A b-5 a^4 C b\right) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left(-4 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a+2 (2 a-b) b F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}+2 b^2 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}}{16 a^3 (a-b)^2 (a+b)^2 d}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(a^4 (8 A-5 C)+9 a^3 b B-a^2 b^2 (29 A+C)-3 a b^3 B+15 A b^4\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-3 a^4 C+7 a^3 b B-a^2 b^2 (11 A+3 C)-a b^3 B+5 A b^4\right)}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-3 a^4 C+7 a^3 b B-a^2 b^2 (11 A+3 C)-a b^3 B+5 A b^4\right)}{4 a^2 b d \left(a^2-b^2\right)^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (8 A-5 C)+9 a^3 b B-a^2 b^2 (29 A+C)-3 a b^3 B+15 A b^4\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(3 a^6 C-15 a^5 b B+5 a^4 b^2 (7 A+2 C)+6 a^3 b^3 B-a^2 b^4 (38 A+C)-3 a b^5 B+15 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 b d (a-b)^2 (a+b)^3}",1,"-1/16*((2*(56*a^4*A*b - 95*a^2*A*b^3 + 45*A*b^5 - 16*a^5*B + 19*a^3*b^2*B - 9*a*b^4*B + 9*a^4*b*C - 3*a^2*b^3*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(16*a^5*A - 80*a^3*A*b^2 + 40*a*A*b^4 + 32*a^4*b*B - 8*a^2*b^3*B - 16*a^5*C - 8*a^3*b^2*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((8*a^4*A*b - 29*a^2*A*b^3 + 15*A*b^5 + 9*a^3*b^2*B - 3*a*b^4*B - 5*a^4*b*C - a^2*b^3*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(a^3*(a - b)^2*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B - 5*a^4*C - a^2*b^2*C)*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2) + (A*b^2*Sin[c + d*x] - a*b*B*Sin[c + d*x] + a^2*C*Sin[c + d*x])/(2*a*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) + (11*a^2*A*b^2*Sin[c + d*x] - 5*A*b^4*Sin[c + d*x] - 7*a^3*b*B*Sin[c + d*x] + a*b^3*B*Sin[c + d*x] + 3*a^4*C*Sin[c + d*x] + 3*a^2*b^2*C*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/d","A",0
1497,1,903,473,7.244471,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^3} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{2 \left(16 A a^4+5 C a^4-9 b B a^3-19 A b^2 a^2+b^2 C a^2+3 b^3 B a+9 A b^4\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(16 B a^4-32 A b a^3-24 b C a^3+8 b^2 B a^2+8 A b^3 a\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(-C a^4+5 b B a^3-9 A b^2 a^2-5 b^2 C a^2+b^3 B a+3 A b^4\right) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left(-4 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a+2 (2 a-b) b F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}+2 b^2 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}}{16 a^2 (a-b)^2 (a+b)^2 d}+\frac{\sqrt{\sec (c+d x)} \left(\frac{\left(C a^4-5 b B a^3+9 A b^2 a^2+5 b^2 C a^2-b^3 B a-3 A b^4\right) \sin (c+d x)}{4 a^2 b \left(a^2-b^2\right)^2}+\frac{C \sin (c+d x) a^2-b B \sin (c+d x) a+A b^2 \sin (c+d x)}{2 b \left(b^2-a^2\right) (a+b \cos (c+d x))^2}+\frac{C \sin (c+d x) a^4+3 b B \sin (c+d x) a^3-7 A b^2 \sin (c+d x) a^2-7 b^2 C \sin (c+d x) a^2+3 b^3 B \sin (c+d x) a+A b^4 \sin (c+d x)}{4 a b \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}\right)}{d}","\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}-\frac{\sin (c+d x) \left(a^4 (-C)+5 a^3 b B-a^2 b^2 (9 A+5 C)+a b^3 B+3 A b^4\right)}{4 a^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 C+3 a^3 b B-7 a^2 b^2 (A+C)+3 a b^3 B+A b^4\right)}{4 a b^2 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (-C)+5 a^3 b B-a^2 b^2 (9 A+5 C)+a b^3 B+3 A b^4\right)}{4 a^2 b d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(a^6 (-C)-3 a^5 b B+5 a^4 b^2 (3 A+2 C)-10 a^3 b^3 B-3 a^2 b^4 (2 A-C)+a b^5 B+3 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b^2 d (a-b)^2 (a+b)^3}",1,"((2*(16*a^4*A - 19*a^2*A*b^2 + 9*A*b^4 - 9*a^3*b*B + 3*a*b^3*B + 5*a^4*C + a^2*b^2*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-32*a^3*A*b + 8*a*A*b^3 + 16*a^4*B + 8*a^2*b^2*B - 24*a^3*b*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-9*a^2*A*b^2 + 3*A*b^4 + 5*a^3*b*B + a*b^3*B - a^4*C - 5*a^2*b^2*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(16*a^2*(a - b)^2*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(((9*a^2*A*b^2 - 3*A*b^4 - 5*a^3*b*B - a*b^3*B + a^4*C + 5*a^2*b^2*C)*Sin[c + d*x])/(4*a^2*b*(a^2 - b^2)^2) + (A*b^2*Sin[c + d*x] - a*b*B*Sin[c + d*x] + a^2*C*Sin[c + d*x])/(2*b*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (-7*a^2*A*b^2*Sin[c + d*x] + A*b^4*Sin[c + d*x] + 3*a^3*b*B*Sin[c + d*x] + 3*a*b^3*B*Sin[c + d*x] + a^4*C*Sin[c + d*x] - 7*a^2*b^2*C*Sin[c + d*x])/(4*a*b*(a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/d","A",0
1498,1,902,478,7.3215309,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{\left(3 C a^4+b B a^3-5 A b^2 a^2-9 b^2 C a^2+5 b^3 B a-A b^4\right) \sin (c+d x)}{4 a b^2 \left(a^2-b^2\right)^2}-\frac{C \sin (c+d x) a^3-b B \sin (c+d x) a^2+A b^2 \sin (c+d x) a}{2 b^2 \left(b^2-a^2\right) (a+b \cos (c+d x))^2}+\frac{-5 C \sin (c+d x) a^4+b B \sin (c+d x) a^3+3 A b^2 \sin (c+d x) a^2+11 b^2 C \sin (c+d x) a^2-7 b^3 B \sin (c+d x) a+3 A b^4 \sin (c+d x)}{4 b^2 \left(b^2-a^2\right)^2 (a+b \cos (c+d x))}\right)}{d}-\frac{\frac{2 \left(C a^4-5 b B a^3+9 A b^2 a^2+5 b^2 C a^2-b^3 B a-3 A b^4\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(-16 A b a^3-8 b C a^3+24 b^2 B a^2-8 A b^3 a-16 b^3 C a\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(3 C a^4+b B a^3-5 A b^2 a^2-9 b^2 C a^2+5 b^3 B a-A b^4\right) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left(-4 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a+2 (2 a-b) b F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}+2 b^2 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}}{16 a (a-b)^2 b (a+b)^2 d}","-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}-\frac{\sin (c+d x) \left(-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right)}{4 a b d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^4 C+a^3 b B+a^2 b^2 (3 A-5 C)-7 a b^3 B+b^4 (3 A+8 C)\right)}{4 b^3 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right)}{4 a b^2 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-3 a^6 C-a^5 b B-3 a^4 b^2 (A-2 C)+10 a^3 b^3 B-5 a^2 b^4 (2 A+3 C)+3 a b^5 B+A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^3 d (a-b)^2 (a+b)^3}",1,"-1/16*((2*(9*a^2*A*b^2 - 3*A*b^4 - 5*a^3*b*B - a*b^3*B + a^4*C + 5*a^2*b^2*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-16*a^3*A*b - 8*a*A*b^3 + 24*a^2*b^2*B - 8*a^3*b*C - 16*a*b^3*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-5*a^2*A*b^2 - A*b^4 + a^3*b*B + 5*a*b^3*B + 3*a^4*C - 9*a^2*b^2*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(a*(a - b)^2*b*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(((-5*a^2*A*b^2 - A*b^4 + a^3*b*B + 5*a*b^3*B + 3*a^4*C - 9*a^2*b^2*C)*Sin[c + d*x])/(4*a*b^2*(a^2 - b^2)^2) - (a*A*b^2*Sin[c + d*x] - a^2*b*B*Sin[c + d*x] + a^3*C*Sin[c + d*x])/(2*b^2*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (3*a^2*A*b^2*Sin[c + d*x] + 3*A*b^4*Sin[c + d*x] + a^3*b*B*Sin[c + d*x] - 7*a*b^3*B*Sin[c + d*x] - 5*a^4*C*Sin[c + d*x] + 11*a^2*b^2*C*Sin[c + d*x])/(4*b^2*(-a^2 + b^2)^2*(a + b*Cos[c + d*x]))))/d","A",0
1499,1,915,483,7.3972631,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)),x]","\frac{\frac{2 \left(5 C a^4-b B a^3+5 A b^2 a^2-7 b^2 C a^2-5 b^3 B a+A b^4+8 b^4 C\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(16 B b^4-24 a A b^3-32 a C b^3+8 a^2 B b^2+8 a^3 C b\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(15 C a^4-3 b B a^3-A b^2 a^2-29 b^2 C a^2+9 b^3 B a-5 A b^4+8 b^4 C\right) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left(-4 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a+2 (2 a-b) b F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}+2 b^2 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}}{16 (a-b)^2 b^2 (a+b)^2 d}+\frac{\sqrt{\sec (c+d x)} \left(-\frac{\left(7 C a^4-3 b B a^3-A b^2 a^2-13 b^2 C a^2+9 b^3 B a-5 A b^4\right) \sin (c+d x)}{4 b^3 \left(a^2-b^2\right)^2}-\frac{-C \sin (c+d x) a^4+b B \sin (c+d x) a^3-A b^2 \sin (c+d x) a^2}{2 b^3 \left(b^2-a^2\right) (a+b \cos (c+d x))^2}+\frac{9 C \sin (c+d x) a^5-5 b B \sin (c+d x) a^4+A b^2 \sin (c+d x) a^3-15 b^2 C \sin (c+d x) a^3+11 b^3 B \sin (c+d x) a^2-7 A b^4 \sin (c+d x) a}{4 b^3 \left(b^2-a^2\right)^2 (a+b \cos (c+d x))}\right)}{d}","-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2}+\frac{\sin (c+d x) \left(-5 a^4 C+a^3 b B+a^2 b^2 (3 A+11 C)-7 a b^3 B+3 A b^4\right)}{4 b^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-15 a^4 C+3 a^3 b B+a^2 b^2 (A+29 C)-9 a b^3 B+b^4 (5 A-8 C)\right)}{4 b^3 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-15 a^5 C+3 a^4 b B+a^3 b^2 (A+33 C)-5 a^2 b^3 B-a b^4 (7 A+24 C)+8 b^5 B\right)}{4 b^4 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(15 a^6 C-3 a^5 b B-a^4 b^2 (A+38 C)+6 a^3 b^3 B+5 a^2 b^4 (2 A+7 C)-15 a b^5 B+3 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d (a-b)^2 (a+b)^3}",1,"((2*(5*a^2*A*b^2 + A*b^4 - a^3*b*B - 5*a*b^3*B + 5*a^4*C - 7*a^2*b^2*C + 8*b^4*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-24*a*A*b^3 + 8*a^2*b^2*B + 16*b^4*B + 8*a^3*b*C - 32*a*b^3*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-(a^2*A*b^2) - 5*A*b^4 - 3*a^3*b*B + 9*a*b^3*B + 15*a^4*C - 29*a^2*b^2*C + 8*b^4*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(16*(a - b)^2*b^2*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(-1/4*((-(a^2*A*b^2) - 5*A*b^4 - 3*a^3*b*B + 9*a*b^3*B + 7*a^4*C - 13*a^2*b^2*C)*Sin[c + d*x])/(b^3*(a^2 - b^2)^2) - (-(a^2*A*b^2*Sin[c + d*x]) + a^3*b*B*Sin[c + d*x] - a^4*C*Sin[c + d*x])/(2*b^3*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (a^3*A*b^2*Sin[c + d*x] - 7*a*A*b^4*Sin[c + d*x] - 5*a^4*b*B*Sin[c + d*x] + 11*a^2*b^3*B*Sin[c + d*x] + 9*a^5*C*Sin[c + d*x] - 15*a^3*b^2*C*Sin[c + d*x])/(4*b^3*(-a^2 + b^2)^2*(a + b*Cos[c + d*x]))))/d","A",0
1500,1,972,596,7.6211982,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{a \left(11 C a^4-7 b B a^3+3 A b^2 a^2-17 b^2 C a^2+13 b^3 B a-9 A b^4\right) \sin (c+d x)}{4 b^4 \left(a^2-b^2\right)^2}-\frac{C \sin (c+d x) a^5-b B \sin (c+d x) a^4+A b^2 \sin (c+d x) a^3}{2 b^4 \left(b^2-a^2\right) (a+b \cos (c+d x))^2}+\frac{-13 C \sin (c+d x) a^6+9 b B \sin (c+d x) a^5-5 A b^2 \sin (c+d x) a^4+19 b^2 C \sin (c+d x) a^4-15 b^3 B \sin (c+d x) a^3+11 A b^4 \sin (c+d x) a^2}{4 b^4 \left(b^2-a^2\right)^2 (a+b \cos (c+d x))}+\frac{C \sin (2 (c+d x))}{3 b^3}\right)}{d}-\frac{\frac{2 \left(35 C a^5-15 b B a^4+3 A b^2 a^3-73 b^2 C a^3+21 b^3 B a^2+15 A b^4 a+56 b^4 C a-24 b^5 B\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(-48 A b^5-16 C b^5+96 a B b^4-24 a^2 A b^3-112 a^2 C b^3-24 a^3 B b^2+56 a^4 C b\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(105 C a^5-45 b B a^4+9 A b^2 a^3-195 b^2 C a^3+87 b^3 B a^2-27 A b^4 a+72 b^4 C a-24 b^5 B\right) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left(-4 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a+2 (2 a-b) b F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}+2 b^2 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}}{48 (a-b)^2 b^3 (a+b)^2 d}","-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac{\sin (c+d x) \left(-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right)}{12 b^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) \left(-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right)}{4 b^2 d \left(a^2-b^2\right)^2 \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right)}{4 b^4 d \left(a^2-b^2\right)^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-105 a^6 C+45 a^5 b B-a^4 b^2 (9 A-223 C)-99 a^3 b^3 B+a^2 b^4 (15 A-128 C)+72 a b^5 B-8 b^6 (3 A+C)\right)}{12 b^5 d \left(a^2-b^2\right)^2}-\frac{a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(35 a^6 C-15 a^5 b B+a^4 b^2 (3 A-86 C)+38 a^3 b^3 B-3 a^2 b^4 (2 A-21 C)-35 a b^5 B+15 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^5 d (a-b)^2 (a+b)^3}",1,"-1/48*((2*(3*a^3*A*b^2 + 15*a*A*b^4 - 15*a^4*b*B + 21*a^2*b^3*B - 24*b^5*B + 35*a^5*C - 73*a^3*b^2*C + 56*a*b^4*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-24*a^2*A*b^3 - 48*A*b^5 - 24*a^3*b^2*B + 96*a*b^4*B + 56*a^4*b*C - 112*a^2*b^3*C - 16*b^5*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((9*a^3*A*b^2 - 27*a*A*b^4 - 45*a^4*b*B + 87*a^2*b^3*B - 24*b^5*B + 105*a^5*C - 195*a^3*b^2*C + 72*a*b^4*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/((a - b)^2*b^3*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*((a*(3*a^2*A*b^2 - 9*A*b^4 - 7*a^3*b*B + 13*a*b^3*B + 11*a^4*C - 17*a^2*b^2*C)*Sin[c + d*x])/(4*b^4*(a^2 - b^2)^2) - (a^3*A*b^2*Sin[c + d*x] - a^4*b*B*Sin[c + d*x] + a^5*C*Sin[c + d*x])/(2*b^4*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (-5*a^4*A*b^2*Sin[c + d*x] + 11*a^2*A*b^4*Sin[c + d*x] + 9*a^5*b*B*Sin[c + d*x] - 15*a^3*b^3*B*Sin[c + d*x] - 13*a^6*C*Sin[c + d*x] + 19*a^4*b^2*C*Sin[c + d*x])/(4*b^4*(-a^2 + b^2)^2*(a + b*Cos[c + d*x])) + (C*Sin[2*(c + d*x)])/(3*b^3)))/d","A",0
1501,1,1059,714,7.9339234,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)),x]","\frac{\frac{2 \left(315 C a^6-175 b B a^5+75 A b^2 a^4-633 b^2 C a^4+365 b^3 B a^3-105 A b^4 a^2+336 b^4 C a^2-280 b^5 B a+120 A b^6+72 b^6 C\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(80 B b^6-480 a A b^5-96 a C b^5+560 a^2 B b^4+120 a^3 A b^3-768 a^3 C b^3-280 a^4 B b^2+504 a^5 C b\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(945 C a^6-525 b B a^5+225 A b^2 a^4-1683 b^2 C a^4+975 b^3 B a^3-435 A b^4 a^2+576 b^4 C a^2-360 b^5 B a+120 A b^6+72 b^6 C\right) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left(-4 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a+2 (2 a-b) b F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}+2 b^2 \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}}{240 (a-b)^2 b^4 (a+b)^2 d}+\frac{\sqrt{\sec (c+d x)} \left(-\frac{\left(75 C a^6-55 b B a^5+35 A b^2 a^4-107 b^2 C a^4+85 b^3 B a^3-65 A b^4 a^2+4 b^4 C a^2-2 b^6 C\right) \sin (c+d x)}{20 b^5 \left(a^2-b^2\right)^2}-\frac{-C \sin (c+d x) a^6+b B \sin (c+d x) a^5-A b^2 \sin (c+d x) a^4}{2 b^5 \left(b^2-a^2\right) (a+b \cos (c+d x))^2}+\frac{17 C \sin (c+d x) a^7-13 b B \sin (c+d x) a^6+9 A b^2 \sin (c+d x) a^5-23 b^2 C \sin (c+d x) a^5+19 b^3 B \sin (c+d x) a^4-15 A b^4 \sin (c+d x) a^3}{4 b^5 \left(b^2-a^2\right)^2 (a+b \cos (c+d x))}+\frac{(b B-3 a C) \sin (2 (c+d x))}{3 b^4}+\frac{C \sin (3 (c+d x))}{10 b^3}\right)}{d}","-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^2}-\frac{\sin (c+d x) \left(-63 a^4 C+35 a^3 b B-a^2 b^2 (15 A-101 C)-65 a b^3 B+b^4 (45 A-8 C)\right)}{20 b^3 d \left(a^2-b^2\right)^2 \sec ^{\frac{3}{2}}(c+d x)}+\frac{\sin (c+d x) \left(-9 a^4 C+5 a^3 b B-a^2 b^2 (A-15 C)-11 a b^3 B+7 A b^4\right)}{4 b^2 d \left(a^2-b^2\right)^2 \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \left(-63 a^5 C+35 a^4 b B-15 a^3 b^2 (A-7 C)-61 a^2 b^3 B+3 a b^4 (11 A-8 C)+8 b^5 B\right)}{12 b^4 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-315 a^6 C+175 a^5 b B-3 a^4 b^2 (25 A-187 C)-325 a^3 b^3 B+a^2 b^4 (145 A-192 C)+120 a b^5 B-8 b^6 (5 A+3 C)\right)}{20 b^5 d \left(a^2-b^2\right)^2}+\frac{a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(63 a^6 C-35 a^5 b B+15 a^4 b^2 (A-10 C)+86 a^3 b^3 B-a^2 b^4 (38 A-99 C)-63 a b^5 B+35 A b^6\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^6 d (a-b)^2 (a+b)^3}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-189 a^7 C+105 a^6 b B-9 a^5 b^2 (5 A-43 C)-223 a^4 b^3 B+3 a^3 b^4 (33 A-64 C)+128 a^2 b^5 B-24 a b^6 (3 A+C)+8 b^7 B\right)}{12 b^6 d \left(a^2-b^2\right)^2}",1,"((2*(75*a^4*A*b^2 - 105*a^2*A*b^4 + 120*A*b^6 - 175*a^5*b*B + 365*a^3*b^3*B - 280*a*b^5*B + 315*a^6*C - 633*a^4*b^2*C + 336*a^2*b^4*C + 72*b^6*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(120*a^3*A*b^3 - 480*a*A*b^5 - 280*a^4*b^2*B + 560*a^2*b^4*B + 80*b^6*B + 504*a^5*b*C - 768*a^3*b^3*C - 96*a*b^5*C)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((225*a^4*A*b^2 - 435*a^2*A*b^4 + 120*A*b^6 - 525*a^5*b*B + 975*a^3*b^3*B - 360*a*b^5*B + 945*a^6*C - 1683*a^4*b^2*C + 576*a^2*b^4*C + 72*b^6*C)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(240*(a - b)^2*b^4*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(-1/20*((35*a^4*A*b^2 - 65*a^2*A*b^4 - 55*a^5*b*B + 85*a^3*b^3*B + 75*a^6*C - 107*a^4*b^2*C + 4*a^2*b^4*C - 2*b^6*C)*Sin[c + d*x])/(b^5*(a^2 - b^2)^2) - (-(a^4*A*b^2*Sin[c + d*x]) + a^5*b*B*Sin[c + d*x] - a^6*C*Sin[c + d*x])/(2*b^5*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (9*a^5*A*b^2*Sin[c + d*x] - 15*a^3*A*b^4*Sin[c + d*x] - 13*a^6*b*B*Sin[c + d*x] + 19*a^4*b^3*B*Sin[c + d*x] + 17*a^7*C*Sin[c + d*x] - 23*a^5*b^2*C*Sin[c + d*x])/(4*b^5*(-a^2 + b^2)^2*(a + b*Cos[c + d*x])) + ((b*B - 3*a*C)*Sin[2*(c + d*x)])/(3*b^4) + (C*Sin[3*(c + d*x)])/(10*b^3)))/d","A",0
1502,1,802,592,20.5915249,"\int \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{2 \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(-\left((a+b) \left(21 (7 A+9 C) a^4+57 b B a^3-6 b^2 (4 A+7 C) a^2+24 b^3 B a-16 A b^4\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)+a (a+b) \left(3 (49 A+25 B+63 C) a^3-6 b (6 A+3 B+7 C) a^2+12 b^2 (A+2 B) a-16 A b^3\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)+\left(21 (7 A+9 C) a^4+57 b B a^3-6 b^2 (4 A+7 C) a^2+24 b^3 B a-16 A b^4\right) \tan \left(\frac{1}{2} (c+d x)\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b\right)\right)}{315 a^4 d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}+\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{2 (A b \sin (c+d x)+9 a B \sin (c+d x)) \sec ^3(c+d x)}{63 a}+\frac{2}{9} A \tan (c+d x) \sec ^3(c+d x)+\frac{2 \left(49 A \sin (c+d x) a^2+63 C \sin (c+d x) a^2+9 b B \sin (c+d x) a-6 A b^2 \sin (c+d x)\right) \sec ^2(c+d x)}{315 a^2}+\frac{2 \left(75 B \sin (c+d x) a^3+13 A b \sin (c+d x) a^2+21 b C \sin (c+d x) a^2-12 b^2 B \sin (c+d x) a+8 A b^3 \sin (c+d x)\right) \sec (c+d x)}{315 a^3}+\frac{2 \left(147 A a^4+189 C a^4+57 b B a^3-24 A b^2 a^2-42 b^2 C a^2+24 b^3 B a-16 A b^4\right) \sin (c+d x)}{315 a^4}\right)}{d}","-\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right) \sqrt{a+b \cos (c+d x)}}{315 a^2 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right) \sqrt{a+b \cos (c+d x)}}{315 a^3 d}-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^3 (49 A-25 B+63 C)+6 a^2 b (6 A-3 B+7 C)+12 a b^2 (A-2 B)+16 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^4 d \sqrt{\sec (c+d x)}}-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-21 a^4 (7 A+9 C)-57 a^3 b B+6 a^2 b^2 (4 A+7 C)-24 a b^3 B+16 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^5 d \sqrt{\sec (c+d x)}}+\frac{2 (9 a B+A b) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{63 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{9 d}",1,"(2*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*((-16*A*b^4 + 57*a^3*b*B + 24*a*b^3*B - 6*a^2*b^2*(4*A + 7*C) + 21*a^4*(7*A + 9*C))*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)*(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2) - (a + b)*(-16*A*b^4 + 57*a^3*b*B + 24*a*b^3*B - 6*a^2*b^2*(4*A + 7*C) + 21*a^4*(7*A + 9*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + a*(a + b)*(-16*A*b^3 + 12*a*b^2*(A + 2*B) - 6*a^2*b*(6*A + 3*B + 7*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(315*a^4*d*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(147*a^4*A - 24*a^2*A*b^2 - 16*A*b^4 + 57*a^3*b*B + 24*a*b^3*B + 189*a^4*C - 42*a^2*b^2*C)*Sin[c + d*x])/(315*a^4) + (2*Sec[c + d*x]^3*(A*b*Sin[c + d*x] + 9*a*B*Sin[c + d*x]))/(63*a) + (2*Sec[c + d*x]^2*(49*a^2*A*Sin[c + d*x] - 6*A*b^2*Sin[c + d*x] + 9*a*b*B*Sin[c + d*x] + 63*a^2*C*Sin[c + d*x]))/(315*a^2) + (2*Sec[c + d*x]*(13*a^2*A*b*Sin[c + d*x] + 8*A*b^3*Sin[c + d*x] + 75*a^3*B*Sin[c + d*x] - 12*a*b^2*B*Sin[c + d*x] + 21*a^2*b*C*Sin[c + d*x]))/(315*a^3) + (2*A*Sec[c + d*x]^3*Tan[c + d*x])/9))/d","A",0
1503,1,3574,487,25.9947096,"\int \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\text{Result too large to show}","-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(-5 a^2 (5 A+7 C)-7 a b B+4 A b^2\right) \sqrt{a+b \cos (c+d x)}}{105 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (25 A-63 B+35 C)+2 a b (3 A-7 B)+8 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(63 a^3 B+a^2 b (19 A+35 C)-14 a b^2 B+8 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 (7 a B+A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{35 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{7 d}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B + 35*a^2*b*C)*Sin[c + d*x])/(105*a^3) + (2*Sec[c + d*x]^2*(A*b*Sin[c + d*x] + 7*a*B*Sin[c + d*x]))/(35*a) + (2*Sec[c + d*x]*(25*a^2*A*Sin[c + d*x] - 4*A*b^2*Sin[c + d*x] + 7*a*b*B*Sin[c + d*x] + 35*a^2*C*Sin[c + d*x]))/(105*a^2) + (2*A*Sec[c + d*x]^2*Tan[c + d*x])/7))/d + (2*((-19*A*b)/(105*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*A*b^3)/(105*a^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (3*a*B)/(5*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b^2*B)/(15*a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (b*C)/(3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (5*a*A*Sqrt[Sec[c + d*x]])/(21*Sqrt[a + b*Cos[c + d*x]]) - (17*A*b^2*Sqrt[Sec[c + d*x]])/(105*a*Sqrt[a + b*Cos[c + d*x]]) - (8*A*b^4*Sqrt[Sec[c + d*x]])/(105*a^3*Sqrt[a + b*Cos[c + d*x]]) - (2*b*B*Sqrt[Sec[c + d*x]])/(15*Sqrt[a + b*Cos[c + d*x]]) + (2*b^3*B*Sqrt[Sec[c + d*x]])/(15*a^2*Sqrt[a + b*Cos[c + d*x]]) + (a*C*Sqrt[Sec[c + d*x]])/(3*Sqrt[a + b*Cos[c + d*x]]) - (b^2*C*Sqrt[Sec[c + d*x]])/(3*a*Sqrt[a + b*Cos[c + d*x]]) - (19*A*b^2*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*a*Sqrt[a + b*Cos[c + d*x]]) - (8*A*b^4*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*a^3*Sqrt[a + b*Cos[c + d*x]]) - (3*b*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*Sqrt[a + b*Cos[c + d*x]]) + (2*b^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*a^2*Sqrt[a + b*Cos[c + d*x]]) - (b^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*a*Sqrt[a + b*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(8*A*b^2 - 2*a*b*(3*A + 7*B) + a^2*(25*A + 63*B + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*a^3*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*((b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(-2*(a + b)*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(8*A*b^2 - 2*a*b*(3*A + 7*B) + a^2*(25*A + 63*B + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*a^3*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(-2*(a + b)*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(8*A*b^2 - 2*a*b*(3*A + 7*B) + a^2*(25*A + 63*B + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1/2*((8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4) - ((a + b)*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (a*(a + b)*(8*A*b^2 - 2*a*b*(3*A + 7*B) + a^2*(25*A + 63*B + 35*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - ((a + b)*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (a*(a + b)*(8*A*b^2 - 2*a*b*(3*A + 7*B) + a^2*(25*A + 63*B + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + b*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (a*(a + b)*(8*A*b^2 - 2*a*b*(3*A + 7*B) + a^2*(25*A + 63*B + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a + b)*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(105*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((-2*(a + b)*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(8*A*b^2 - 2*a*b*(3*A + 7*B) + a^2*(25*A + 63*B + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(105*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
1504,1,466,400,19.5776341,"\int \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{\sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \sin (c+d x) \left(9 a^2 A+15 a^2 C+5 a b B-2 A b^2\right)}{15 a^2}+\frac{2 \sec (c+d x) (5 a B \sin (c+d x)+A b \sin (c+d x))}{15 a}+\frac{2}{5} A \tan (c+d x) \sec (c+d x)\right)}{d}+\frac{2 \sqrt{2} \sqrt{\frac{\cos (c+d x)}{(\cos (c+d x)+1)^2}} \sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \left(-\left(\cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(3 a^2 (3 A+5 C)+5 a b B-2 A b^2\right) (a+b \cos (c+d x))\right)-(a+b) \sec (c+d x) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2} \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))}{a+b}} \left(\left(3 a^2 (3 A+5 C)+5 a b B-2 A b^2\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-a (9 a A+5 a (B+3 C)-2 A b) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)\right)}{15 a^2 d \sqrt{\frac{1}{\cos (c+d x)+1}} \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} \sqrt{a+b \cos (c+d x)}}","-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) (a (9 A-5 B+15 C)+2 A b) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^2 d \sqrt{\sec (c+d x)}}-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-3 a^2 (3 A+5 C)-5 a b B+2 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (5 a B+A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{15 a d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}",1,"(2*Sqrt[2]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])^2]*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(-((a + b)*((-2*A*b^2 + 5*a*b*B + 3*a^2*(3*A + 5*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - a*(9*a*A - 2*A*b + 5*a*(B + 3*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]) - (-2*A*b^2 + 5*a*b*B + 3*a^2*(3*A + 5*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(15*a^2*d*Sqrt[(1 + Cos[c + d*x])^(-1)]*Sqrt[a + b*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(9*a^2*A - 2*A*b^2 + 5*a*b*B + 15*a^2*C)*Sin[c + d*x])/(15*a^2) + (2*Sec[c + d*x]*(A*b*Sin[c + d*x] + 5*a*B*Sin[c + d*x]))/(15*a) + (2*A*Sec[c + d*x]*Tan[c + d*x])/5))/d","A",0
1505,1,5171,467,23.770864,"\int \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\text{Result too large to show}","\frac{2 (a-b) \sqrt{a+b} (3 a B+A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) (b (A-3 B)-a (A-3 B+3 C)) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}-\frac{2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}",1,"Result too large to show","B",0
1506,1,896,509,16.2772695,"\int \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{2 A \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(2 a A \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)-a C \tan ^5\left(\frac{1}{2} (c+d x)\right)+b C \tan ^5\left(\frac{1}{2} (c+d x)\right)+4 A b \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 b C \tan ^3\left(\frac{1}{2} (c+d x)\right)+4 b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+2 a C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-2 a A \tan \left(\frac{1}{2} (c+d x)\right)-2 A b \tan \left(\frac{1}{2} (c+d x)\right)+a C \tan \left(\frac{1}{2} (c+d x)\right)+b C \tan \left(\frac{1}{2} (c+d x)\right)-(a+b) (2 A-C) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 (b (A-B)+a (A+B-C)) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+4 b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 a C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) (2 A b-a (2 A-2 B-C)) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{\sec (c+d x)}}-\frac{(2 A-C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}+\frac{(a-b) \sqrt{a+b} (2 A-C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{\sqrt{a+b} (a C+2 b B) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d \sqrt{\sec (c+d x)}}",1,"(2*A*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-2*a*A*Tan[(c + d*x)/2] - 2*A*b*Tan[(c + d*x)/2] + a*C*Tan[(c + d*x)/2] + b*C*Tan[(c + d*x)/2] + 4*A*b*Tan[(c + d*x)/2]^3 - 2*b*C*Tan[(c + d*x)/2]^3 + 2*a*A*Tan[(c + d*x)/2]^5 - 2*A*b*Tan[(c + d*x)/2]^5 - a*C*Tan[(c + d*x)/2]^5 + b*C*Tan[(c + d*x)/2]^5 + 4*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*a*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 4*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*a*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (a + b)*(2*A - C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*(b*(A - B) + a*(A + B - C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(d*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","A",0
1507,1,1816,543,19.372417,"\int \sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\frac{C \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (2 (c+d x))}{4 d}+\frac{-4 b^2 \sqrt{\frac{a-b}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)+4 a b \sqrt{\frac{a-b}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)+a^2 \sqrt{\frac{a-b}{a+b}} C \tan ^5\left(\frac{1}{2} (c+d x)\right)-a b \sqrt{\frac{a-b}{a+b}} C \tan ^5\left(\frac{1}{2} (c+d x)\right)+8 b^2 \sqrt{\frac{a-b}{a+b}} B \tan ^3\left(\frac{1}{2} (c+d x)\right)+2 a b \sqrt{\frac{a-b}{a+b}} C \tan ^3\left(\frac{1}{2} (c+d x)\right)+16 i A b^2 \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+8 i a b B \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-2 i a^2 C \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+8 i b^2 C \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-4 b^2 \sqrt{\frac{a-b}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)-4 a b \sqrt{\frac{a-b}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)-a^2 \sqrt{\frac{a-b}{a+b}} C \tan \left(\frac{1}{2} (c+d x)\right)-a b \sqrt{\frac{a-b}{a+b}} C \tan \left(\frac{1}{2} (c+d x)\right)-i (a-b) (4 b B+a C) E\left(i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 i (a-b) (4 A b+(a+2 b) C) F\left(i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+16 i A b^2 \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+8 i a b B \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 i a^2 C \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+8 i b^2 C \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}}{4 b \sqrt{\frac{a-b}{a+b}} d \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (-C)+4 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) (a C+8 A b+2 b (2 B+C)) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d \sqrt{\sec (c+d x)}}+\frac{(a C+4 b B) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}-\frac{(a-b) \sqrt{a+b} (a C+4 b B) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}",1,"(C*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[2*(c + d*x)])/(4*d) + (-4*a*b*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2] - 4*b^2*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2] - a^2*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2] - a*b*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2] + 8*b^2*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2]^3 + 2*a*b*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2]^3 + 4*a*b*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - 4*b^2*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2]^5 + a^2*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2]^5 - a*b*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2]^5 + (16*I)*A*b^2*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*I)*a*b*B*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*a^2*C*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*I)*b^2*C*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (16*I)*A*b^2*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*I)*a*b*B*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*a^2*C*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*I)*b^2*C*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(a - b)*(4*b*B + a*C)*EllipticE[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*(a - b)*(4*A*b + (a + 2*b)*C)*EllipticF[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)])/(4*b*Sqrt[(a - b)/(a + b)]*d*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","C",0
1508,1,1828,646,15.5815123,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{1}{12} C \sin (c+d x)+\frac{(6 b B+a C) \sin (2 (c+d x))}{24 b}+\frac{1}{12} C \sin (3 (c+d x))\right)}{d}+\frac{\sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(24 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-24 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+6 a b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-6 a^2 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 a^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+16 b^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 a b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 a^2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-48 A b^3 \tan ^3\left(\frac{1}{2} (c+d x)\right)-12 a b^2 B \tan ^3\left(\frac{1}{2} (c+d x)\right)-32 b^3 C \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 a^2 b C \tan ^3\left(\frac{1}{2} (c+d x)\right)+48 a A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+48 b^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-12 a^2 b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+6 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+24 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+24 A b^3 \tan \left(\frac{1}{2} (c+d x)\right)+24 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+6 a b^2 B \tan \left(\frac{1}{2} (c+d x)\right)+6 a^2 b B \tan \left(\frac{1}{2} (c+d x)\right)-3 a^3 C \tan \left(\frac{1}{2} (c+d x)\right)+16 b^3 C \tan \left(\frac{1}{2} (c+d x)\right)+16 a b^2 C \tan \left(\frac{1}{2} (c+d x)\right)-3 a^2 b C \tan \left(\frac{1}{2} (c+d x)\right)-(a+b) \left(3 C a^2-6 b B a-24 A b^2-16 b^2 C\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 b \left(C a^2-2 b (12 A-3 B+7 C) a-12 b^2 B\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+48 a A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+48 b^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-12 a^2 b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+6 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+24 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{24 b^2 d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \sqrt{\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^3 (-C)+2 a^2 b B-4 a b^2 (2 A+C)-8 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^3 d \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(3 a (2 b B-a C)+8 b^2 (3 A+2 C)\right) \sqrt{a+b \cos (c+d x)}}{24 b^2 d}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left((a+2 b) (-3 a C+6 b B+8 b C)+24 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^2 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a (2 b B-a C)+8 b^2 (3 A+2 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^2 d \sqrt{\sec (c+d x)}}+\frac{(2 b B-a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 b d \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((C*Sin[c + d*x])/12 + ((6*b*B + a*C)*Sin[2*(c + d*x)])/(24*b) + (C*Sin[3*(c + d*x)])/12))/d + (Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(24*a*A*b^2*Tan[(c + d*x)/2] + 24*A*b^3*Tan[(c + d*x)/2] + 6*a^2*b*B*Tan[(c + d*x)/2] + 6*a*b^2*B*Tan[(c + d*x)/2] - 3*a^3*C*Tan[(c + d*x)/2] - 3*a^2*b*C*Tan[(c + d*x)/2] + 16*a*b^2*C*Tan[(c + d*x)/2] + 16*b^3*C*Tan[(c + d*x)/2] - 48*A*b^3*Tan[(c + d*x)/2]^3 - 12*a*b^2*B*Tan[(c + d*x)/2]^3 + 6*a^2*b*C*Tan[(c + d*x)/2]^3 - 32*b^3*C*Tan[(c + d*x)/2]^3 - 24*a*A*b^2*Tan[(c + d*x)/2]^5 + 24*A*b^3*Tan[(c + d*x)/2]^5 - 6*a^2*b*B*Tan[(c + d*x)/2]^5 + 6*a*b^2*B*Tan[(c + d*x)/2]^5 + 3*a^3*C*Tan[(c + d*x)/2]^5 - 3*a^2*b*C*Tan[(c + d*x)/2]^5 - 16*a*b^2*C*Tan[(c + d*x)/2]^5 + 16*b^3*C*Tan[(c + d*x)/2]^5 + 48*a*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 12*a^2*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*b^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 24*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 12*a^2*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*b^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 24*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (a + b)*(-24*A*b^2 - 6*a*b*B + 3*a^2*C - 16*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*b*(-12*b^2*B + a^2*C - 2*a*b*(12*A - 3*B + 7*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(24*b^2*d*(-1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(b*(-1 + Tan[(c + d*x)/2]^2) - a*(1 + Tan[(c + d*x)/2]^2)))","B",1
1509,1,852,766,15.136429,"\int \frac{\sqrt{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{(8 b B+a C) \sin (c+d x)}{96 b}+\frac{\left(-5 C a^2+8 b B a+48 A b^2+48 b^2 C\right) \sin (2 (c+d x))}{192 b^2}+\frac{(8 b B+a C) \sin (3 (c+d x))}{96 b}+\frac{1}{32} C \sin (4 (c+d x))\right)}{d}-\frac{-b (b-a) (a+b) \left(15 C a^3-24 b B a^2+4 b^2 (12 A+7 C) a+128 b^3 B\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)+a (a-b) (a+b) \left(15 C a^3-6 b (4 B+5 C) a^2+4 b^2 (12 A+12 B+11 C) a-8 b^3 (12 A+16 B+9 C)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)-3 (a-b) \left(5 C a^4-8 b B a^3+8 b^2 (2 A+C) a^2-32 b^3 B a-16 b^4 (4 A+3 C)\right) \left((a-b) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+2 b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)-(a-b) b \left(15 C a^3-24 b B a^2+4 b^2 (12 A+7 C) a+128 b^3 B\right) \tan \left(\frac{1}{2} (c+d x)\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b\right)}{192 (a-b) b^4 d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\sin (c+d x) \left(5 a^2 C-8 a b B+16 A b^2+12 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{32 b^2 d \sqrt{\sec (c+d x)}}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-15 a^3 C+24 a^2 b B-4 a b^2 (12 A+7 C)-128 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{192 b^3 d}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(15 a^3 C-2 a^2 b (12 B+5 C)+4 a b^2 (12 A+4 B+7 C)+8 b^3 (12 A+16 B+9 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b^3 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-15 a^3 C+24 a^2 b B-4 a b^2 (12 A+7 C)-128 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 a b^3 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-5 a^4 C+8 a^3 b B-8 a^2 b^2 (2 A+C)+32 a b^3 B+16 b^4 (4 A+3 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^4 d \sqrt{\sec (c+d x)}}+\frac{(8 b B-5 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{4 b d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(((8*b*B + a*C)*Sin[c + d*x])/(96*b) + ((48*A*b^2 + 8*a*b*B - 5*a^2*C + 48*b^2*C)*Sin[2*(c + d*x)])/(192*b^2) + ((8*b*B + a*C)*Sin[3*(c + d*x)])/(96*b) + (C*Sin[4*(c + d*x)])/32))/d - (-((a - b)*b*(-24*a^2*b*B + 128*b^3*B + 15*a^3*C + 4*a*b^2*(12*A + 7*C))*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)*(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)) - b*(-a + b)*(a + b)*(-24*a^2*b*B + 128*b^3*B + 15*a^3*C + 4*a*b^2*(12*A + 7*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + a*(a - b)*(a + b)*(15*a^3*C - 6*a^2*b*(4*B + 5*C) - 8*b^3*(12*A + 16*B + 9*C) + 4*a*b^2*(12*A + 12*B + 11*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 3*(a - b)*(-8*a^3*b*B - 32*a*b^3*B + 5*a^4*C + 8*a^2*b^2*(2*A + C) - 16*b^4*(4*A + 3*C))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)])/(192*(a - b)*b^4*d*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","A",1
1510,1,801,590,20.6472783,"\int (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{2 \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(-\left((a+b) \left(21 (7 A+9 C) a^4+246 b B a^3+3 b^2 (11 A+21 C) a^2-18 b^3 B a+8 A b^4\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)+a (a+b) \left(3 (49 A+25 B+63 C) a^3+3 b (13 A+57 B+21 C) a^2-6 b^2 (A+3 B) a+8 A b^3\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)+\left(21 (7 A+9 C) a^4+246 b B a^3+3 b^2 (11 A+21 C) a^2-18 b^3 B a+8 A b^4\right) \tan \left(\frac{1}{2} (c+d x)\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b\right)\right)}{315 a^3 d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}+\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{2}{63} (10 A b \sin (c+d x)+9 a B \sin (c+d x)) \sec ^3(c+d x)+\frac{2}{9} a A \tan (c+d x) \sec ^3(c+d x)+\frac{2 \left(49 A \sin (c+d x) a^2+63 C \sin (c+d x) a^2+72 b B \sin (c+d x) a+3 A b^2 \sin (c+d x)\right) \sec ^2(c+d x)}{315 a}+\frac{2 \left(75 B \sin (c+d x) a^3+88 A b \sin (c+d x) a^2+126 b C \sin (c+d x) a^2+9 b^2 B \sin (c+d x) a-4 A b^3 \sin (c+d x)\right) \sec (c+d x)}{315 a^2}+\frac{2 \left(147 A a^4+189 C a^4+246 b B a^3+33 A b^2 a^2+63 b^2 C a^2-18 b^3 B a+8 A b^4\right) \sin (c+d x)}{315 a^3}\right)}{d}","\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)}}{315 a d}-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(-75 a^3 B-2 a^2 b (44 A+63 C)-9 a b^2 B+4 A b^3\right) \sqrt{a+b \cos (c+d x)}}{315 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-3 a^3 (49 A-25 B+63 C)+3 a^2 b (13 A-57 B+21 C)+6 a b^2 (A-3 B)+8 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(21 a^4 (7 A+9 C)+246 a^3 b B+3 a^2 b^2 (11 A+21 C)-18 a b^3 B+8 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 (3 a B+A b) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{21 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{9 d}",1,"(2*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*((8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)*(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2) - (a + b)*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + a*(a + b)*(8*A*b^3 - 6*a*b^2*(A + 3*B) + 3*a^2*b*(13*A + 57*B + 21*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(315*a^3*d*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 189*a^4*C + 63*a^2*b^2*C)*Sin[c + d*x])/(315*a^3) + (2*Sec[c + d*x]^3*(10*A*b*Sin[c + d*x] + 9*a*B*Sin[c + d*x]))/63 + (2*Sec[c + d*x]^2*(49*a^2*A*Sin[c + d*x] + 3*A*b^2*Sin[c + d*x] + 72*a*b*B*Sin[c + d*x] + 63*a^2*C*Sin[c + d*x]))/(315*a) + (2*Sec[c + d*x]*(88*a^2*A*b*Sin[c + d*x] - 4*A*b^3*Sin[c + d*x] + 75*a^3*B*Sin[c + d*x] + 9*a*b^2*B*Sin[c + d*x] + 126*a^2*b*C*Sin[c + d*x]))/(315*a^2) + (2*a*A*Sec[c + d*x]^3*Tan[c + d*x])/9))/d","A",0
1511,1,3611,490,26.1608261,"\int (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\text{Result too large to show}","\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(5 a^2 (5 A+7 C)+42 a b B+3 A b^2\right) \sqrt{a+b \cos (c+d x)}}{105 a d}-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-\left(a^2 (25 A-63 B+35 C)\right)+3 a b (19 A-7 B+35 C)+6 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d \sqrt{\sec (c+d x)}}-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-63 a^3 B-2 a^2 b (41 A+70 C)-21 a b^2 B+6 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (7 a B+3 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{7 d}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((-2*(-82*a^2*A*b + 6*A*b^3 - 63*a^3*B - 21*a*b^2*B - 140*a^2*b*C)*Sin[c + d*x])/(105*a^2) + (2*Sec[c + d*x]^2*(8*A*b*Sin[c + d*x] + 7*a*B*Sin[c + d*x]))/35 + (2*Sec[c + d*x]*(25*a^2*A*Sin[c + d*x] + 3*A*b^2*Sin[c + d*x] + 42*a*b*B*Sin[c + d*x] + 35*a^2*C*Sin[c + d*x]))/(105*a) + (2*a*A*Sec[c + d*x]^2*Tan[c + d*x])/7))/d + (2*((-82*a*A*b)/(105*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*A*b^3)/(35*a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (3*a^2*B)/(5*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (b^2*B)/(5*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*a*b*C)/(3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (5*a^2*A*Sqrt[Sec[c + d*x]])/(21*Sqrt[a + b*Cos[c + d*x]]) - (31*A*b^2*Sqrt[Sec[c + d*x]])/(105*Sqrt[a + b*Cos[c + d*x]]) + (2*A*b^4*Sqrt[Sec[c + d*x]])/(35*a^2*Sqrt[a + b*Cos[c + d*x]]) + (a*b*B*Sqrt[Sec[c + d*x]])/(5*Sqrt[a + b*Cos[c + d*x]]) - (b^3*B*Sqrt[Sec[c + d*x]])/(5*a*Sqrt[a + b*Cos[c + d*x]]) + (a^2*C*Sqrt[Sec[c + d*x]])/(3*Sqrt[a + b*Cos[c + d*x]]) - (b^2*C*Sqrt[Sec[c + d*x]])/(3*Sqrt[a + b*Cos[c + d*x]]) - (82*A*b^2*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*Sqrt[a + b*Cos[c + d*x]]) + (2*A*b^4*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(35*a^2*Sqrt[a + b*Cos[c + d*x]]) - (3*a*b*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*Sqrt[a + b*Cos[c + d*x]]) - (b^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*a*Sqrt[a + b*Cos[c + d*x]]) - (4*b^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*Sqrt[a + b*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(-6*A*b^2 + a^2*(25*A + 63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*a^2*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*((b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(-2*(a + b)*(-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(-6*A*b^2 + a^2*(25*A + 63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*a^2*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(-2*(a + b)*(-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(-6*A*b^2 + a^2*(25*A + 63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*a^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1/2*((-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4) - ((a + b)*(-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (a*(a + b)*(-6*A*b^2 + a^2*(25*A + 63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*C)))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - ((a + b)*(-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (a*(a + b)*(-6*A*b^2 + a^2*(25*A + 63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + b*(-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (a*(a + b)*(-6*A*b^2 + a^2*(25*A + 63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a + b)*(-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(105*a^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((-2*(a + b)*(-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(-6*A*b^2 + a^2*(25*A + 63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(105*a^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
1512,1,6826,550,26.3633569,"\int (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\text{Result too large to show}","-\frac{2 \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (9 A-5 B+15 C)-2 a b (6 A-10 B+15 C)+3 b^2 (A-5 B)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^2 (3 A+5 C)+20 a b B+3 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 (5 a B+3 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac{2 b C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}",1,"Result too large to show","B",0
1513,1,7536,588,26.1227354,"\int (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\text{Result too large to show}","\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(2 a^2 (A-3 B+3 C)-a b (8 A-3 (4 B+C))+6 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d \sqrt{\sec (c+d x)}}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} (6 a B+8 A b-3 b C) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) (6 a B+8 A b-3 b C) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d \sqrt{\sec (c+d x)}}+\frac{2 (a B+A b) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}-\frac{\sqrt{a+b} (3 a C+2 b B) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}",1,"Result too large to show","B",0
1514,1,1453,595,19.3861164,"\int (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(2 a A \sin (c+d x)+\frac{1}{4} b C \sin (2 (c+d x))\right)}{d}+\frac{-8 a^2 A \tan ^5\left(\frac{1}{2} (c+d x)\right)+8 a A b \tan ^5\left(\frac{1}{2} (c+d x)\right)-4 b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+4 a b B \tan ^5\left(\frac{1}{2} (c+d x)\right)+5 a^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-5 a b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 a A b \tan ^3\left(\frac{1}{2} (c+d x)\right)+8 b^2 B \tan ^3\left(\frac{1}{2} (c+d x)\right)+10 a b C \tan ^3\left(\frac{1}{2} (c+d x)\right)-16 A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-24 a b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-6 a^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-8 b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+8 a^2 A \tan \left(\frac{1}{2} (c+d x)\right)+8 a A b \tan \left(\frac{1}{2} (c+d x)\right)-4 b^2 B \tan \left(\frac{1}{2} (c+d x)\right)-4 a b B \tan \left(\frac{1}{2} (c+d x)\right)-5 a^2 C \tan \left(\frac{1}{2} (c+d x)\right)-5 a b C \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) (8 a A-4 b B-5 a C) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 \left(4 (A+B-C) a^2+b (8 A-8 B+C) a-2 b^2 (2 A+C)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-16 A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-24 a b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-6 a^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-8 b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}}{4 d \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^2 C+12 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d \sqrt{\sec (c+d x)}}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} (8 a A-5 a C-4 b B) \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) (a (8 A-8 B-5 C)-2 b (8 A+2 B+C)) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) (8 a A-5 a C-4 b B) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a d \sqrt{\sec (c+d x)}}-\frac{b (4 A-C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}{d}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(2*a*A*Sin[c + d*x] + (b*C*Sin[2*(c + d*x)])/4))/d + (8*a^2*A*Tan[(c + d*x)/2] + 8*a*A*b*Tan[(c + d*x)/2] - 4*a*b*B*Tan[(c + d*x)/2] - 4*b^2*B*Tan[(c + d*x)/2] - 5*a^2*C*Tan[(c + d*x)/2] - 5*a*b*C*Tan[(c + d*x)/2] - 16*a*A*b*Tan[(c + d*x)/2]^3 + 8*b^2*B*Tan[(c + d*x)/2]^3 + 10*a*b*C*Tan[(c + d*x)/2]^3 - 8*a^2*A*Tan[(c + d*x)/2]^5 + 8*a*A*b*Tan[(c + d*x)/2]^5 + 4*a*b*B*Tan[(c + d*x)/2]^5 - 4*b^2*B*Tan[(c + d*x)/2]^5 + 5*a^2*C*Tan[(c + d*x)/2]^5 - 5*a*b*C*Tan[(c + d*x)/2]^5 - 16*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 24*a*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 6*a^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 8*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 16*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 24*a*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 6*a^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 8*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(8*a*A - 4*b*B - 5*a*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*(4*a^2*(A + B - C) - 2*b^2*(2*A + C) + a*b*(8*A - 8*B + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)])/(4*d*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",1
1515,1,4952,647,23.8106567,"\int (a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\text{Result too large to show}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(3 a^2 C+30 a b B+24 A b^2+16 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{24 b d}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^2 C+2 a b (24 A+15 B+7 C)+4 b^2 (6 A+3 B+4 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^2 C+30 a b B+24 A b^2+16 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^3 (-C)+6 a^2 b B+12 a b^2 (2 A+C)+8 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^2 d \sqrt{\sec (c+d x)}}+\frac{(a C+2 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((b*C*Sin[c + d*x])/12 + ((6*b*B + 7*a*C)*Sin[2*(c + d*x)])/24 + (b*C*Sin[3*(c + d*x)])/12))/d + (Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*((2*a*A*b)/(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^2*B)/(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (b^2*B)/(2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (13*a*b*C)/(12*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^2*A*Sqrt[Sec[c + d*x]])/Sqrt[a + b*Cos[c + d*x]] + (A*b^2*Sqrt[Sec[c + d*x]])/(2*Sqrt[a + b*Cos[c + d*x]]) + (7*a*b*B*Sqrt[Sec[c + d*x]])/(8*Sqrt[a + b*Cos[c + d*x]]) + (17*a^2*C*Sqrt[Sec[c + d*x]])/(48*Sqrt[a + b*Cos[c + d*x]]) + (b^2*C*Sqrt[Sec[c + d*x]])/(3*Sqrt[a + b*Cos[c + d*x]]) + (A*b^2*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(2*Sqrt[a + b*Cos[c + d*x]]) + (5*a*b*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(8*Sqrt[a + b*Cos[c + d*x]]) + (a^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(16*Sqrt[a + b*Cos[c + d*x]]) + (b^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*Sqrt[a + b*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(b*(a + b)*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + a*(a + b)*(-24*A*b^2 + 3*a^2*C - 6*a*b*(3*B + C) - 4*b^2*(3*B + 4*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 3*(6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*(2*A + C))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + b*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(24*b^2*d*Sqrt[a + b*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)*((Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(b*(a + b)*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + a*(a + b)*(-24*A*b^2 + 3*a^2*C - 6*a*b*(3*B + C) - 4*b^2*(3*B + 4*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 3*(6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*(2*A + C))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + b*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(48*b*(a + b*Cos[c + d*x])^(3/2)*(Sec[(c + d*x)/2]^2)^(3/2)) - (Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(b*(a + b)*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + a*(a + b)*(-24*A*b^2 + 3*a^2*C - 6*a*b*(3*B + C) - 4*b^2*(3*B + 4*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 3*(6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*(2*A + C))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + b*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(16*b^2*Sqrt[a + b*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) + (Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(b*(a + b)*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + a*(a + b)*(-24*A*b^2 + 3*a^2*C - 6*a*b*(3*B + C) - 4*b^2*(3*B + 4*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 3*(6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*(2*A + C))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + b*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(48*b^2*Sqrt[a + b*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) + (Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*(b*(a + b)*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + a*(a + b)*(-24*A*b^2 + 3*a^2*C - 6*a*b*(3*B + C) - 4*b^2*(3*B + 4*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 3*(6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*(2*A + C))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + b*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(48*b^2*Sqrt[a + b*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]) + (Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((b*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x])/2 + b*(a + b)*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + a*(a + b)*(-24*A*b^2 + 3*a^2*C - 6*a*b*(3*B + C) - 4*b^2*(3*B + 4*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + 3*(6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*(2*A + C))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + (3*b*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sec[c + d*x]*Tan[(c + d*x)/2]*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/2 + (b*(a + b)*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((b*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/(2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + (a*(a + b)*(-24*A*b^2 + 3*a^2*C - 6*a*b*(3*B + C) - 4*b^2*(3*B + 4*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((b*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/(2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + (3*(6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*(2*A + C))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])*Sec[(c + d*x)/2]^2*(-((b*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/(2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + (a*(a + b)*(-24*A*b^2 + 3*a^2*C - 6*a*b*(3*B + C) - 4*b^2*(3*B + 4*C))*Sec[(c + d*x)/2]^4*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) + (b*(a + b)*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*Sec[(c + d*x)/2]^4*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]) + 3*(6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*(2*A + C))*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*(((a - b)*Sec[(c + d*x)/2]^2)/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) + (b*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])) - b^2*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]*Tan[c + d*x] + b*(24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*(a + b*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]*Tan[c + d*x]))/(24*b^2*Sqrt[a + b*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2))))","B",0
1516,1,601,764,15.7366568,"\int \frac{(a+b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{\frac{2 \tan (c+d x) (a+b \cos (c+d x)) \left(3 a^2 C+4 b (9 a C+8 b B) \cos (c+d x)+56 a b B+48 A b^2+12 b^2 C \cos (2 (c+d x))+48 b^2 C\right)}{b}-\frac{-b \tan \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2} (a+b \cos (c+d x))+a (a+b) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(9 a^3 C-6 a^2 b (4 B+3 C)+12 a b^2 (12 A+4 B+7 C)+8 b^3 (12 A+16 B+9 C)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))}{a+b}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-b (a+b) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+3 \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(-3 a^4 C+8 a^3 b B-24 a^2 b^2 (2 A+C)-96 a b^3 B-16 b^4 (4 A+3 C)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))}{a+b}} \left((a-b) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+2 b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{b^3 \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2}}}{192 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{192 b^2 d}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(9 a^3 C-6 a^2 b (4 B+C)-4 a b^2 (60 A+28 B+39 C)-8 b^3 (12 A+16 B+9 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b^2 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 a b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-3 a^4 C+8 a^3 b B-24 a^2 b^2 (2 A+C)-96 a b^3 B-16 b^4 (4 A+3 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^3 d \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) \left(a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right) \sqrt{a+b \cos (c+d x)}}{32 b d \sqrt{\sec (c+d x)}}+\frac{(8 b B-3 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 b d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{4 b d \sqrt{\sec (c+d x)}}",1,"(-((-(b*(a + b)*(24*a^2*b*B + 128*b^3*B - 9*a^3*C + 12*a*b^2*(20*A + 13*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + a*(a + b)*(9*a^3*C - 6*a^2*b*(4*B + 3*C) + 12*a*b^2*(12*A + 4*B + 7*C) + 8*b^3*(12*A + 16*B + 9*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 3*(8*a^3*b*B - 96*a*b^3*B - 3*a^4*C - 24*a^2*b^2*(2*A + C) - 16*b^4*(4*A + 3*C))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - b*(24*a^2*b*B + 128*b^3*B - 9*a^3*C + 12*a*b^2*(20*A + 13*C))*(a + b*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2])/(b^3*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2))) + (2*(a + b*Cos[c + d*x])*(48*A*b^2 + 56*a*b*B + 3*a^2*C + 48*b^2*C + 4*b*(8*b*B + 9*a*C)*Cos[c + d*x] + 12*b^2*C*Cos[2*(c + d*x)])*Tan[c + d*x])/b)/(192*d*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2))","A",1
1517,1,959,705,21.7206739,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{13}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(13/2),x]","\frac{2 \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(-\left((a+b) \left(1617 B a^5+15 b (247 A+319 C) a^4+3069 b^2 B a^3+15 b^3 (17 A+33 C) a^2-110 b^4 B a+40 A b^5\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)+a (a+b) \left(3 (225 A+539 B+275 C) a^4+6 b (505 A+209 B+660 C) a^3+15 b^2 (19 A+121 B+33 C) a^2-10 b^3 (3 A+11 B) a+40 A b^4\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)+\left(1617 B a^5+15 b (247 A+319 C) a^4+3069 b^2 B a^3+15 b^3 (17 A+33 C) a^2-110 b^4 B a+40 A b^5\right) \tan \left(\frac{1}{2} (c+d x)\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b\right)\right)}{3465 a^3 d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}+\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{2}{99} \left(11 B \sin (c+d x) a^2+23 A b \sin (c+d x) a\right) \sec ^4(c+d x)+\frac{2}{11} a^2 A \tan (c+d x) \sec ^4(c+d x)+\frac{2}{693} \left(81 A \sin (c+d x) a^2+99 C \sin (c+d x) a^2+209 b B \sin (c+d x) a+113 A b^2 \sin (c+d x)\right) \sec ^3(c+d x)+\frac{2 \left(539 B \sin (c+d x) a^3+1145 A b \sin (c+d x) a^2+1485 b C \sin (c+d x) a^2+825 b^2 B \sin (c+d x) a+15 A b^3 \sin (c+d x)\right) \sec ^2(c+d x)}{3465 a}+\frac{2 \left(675 A \sin (c+d x) a^4+825 C \sin (c+d x) a^4+1793 b B \sin (c+d x) a^3+1025 A b^2 \sin (c+d x) a^2+1485 b^2 C \sin (c+d x) a^2+55 b^3 B \sin (c+d x) a-20 A b^4 \sin (c+d x)\right) \sec (c+d x)}{3465 a^2}+\frac{2 \left(1617 B a^5+3705 A b a^4+4785 b C a^4+3069 b^2 B a^3+255 A b^3 a^2+495 b^3 C a^2-110 b^4 B a+40 A b^5\right) \sin (c+d x)}{3465 a^3}\right)}{d}","\frac{2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right) \sqrt{a+b \cos (c+d x)}}{231 d}+\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right) \sqrt{a+b \cos (c+d x)}}{3465 a d}-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right) \sqrt{a+b \cos (c+d x)}}{3465 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^4 (225 A-539 B+275 C)-6 a^3 b (505 A-209 B+660 C)+15 a^2 b^2 (19 A-121 B+33 C)+10 a b^3 (3 A-11 B)+40 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3465 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3465 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 (11 a B+5 A b) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{99 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{11 d}",1,"(2*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*((40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A + 33*C) + 15*a^4*b*(247*A + 319*C))*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)*(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2) - (a + b)*(40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A + 33*C) + 15*a^4*b*(247*A + 319*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + a*(a + b)*(40*A*b^4 - 10*a*b^3*(3*A + 11*B) + 15*a^2*b^2*(19*A + 121*B + 33*C) + 3*a^4*(225*A + 539*B + 275*C) + 6*a^3*b*(505*A + 209*B + 660*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(3465*a^3*d*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(3705*a^4*A*b + 255*a^2*A*b^3 + 40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 4785*a^4*b*C + 495*a^2*b^3*C)*Sin[c + d*x])/(3465*a^3) + (2*Sec[c + d*x]^4*(23*a*A*b*Sin[c + d*x] + 11*a^2*B*Sin[c + d*x]))/99 + (2*Sec[c + d*x]^3*(81*a^2*A*Sin[c + d*x] + 113*A*b^2*Sin[c + d*x] + 209*a*b*B*Sin[c + d*x] + 99*a^2*C*Sin[c + d*x]))/693 + (2*Sec[c + d*x]^2*(1145*a^2*A*b*Sin[c + d*x] + 15*A*b^3*Sin[c + d*x] + 539*a^3*B*Sin[c + d*x] + 825*a*b^2*B*Sin[c + d*x] + 1485*a^2*b*C*Sin[c + d*x]))/(3465*a) + (2*Sec[c + d*x]*(675*a^4*A*Sin[c + d*x] + 1025*a^2*A*b^2*Sin[c + d*x] - 20*A*b^4*Sin[c + d*x] + 1793*a^3*b*B*Sin[c + d*x] + 55*a*b^3*B*Sin[c + d*x] + 825*a^4*C*Sin[c + d*x] + 1485*a^2*b^2*C*Sin[c + d*x]))/(3465*a^2) + (2*a^2*A*Sec[c + d*x]^4*Tan[c + d*x])/11))/d","A",0
1518,1,809,592,20.8586112,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(11/2),x]","\frac{2 \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(-\left((a+b) \left(21 (7 A+9 C) a^4+435 b B a^3+3 b^2 (93 A+161 C) a^2+45 b^3 B a-10 A b^4\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)+a (a+b) \left(3 (49 A+25 B+63 C) a^3+6 b (19 A+60 B+28 C) a^2+15 b^2 (11 A+3 (B+7 C)) a-10 A b^3\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)+\left(21 (7 A+9 C) a^4+435 b B a^3+3 b^2 (93 A+161 C) a^2+45 b^3 B a-10 A b^4\right) \tan \left(\frac{1}{2} (c+d x)\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b\right)\right)}{315 a^2 d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}+\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{2}{63} \left(9 B \sin (c+d x) a^2+19 A b \sin (c+d x) a\right) \sec ^3(c+d x)+\frac{2}{9} a^2 A \tan (c+d x) \sec ^3(c+d x)+\frac{2}{315} \left(49 A \sin (c+d x) a^2+63 C \sin (c+d x) a^2+135 b B \sin (c+d x) a+75 A b^2 \sin (c+d x)\right) \sec ^2(c+d x)+\frac{2 \left(75 B \sin (c+d x) a^3+163 A b \sin (c+d x) a^2+231 b C \sin (c+d x) a^2+135 b^2 B \sin (c+d x) a+5 A b^3 \sin (c+d x)\right) \sec (c+d x)}{315 a}-\frac{2 \left(-147 A a^4-189 C a^4-435 b B a^3-279 A b^2 a^2-483 b^2 C a^2-45 b^3 B a+10 A b^4\right) \sin (c+d x)}{315 a^2}\right)}{d}","\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right) \sqrt{a+b \cos (c+d x)}}{315 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(75 a^3 B+a^2 b (163 A+231 C)+135 a b^2 B+5 A b^3\right) \sqrt{a+b \cos (c+d x)}}{315 a d}-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^3 (49 A-25 B+63 C)-6 a^2 b (19 A-60 B+28 C)+15 a b^2 (11 A-3 B+21 C)+10 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^2 d \sqrt{\sec (c+d x)}}-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-21 a^4 (7 A+9 C)-435 a^3 b B-3 a^2 b^2 (93 A+161 C)-45 a b^3 B+10 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (9 a B+5 A b) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{63 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{9 d}",1,"(2*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*((-10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(93*A + 161*C))*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)*(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2) - (a + b)*(-10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(93*A + 161*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + a*(a + b)*(-10*A*b^3 + 6*a^2*b*(19*A + 60*B + 28*C) + 3*a^3*(49*A + 25*B + 63*C) + 15*a*b^2*(11*A + 3*(B + 7*C)))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(315*a^2*d*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((-2*(-147*a^4*A - 279*a^2*A*b^2 + 10*A*b^4 - 435*a^3*b*B - 45*a*b^3*B - 189*a^4*C - 483*a^2*b^2*C)*Sin[c + d*x])/(315*a^2) + (2*Sec[c + d*x]^3*(19*a*A*b*Sin[c + d*x] + 9*a^2*B*Sin[c + d*x]))/63 + (2*Sec[c + d*x]^2*(49*a^2*A*Sin[c + d*x] + 75*A*b^2*Sin[c + d*x] + 135*a*b*B*Sin[c + d*x] + 63*a^2*C*Sin[c + d*x]))/315 + (2*Sec[c + d*x]*(163*a^2*A*b*Sin[c + d*x] + 5*A*b^3*Sin[c + d*x] + 75*a^3*B*Sin[c + d*x] + 135*a*b^2*B*Sin[c + d*x] + 231*a^2*b*C*Sin[c + d*x]))/(315*a) + (2*a^2*A*Sec[c + d*x]^3*Tan[c + d*x])/9))/d","A",0
1519,1,1257,640,20.9787033,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2),x]","\frac{2 \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(-15 A b^4 \tan ^5\left(\frac{1}{2} (c+d x)\right)+15 a A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-145 a^2 A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+145 a^3 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)+63 a^4 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-161 a b^3 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+161 a^2 b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-63 a^3 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)-245 a^2 b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+245 a^3 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)+30 A b^4 \tan ^3\left(\frac{1}{2} (c+d x)\right)+290 a^2 A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)+322 a b^3 B \tan ^3\left(\frac{1}{2} (c+d x)\right)+126 a^3 b B \tan ^3\left(\frac{1}{2} (c+d x)\right)+490 a^2 b^2 C \tan ^3\left(\frac{1}{2} (c+d x)\right)+210 a b^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-15 A b^4 \tan \left(\frac{1}{2} (c+d x)\right)-15 a A b^3 \tan \left(\frac{1}{2} (c+d x)\right)-145 a^2 A b^2 \tan \left(\frac{1}{2} (c+d x)\right)-145 a^3 A b \tan \left(\frac{1}{2} (c+d x)\right)-63 a^4 B \tan \left(\frac{1}{2} (c+d x)\right)-161 a b^3 B \tan \left(\frac{1}{2} (c+d x)\right)-161 a^2 b^2 B \tan \left(\frac{1}{2} (c+d x)\right)-63 a^3 b B \tan \left(\frac{1}{2} (c+d x)\right)-245 a^2 b^2 C \tan \left(\frac{1}{2} (c+d x)\right)-245 a^3 b C \tan \left(\frac{1}{2} (c+d x)\right)-(a+b) \left(63 B a^3+5 b (29 A+49 C) a^2+161 b^2 B a+15 A b^3\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+a \left((25 A+63 B+35 C) a^3+b (145 A+119 B+245 C) a^2+b^2 (135 A+161 B+315 C) a+15 b^3 (A+7 B-7 C)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+210 a b^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{105 a d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}+\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{2}{35} \left(7 B \sin (c+d x) a^2+15 A b \sin (c+d x) a\right) \sec ^2(c+d x)+\frac{2}{7} a^2 A \tan (c+d x) \sec ^2(c+d x)+\frac{2}{105} \left(25 A \sin (c+d x) a^2+35 C \sin (c+d x) a^2+77 b B \sin (c+d x) a+45 A b^2 \sin (c+d x)\right) \sec (c+d x)+\frac{2 \left(63 B a^3+145 A b a^2+245 b C a^2+161 b^2 B a+15 A b^3\right) \sin (c+d x)}{105 a}\right)}{d}","\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right) \sqrt{a+b \cos (c+d x)}}{105 d}-\frac{2 \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-\left(a^3 (25 A-63 B+35 C)\right)+a^2 b (145 A-119 B+245 C)-a b^2 (135 A-161 B+315 C)+15 b^3 (A-7 B)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(63 a^3 B+5 a^2 b (29 A+49 C)+161 a b^2 B+15 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 (7 a B+5 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{35 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}-\frac{2 b^2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}",1,"(2*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-145*a^3*A*b*Tan[(c + d*x)/2] - 145*a^2*A*b^2*Tan[(c + d*x)/2] - 15*a*A*b^3*Tan[(c + d*x)/2] - 15*A*b^4*Tan[(c + d*x)/2] - 63*a^4*B*Tan[(c + d*x)/2] - 63*a^3*b*B*Tan[(c + d*x)/2] - 161*a^2*b^2*B*Tan[(c + d*x)/2] - 161*a*b^3*B*Tan[(c + d*x)/2] - 245*a^3*b*C*Tan[(c + d*x)/2] - 245*a^2*b^2*C*Tan[(c + d*x)/2] + 290*a^2*A*b^2*Tan[(c + d*x)/2]^3 + 30*A*b^4*Tan[(c + d*x)/2]^3 + 126*a^3*b*B*Tan[(c + d*x)/2]^3 + 322*a*b^3*B*Tan[(c + d*x)/2]^3 + 490*a^2*b^2*C*Tan[(c + d*x)/2]^3 + 145*a^3*A*b*Tan[(c + d*x)/2]^5 - 145*a^2*A*b^2*Tan[(c + d*x)/2]^5 + 15*a*A*b^3*Tan[(c + d*x)/2]^5 - 15*A*b^4*Tan[(c + d*x)/2]^5 + 63*a^4*B*Tan[(c + d*x)/2]^5 - 63*a^3*b*B*Tan[(c + d*x)/2]^5 + 161*a^2*b^2*B*Tan[(c + d*x)/2]^5 - 161*a*b^3*B*Tan[(c + d*x)/2]^5 + 245*a^3*b*C*Tan[(c + d*x)/2]^5 - 245*a^2*b^2*C*Tan[(c + d*x)/2]^5 + 210*a*b^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 210*a*b^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (a + b)*(15*A*b^3 + 63*a^3*B + 161*a*b^2*B + 5*a^2*b*(29*A + 49*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + a*(15*b^3*(A + 7*B - 7*C) + a^3*(25*A + 63*B + 35*C) + a^2*b*(145*A + 119*B + 245*C) + a*b^2*(135*A + 161*B + 315*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(105*a*d*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B + 245*a^2*b*C)*Sin[c + d*x])/(105*a) + (2*Sec[c + d*x]^2*(15*a*A*b*Sin[c + d*x] + 7*a^2*B*Sin[c + d*x]))/35 + (2*Sec[c + d*x]*(25*a^2*A*Sin[c + d*x] + 45*A*b^2*Sin[c + d*x] + 77*a*b*B*Sin[c + d*x] + 35*a^2*C*Sin[c + d*x]))/105 + (2*a^2*A*Sec[c + d*x]^2*Tan[c + d*x])/7))/d","A",0
1520,1,1364,703,20.3643388,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2),x]","\frac{46 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-46 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)-18 a^3 A \tan ^5\left(\frac{1}{2} (c+d x)\right)+18 a^2 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)+70 a b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-70 a^2 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)-30 a^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-15 b^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+15 a b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+30 a^2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-92 A b^3 \tan ^3\left(\frac{1}{2} (c+d x)\right)-36 a^2 A b \tan ^3\left(\frac{1}{2} (c+d x)\right)-140 a b^2 B \tan ^3\left(\frac{1}{2} (c+d x)\right)+30 b^3 C \tan ^3\left(\frac{1}{2} (c+d x)\right)-60 a^2 b C \tan ^3\left(\frac{1}{2} (c+d x)\right)-60 b^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-150 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+46 A b^3 \tan \left(\frac{1}{2} (c+d x)\right)+46 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+18 a^3 A \tan \left(\frac{1}{2} (c+d x)\right)+18 a^2 A b \tan \left(\frac{1}{2} (c+d x)\right)+70 a b^2 B \tan \left(\frac{1}{2} (c+d x)\right)+70 a^2 b B \tan \left(\frac{1}{2} (c+d x)\right)+30 a^3 C \tan \left(\frac{1}{2} (c+d x)\right)-15 b^3 C \tan \left(\frac{1}{2} (c+d x)\right)-15 a b^2 C \tan \left(\frac{1}{2} (c+d x)\right)+30 a^2 b C \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(6 (3 A+5 C) a^2+70 b B a+b^2 (46 A-15 C)\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 \left((9 A+5 (B+3 C)) a^3+b (17 A+35 B+45 C) a^2+b^2 (23 A+45 (B-C)) a+15 b^3 (A-B)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-60 b^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-150 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}}{15 d \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}+\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{2}{5} A \sec (c+d x) \tan (c+d x) a^2+\frac{2}{15} \left(9 A a^2+15 C a^2+35 b B a+23 A b^2\right) \sin (c+d x)+\frac{2}{15} \sec (c+d x) \left(5 B \sin (c+d x) a^2+11 A b \sin (c+d x) a\right)\right)}{d}","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 (3 A+5 C)+10 a b B+5 A b^2\right) \sqrt{a+b \cos (c+d x)}}{5 d}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-2 a^3 (9 A-5 B+15 C)+2 a^2 b (17 A-35 B+45 C)-a b^2 (46 A-15 (6 B+C))+30 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a d \sqrt{\sec (c+d x)}}+\frac{2 (a B+A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{5 d}-\frac{b \sqrt{a+b} (5 a C+2 b B) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}",1,"(18*a^3*A*Tan[(c + d*x)/2] + 18*a^2*A*b*Tan[(c + d*x)/2] + 46*a*A*b^2*Tan[(c + d*x)/2] + 46*A*b^3*Tan[(c + d*x)/2] + 70*a^2*b*B*Tan[(c + d*x)/2] + 70*a*b^2*B*Tan[(c + d*x)/2] + 30*a^3*C*Tan[(c + d*x)/2] + 30*a^2*b*C*Tan[(c + d*x)/2] - 15*a*b^2*C*Tan[(c + d*x)/2] - 15*b^3*C*Tan[(c + d*x)/2] - 36*a^2*A*b*Tan[(c + d*x)/2]^3 - 92*A*b^3*Tan[(c + d*x)/2]^3 - 140*a*b^2*B*Tan[(c + d*x)/2]^3 - 60*a^2*b*C*Tan[(c + d*x)/2]^3 + 30*b^3*C*Tan[(c + d*x)/2]^3 - 18*a^3*A*Tan[(c + d*x)/2]^5 + 18*a^2*A*b*Tan[(c + d*x)/2]^5 - 46*a*A*b^2*Tan[(c + d*x)/2]^5 + 46*A*b^3*Tan[(c + d*x)/2]^5 - 70*a^2*b*B*Tan[(c + d*x)/2]^5 + 70*a*b^2*B*Tan[(c + d*x)/2]^5 - 30*a^3*C*Tan[(c + d*x)/2]^5 + 30*a^2*b*C*Tan[(c + d*x)/2]^5 + 15*a*b^2*C*Tan[(c + d*x)/2]^5 - 15*b^3*C*Tan[(c + d*x)/2]^5 - 60*b^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 150*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 60*b^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 150*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(70*a*b*B + b^2*(46*A - 15*C) + 6*a^2*(3*A + 5*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*(15*b^3*(A - B) + a*b^2*(23*A + 45*(B - C)) + a^2*b*(17*A + 35*B + 45*C) + a^3*(9*A + 5*(B + 3*C)))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)])/(15*d*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(9*a^2*A + 23*A*b^2 + 35*a*b*B + 15*a^2*C)*Sin[c + d*x])/15 + (2*Sec[c + d*x]*(11*a*A*b*Sin[c + d*x] + 5*a^2*B*Sin[c + d*x]))/15 + (2*a^2*A*Sec[c + d*x]*Tan[c + d*x])/5))/d","A",1
1521,1,1624,682,20.4388385,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2),x]","\frac{56 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)-56 a^2 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)-24 a^3 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-12 b^3 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+12 a b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+24 a^2 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)-27 a b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+27 a^2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-112 a A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)+24 b^3 B \tan ^3\left(\frac{1}{2} (c+d x)\right)-48 a^2 b B \tan ^3\left(\frac{1}{2} (c+d x)\right)+54 a b^2 C \tan ^3\left(\frac{1}{2} (c+d x)\right)-48 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-120 a b^2 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-24 b^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-90 a^2 b C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+56 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+56 a^2 A b \tan \left(\frac{1}{2} (c+d x)\right)+24 a^3 B \tan \left(\frac{1}{2} (c+d x)\right)-12 b^3 B \tan \left(\frac{1}{2} (c+d x)\right)-12 a b^2 B \tan \left(\frac{1}{2} (c+d x)\right)+24 a^2 b B \tan \left(\frac{1}{2} (c+d x)\right)-27 a b^2 C \tan \left(\frac{1}{2} (c+d x)\right)-27 a^2 b C \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(24 B a^2+b (56 A-27 C) a-12 b^2 B\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 \left(4 (A+3 (B+C)) a^3+4 b (7 A+9 B-9 C) a^2+3 b^2 (12 A-12 B+C) a-6 b^3 (2 A+C)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-48 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-120 a b^2 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-24 b^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-90 a^2 b C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}}{12 d \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}+\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{2}{3} A \tan (c+d x) a^2+\frac{2}{3} (7 A b+3 a B) \sin (c+d x) a+\frac{1}{4} b^2 C \sin (2 (c+d x))\right)}{d}","-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(24 a^2 B+a b (56 A-27 C)-12 b^2 B\right) \sqrt{a+b \cos (c+d x)}}{12 d}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-8 a^2 (A-3 B+3 C)+a b (56 A-72 B-27 C)-6 b^2 (12 A+2 B+C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{12 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(24 a^2 B+a b (56 A-27 C)-12 b^2 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{12 a d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d \sqrt{\sec (c+d x)}}-\frac{b \sin (c+d x) (4 a B+8 A b-b C) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 (3 a B+5 A b) \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}}{3 d}",1,"(56*a^2*A*b*Tan[(c + d*x)/2] + 56*a*A*b^2*Tan[(c + d*x)/2] + 24*a^3*B*Tan[(c + d*x)/2] + 24*a^2*b*B*Tan[(c + d*x)/2] - 12*a*b^2*B*Tan[(c + d*x)/2] - 12*b^3*B*Tan[(c + d*x)/2] - 27*a^2*b*C*Tan[(c + d*x)/2] - 27*a*b^2*C*Tan[(c + d*x)/2] - 112*a*A*b^2*Tan[(c + d*x)/2]^3 - 48*a^2*b*B*Tan[(c + d*x)/2]^3 + 24*b^3*B*Tan[(c + d*x)/2]^3 + 54*a*b^2*C*Tan[(c + d*x)/2]^3 - 56*a^2*A*b*Tan[(c + d*x)/2]^5 + 56*a*A*b^2*Tan[(c + d*x)/2]^5 - 24*a^3*B*Tan[(c + d*x)/2]^5 + 24*a^2*b*B*Tan[(c + d*x)/2]^5 + 12*a*b^2*B*Tan[(c + d*x)/2]^5 - 12*b^3*B*Tan[(c + d*x)/2]^5 + 27*a^2*b*C*Tan[(c + d*x)/2]^5 - 27*a*b^2*C*Tan[(c + d*x)/2]^5 - 48*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 120*a*b^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 90*a^2*b*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 24*b^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 48*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 120*a*b^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 90*a^2*b*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 24*b^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(24*a^2*B - 12*b^2*B + a*b*(56*A - 27*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*(4*a^2*b*(7*A + 9*B - 9*C) - 6*b^3*(2*A + C) + 3*a*b^2*(12*A - 12*B + C) + 4*a^3*(A + 3*(B + C)))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)])/(12*d*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*a*(7*A*b + 3*a*B)*Sin[c + d*x])/3 + (b^2*C*Sin[2*(c + d*x)])/4 + (2*a^2*A*Tan[c + d*x])/3))/d","B",1
1522,1,1920,707,20.6720006,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2),x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{1}{12} C \sin (3 (c+d x)) b^2+\frac{1}{24} (6 b B+13 a C) \sin (2 (c+d x)) b+\frac{1}{12} \left(24 A a^2+b^2 C\right) \sin (c+d x)\right)}{d}+\frac{-24 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)+24 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)-48 a^3 A \tan ^5\left(\frac{1}{2} (c+d x)\right)+48 a^2 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)-54 a b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+54 a^2 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)+33 a^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 b^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+16 a b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-33 a^2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)+48 A b^3 \tan ^3\left(\frac{1}{2} (c+d x)\right)-96 a^2 A b \tan ^3\left(\frac{1}{2} (c+d x)\right)+108 a b^2 B \tan ^3\left(\frac{1}{2} (c+d x)\right)+32 b^3 C \tan ^3\left(\frac{1}{2} (c+d x)\right)+66 a^2 b C \tan ^3\left(\frac{1}{2} (c+d x)\right)-240 a A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-48 b^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-180 a^2 b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-30 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-120 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-24 A b^3 \tan \left(\frac{1}{2} (c+d x)\right)-24 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+48 a^3 A \tan \left(\frac{1}{2} (c+d x)\right)+48 a^2 A b \tan \left(\frac{1}{2} (c+d x)\right)-54 a b^2 B \tan \left(\frac{1}{2} (c+d x)\right)-54 a^2 b B \tan \left(\frac{1}{2} (c+d x)\right)-33 a^3 C \tan \left(\frac{1}{2} (c+d x)\right)-16 b^3 C \tan \left(\frac{1}{2} (c+d x)\right)-16 a b^2 C \tan \left(\frac{1}{2} (c+d x)\right)-33 a^2 b C \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left((48 A-33 C) a^2-54 b B a-8 b^2 (3 A+2 C)\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 \left(24 (A+B-C) a^3+b (72 A-72 B+13 C) a^2-2 b^2 (36 A-3 B+19 C) a-12 b^3 B\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-240 a A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-48 b^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-180 a^2 b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-30 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-120 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}}{24 d \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-\left(a^2 (48 A-33 C)\right)+54 a b B+8 b^2 (3 A+2 C)\right) \sqrt{a+b \cos (c+d x)}}{24 d}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (48 A-48 B-33 C)-2 a b (72 A+27 B+13 C)-4 b^2 (6 A+3 B+4 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-\left(a^2 (48 A-33 C)\right)+54 a b B+8 b^2 (3 A+2 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b d \sqrt{\sec (c+d x)}}-\frac{b \sin (c+d x) (8 a A-3 a C-2 b B) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\sec (c+d x)}}-\frac{b (6 A-C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{5/2}}{d}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(((24*a^2*A + b^2*C)*Sin[c + d*x])/12 + (b*(6*b*B + 13*a*C)*Sin[2*(c + d*x)])/24 + (b^2*C*Sin[3*(c + d*x)])/12))/d + (48*a^3*A*Tan[(c + d*x)/2] + 48*a^2*A*b*Tan[(c + d*x)/2] - 24*a*A*b^2*Tan[(c + d*x)/2] - 24*A*b^3*Tan[(c + d*x)/2] - 54*a^2*b*B*Tan[(c + d*x)/2] - 54*a*b^2*B*Tan[(c + d*x)/2] - 33*a^3*C*Tan[(c + d*x)/2] - 33*a^2*b*C*Tan[(c + d*x)/2] - 16*a*b^2*C*Tan[(c + d*x)/2] - 16*b^3*C*Tan[(c + d*x)/2] - 96*a^2*A*b*Tan[(c + d*x)/2]^3 + 48*A*b^3*Tan[(c + d*x)/2]^3 + 108*a*b^2*B*Tan[(c + d*x)/2]^3 + 66*a^2*b*C*Tan[(c + d*x)/2]^3 + 32*b^3*C*Tan[(c + d*x)/2]^3 - 48*a^3*A*Tan[(c + d*x)/2]^5 + 48*a^2*A*b*Tan[(c + d*x)/2]^5 + 24*a*A*b^2*Tan[(c + d*x)/2]^5 - 24*A*b^3*Tan[(c + d*x)/2]^5 + 54*a^2*b*B*Tan[(c + d*x)/2]^5 - 54*a*b^2*B*Tan[(c + d*x)/2]^5 + 33*a^3*C*Tan[(c + d*x)/2]^5 - 33*a^2*b*C*Tan[(c + d*x)/2]^5 + 16*a*b^2*C*Tan[(c + d*x)/2]^5 - 16*b^3*C*Tan[(c + d*x)/2]^5 - 240*a*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 180*a^2*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 48*b^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 30*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 120*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 240*a*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 180*a^2*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 48*b^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 30*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 120*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(-54*a*b*B + a^2*(48*A - 33*C) - 8*b^2*(3*A + 2*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*(-12*b^3*B + 24*a^3*(A + B - C) + a^2*b*(72*A - 72*B + 13*C) - 2*a*b^2*(36*A - 3*B + 19*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)])/(24*d*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",1
1523,1,5541,760,25.9514787,"\int (a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]","\text{Result too large to show}","\frac{\sin (c+d x) \left(5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{32 d \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{192 b d}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(15 a^3 C+2 a^2 b (192 A+132 B+59 C)+4 a b^2 (108 A+52 B+71 C)+8 b^3 (12 A+16 B+9 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{192 a b d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d \sqrt{\sec (c+d x)}}+\frac{(5 a C+8 b B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{4 d \sqrt{\sec (c+d x)}}",1,"Result too large to show","B",0
1524,1,803,894,20.4843653,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{1}{80} C \sin (5 (c+d x)) b^2+\frac{1}{320} (10 b B+21 a C) \sin (4 (c+d x)) b+\frac{1}{960} \left(93 C a^2+170 b B a+80 A b^2+88 b^2 C\right) \sin (c+d x)+\frac{1}{960} \left(93 C a^2+170 b B a+80 A b^2+100 b^2 C\right) \sin (3 (c+d x))+\frac{\left(15 C a^3+590 b B a^2+1040 A b^2 a+1024 b^2 C a+480 b^3 B\right) \sin (2 (c+d x))}{1920 b}\right)}{d}+\frac{\sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(\left(-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right) \tan \left(\frac{1}{2} (c+d x)\right)+\frac{i \left((a-b) \left(45 C a^4-150 b B a^3-12 b^2 (220 A+141 C) a^2-2840 b^3 B a-256 b^4 (5 A+4 C)\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right)-2 (a-b) \left(45 C a^4-30 b (5 B-C) a^3-4 b^2 (180 A+185 B+129 C) a^2-8 b^3 (220 A+45 B+161 C) a-720 b^4 B\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right)+30 \left(3 C a^5-10 b B a^4+40 b^2 (2 A+C) a^3+240 b^3 B a^2+80 b^4 (4 A+3 C) a+96 b^5 B\right) \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right)\right) \left(-\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}}{\sqrt{\frac{a-b}{a+b}} \left(-a \tan ^4\left(\frac{1}{2} (c+d x)\right)+b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)^2+a\right)}\right)}{1920 b^2 d \sqrt{\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}}}","\frac{C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d \sqrt{\sec (c+d x)}}+\frac{(10 b B-3 a C) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{40 b d \sqrt{\sec (c+d x)}}+\frac{\left(-15 C a^2+50 b B a+80 A b^2+64 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{240 b d \sqrt{\sec (c+d x)}}+\frac{\left(-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right) \sqrt{\sec (c+d x)} \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d}+\frac{\left(-15 C a^3+50 b B a^2+4 b^2 (60 A+43 C) a+120 b^3 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{320 b d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 a b^2 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left(45 C a^4-30 b (5 B+C) a^3-4 b^2 (660 A+295 B+423 C) a^2-8 b^3 (260 A+355 B+193 C) a-16 b^4 (80 A+45 B+64 C)\right) \sqrt{\cos (c+d x)} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left(-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right) \sqrt{\cos (c+d x)} \csc (c+d x) \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{128 b^3 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(((80*A*b^2 + 170*a*b*B + 93*a^2*C + 88*b^2*C)*Sin[c + d*x])/960 + ((1040*a*A*b^2 + 590*a^2*b*B + 480*b^3*B + 15*a^3*C + 1024*a*b^2*C)*Sin[2*(c + d*x)])/(1920*b) + ((80*A*b^2 + 170*a*b*B + 93*a^2*C + 100*b^2*C)*Sin[3*(c + d*x)])/960 + (b*(10*b*B + 21*a*C)*Sin[4*(c + d*x)])/320 + (b^2*C*Sin[5*(c + d*x)])/80))/d + (Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*((150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 256*b^4*(5*A + 4*C) + 12*a^2*b^2*(220*A + 141*C))*Tan[(c + d*x)/2] + (I*((a - b)*(-150*a^3*b*B - 2840*a*b^3*B + 45*a^4*C - 256*b^4*(5*A + 4*C) - 12*a^2*b^2*(220*A + 141*C))*EllipticE[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))] - 2*(a - b)*(-720*b^4*B - 30*a^3*b*(5*B - C) + 45*a^4*C - 4*a^2*b^2*(180*A + 185*B + 129*C) - 8*a*b^3*(220*A + 45*B + 161*C))*EllipticF[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))] + 30*(-10*a^4*b*B + 240*a^2*b^3*B + 96*b^5*B + 3*a^5*C + 40*a^3*b^2*(2*A + C) + 80*a*b^4*(4*A + 3*C))*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))])*(-1 - Tan[(c + d*x)/2]^2)*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)])/(Sqrt[(a - b)/(a + b)]*(a - a*Tan[(c + d*x)/2]^4 + b*(-1 + Tan[(c + d*x)/2]^2)^2))))/(1920*b^2*d*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)])","C",0
1525,1,3704,506,27.1031136,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{9}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(9/2))/Sqrt[a + b*Cos[c + d*x]],x]","\text{Result too large to show}","-\frac{2 (6 A b-7 a B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{35 a^2 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(5 a^2 (5 A+7 C)-28 a b B+24 A b^2\right) \sqrt{a+b \cos (c+d x)}}{105 a^3 d}-\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-63 a^3 B+a^2 (44 A b+70 b C)-56 a b^2 B+48 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^5 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^3 (25 A-63 B+35 C)+2 a^2 b (22 A+7 (B+5 C))-4 a b^2 (3 A+14 B)+48 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{7 a d}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(-44*a^2*A*b - 48*A*b^3 + 63*a^3*B + 56*a*b^2*B - 70*a^2*b*C)*Sin[c + d*x])/(105*a^4) + (2*Sec[c + d*x]^2*(-6*A*b*Sin[c + d*x] + 7*a*B*Sin[c + d*x]))/(35*a^2) + (2*Sec[c + d*x]*(25*a^2*A*Sin[c + d*x] + 24*A*b^2*Sin[c + d*x] - 28*a*b*B*Sin[c + d*x] + 35*a^2*C*Sin[c + d*x]))/(105*a^3) + (2*A*Sec[c + d*x]^2*Tan[c + d*x])/(7*a)))/d + (2*((44*A*b)/(105*a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*A*b^3)/(35*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (3*B)/(5*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*b^2*B)/(15*a^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b*C)/(3*a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (5*A*Sqrt[Sec[c + d*x]])/(21*Sqrt[a + b*Cos[c + d*x]]) + (32*A*b^2*Sqrt[Sec[c + d*x]])/(105*a^2*Sqrt[a + b*Cos[c + d*x]]) + (16*A*b^4*Sqrt[Sec[c + d*x]])/(35*a^4*Sqrt[a + b*Cos[c + d*x]]) - (7*b*B*Sqrt[Sec[c + d*x]])/(15*a*Sqrt[a + b*Cos[c + d*x]]) - (8*b^3*B*Sqrt[Sec[c + d*x]])/(15*a^3*Sqrt[a + b*Cos[c + d*x]]) + (C*Sqrt[Sec[c + d*x]])/(3*Sqrt[a + b*Cos[c + d*x]]) + (2*b^2*C*Sqrt[Sec[c + d*x]])/(3*a^2*Sqrt[a + b*Cos[c + d*x]]) + (44*A*b^2*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*a^2*Sqrt[a + b*Cos[c + d*x]]) + (16*A*b^4*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(35*a^4*Sqrt[a + b*Cos[c + d*x]]) - (3*b*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*a*Sqrt[a + b*Cos[c + d*x]]) - (8*b^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*a^3*Sqrt[a + b*Cos[c + d*x]]) + (2*b^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*a^2*Sqrt[a + b*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(-48*A*b^3 + 63*a^3*B + 56*a*b^2*B - 2*a^2*b*(22*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(-48*A*b^3 + 4*a*b^2*(-3*A + 14*B) - 2*a^2*b*(22*A - 7*B + 35*C) + a^3*(25*A + 63*B + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (48*A*b^3 - 63*a^3*B - 56*a*b^2*B + a^2*(44*A*b + 70*b*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*a^4*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*((b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(-2*(a + b)*(-48*A*b^3 + 63*a^3*B + 56*a*b^2*B - 2*a^2*b*(22*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(-48*A*b^3 + 4*a*b^2*(-3*A + 14*B) - 2*a^2*b*(22*A - 7*B + 35*C) + a^3*(25*A + 63*B + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (48*A*b^3 - 63*a^3*B - 56*a*b^2*B + a^2*(44*A*b + 70*b*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*a^4*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(-2*(a + b)*(-48*A*b^3 + 63*a^3*B + 56*a*b^2*B - 2*a^2*b*(22*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(-48*A*b^3 + 4*a*b^2*(-3*A + 14*B) - 2*a^2*b*(22*A - 7*B + 35*C) + a^3*(25*A + 63*B + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (48*A*b^3 - 63*a^3*B - 56*a*b^2*B + a^2*(44*A*b + 70*b*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*a^4*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((48*A*b^3 - 63*a^3*B - 56*a*b^2*B + a^2*(44*A*b + 70*b*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 - ((a + b)*(-48*A*b^3 + 63*a^3*B + 56*a*b^2*B - 2*a^2*b*(22*A + 35*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (a*(-48*A*b^3 + 4*a*b^2*(-3*A + 14*B) - 2*a^2*b*(22*A - 7*B + 35*C) + a^3*(25*A + 63*B + 35*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - ((a + b)*(-48*A*b^3 + 63*a^3*B + 56*a*b^2*B - 2*a^2*b*(22*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (a*(-48*A*b^3 + 4*a*b^2*(-3*A + 14*B) - 2*a^2*b*(22*A - 7*B + 35*C) + a^3*(25*A + 63*B + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - b*(48*A*b^3 - 63*a^3*B - 56*a*b^2*B + a^2*(44*A*b + 70*b*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (48*A*b^3 - 63*a^3*B - 56*a*b^2*B + a^2*(44*A*b + 70*b*C))*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (48*A*b^3 - 63*a^3*B - 56*a*b^2*B + a^2*(44*A*b + 70*b*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (a*(-48*A*b^3 + 4*a*b^2*(-3*A + 14*B) - 2*a^2*b*(22*A - 7*B + 35*C) + a^3*(25*A + 63*B + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a + b)*(-48*A*b^3 + 63*a^3*B + 56*a*b^2*B - 2*a^2*b*(22*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(105*a^4*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((-2*(a + b)*(-48*A*b^3 + 63*a^3*B + 56*a*b^2*B - 2*a^2*b*(22*A + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(-48*A*b^3 + 4*a*b^2*(-3*A + 14*B) - 2*a^2*b*(22*A - 7*B + 35*C) + a^3*(25*A + 63*B + 35*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (48*A*b^3 - 63*a^3*B - 56*a*b^2*B + a^2*(44*A*b + 70*b*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(105*a^4*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
1526,1,3208,412,25.6705294,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/Sqrt[a + b*Cos[c + d*x]],x]","\text{Result too large to show}","-\frac{2 (4 A b-5 a B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{15 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^2 (3 A+5 C)-10 a b B+8 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^4 d \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (9 A-5 B+15 C)-2 a b (A+5 B)+8 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 a d}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(9*a^2*A + 8*A*b^2 - 10*a*b*B + 15*a^2*C)*Sin[c + d*x])/(15*a^3) + (2*Sec[c + d*x]*(-4*A*b*Sin[c + d*x] + 5*a*B*Sin[c + d*x]))/(15*a^2) + (2*A*Sec[c + d*x]*Tan[c + d*x])/(5*a)))/d + (2*((-3*A)/(5*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*A*b^2)/(15*a^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b*B)/(3*a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - C/(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (7*A*b*Sqrt[Sec[c + d*x]])/(15*a*Sqrt[a + b*Cos[c + d*x]]) - (8*A*b^3*Sqrt[Sec[c + d*x]])/(15*a^3*Sqrt[a + b*Cos[c + d*x]]) + (B*Sqrt[Sec[c + d*x]])/(3*Sqrt[a + b*Cos[c + d*x]]) + (2*b^2*B*Sqrt[Sec[c + d*x]])/(3*a^2*Sqrt[a + b*Cos[c + d*x]]) - (b*C*Sqrt[Sec[c + d*x]])/(a*Sqrt[a + b*Cos[c + d*x]]) - (3*A*b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*a*Sqrt[a + b*Cos[c + d*x]]) - (8*A*b^3*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*a^3*Sqrt[a + b*Cos[c + d*x]]) + (2*b^2*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*a^2*Sqrt[a + b*Cos[c + d*x]]) - (b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(a*Sqrt[a + b*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(8*A*b^2 - 10*a*b*B + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(8*A*b^2 + 2*a*b*(A - 5*B) + a^2*(9*A + 5*(B + 3*C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (8*A*b^2 - 10*a*b*B + 3*a^2*(3*A + 5*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*a^3*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*((b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(-2*(a + b)*(8*A*b^2 - 10*a*b*B + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(8*A*b^2 + 2*a*b*(A - 5*B) + a^2*(9*A + 5*(B + 3*C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (8*A*b^2 - 10*a*b*B + 3*a^2*(3*A + 5*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*a^3*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(-2*(a + b)*(8*A*b^2 - 10*a*b*B + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(8*A*b^2 + 2*a*b*(A - 5*B) + a^2*(9*A + 5*(B + 3*C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (8*A*b^2 - 10*a*b*B + 3*a^2*(3*A + 5*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1/2*((8*A*b^2 - 10*a*b*B + 3*a^2*(3*A + 5*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4) - ((a + b)*(8*A*b^2 - 10*a*b*B + 3*a^2*(3*A + 5*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (a*(8*A*b^2 + 2*a*b*(A - 5*B) + a^2*(9*A + 5*(B + 3*C)))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - ((a + b)*(8*A*b^2 - 10*a*b*B + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (a*(8*A*b^2 + 2*a*b*(A - 5*B) + a^2*(9*A + 5*(B + 3*C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + b*(8*A*b^2 - 10*a*b*B + 3*a^2*(3*A + 5*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (8*A*b^2 - 10*a*b*B + 3*a^2*(3*A + 5*C))*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (8*A*b^2 - 10*a*b*B + 3*a^2*(3*A + 5*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (a*(8*A*b^2 + 2*a*b*(A - 5*B) + a^2*(9*A + 5*(B + 3*C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a + b)*(8*A*b^2 - 10*a*b*B + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(15*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((-2*(a + b)*(8*A*b^2 - 10*a*b*B + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(8*A*b^2 + 2*a*b*(A - 5*B) + a^2*(9*A + 5*(B + 3*C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (8*A*b^2 - 10*a*b*B + 3*a^2*(3*A + 5*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(15*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
1527,1,2616,333,20.6312831,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/Sqrt[a + b*Cos[c + d*x]],x]","\text{Result too large to show}","-\frac{2 (a-b) \sqrt{a+b} (2 A b-3 a B) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) (a (A-3 B+3 C)+2 A b) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 a d}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(-2*A*b + 3*a*B)*Sin[c + d*x])/(3*a^2) + (2*A*Tan[c + d*x])/(3*a)))/d + (2*((2*A*b)/(3*a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - B/(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (A*Sqrt[Sec[c + d*x]])/(3*Sqrt[a + b*Cos[c + d*x]]) + (2*A*b^2*Sqrt[Sec[c + d*x]])/(3*a^2*Sqrt[a + b*Cos[c + d*x]]) - (b*B*Sqrt[Sec[c + d*x]])/(a*Sqrt[a + b*Cos[c + d*x]]) + (C*Sqrt[Sec[c + d*x]])/Sqrt[a + b*Cos[c + d*x]] + (2*A*b^2*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*a^2*Sqrt[a + b*Cos[c + d*x]]) - (b*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(a*Sqrt[a + b*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(-2*A*b + 3*a*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(-2*A*b + a*(A + 3*(B + C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (2*A*b - 3*a*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*a^2*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*((b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(-2*(a + b)*(-2*A*b + 3*a*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(-2*A*b + a*(A + 3*(B + C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (2*A*b - 3*a*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*a^2*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(-2*(a + b)*(-2*A*b + 3*a*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(-2*A*b + a*(A + 3*(B + C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (2*A*b - 3*a*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*a^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((2*A*b - 3*a*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 - ((a + b)*(-2*A*b + 3*a*B)*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (a*(-2*A*b + a*(A + 3*(B + C)))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - ((a + b)*(-2*A*b + 3*a*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (a*(-2*A*b + a*(A + 3*(B + C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - b*(2*A*b - 3*a*B)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (2*A*b - 3*a*B)*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (2*A*b - 3*a*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (a*(-2*A*b + a*(A + 3*(B + C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a + b)*(-2*A*b + 3*a*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(3*a^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((-2*(a + b)*(-2*A*b + 3*a*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(-2*A*b + a*(A + 3*(B + C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (2*A*b - 3*a*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(3*a^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
1528,1,621,407,17.0941854,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(a (A+B-C) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-A (a+b) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+a A \tan ^5\left(\frac{1}{2} (c+d x)\right)-a A \tan \left(\frac{1}{2} (c+d x)\right)+2 a C \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+2 a C \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-A b \tan ^5\left(\frac{1}{2} (c+d x)\right)+2 A b \tan ^3\left(\frac{1}{2} (c+d x)\right)-A b \tan \left(\frac{1}{2} (c+d x)\right)\right)}{a d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{a d}","\frac{2 A (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{a+b} (A-B) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{\sec (c+d x)}}-\frac{2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d \sqrt{\sec (c+d x)}}",1,"(2*A*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) + (2*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-(a*A*Tan[(c + d*x)/2]) - A*b*Tan[(c + d*x)/2] + 2*A*b*Tan[(c + d*x)/2]^3 + a*A*Tan[(c + d*x)/2]^5 - A*b*Tan[(c + d*x)/2]^5 + 2*a*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*a*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - A*(a + b)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + a*(A + B - C)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(a*d*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","A",0
1529,1,769,461,16.7985528,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(2 b (A-B) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+4 b B \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+4 b B \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+C (a+b) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-2 a C \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-2 a C \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-a C \tan ^5\left(\frac{1}{2} (c+d x)\right)+a C \tan \left(\frac{1}{2} (c+d x)\right)+b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 b C \tan ^3\left(\frac{1}{2} (c+d x)\right)+b C \tan \left(\frac{1}{2} (c+d x)\right)\right)}{b d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\sqrt{a+b} (a C+2 A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} (2 b B-a C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{b d}-\frac{C (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{\sec (c+d x)}}",1,"(Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(a*C*Tan[(c + d*x)/2] + b*C*Tan[(c + d*x)/2] - 2*b*C*Tan[(c + d*x)/2]^3 - a*C*Tan[(c + d*x)/2]^5 + b*C*Tan[(c + d*x)/2]^5 + 4*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 4*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*C*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*b*(A - B)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(b*d*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","A",0
1530,1,1360,545,20.7910018,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","\frac{C \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (2 (c+d x))}{4 b d}+\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(-4 b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+4 a b B \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 a^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 a b C \tan ^5\left(\frac{1}{2} (c+d x)\right)+8 b^2 B \tan ^3\left(\frac{1}{2} (c+d x)\right)-6 a b C \tan ^3\left(\frac{1}{2} (c+d x)\right)-16 A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+8 a b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-6 a^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-8 b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-4 b^2 B \tan \left(\frac{1}{2} (c+d x)\right)-4 a b B \tan \left(\frac{1}{2} (c+d x)\right)+3 a^2 C \tan \left(\frac{1}{2} (c+d x)\right)+3 a b C \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) (3 a C-4 b B) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 b (4 A b+2 C b-a C) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-16 A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+8 a b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-6 a^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-8 b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{4 b^2 d \sqrt{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1} \left(b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^2 C-4 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{\sec (c+d x)}}+\frac{(4 b B-3 a C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b^2 d}-\frac{\sqrt{a+b} (3 a C-2 b (2 B+C)) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} (4 b B-3 a C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 b d \sqrt{\sec (c+d x)}}",1,"(C*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[2*(c + d*x)])/(4*b*d) + (Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(-4*a*b*B*Tan[(c + d*x)/2] - 4*b^2*B*Tan[(c + d*x)/2] + 3*a^2*C*Tan[(c + d*x)/2] + 3*a*b*C*Tan[(c + d*x)/2] + 8*b^2*B*Tan[(c + d*x)/2]^3 - 6*a*b*C*Tan[(c + d*x)/2]^3 + 4*a*b*B*Tan[(c + d*x)/2]^5 - 4*b^2*B*Tan[(c + d*x)/2]^5 - 3*a^2*C*Tan[(c + d*x)/2]^5 + 3*a*b*C*Tan[(c + d*x)/2]^5 - 16*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 8*a*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 6*a^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 8*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 16*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 8*a*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 6*a^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 8*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(-4*b*B + 3*a*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*b*(4*A*b - a*C + 2*b*C)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(4*b^2*d*Sqrt[1 + Tan[(c + d*x)/2]^2]*(b*(-1 + Tan[(c + d*x)/2]^2) - a*(1 + Tan[(c + d*x)/2]^2)))","B",0
1531,1,1818,653,21.5375343,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)),x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{C \sin (c+d x)}{12 b}+\frac{(6 b B-5 a C) \sin (2 (c+d x))}{24 b^2}+\frac{C \sin (3 (c+d x))}{12 b}\right)}{d}+\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(-24 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)+24 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+18 a b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-18 a^2 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)+15 a^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 b^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+16 a b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-15 a^2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)+48 A b^3 \tan ^3\left(\frac{1}{2} (c+d x)\right)-36 a b^2 B \tan ^3\left(\frac{1}{2} (c+d x)\right)+32 b^3 C \tan ^3\left(\frac{1}{2} (c+d x)\right)+30 a^2 b C \tan ^3\left(\frac{1}{2} (c+d x)\right)+48 a A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-48 b^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-36 a^2 b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+30 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+24 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-24 A b^3 \tan \left(\frac{1}{2} (c+d x)\right)-24 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+18 a b^2 B \tan \left(\frac{1}{2} (c+d x)\right)+18 a^2 b B \tan \left(\frac{1}{2} (c+d x)\right)-15 a^3 C \tan \left(\frac{1}{2} (c+d x)\right)-16 b^3 C \tan \left(\frac{1}{2} (c+d x)\right)-16 a b^2 C \tan \left(\frac{1}{2} (c+d x)\right)-15 a^2 b C \tan \left(\frac{1}{2} (c+d x)\right)-(a+b) \left(15 C a^2-18 b B a+24 A b^2+16 b^2 C\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 b \left(5 C a^2+2 b (C-3 B) a+12 b^2 B\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+48 a A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-48 b^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-36 a^2 b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+30 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+24 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{24 b^3 d \sqrt{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1} \left(b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(15 a^2 C-18 a b B+24 A b^2+16 b^2 C\right) \sqrt{a+b \cos (c+d x)}}{24 b^3 d}+\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(15 a^2 C-18 a b B-10 a b C+24 A b^2+12 b^2 B+16 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 b^3 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(15 a^2 C-18 a b B+24 A b^2+16 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{24 a b^3 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(-5 a^3 C+6 a^2 b B-4 a b^2 (2 A+C)+8 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^4 d \sqrt{\sec (c+d x)}}+\frac{(6 b B-5 a C) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{12 b^2 d \sqrt{\sec (c+d x)}}+\frac{C \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((C*Sin[c + d*x])/(12*b) + ((6*b*B - 5*a*C)*Sin[2*(c + d*x)])/(24*b^2) + (C*Sin[3*(c + d*x)])/(12*b)))/d + (Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(-24*a*A*b^2*Tan[(c + d*x)/2] - 24*A*b^3*Tan[(c + d*x)/2] + 18*a^2*b*B*Tan[(c + d*x)/2] + 18*a*b^2*B*Tan[(c + d*x)/2] - 15*a^3*C*Tan[(c + d*x)/2] - 15*a^2*b*C*Tan[(c + d*x)/2] - 16*a*b^2*C*Tan[(c + d*x)/2] - 16*b^3*C*Tan[(c + d*x)/2] + 48*A*b^3*Tan[(c + d*x)/2]^3 - 36*a*b^2*B*Tan[(c + d*x)/2]^3 + 30*a^2*b*C*Tan[(c + d*x)/2]^3 + 32*b^3*C*Tan[(c + d*x)/2]^3 + 24*a*A*b^2*Tan[(c + d*x)/2]^5 - 24*A*b^3*Tan[(c + d*x)/2]^5 - 18*a^2*b*B*Tan[(c + d*x)/2]^5 + 18*a*b^2*B*Tan[(c + d*x)/2]^5 + 15*a^3*C*Tan[(c + d*x)/2]^5 - 15*a^2*b*C*Tan[(c + d*x)/2]^5 + 16*a*b^2*C*Tan[(c + d*x)/2]^5 - 16*b^3*C*Tan[(c + d*x)/2]^5 + 48*a*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 36*a^2*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 48*b^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 24*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 36*a^2*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 48*b^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 24*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (a + b)*(24*A*b^2 - 18*a*b*B + 15*a^2*C + 16*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*b*(12*b^2*B + 5*a^2*C + 2*a*b*(-3*B + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(24*b^3*d*Sqrt[1 + Tan[(c + d*x)/2]^2]*(b*(-1 + Tan[(c + d*x)/2]^2) - a*(1 + Tan[(c + d*x)/2]^2)))","B",0
1532,1,787,445,6.1504029,"\int \frac{\left(a A+(A b+a B) \cos (c+d x)+b B \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((a*A + (A*b + a*B)*Cos[c + d*x] + b*B*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 (a (B-A)+A b) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-4 A b \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-4 A b \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-B (a+b) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-2 a B \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-2 a B \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+a B \tan ^5\left(\frac{1}{2} (c+d x)\right)-a B \tan \left(\frac{1}{2} (c+d x)\right)-b B \tan ^5\left(\frac{1}{2} (c+d x)\right)+2 b B \tan ^3\left(\frac{1}{2} (c+d x)\right)-b B \tan \left(\frac{1}{2} (c+d x)\right)}{d \sqrt{\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^4\left(\frac{1}{2} (c+d x)\right)-1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\sqrt{a+b} (2 A+B) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} (a B+2 A b) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b d \sqrt{\sec (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{B (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{\sec (c+d x)}}",1,"(-(a*B*Tan[(c + d*x)/2]) - b*B*Tan[(c + d*x)/2] + 2*b*B*Tan[(c + d*x)/2]^3 + a*B*Tan[(c + d*x)/2]^5 - b*B*Tan[(c + d*x)/2]^5 - 4*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 4*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (a + b)*B*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*(A*b + a*(-A + B))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)])/(d*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(-1 + Tan[(c + d*x)/2]^4))","A",1
1533,1,752,585,21.1653152,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{7}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(7/2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \sec (c+d x) (5 a B \sin (c+d x)-9 A b \sin (c+d x))}{15 a^3}+\frac{2 A \tan (c+d x) \sec (c+d x)}{5 a^2}+\frac{2 \left(a^2 b^2 C \sin (c+d x)-a b^3 B \sin (c+d x)+A b^4 \sin (c+d x)\right)}{a^3 \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{2 \sin (c+d x) \left(9 a^4 A+15 a^4 C-25 a^3 b B+24 a^2 A b^2-30 a^2 b^2 C+40 a b^3 B-48 A b^4\right)}{15 a^4 \left(a^2-b^2\right)}\right)}{d}+\frac{2 \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(a (a+b) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \left(a^3 (9 A+5 (B+3 C))-6 a^2 b (2 A+5 (B+C))+4 a b^2 (9 A+10 B)-48 A b^3\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+\tan \left(\frac{1}{2} (c+d x)\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(3 a^4 (3 A+5 C)-25 a^3 b B+6 a^2 b^2 (4 A-5 C)+40 a b^3 B-48 A b^4\right) \left(a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b\right)-\left((a+b) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \left(3 a^4 (3 A+5 C)-25 a^3 b B+6 a^2 b^2 (4 A-5 C)+40 a b^3 B-48 A b^4\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)\right)}{15 a^4 d \left(a^2-b^2\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","-\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(-\left(a^2 (A-5 C)\right)-5 a b B+6 A b^2\right) \sqrt{a+b \cos (c+d x)}}{5 a^2 d \left(a^2-b^2\right)}+\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(5 a^3 B-a^2 (9 A b-15 b C)-20 a b^2 B+24 A b^3\right) \sqrt{a+b \cos (c+d x)}}{15 a^3 d \left(a^2-b^2\right)}-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^3 (9 A-5 B+15 C)+6 a^2 b (2 A-5 B+5 C)+4 a b^2 (9 A-10 B)+48 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^4 d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(-3 a^4 (3 A+5 C)+25 a^3 b B-6 a^2 b^2 (4 A-5 C)-40 a b^3 B+48 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{15 a^5 d \sqrt{a+b} \sqrt{\sec (c+d x)}}",1,"(2*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*((-48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B + 6*a^2*b^2*(4*A - 5*C) + 3*a^4*(3*A + 5*C))*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)*(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2) - (a + b)*(-48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B + 6*a^2*b^2*(4*A - 5*C) + 3*a^4*(3*A + 5*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + a*(a + b)*(-48*A*b^3 + 4*a*b^2*(9*A + 10*B) - 6*a^2*b*(2*A + 5*(B + C)) + a^3*(9*A + 5*(B + 3*C)))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(15*a^4*(a^2 - b^2)*d*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(9*a^4*A + 24*a^2*A*b^2 - 48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B + 15*a^4*C - 30*a^2*b^2*C)*Sin[c + d*x])/(15*a^4*(a^2 - b^2)) + (2*Sec[c + d*x]*(-9*A*b*Sin[c + d*x] + 5*a*B*Sin[c + d*x]))/(15*a^3) + (2*(A*b^4*Sin[c + d*x] - a*b^3*B*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(a^3*(a^2 - b^2)*(a + b*Cos[c + d*x])) + (2*A*Sec[c + d*x]*Tan[c + d*x])/(5*a^2)))/d","A",0
1534,1,3736,464,26.6439115,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^(3/2),x]","\text{Result too large to show}","-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(-\left(a^2 (A-3 C)\right)-3 a b B+4 A b^2\right) \sqrt{a+b \cos (c+d x)}}{3 a^2 d \left(a^2-b^2\right)}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(a^2 (A-3 B+3 C)+6 a b (A-B)+8 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^3 B-a^2 (5 A b-3 b C)-6 a b^2 B+8 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(-5*a^2*A*b + 8*A*b^3 + 3*a^3*B - 6*a*b^2*B + 3*a^2*b*C)*Sin[c + d*x])/(3*a^3*(a^2 - b^2)) - (2*(A*b^3*Sin[c + d*x] - a*b^2*B*Sin[c + d*x] + a^2*b*C*Sin[c + d*x]))/(a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])) + (2*A*Tan[c + d*x])/(3*a^2)))/d + (2*((5*A*b)/(3*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*A*b^3)/(3*a^2*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a*B)/((a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b^2*B)/(a*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (b*C)/((a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a*A*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]) + (7*A*b^2*Sqrt[Sec[c + d*x]])/(3*a*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]) - (8*A*b^4*Sqrt[Sec[c + d*x]])/(3*a^3*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]) - (2*b*B*Sqrt[Sec[c + d*x]])/((a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]) + (2*b^3*B*Sqrt[Sec[c + d*x]])/(a^2*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]) + (a*C*Sqrt[Sec[c + d*x]])/((a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]) - (b^2*C*Sqrt[Sec[c + d*x]])/(a*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]) + (5*A*b^2*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*a*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]) - (8*A*b^4*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*a^3*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]) - (b*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]) + (2*b^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(a^2*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]) - (b^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(a*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(8*A*b^2 - 6*a*b*(A + B) + a^2*(A + 3*(B + C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*a^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*((b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(-2*(a + b)*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(8*A*b^2 - 6*a*b*(A + B) + a^2*(A + 3*(B + C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*a^3*(a^2 - b^2)*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(-2*(a + b)*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(8*A*b^2 - 6*a*b*(A + B) + a^2*(A + 3*(B + C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*a^3*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1/2*((8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4) - ((a + b)*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (a*(a + b)*(8*A*b^2 - 6*a*b*(A + B) + a^2*(A + 3*(B + C)))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - ((a + b)*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (a*(a + b)*(8*A*b^2 - 6*a*b*(A + B) + a^2*(A + 3*(B + C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + b*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (a*(a + b)*(8*A*b^2 - 6*a*b*(A + B) + a^2*(A + 3*(B + C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a + b)*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(3*a^3*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((-2*(a + b)*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(8*A*b^2 - 6*a*b*(A + B) + a^2*(A + 3*(B + C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(3*a^3*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
1535,1,482,362,20.2621096,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \sin (c+d x) \left(a^2 A-a^2 C+a b B-2 A b^2\right)}{a^2 \left(a^2-b^2\right)}+\frac{2 \left(a^2 C \sin (c+d x)-a b B \sin (c+d x)+A b^2 \sin (c+d x)\right)}{a \left(a^2-b^2\right) (a+b \cos (c+d x))}\right)}{d}+\frac{2 \sqrt{2} \sqrt{\frac{\cos (c+d x)}{(\cos (c+d x)+1)^2}} \sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \left(\cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(a^2 (C-A)-a b B+2 A b^2\right) (a+b \cos (c+d x))-(a+b) \sec (c+d x) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2} \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))}{a+b}} \left(\left(a^2 (A-C)+a b B-2 A b^2\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+a (2 A b-a (A+B-C)) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\frac{1}{\cos (c+d x)+1}} \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} \sqrt{a+b \cos (c+d x)}}","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) (a (A-B-C)+2 A b) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(-\left(a^2 (A-C)\right)-a b B+2 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(a^2*A - 2*A*b^2 + a*b*B - a^2*C)*Sin[c + d*x])/(a^2*(a^2 - b^2)) + (2*(A*b^2*Sin[c + d*x] - a*b*B*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(a*(a^2 - b^2)*(a + b*Cos[c + d*x]))))/d + (2*Sqrt[2]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])^2]*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(-((a + b)*((-2*A*b^2 + a*b*B + a^2*(A - C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*(2*A*b - a*(A + B - C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]) + (2*A*b^2 - a*b*B + a^2*(-A + C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(a^2*(a^2 - b^2)*d*Sqrt[(1 + Cos[c + d*x])^(-1)]*Sqrt[a + b*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2))","A",0
1536,1,7547,496,25.6589637,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^(3/2),x]","\text{Result too large to show}","-\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(A b^2-a (b B-a C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a^2 b d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) (-a C+A b+b B) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{2 C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d \sqrt{\sec (c+d x)}}",1,"Result too large to show","B",0
1537,1,1667,595,20.5452606,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]),x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{2 \left(C a^2-b B a+A b^2\right) \sin (c+d x)}{b^2 \left(b^2-a^2\right)}-\frac{2 \left(C \sin (c+d x) a^3-b B \sin (c+d x) a^2+A b^2 \sin (c+d x) a\right)}{b^2 \left(b^2-a^2\right) (a+b \cos (c+d x))}\right)}{d}-\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(-2 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)+2 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+2 a b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a^2 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 a^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+b^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-a b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 a^2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)+4 A b^3 \tan ^3\left(\frac{1}{2} (c+d x)\right)-4 a b^2 B \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 b^3 C \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 a^2 b C \tan ^3\left(\frac{1}{2} (c+d x)\right)+4 b^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-4 a^2 b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+6 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-6 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-2 A b^3 \tan \left(\frac{1}{2} (c+d x)\right)-2 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+2 a b^2 B \tan \left(\frac{1}{2} (c+d x)\right)+2 a^2 b B \tan \left(\frac{1}{2} (c+d x)\right)-3 a^3 C \tan \left(\frac{1}{2} (c+d x)\right)+b^3 C \tan \left(\frac{1}{2} (c+d x)\right)+a b^2 C \tan \left(\frac{1}{2} (c+d x)\right)-3 a^2 b C \tan \left(\frac{1}{2} (c+d x)\right)-(a+b) \left(3 C a^2-2 b B a+2 A b^2-b^2 C\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 b (a+b) (A b-B b+a C) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+4 b^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-4 a^2 b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+6 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-6 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{b^2 \left(b^2-a^2\right) d \sqrt{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1} \left(b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(3 a^2 C-2 a b B+2 A b^2-b^2 C\right) \sqrt{a+b \cos (c+d x)}}{b^2 d \left(a^2-b^2\right)}-\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^2 C-2 a b B+2 A b^2-b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{\sqrt{\cos (c+d x)} \csc (c+d x) \left(2 A b^2-a (b (2 B-C)-3 a C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} (2 b B-3 a C) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(b^2*(-a^2 + b^2)) - (2*(a*A*b^2*Sin[c + d*x] - a^2*b*B*Sin[c + d*x] + a^3*C*Sin[c + d*x]))/(b^2*(-a^2 + b^2)*(a + b*Cos[c + d*x]))))/d - (Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(-2*a*A*b^2*Tan[(c + d*x)/2] - 2*A*b^3*Tan[(c + d*x)/2] + 2*a^2*b*B*Tan[(c + d*x)/2] + 2*a*b^2*B*Tan[(c + d*x)/2] - 3*a^3*C*Tan[(c + d*x)/2] - 3*a^2*b*C*Tan[(c + d*x)/2] + a*b^2*C*Tan[(c + d*x)/2] + b^3*C*Tan[(c + d*x)/2] + 4*A*b^3*Tan[(c + d*x)/2]^3 - 4*a*b^2*B*Tan[(c + d*x)/2]^3 + 6*a^2*b*C*Tan[(c + d*x)/2]^3 - 2*b^3*C*Tan[(c + d*x)/2]^3 + 2*a*A*b^2*Tan[(c + d*x)/2]^5 - 2*A*b^3*Tan[(c + d*x)/2]^5 - 2*a^2*b*B*Tan[(c + d*x)/2]^5 + 2*a*b^2*B*Tan[(c + d*x)/2]^5 + 3*a^3*C*Tan[(c + d*x)/2]^5 - 3*a^2*b*C*Tan[(c + d*x)/2]^5 - a*b^2*C*Tan[(c + d*x)/2]^5 + b^3*C*Tan[(c + d*x)/2]^5 - 4*a^2*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 4*b^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 6*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 4*a^2*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 4*b^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 6*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (a + b)*(2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*b*(a + b)*(A*b - b*B + a*C)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(b^2*(-a^2 + b^2)*d*Sqrt[1 + Tan[(c + d*x)/2]^2]*(b*(-1 + Tan[(c + d*x)/2]^2) - a*(1 + Tan[(c + d*x)/2]^2)))","B",0
1538,1,3353,720,21.1297575,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)),x]","\text{Result too large to show}","-\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}+\frac{\sin (c+d x) \left(5 a^2 C-4 a b B+4 A b^2-b^2 C\right) \sqrt{a+b \cos (c+d x)}}{2 b^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \left(15 a^2 C-12 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^4 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \csc (c+d x) \left(15 a^2 C-a b (12 B-5 C)+8 A b^2-2 b^2 (2 B+C)\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-15 a^3 C+12 a^2 b B-a b^2 (8 A-7 C)-4 b^3 B\right) \sqrt{a+b \cos (c+d x)}}{4 b^3 d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \csc (c+d x) \left(-15 a^3 C+12 a^2 b B-a b^2 (8 A-7 C)-4 b^3 B\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 a b^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*a*(A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(b^3*(a^2 - b^2)) + (2*(a^2*A*b^2*Sin[c + d*x] - a^3*b*B*Sin[c + d*x] + a^4*C*Sin[c + d*x]))/(b^3*(-a^2 + b^2)*(a + b*Cos[c + d*x])) + (C*Sin[2*(c + d*x)])/(4*b^2)))/d + (Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(-8*a^2*A*b^2*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2] - 8*a*A*b^3*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2] + 12*a^3*b*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2] + 12*a^2*b^2*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2] - 4*a*b^3*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2] - 4*b^4*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2] - 15*a^4*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2] - 15*a^3*b*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2] + 7*a^2*b^2*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2] + 7*a*b^3*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2] + 16*a*A*b^3*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^3 - 24*a^2*b^2*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2]^3 + 8*b^4*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2]^3 + 30*a^3*b*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2]^3 - 14*a*b^3*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2]^3 + 8*a^2*A*b^2*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^5 - 8*a*A*b^3*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^5 - 12*a^3*b*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2]^5 + 12*a^2*b^2*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2]^5 + 4*a*b^3*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - 4*b^4*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2]^5 + 15*a^4*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2]^5 - 15*a^3*b*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2]^5 - 7*a^2*b^2*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2]^5 + 7*a*b^3*Sqrt[(a - b)/(a + b)]*C*Tan[(c + d*x)/2]^5 - (16*I)*a^2*A*b^2*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (16*I)*A*b^4*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (24*I)*a^3*b*B*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (24*I)*a*b^3*B*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (30*I)*a^4*C*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (22*I)*a^2*b^2*C*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*I)*b^4*C*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (16*I)*a^2*A*b^2*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (16*I)*A*b^4*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (24*I)*a^3*b*B*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (24*I)*a*b^3*B*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (30*I)*a^4*C*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (22*I)*a^2*b^2*C*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*I)*b^4*C*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(a - b)*(-12*a^2*b*B + 4*b^3*B + a*b^2*(8*A - 7*C) + 15*a^3*C)*EllipticE[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*(a - b)*(-2*a^2*b*(6*B - 5*C) + 15*a^3*C + 2*b^3*(2*A + C) + a*b^2*(8*A - 8*B + C))*EllipticF[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(4*b^3*Sqrt[(a - b)/(a + b)]*(a^2 - b^2)*d*(-1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(b*(-1 + Tan[(c + d*x)/2]^2) - a*(1 + Tan[(c + d*x)/2]^2)))","C",0
1539,1,867,660,22.246893,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(-\left((a+b) \left(3 B a^5+(6 b C-8 A b) a^4-15 b^2 B a^3+2 b^3 (14 A-C) a^2+8 b^4 B a-16 A b^5\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)+a (a+b) \left((A+3 (B+C)) a^4+3 b (-3 A-3 B+C) a^3+2 b^2 (8 A-3 B-C) a^2+4 b^3 (3 A+2 B) a-16 A b^4\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)+\left(3 B a^5+(6 b C-8 A b) a^4-15 b^2 B a^3+2 b^3 (14 A-C) a^2+8 b^4 B a-16 A b^5\right) \tan \left(\frac{1}{2} (c+d x)\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b\right)\right)}{3 a^4 \left(a^2-b^2\right)^2 d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}+\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{2 \left(3 B a^5-8 A b a^4+6 b C a^4-15 b^2 B a^3+28 A b^3 a^2-2 b^3 C a^2+8 b^4 B a-16 A b^5\right) \sin (c+d x)}{3 a^4 \left(a^2-b^2\right)^2}-\frac{2 \left(A \sin (c+d x) b^3-a B \sin (c+d x) b^2+a^2 C \sin (c+d x) b\right)}{3 a^2 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{2 \left(-7 A \sin (c+d x) b^5+4 a B \sin (c+d x) b^4+11 a^2 A \sin (c+d x) b^3-a^2 C \sin (c+d x) b^3-8 a^3 B \sin (c+d x) b^2+5 a^4 C \sin (c+d x) b\right)}{3 a^3 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{2 A \tan (c+d x)}{3 a^3}\right)}{d}","\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^4 (A-5 C)+8 a^3 b B-a^2 b^2 (13 A-C)-4 a b^3 B+8 A b^4\right) \sqrt{a+b \cos (c+d x)}}{3 a^3 d \left(a^2-b^2\right)^2}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(4 a^4 C-7 a^3 b B+10 a^2 A b^2+3 a b^3 B-6 A b^4\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(-\left(a^4 (A-3 B+3 C)\right)-3 a^3 b (3 A-3 B-C)-2 a^2 b^2 (8 A+3 B-C)+4 a b^3 (3 A-2 B)+16 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(-3 a^5 B+a^4 (8 A b-6 b C)+15 a^3 b^2 B-2 a^2 b^3 (14 A-C)-8 a b^4 B+16 A b^5\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^5 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}",1,"(2*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*((-16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B + 2*a^2*b^3*(14*A - C) + a^4*(-8*A*b + 6*b*C))*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)*(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2) - (a + b)*(-16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B + 2*a^2*b^3*(14*A - C) + a^4*(-8*A*b + 6*b*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + a*(a + b)*(-16*A*b^4 + 4*a*b^3*(3*A + 2*B) + 2*a^2*b^2*(8*A - 3*B - C) + 3*a^3*b*(-3*A - 3*B + C) + a^4*(A + 3*(B + C)))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(3*a^4*(a^2 - b^2)^2*d*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(-8*a^4*A*b + 28*a^2*A*b^3 - 16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B + 6*a^4*b*C - 2*a^2*b^3*C)*Sin[c + d*x])/(3*a^4*(a^2 - b^2)^2) - (2*(A*b^3*Sin[c + d*x] - a*b^2*B*Sin[c + d*x] + a^2*b*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) - (2*(11*a^2*A*b^3*Sin[c + d*x] - 7*A*b^5*Sin[c + d*x] - 8*a^3*b^2*B*Sin[c + d*x] + 4*a*b^4*B*Sin[c + d*x] + 5*a^4*b*C*Sin[c + d*x] - a^2*b^3*C*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) + (2*A*Tan[c + d*x])/(3*a^3)))/d","A",0
1540,1,790,535,21.0237052,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \left(a^2 C \sin (c+d x)-a b B \sin (c+d x)+A b^2 \sin (c+d x)\right)}{3 a \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{2 \sin (c+d x) \left(3 a^4 A-3 a^4 C+6 a^3 b B-15 a^2 A b^2-a^2 b^2 C-2 a b^3 B+8 A b^4\right)}{3 a^3 \left(a^2-b^2\right)^2}+\frac{2 \left(2 a^4 C \sin (c+d x)-5 a^3 b B \sin (c+d x)+8 a^2 A b^2 \sin (c+d x)+2 a^2 b^2 C \sin (c+d x)+a b^3 B \sin (c+d x)-4 A b^4 \sin (c+d x)\right)}{3 a^2 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}\right)}{d}+\frac{2 \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(a (a+b) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \left(3 a^3 (A+B-C)-a^2 b (9 A-3 B+C)-2 a b^2 (3 A+B)+8 A b^3\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+\tan \left(\frac{1}{2} (c+d x)\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(3 a^4 (A-C)+6 a^3 b B-a^2 b^2 (15 A+C)-2 a b^3 B+8 A b^4\right) \left(a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b\right)-\left((a+b) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \left(3 a^4 (A-C)+6 a^3 b B-a^2 b^2 (15 A+C)-2 a b^3 B+8 A b^4\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)\right)}{3 a^3 d \left(a^2-b^2\right)^2 \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(-3 a^3 (A-B-C)-a^2 b (9 A+3 B+C)+2 a b^2 (3 A-B)+8 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(-2 a^4 C+5 a^3 b B-2 a^2 b^2 (4 A+C)-a b^3 B+4 A b^4\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^4 (A-C)+6 a^3 b B-a^2 b^2 (15 A+C)-2 a b^3 B+8 A b^4\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^4 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B - 3*a^4*C - a^2*b^2*C)*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2) + (2*(A*b^2*Sin[c + d*x] - a*b*B*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(3*a*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) + (2*(8*a^2*A*b^2*Sin[c + d*x] - 4*A*b^4*Sin[c + d*x] - 5*a^3*b*B*Sin[c + d*x] + a*b^3*B*Sin[c + d*x] + 2*a^4*C*Sin[c + d*x] + 2*a^2*b^2*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/d + (2*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*((8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)*(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2) - (a + b)*(8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + a*(a + b)*(8*A*b^3 - 2*a*b^2*(3*A + B) + 3*a^3*(A + B - C) - a^2*b*(9*A - 3*B + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(3*a^3*(a^2 - b^2)^2*d*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","A",0
1541,1,3853,495,26.5931675,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(-\left(a^2 (3 A+3 B+C)\right)+a b (3 A+B+3 C)+2 A b^2\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(3 a^3 B-2 a^2 b (3 A+2 C)+a b^2 B+2 A b^3\right)}{3 a d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \left(3 a^3 B-2 a^2 b (3 A+2 C)+a b^2 B+2 A b^3\right) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((-2*(-6*a^2*A*b + 2*A*b^3 + 3*a^3*B + a*b^2*B - 4*a^2*b*C)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2) + (2*(A*b^2*Sin[c + d*x] - a*b*B*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(3*b*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (2*(-5*a^2*A*b^2*Sin[c + d*x] + A*b^4*Sin[c + d*x] + 2*a^3*b*B*Sin[c + d*x] + 2*a*b^3*B*Sin[c + d*x] + a^4*C*Sin[c + d*x] - 5*a^2*b^2*C*Sin[c + d*x]))/(3*a*b*(a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/d + (2*((-2*a*A*b)/((a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*A*b^3)/(3*a*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^2*B)/((a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (b^2*B)/(3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*a*b*C)/(3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^2*A*Sqrt[Sec[c + d*x]])/((a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) - (5*A*b^2*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) + (2*A*b^4*Sqrt[Sec[c + d*x]])/(3*a^2*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) - (a*b*B*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) + (b^3*B*Sqrt[Sec[c + d*x]])/(3*a*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) + (a^2*C*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) - (b^2*C*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) - (2*A*b^2*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) + (2*A*b^4*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*a^2*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) + (a*b*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) + (b^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*a*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) - (4*b^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a + b)*(2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(-2*A*b^2 + a^2*(3*A - 3*B + C) + a*b*(3*A - B + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(a^3 - a*b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*((b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(-2*A*b^2 + a^2*(3*A - 3*B + C) + a*b*(3*A - B + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(a^3 - a*b^2)^2*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(-2*A*b^2 + a^2*(3*A - 3*B + C) + a*b*(3*A - B + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(a^3 - a*b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (a*(a + b)*(-2*A*b^2 + a^2*(3*A - 3*B + C) + a*b*(3*A - B + 3*C))*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (a*(a + b)*(-2*A*b^2 + a^2*(3*A - 3*B + C) + a*b*(3*A - B + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - b*(2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (a*(a + b)*(-2*A*b^2 + a^2*(3*A - 3*B + C) + a*b*(3*A - B + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(3*(a^3 - a*b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((2*(a + b)*(2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(a + b)*(-2*A*b^2 + a^2*(3*A - 3*B + C) + a*b*(3*A - B + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(3*(a^3 - a*b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0