1,1,77,125,0.2745225,"\int \cos ^3(c+d x) (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","\frac{a \left(-160 (A+2 B) \sin ^3(c+d x)+480 (A+B) \sin (c+d x)+15 (A+B) (12 (c+d x)+8 \sin (2 (c+d x))+\sin (4 (c+d x)))+96 B \sin ^5(c+d x)\right)}{480 d}","-\frac{a (5 A+4 B) \sin ^3(c+d x)}{15 d}+\frac{a (5 A+4 B) \sin (c+d x)}{5 d}+\frac{a (A+B) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a (A+B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3}{8} a x (A+B)+\frac{a B \sin (c+d x) \cos ^4(c+d x)}{5 d}",1,"(a*(480*(A + B)*Sin[c + d*x] - 160*(A + 2*B)*Sin[c + d*x]^3 + 96*B*Sin[c + d*x]^5 + 15*(A + B)*(12*(c + d*x) + 8*Sin[2*(c + d*x)] + Sin[4*(c + d*x)])))/(480*d)","A",1
2,1,75,97,0.2587067,"\int \cos ^2(c+d x) (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","\frac{a \left(-32 (A+B) \sin ^3(c+d x)+96 (A+B) \sin (c+d x)+24 (A+B) \sin (2 (c+d x))+48 A c+48 A d x+3 B \sin (4 (c+d x))+36 B c+36 B d x\right)}{96 d}","-\frac{a (A+B) \sin ^3(c+d x)}{3 d}+\frac{a (A+B) \sin (c+d x)}{d}+\frac{a (4 A+3 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (4 A+3 B)+\frac{a B \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"(a*(48*A*c + 36*B*c + 48*A*d*x + 36*B*d*x + 96*(A + B)*Sin[c + d*x] - 32*(A + B)*Sin[c + d*x]^3 + 24*(A + B)*Sin[2*(c + d*x)] + 3*B*Sin[4*(c + d*x)]))/(96*d)","A",1
3,1,65,77,0.1939115,"\int \cos (c+d x) (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","\frac{a (3 (4 A+3 B) \sin (c+d x)+3 (A+B) \sin (2 (c+d x))+6 A c+6 A d x+B \sin (3 (c+d x))+6 B c+6 B d x)}{12 d}","\frac{a (3 A+2 B) \sin (c+d x)}{3 d}+\frac{a (A+B) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} a x (A+B)+\frac{a B \sin (c+d x) \cos ^2(c+d x)}{3 d}",1,"(a*(6*A*c + 6*B*c + 6*A*d*x + 6*B*d*x + 3*(4*A + 3*B)*Sin[c + d*x] + 3*(A + B)*Sin[2*(c + d*x)] + B*Sin[3*(c + d*x)]))/(12*d)","A",1
4,1,44,47,0.1031594,"\int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","\frac{a (4 (A+B) \sin (c+d x)+4 A d x+B \sin (2 (c+d x))+2 B c+2 B d x)}{4 d}","\frac{a (A+B) \sin (c+d x)}{d}+\frac{1}{2} a x (2 A+B)+\frac{a B \sin (c+d x) \cos (c+d x)}{2 d}",1,"(a*(2*B*c + 4*A*d*x + 2*B*d*x + 4*(A + B)*Sin[c + d*x] + B*Sin[2*(c + d*x)]))/(4*d)","A",1
5,1,46,32,0.0272407,"\int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+a A x+\frac{a B \sin (c) \cos (d x)}{d}+\frac{a B \cos (c) \sin (d x)}{d}+a B x","a x (A+B)+\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a B \sin (c+d x)}{d}",1,"a*A*x + a*B*x + (a*A*ArcTanh[Sin[c + d*x]])/d + (a*B*Cos[d*x]*Sin[c])/d + (a*B*Cos[c]*Sin[d*x])/d","A",1
6,1,43,32,0.020909,"\int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*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 (A+B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A \tan (c+d x)}{d}+a B x",1,"a*B*x + (a*A*ArcTanh[Sin[c + d*x]])/d + (a*B*ArcTanh[Sin[c + d*x]])/d + (a*A*Tan[c + d*x])/d","A",1
7,1,75,56,0.0276002,"\int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*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 (A+B) \tan (c+d x)}{d}+\frac{a (A+2 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}",1,"(a*A*ArcTanh[Sin[c + d*x]])/(2*d) + (a*B*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
8,1,56,86,0.3520637,"\int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{a \left(3 (A+B) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(3 (A+B) \sec (c+d x)+6 (A+B)+2 A \tan ^2(c+d x)\right)\right)}{6 d}","\frac{a (2 A+3 B) \tan (c+d x)}{3 d}+\frac{a (A+B) \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)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(6*(A + B) + 3*(A + B)*Sec[c + d*x] + 2*A*Tan[c + d*x]^2)))/(6*d)","A",1
9,1,77,106,0.4185896,"\int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^5,x]","\frac{a \left(3 (3 A+4 B) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x) \left(8 (A+B) (\cos (2 (c+d x))+2) \sec (c+d x)+6 A \sec ^2(c+d x)+9 A+12 B\right)\right)}{24 d}","\frac{a (A+B) \tan ^3(c+d x)}{3 d}+\frac{a (A+B) \tan (c+d x)}{d}+\frac{a (3 A+4 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (3 A+4 B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"(a*(3*(3*A + 4*B)*ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*(9*A + 12*B + 8*(A + B)*(2 + Cos[2*(c + d*x)])*Sec[c + d*x] + 6*A*Sec[c + d*x]^2)*Tan[c + d*x]))/(24*d)","A",1
10,1,134,191,0.6492774,"\int \cos ^3(c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]","\frac{a^2 (120 (11 A+10 B) \sin (c+d x)+15 (32 A+31 B) \sin (2 (c+d x))+180 A \sin (3 (c+d x))+60 A \sin (4 (c+d x))+12 A \sin (5 (c+d x))+720 A d x+200 B \sin (3 (c+d x))+75 B \sin (4 (c+d x))+24 B \sin (5 (c+d x))+5 B \sin (6 (c+d x))+660 B c+660 B d x)}{960 d}","-\frac{a^2 (9 A+8 B) \sin ^3(c+d x)}{15 d}+\frac{a^2 (9 A+8 B) \sin (c+d x)}{5 d}+\frac{a^2 (6 A+7 B) \sin (c+d x) \cos ^4(c+d x)}{30 d}+\frac{a^2 (12 A+11 B) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{a^2 (12 A+11 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^2 x (12 A+11 B)+\frac{B \sin (c+d x) \cos ^4(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{6 d}",1,"(a^2*(660*B*c + 720*A*d*x + 660*B*d*x + 120*(11*A + 10*B)*Sin[c + d*x] + 15*(32*A + 31*B)*Sin[2*(c + d*x)] + 180*A*Sin[3*(c + d*x)] + 200*B*Sin[3*(c + d*x)] + 60*A*Sin[4*(c + d*x)] + 75*B*Sin[4*(c + d*x)] + 12*A*Sin[5*(c + d*x)] + 24*B*Sin[5*(c + d*x)] + 5*B*Sin[6*(c + d*x)]))/(960*d)","A",1
11,1,108,160,0.4488863,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]","\frac{a^2 (60 (12 A+11 B) \sin (c+d x)+240 (A+B) \sin (2 (c+d x))+80 A \sin (3 (c+d x))+15 A \sin (4 (c+d x))+420 A d x+90 B \sin (3 (c+d x))+30 B \sin (4 (c+d x))+6 B \sin (5 (c+d x))+360 B c+360 B d x)}{480 d}","-\frac{a^2 (10 A+9 B) \sin ^3(c+d x)}{15 d}+\frac{a^2 (10 A+9 B) \sin (c+d x)}{5 d}+\frac{a^2 (5 A+6 B) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{a^2 (7 A+6 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a^2 x (7 A+6 B)+\frac{B \sin (c+d x) \cos ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{5 d}",1,"(a^2*(360*B*c + 420*A*d*x + 360*B*d*x + 60*(12*A + 11*B)*Sin[c + d*x] + 240*(A + B)*Sin[2*(c + d*x)] + 80*A*Sin[3*(c + d*x)] + 90*B*Sin[3*(c + d*x)] + 15*A*Sin[4*(c + d*x)] + 30*B*Sin[4*(c + d*x)] + 6*B*Sin[5*(c + d*x)]))/(480*d)","A",1
12,1,86,129,0.3723843,"\int \cos (c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]","\frac{a^2 (24 (7 A+6 B) \sin (c+d x)+48 (A+B) \sin (2 (c+d x))+8 A \sin (3 (c+d x))+96 A d x+16 B \sin (3 (c+d x))+3 B \sin (4 (c+d x))+84 B c+84 B d x)}{96 d}","\frac{a^2 (8 A+7 B) \sin (c+d x)}{6 d}+\frac{a^2 (8 A+7 B) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} a^2 x (8 A+7 B)+\frac{(4 A-B) \sin (c+d x) (a \cos (c+d x)+a)^2}{12 d}+\frac{B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 a d}",1,"(a^2*(84*B*c + 96*A*d*x + 84*B*d*x + 24*(7*A + 6*B)*Sin[c + d*x] + 48*(A + B)*Sin[2*(c + d*x)] + 8*A*Sin[3*(c + d*x)] + 16*B*Sin[3*(c + d*x)] + 3*B*Sin[4*(c + d*x)]))/(96*d)","A",1
13,1,61,94,0.204894,"\int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]","\frac{a^2 (3 (8 A+7 B) \sin (c+d x)+3 (A+2 B) \sin (2 (c+d x))+18 A d x+B \sin (3 (c+d x))+12 B d x)}{12 d}","\frac{2 a^2 (3 A+2 B) \sin (c+d x)}{3 d}+\frac{a^2 (3 A+2 B) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{1}{2} a^2 x (3 A+2 B)+\frac{B \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}",1,"(a^2*(18*A*d*x + 12*B*d*x + 3*(8*A + 7*B)*Sin[c + d*x] + 3*(A + 2*B)*Sin[2*(c + d*x)] + B*Sin[3*(c + d*x)]))/(12*d)","A",1
14,1,96,82,0.2040773,"\int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{a^2 \left(4 (A+2 B) \sin (c+d x)-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)+8 A d x+B \sin (2 (c+d x))+6 B d x\right)}{4 d}","\frac{a^2 (2 A+3 B) \sin (c+d x)}{2 d}+\frac{1}{2} a^2 x (4 A+3 B)+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{B \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{2 d}",1,"(a^2*(8*A*d*x + 6*B*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*(A + 2*B)*Sin[c + d*x] + B*Sin[2*(c + d*x)]))/(4*d)","A",1
15,1,143,74,0.3685308,"\int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{a^2 \left(A \tan (c+d x)-2 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+A c+A d x+B \sin (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)+2 B c+2 B d x\right)}{d}","-\frac{a^2 (A-B) \sin (c+d x)}{d}+\frac{a^2 (2 A+B) \tanh ^{-1}(\sin (c+d x))}{d}+a^2 x (A+2 B)+\frac{A \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{d}",1,"(a^2*(A*c + 2*B*c + A*d*x + 2*B*d*x - 2*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + B*Sin[c + d*x] + A*Tan[c + d*x]))/d","A",1
16,1,277,88,1.3573734,"\int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*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{4 (2 A+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 (2 A+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)}-\frac{2 (3 A+4 B) \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+4 B) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\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}+4 B x\right)","\frac{a^2 (3 A+2 B) \tan (c+d x)}{2 d}+\frac{a^2 (3 A+4 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A \tan (c+d x) \sec (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{2 d}+a^2 B x",1,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(4*B*x - (2*(3*A + 4*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (2*(3*A + 4*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + A/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (4*(2*A + B)*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) + (4*(2*A + B)*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/16","B",1
17,1,451,113,6.3550712,"\int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*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(\frac{4 (5 A+6 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 (5 A+6 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)}+\frac{(7 A+3 B) \cos \left(\frac{c}{2}\right)-(5 A+3 B) \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 A+3 B) \sin \left(\frac{c}{2}\right)+(7 A+3 B) \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 A+3 B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 (2 A+3 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 \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 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)^3}\right)}{48 d}","\frac{a^2 (5 A+6 B) \tan (c+d x)}{3 d}+\frac{a^2 (2 A+3 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 (4 A+3 B) \tan (c+d x) \sec (c+d x)}{6 d}+\frac{A \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*A + 3*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*(2*A + 3*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*A*Sin[(d*x)/2])/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + ((7*A + 3*B)*Cos[c/2] - (5*A + 3*B)*Sin[c/2])/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (4*(5*A + 6*B)*Sin[(d*x)/2])/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (2*A*Sin[(d*x)/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - ((7*A + 3*B)*Cos[c/2] + (5*A + 3*B)*Sin[c/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (4*(5*A + 6*B)*Sin[(d*x)/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(48*d)","B",1
18,1,262,144,1.2550876,"\int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*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+8 B) \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 A+5 B) \sin (c)+3 (15 A+8 B) \sin (d x)+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)+24 B \sin (2 c+d x)+136 B \sin (c+2 d x)-24 B \sin (3 c+2 d x)+24 B \sin (2 c+3 d x)+24 B \sin (4 c+3 d x)+40 B \sin (3 c+4 d x))\right)}{768 d}","\frac{a^2 (4 A+5 B) \tan (c+d x)}{3 d}+\frac{a^2 (7 A+8 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (5 A+4 B) \tan (c+d x) \sec ^2(c+d x)}{12 d}+\frac{a^2 (7 A+8 B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{A \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*A + 8*B)*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*A + 5*B)*Sin[c] + 3*(15*A + 8*B)*Sin[d*x] + 45*A*Sin[2*c + d*x] + 24*B*Sin[2*c + d*x] + 128*A*Sin[c + 2*d*x] + 136*B*Sin[c + 2*d*x] - 24*B*Sin[3*c + 2*d*x] + 21*A*Sin[2*c + 3*d*x] + 24*B*Sin[2*c + 3*d*x] + 21*A*Sin[4*c + 3*d*x] + 24*B*Sin[4*c + 3*d*x] + 32*A*Sin[3*c + 4*d*x] + 40*B*Sin[3*c + 4*d*x])))/d","A",1
19,1,134,201,0.5899128,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]),x]","\frac{a^3 (120 (23 A+21 B) \sin (c+d x)+15 (64 A+63 B) \sin (2 (c+d x))+340 A \sin (3 (c+d x))+90 A \sin (4 (c+d x))+12 A \sin (5 (c+d x))+1560 A d x+380 B \sin (3 (c+d x))+135 B \sin (4 (c+d x))+36 B \sin (5 (c+d x))+5 B \sin (6 (c+d x))+1380 B c+1380 B d x)}{960 d}","-\frac{a^3 (19 A+17 B) \sin ^3(c+d x)}{15 d}+\frac{a^3 (19 A+17 B) \sin (c+d x)}{5 d}+\frac{a^3 (22 A+21 B) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{(3 A+4 B) \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) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^3 x (26 A+23 B)+\frac{a B \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}",1,"(a^3*(1380*B*c + 1560*A*d*x + 1380*B*d*x + 120*(23*A + 21*B)*Sin[c + d*x] + 15*(64*A + 63*B)*Sin[2*(c + d*x)] + 340*A*Sin[3*(c + d*x)] + 380*B*Sin[3*(c + d*x)] + 90*A*Sin[4*(c + d*x)] + 135*B*Sin[4*(c + d*x)] + 12*A*Sin[5*(c + d*x)] + 36*B*Sin[5*(c + d*x)] + 5*B*Sin[6*(c + d*x)]))/(960*d)","A",1
20,1,108,154,0.4570543,"\int \cos (c+d x) (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]),x]","\frac{a^3 (60 (26 A+23 B) \sin (c+d x)+480 (A+B) \sin (2 (c+d x))+120 A \sin (3 (c+d x))+15 A \sin (4 (c+d x))+900 A d x+170 B \sin (3 (c+d x))+45 B \sin (4 (c+d x))+6 B \sin (5 (c+d x))+780 B c+780 B d x)}{480 d}","-\frac{a^3 (15 A+13 B) \sin ^3(c+d x)}{60 d}+\frac{a^3 (15 A+13 B) \sin (c+d x)}{5 d}+\frac{3 a^3 (15 A+13 B) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{1}{8} a^3 x (15 A+13 B)+\frac{(5 A-B) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 d}+\frac{B \sin (c+d x) (a \cos (c+d x)+a)^4}{5 a d}",1,"(a^3*(780*B*c + 900*A*d*x + 780*B*d*x + 60*(26*A + 23*B)*Sin[c + d*x] + 480*(A + B)*Sin[2*(c + d*x)] + 120*A*Sin[3*(c + d*x)] + 170*B*Sin[3*(c + d*x)] + 15*A*Sin[4*(c + d*x)] + 45*B*Sin[4*(c + d*x)] + 6*B*Sin[5*(c + d*x)]))/(480*d)","A",1
21,1,86,116,0.3334716,"\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]),x]","\frac{a^3 (24 (15 A+13 B) \sin (c+d x)+24 (3 A+4 B) \sin (2 (c+d x))+8 A \sin (3 (c+d x))+240 A d x+24 B \sin (3 (c+d x))+3 B \sin (4 (c+d x))+180 B d x)}{96 d}","-\frac{a^3 (4 A+3 B) \sin ^3(c+d x)}{12 d}+\frac{a^3 (4 A+3 B) \sin (c+d x)}{d}+\frac{3 a^3 (4 A+3 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5}{8} a^3 x (4 A+3 B)+\frac{B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}",1,"(a^3*(240*A*d*x + 180*B*d*x + 24*(15*A + 13*B)*Sin[c + d*x] + 24*(3*A + 4*B)*Sin[2*(c + d*x)] + 8*A*Sin[3*(c + d*x)] + 24*B*Sin[3*(c + d*x)] + 3*B*Sin[4*(c + d*x)]))/(96*d)","A",1
22,1,113,111,0.2821659,"\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{a^3 \left(9 (4 A+5 B) \sin (c+d x)+3 (A+3 B) \sin (2 (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)+42 A d x+B \sin (3 (c+d x))+30 B d x\right)}{12 d}","\frac{5 a^3 (A+B) \sin (c+d x)}{2 d}+\frac{(3 A+5 B) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{6 d}+\frac{1}{2} a^3 x (7 A+5 B)+\frac{a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a B \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d}",1,"(a^3*(42*A*d*x + 30*B*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]] + 9*(4*A + 5*B)*Sin[c + d*x] + 3*(A + 3*B)*Sin[2*(c + d*x)] + B*Sin[3*(c + d*x)]))/(12*d)","A",1
23,1,272,110,1.8533789,"\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,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 (A+3 B) \sin (c) \cos (d x)}{d}+\frac{4 (A+3 B) \cos (c) \sin (d x)}{d}-\frac{4 (3 A+B) \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 A+B) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+2 x (6 A+7 B)+\frac{4 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{4 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{B \sin (2 c) \cos (2 d x)}{d}+\frac{B \cos (2 c) \sin (2 d x)}{d}\right)","\frac{a^3 (3 A+B) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{(2 A-B) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{2 d}+\frac{1}{2} a^3 x (6 A+7 B)+\frac{5 a^3 B \sin (c+d x)}{2 d}+\frac{a A \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*A + 7*B)*x - (4*(3*A + B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (4*(3*A + B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (4*(A + 3*B)*Cos[d*x]*Sin[c])/d + (B*Cos[2*d*x]*Sin[2*c])/d + (4*(A + 3*B)*Cos[c]*Sin[d*x])/d + (B*Cos[2*c]*Sin[2*d*x])/d + (4*A*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (4*A*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/32","B",1
24,1,208,114,2.0045149,"\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{a^3 \left(4 (3 A+B) \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+4 B \sin (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)+12 B c+12 B d x\right)}{4 d}","\frac{a^3 (7 A+6 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(2 A+B) \tan (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{d}+a^3 x (A+3 B)-\frac{5 a^3 A \sin (c+d x)}{2 d}+\frac{a A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}",1,"(a^3*(4*A*c + 12*B*c + 4*A*d*x + 12*B*d*x - 14*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 12*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 14*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 12*B*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*B*Sin[c + d*x] + 4*(3*A + B)*Tan[c + d*x]))/(4*d)","A",1
25,1,786,125,6.4063833,"\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{\sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \left(11 A \sin \left(\frac{d x}{2}\right)+9 B \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 A \sin \left(\frac{d x}{2}\right)+9 B \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 A \sin \left(\frac{c}{2}\right)+10 A \cos \left(\frac{c}{2}\right)-3 B \sin \left(\frac{c}{2}\right)+3 B \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 A \sin \left(\frac{c}{2}\right)-10 A \cos \left(\frac{c}{2}\right)-3 B \sin \left(\frac{c}{2}\right)-3 B \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 A-7 B) \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 A+7 B) \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{A \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{A \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} B x \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3","\frac{5 a^3 (A+B) \tan (c+d x)}{2 d}+\frac{a^3 (5 A+7 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(5 A+3 B) \tan (c+d x) \sec (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{6 d}+a^3 B x+\frac{a A \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^2}{3 d}",1,"(B*x*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6)/8 + ((-5*A - 7*B)*(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 + 7*B)*(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) + (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*(10*A*Cos[c/2] + 3*B*Cos[c/2] - 8*A*Sin[c/2] - 3*B*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*A*Sin[(d*x)/2] + 9*B*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*(-10*A*Cos[c/2] - 3*B*Cos[c/2] - 8*A*Sin[c/2] - 3*B*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*A*Sin[(d*x)/2] + 9*B*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
26,1,273,154,1.385932,"\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*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(120 (3 A+4 B) \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 A+11 B) \sin (c)+(69 A+36 B) \sin (d x)+69 A \sin (2 c+d x)+264 A \sin (c+2 d x)-24 A \sin (3 c+2 d x)+45 A \sin (2 c+3 d x)+45 A \sin (4 c+3 d x)+72 A \sin (3 c+4 d x)+36 B \sin (2 c+d x)+280 B \sin (c+2 d x)-72 B \sin (3 c+2 d x)+36 B \sin (2 c+3 d x)+36 B \sin (4 c+3 d x)+88 B \sin (3 c+4 d x))\right)}{1536 d}","\frac{a^3 (9 A+11 B) \tan (c+d x)}{3 d}+\frac{5 a^3 (3 A+4 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 (27 A+28 B) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{(3 A+2 B) \tan (c+d x) \sec ^2(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{6 d}+\frac{a A \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*A + 4*B)*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*A + 11*B)*Sin[c] + (69*A + 36*B)*Sin[d*x] + 69*A*Sin[2*c + d*x] + 36*B*Sin[2*c + d*x] + 264*A*Sin[c + 2*d*x] + 280*B*Sin[c + 2*d*x] - 24*A*Sin[3*c + 2*d*x] - 72*B*Sin[3*c + 2*d*x] + 45*A*Sin[2*c + 3*d*x] + 36*B*Sin[2*c + 3*d*x] + 45*A*Sin[4*c + 3*d*x] + 36*B*Sin[4*c + 3*d*x] + 72*A*Sin[3*c + 4*d*x] + 88*B*Sin[3*c + 4*d*x])))/d","A",1
27,1,294,185,1.6041417,"\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^6(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*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+15 B) \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+5 B) \sin (2 c+d x)+80 (29 A+30 B) \sin (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)+570 B \sin (c+2 d x)+570 B \sin (3 c+2 d x)+1680 B \sin (2 c+3 d x)-120 B \sin (4 c+3 d x)+225 B \sin (3 c+4 d x)+225 B \sin (5 c+4 d x)+360 B \sin (4 c+5 d x))\right)}{15360 d}","\frac{a^3 (38 A+45 B) \tan (c+d x)}{15 d}+\frac{a^3 (13 A+15 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 (43 A+45 B) \tan (c+d x) \sec ^2(c+d x)}{60 d}+\frac{a^3 (13 A+15 B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{(7 A+5 B) \tan (c+d x) \sec ^3(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{20 d}+\frac{a A \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*A + 15*B)*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 + 30*B)*Sin[d*x] - 240*(3*A + 5*B)*Sin[2*c + d*x] + 750*A*Sin[c + 2*d*x] + 570*B*Sin[c + 2*d*x] + 750*A*Sin[3*c + 2*d*x] + 570*B*Sin[3*c + 2*d*x] + 1520*A*Sin[2*c + 3*d*x] + 1680*B*Sin[2*c + 3*d*x] - 120*B*Sin[4*c + 3*d*x] + 195*A*Sin[3*c + 4*d*x] + 225*B*Sin[3*c + 4*d*x] + 195*A*Sin[5*c + 4*d*x] + 225*B*Sin[5*c + 4*d*x] + 304*A*Sin[4*c + 5*d*x] + 360*B*Sin[4*c + 5*d*x])))/d","A",1
28,1,156,241,0.8703714,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x]),x]","\frac{a^4 (105 (352 A+323 B) \sin (c+d x)+105 (127 A+124 B) \sin (2 (c+d x))+5040 A \sin (3 (c+d x))+1575 A \sin (4 (c+d x))+336 A \sin (5 (c+d x))+35 A \sin (6 (c+d x))+20580 A d x+5495 B \sin (3 (c+d x))+2100 B \sin (4 (c+d x))+651 B \sin (5 (c+d x))+140 B \sin (6 (c+d x))+15 B \sin (7 (c+d x))+18480 B c+18480 B d x)}{6720 d}","-\frac{a^4 (252 A+227 B) \sin ^3(c+d x)}{105 d}+\frac{a^4 (252 A+227 B) \sin (c+d x)}{35 d}+\frac{a^4 (301 A+276 B) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{7 (A+B) \sin (c+d x) \cos ^3(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{15 d}+\frac{a^4 (49 A+44 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a^4 x (49 A+44 B)+\frac{(7 A+10 B) \sin (c+d x) \cos ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{42 d}+\frac{a B \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}",1,"(a^4*(18480*B*c + 20580*A*d*x + 18480*B*d*x + 105*(352*A + 323*B)*Sin[c + d*x] + 105*(127*A + 124*B)*Sin[2*(c + d*x)] + 5040*A*Sin[3*(c + d*x)] + 5495*B*Sin[3*(c + d*x)] + 1575*A*Sin[4*(c + d*x)] + 2100*B*Sin[4*(c + d*x)] + 336*A*Sin[5*(c + d*x)] + 651*B*Sin[5*(c + d*x)] + 35*A*Sin[6*(c + d*x)] + 140*B*Sin[6*(c + d*x)] + 15*B*Sin[7*(c + d*x)]))/(6720*d)","A",1
29,1,134,185,0.5551381,"\int \cos (c+d x) (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x]),x]","\frac{a^4 (120 (49 A+44 B) \sin (c+d x)+15 (128 A+127 B) \sin (2 (c+d x))+580 A \sin (3 (c+d x))+120 A \sin (4 (c+d x))+12 A \sin (5 (c+d x))+3360 A d x+720 B \sin (3 (c+d x))+225 B \sin (4 (c+d x))+48 B \sin (5 (c+d x))+5 B \sin (6 (c+d x))+2940 B c+2940 B d x)}{960 d}","-\frac{2 a^4 (8 A+7 B) \sin ^3(c+d x)}{15 d}+\frac{4 a^4 (8 A+7 B) \sin (c+d x)}{5 d}+\frac{a^4 (8 A+7 B) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{27 a^4 (8 A+7 B) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{7}{16} a^4 x (8 A+7 B)+\frac{(6 A-B) \sin (c+d x) (a \cos (c+d x)+a)^4}{30 d}+\frac{B \sin (c+d x) (a \cos (c+d x)+a)^5}{6 a d}",1,"(a^4*(2940*B*c + 3360*A*d*x + 2940*B*d*x + 120*(49*A + 44*B)*Sin[c + d*x] + 15*(128*A + 127*B)*Sin[2*(c + d*x)] + 580*A*Sin[3*(c + d*x)] + 720*B*Sin[3*(c + d*x)] + 120*A*Sin[4*(c + d*x)] + 225*B*Sin[4*(c + d*x)] + 12*A*Sin[5*(c + d*x)] + 48*B*Sin[5*(c + d*x)] + 5*B*Sin[6*(c + d*x)]))/(960*d)","A",1
30,1,108,150,0.3770099,"\int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x]),x]","\frac{a^4 (420 (8 A+7 B) \sin (c+d x)+120 (7 A+8 B) \sin (2 (c+d x))+160 A \sin (3 (c+d x))+15 A \sin (4 (c+d x))+2100 A d x+290 B \sin (3 (c+d x))+60 B \sin (4 (c+d x))+6 B \sin (5 (c+d x))+1680 B d x)}{480 d}","-\frac{4 a^4 (5 A+4 B) \sin ^3(c+d x)}{15 d}+\frac{8 a^4 (5 A+4 B) \sin (c+d x)}{5 d}+\frac{a^4 (5 A+4 B) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{27 a^4 (5 A+4 B) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{7}{8} a^4 x (5 A+4 B)+\frac{B \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}",1,"(a^4*(2100*A*d*x + 1680*B*d*x + 420*(8*A + 7*B)*Sin[c + d*x] + 120*(7*A + 8*B)*Sin[2*(c + d*x)] + 160*A*Sin[3*(c + d*x)] + 290*B*Sin[3*(c + d*x)] + 15*A*Sin[4*(c + d*x)] + 60*B*Sin[4*(c + d*x)] + 6*B*Sin[5*(c + d*x)]))/(480*d)","A",1
31,1,138,151,0.4023854,"\int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{a^4 \left(24 (27 A+28 B) \sin (c+d x)+24 (4 A+7 B) \sin (2 (c+d x))+8 A \sin (3 (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)+576 A d x+32 B \sin (3 (c+d x))+3 B \sin (4 (c+d x))+420 B d x\right)}{96 d}","\frac{5 a^4 (8 A+7 B) \sin (c+d x)}{8 d}+\frac{(32 A+35 B) \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)+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(4 A+7 B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{12 d}+\frac{a B \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}",1,"(a^4*(576*A*d*x + 420*B*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*(27*A + 28*B)*Sin[c + d*x] + 24*(4*A + 7*B)*Sin[2*(c + d*x)] + 8*A*Sin[3*(c + d*x)] + 32*B*Sin[3*(c + d*x)] + 3*B*Sin[4*(c + d*x)]))/(96*d)","A",1
32,1,312,150,1.706011,"\int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{1}{192} a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \left(\frac{3 (16 A+27 B) \sin (c) \cos (d x)}{d}+\frac{3 (A+4 B) \sin (2 c) \cos (2 d x)}{d}+\frac{3 (16 A+27 B) \cos (c) \sin (d x)}{d}+\frac{3 (A+4 B) \cos (2 c) \sin (2 d x)}{d}-\frac{12 (4 A+B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{12 (4 A+B) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{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)}+78 A x+\frac{B \sin (3 c) \cos (3 d x)}{d}+\frac{B \cos (3 c) \sin (3 d x)}{d}+72 B x\right)","\frac{5 a^4 (A+2 B) \sin (c+d x)}{2 d}+\frac{a^4 (4 A+B) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{(3 A-8 B) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{6 d}+\frac{1}{2} a^4 x (13 A+12 B)-\frac{(3 A-B) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{3 d}+\frac{a A \tan (c+d x) (a \cos (c+d x)+a)^3}{d}",1,"(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*(78*A*x + 72*B*x - (12*(4*A + B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (12*(4*A + B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (3*(16*A + 27*B)*Cos[d*x]*Sin[c])/d + (3*(A + 4*B)*Cos[2*d*x]*Sin[2*c])/d + (B*Cos[3*d*x]*Sin[3*c])/d + (3*(16*A + 27*B)*Cos[c]*Sin[d*x])/d + (3*(A + 4*B)*Cos[2*c]*Sin[2*d*x])/d + (B*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]))))/192","B",1
33,1,343,162,4.6737453,"\int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{1}{64} a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \left(\frac{4 (A+4 B) \sin (c) \cos (d x)}{d}+\frac{4 (A+4 B) \cos (c) \sin (d x)}{d}+\frac{4 (4 A+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 (4 A+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)}-\frac{2 (13 A+8 B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{2 (13 A+8 B) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+2 x (8 A+13 B)+\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{B \sin (2 c) \cos (2 d x)}{d}+\frac{B \cos (2 c) \sin (2 d x)}{d}\right)","-\frac{5 a^4 (A-B) \sin (c+d x)}{2 d}+\frac{a^4 (13 A+8 B) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(6 A+B) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{2 d}+\frac{1}{2} a^4 x (8 A+13 B)+\frac{(5 A+2 B) \tan (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{2 d}+\frac{a A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^3}{2 d}",1,"(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*(2*(8*A + 13*B)*x - (2*(13*A + 8*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (2*(13*A + 8*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (4*(A + 4*B)*Cos[d*x]*Sin[c])/d + (B*Cos[2*d*x]*Sin[2*c])/d + (4*(A + 4*B)*Cos[c]*Sin[d*x])/d + (B*Cos[2*c]*Sin[2*d*x])/d + A/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (4*(4*A + B)*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) + (4*(4*A + B)*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/64","B",1
34,1,380,165,6.2186458,"\int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","a^4 \left(\frac{(A+4 B) (c+d x)}{d}+\frac{-13 A-3 B}{12 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{4 \left(5 A \sin \left(\frac{1}{2} (c+d x)\right)+3 B \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(5 A \sin \left(\frac{1}{2} (c+d x)\right)+3 B \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{13 A+3 B}{12 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{(-12 A-13 B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}+\frac{(12 A+13 B) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 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{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{B \sin (c+d x)}{d}\right)","-\frac{5 a^4 (2 A+B) \sin (c+d x)}{2 d}+\frac{a^4 (12 A+13 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(11 A+9 B) \tan (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{3 d}+a^4 x (A+4 B)+\frac{(2 A+B) \tan (c+d x) \sec (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{2 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*(((A + 4*B)*(c + d*x))/d + ((-12*A - 13*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(2*d) + ((12*A + 13*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(2*d) + (13*A + 3*B)/(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 - 3*B)/(12*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (4*(5*A*Sin[(c + d*x)/2] + 3*B*Sin[(c + d*x)/2]))/(3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (4*(5*A*Sin[(c + d*x)/2] + 3*B*Sin[(c + d*x)/2]))/(3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (B*Sin[c + d*x])/d)","B",1
35,1,326,173,2.0250943,"\int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*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)+48 B \sin (2 c+d x)+496 B \sin (c+2 d x)-144 B \sin (3 c+2 d x)+48 B \sin (2 c+3 d x)+48 B \sin (4 c+3 d x)+160 B \sin (3 c+4 d x)+72 B d x \cos (c)+48 B d x \cos (c+2 d x)+48 B d x \cos (3 c+2 d x)+12 B d x \cos (3 c+4 d x)+12 B d x \cos (5 c+4 d x)-480 B \sin (c)+48 B \sin (d x))-24 (35 A+48 B) \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+8 B) \tan (c+d x)}{8 d}+\frac{a^4 (35 A+48 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(35 A+32 B) \tan (c+d x) \sec (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{24 d}+a^4 B x+\frac{(7 A+4 B) \tan (c+d x) \sec ^2(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{12 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^3}{4 d}",1,"(a^4*Sec[(c + d*x)/2]^8*(1 + Sec[c + d*x])^4*(-24*(35*A + 48*B)*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]*(72*B*d*x*Cos[c] + 48*B*d*x*Cos[c + 2*d*x] + 48*B*d*x*Cos[3*c + 2*d*x] + 12*B*d*x*Cos[3*c + 4*d*x] + 12*B*d*x*Cos[5*c + 4*d*x] - 480*A*Sin[c] - 480*B*Sin[c] + 105*A*Sin[d*x] + 48*B*Sin[d*x] + 105*A*Sin[2*c + d*x] + 48*B*Sin[2*c + d*x] + 544*A*Sin[c + 2*d*x] + 496*B*Sin[c + 2*d*x] - 96*A*Sin[3*c + 2*d*x] - 144*B*Sin[3*c + 2*d*x] + 81*A*Sin[2*c + 3*d*x] + 48*B*Sin[2*c + 3*d*x] + 81*A*Sin[4*c + 3*d*x] + 48*B*Sin[4*c + 3*d*x] + 160*A*Sin[3*c + 4*d*x] + 160*B*Sin[3*c + 4*d*x])))/(3072*d)","A",1
36,1,306,198,1.753324,"\int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^6(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*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(1680 (4 A+5 B) \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) (-960 (2 A+3 B) \sin (2 c+d x)+80 (59 A+64 B) \sin (d x)+1320 A \sin (c+2 d x)+1320 A \sin (3 c+2 d x)+3200 A \sin (2 c+3 d x)-120 A \sin (4 c+3 d x)+420 A \sin (3 c+4 d x)+420 A \sin (5 c+4 d x)+664 A \sin (4 c+5 d x)+930 B \sin (c+2 d x)+930 B \sin (3 c+2 d x)+3520 B \sin (2 c+3 d x)-480 B \sin (4 c+3 d x)+405 B \sin (3 c+4 d x)+405 B \sin (5 c+4 d x)+800 B \sin (4 c+5 d x))\right)}{30720 d}","\frac{a^4 (83 A+100 B) \tan (c+d x)}{15 d}+\frac{7 a^4 (4 A+5 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 (244 A+275 B) \tan (c+d x) \sec (c+d x)}{120 d}+\frac{(26 A+25 B) \tan (c+d x) \sec ^2(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{30 d}+\frac{(8 A+5 B) \tan (c+d x) \sec ^3(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{20 d}+\frac{a A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^3}{5 d}",1,"-1/30720*(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*Sec[c + d*x]^5*(1680*(4*A + 5*B)*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*(59*A + 64*B)*Sin[d*x] - 960*(2*A + 3*B)*Sin[2*c + d*x] + 1320*A*Sin[c + 2*d*x] + 930*B*Sin[c + 2*d*x] + 1320*A*Sin[3*c + 2*d*x] + 930*B*Sin[3*c + 2*d*x] + 3200*A*Sin[2*c + 3*d*x] + 3520*B*Sin[2*c + 3*d*x] - 120*A*Sin[4*c + 3*d*x] - 480*B*Sin[4*c + 3*d*x] + 420*A*Sin[3*c + 4*d*x] + 405*B*Sin[3*c + 4*d*x] + 420*A*Sin[5*c + 4*d*x] + 405*B*Sin[5*c + 4*d*x] + 664*A*Sin[4*c + 5*d*x] + 800*B*Sin[4*c + 5*d*x])))/d","A",1
37,1,358,229,2.3539961,"\int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^7(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*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+8 B) \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 (72 A+83 B) \sin (c)+30 (125 A+88 B) \sin (d x)+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)+2640 B \sin (2 c+d x)+15840 B \sin (c+2 d x)-4080 B \sin (3 c+2 d x)+3480 B \sin (2 c+3 d x)+3480 B \sin (4 c+3 d x)+7728 B \sin (3 c+4 d x)-240 B \sin (5 c+4 d x)+840 B \sin (4 c+5 d x)+840 B \sin (6 c+5 d x)+1328 B \sin (5 c+6 d x))\right)}{122880 d}","\frac{a^4 (72 A+83 B) \tan (c+d x)}{15 d}+\frac{7 a^4 (7 A+8 B) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 (159 A+176 B) \tan (c+d x) \sec ^2(c+d x)}{120 d}+\frac{7 a^4 (7 A+8 B) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(73 A+72 B) \tan (c+d x) \sec ^3(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{120 d}+\frac{(3 A+2 B) \tan (c+d x) \sec ^4(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{10 d}+\frac{a A \tan (c+d x) \sec ^5(c+d x) (a \cos (c+d x)+a)^3}{6 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 + 8*B)*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*(72*A + 83*B)*Sin[c] + 30*(125*A + 88*B)*Sin[d*x] + 3750*A*Sin[2*c + d*x] + 2640*B*Sin[2*c + d*x] + 15360*A*Sin[c + 2*d*x] + 15840*B*Sin[c + 2*d*x] - 1920*A*Sin[3*c + 2*d*x] - 4080*B*Sin[3*c + 2*d*x] + 3845*A*Sin[2*c + 3*d*x] + 3480*B*Sin[2*c + 3*d*x] + 3845*A*Sin[4*c + 3*d*x] + 3480*B*Sin[4*c + 3*d*x] + 6912*A*Sin[3*c + 4*d*x] + 7728*B*Sin[3*c + 4*d*x] - 240*B*Sin[5*c + 4*d*x] + 735*A*Sin[4*c + 5*d*x] + 840*B*Sin[4*c + 5*d*x] + 735*A*Sin[6*c + 5*d*x] + 840*B*Sin[6*c + 5*d*x] + 1152*A*Sin[5*c + 6*d*x] + 1328*B*Sin[5*c + 6*d*x])))/d","A",1
38,1,311,153,0.7038411,"\int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{a+a \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^4*(A + B*Cos[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(-72 d x (4 A-5 B) \cos \left(c+\frac{d x}{2}\right)-72 d x (4 A-5 B) \cos \left(\frac{d x}{2}\right)+168 A \sin \left(c+\frac{d x}{2}\right)+144 A \sin \left(c+\frac{3 d x}{2}\right)+144 A \sin \left(2 c+\frac{3 d x}{2}\right)-16 A \sin \left(2 c+\frac{5 d x}{2}\right)-16 A \sin \left(3 c+\frac{5 d x}{2}\right)+8 A \sin \left(3 c+\frac{7 d x}{2}\right)+8 A \sin \left(4 c+\frac{7 d x}{2}\right)+552 A \sin \left(\frac{d x}{2}\right)-168 B \sin \left(c+\frac{d x}{2}\right)-120 B \sin \left(c+\frac{3 d x}{2}\right)-120 B \sin \left(2 c+\frac{3 d x}{2}\right)+40 B \sin \left(2 c+\frac{5 d x}{2}\right)+40 B \sin \left(3 c+\frac{5 d x}{2}\right)-5 B \sin \left(3 c+\frac{7 d x}{2}\right)-5 B \sin \left(4 c+\frac{7 d x}{2}\right)+3 B \sin \left(4 c+\frac{9 d x}{2}\right)+3 B \sin \left(5 c+\frac{9 d x}{2}\right)-552 B \sin \left(\frac{d x}{2}\right)\right)}{192 a d (\cos (c+d x)+1)}","-\frac{4 (A-B) \sin ^3(c+d x)}{3 a d}+\frac{4 (A-B) \sin (c+d x)}{a d}+\frac{(A-B) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(4 A-5 B) \sin (c+d x) \cos ^3(c+d x)}{4 a d}-\frac{3 (4 A-5 B) \sin (c+d x) \cos (c+d x)}{8 a d}-\frac{3 x (4 A-5 B)}{8 a}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(-72*(4*A - 5*B)*d*x*Cos[(d*x)/2] - 72*(4*A - 5*B)*d*x*Cos[c + (d*x)/2] + 552*A*Sin[(d*x)/2] - 552*B*Sin[(d*x)/2] + 168*A*Sin[c + (d*x)/2] - 168*B*Sin[c + (d*x)/2] + 144*A*Sin[c + (3*d*x)/2] - 120*B*Sin[c + (3*d*x)/2] + 144*A*Sin[2*c + (3*d*x)/2] - 120*B*Sin[2*c + (3*d*x)/2] - 16*A*Sin[2*c + (5*d*x)/2] + 40*B*Sin[2*c + (5*d*x)/2] - 16*A*Sin[3*c + (5*d*x)/2] + 40*B*Sin[3*c + (5*d*x)/2] + 8*A*Sin[3*c + (7*d*x)/2] - 5*B*Sin[3*c + (7*d*x)/2] + 8*A*Sin[4*c + (7*d*x)/2] - 5*B*Sin[4*c + (7*d*x)/2] + 3*B*Sin[4*c + (9*d*x)/2] + 3*B*Sin[5*c + (9*d*x)/2]))/(192*a*d*(1 + Cos[c + d*x]))","B",1
39,1,249,122,0.6405586,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{a+a \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[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(36 d x (A-B) \cos \left(c+\frac{d x}{2}\right)+36 d x (A-B) \cos \left(\frac{d x}{2}\right)-12 A \sin \left(c+\frac{d x}{2}\right)-9 A \sin \left(c+\frac{3 d x}{2}\right)-9 A \sin \left(2 c+\frac{3 d x}{2}\right)+3 A \sin \left(2 c+\frac{5 d x}{2}\right)+3 A \sin \left(3 c+\frac{5 d x}{2}\right)-60 A \sin \left(\frac{d x}{2}\right)+21 B \sin \left(c+\frac{d x}{2}\right)+18 B \sin \left(c+\frac{3 d x}{2}\right)+18 B \sin \left(2 c+\frac{3 d x}{2}\right)-2 B \sin \left(2 c+\frac{5 d x}{2}\right)-2 B \sin \left(3 c+\frac{5 d x}{2}\right)+B \sin \left(3 c+\frac{7 d x}{2}\right)+B \sin \left(4 c+\frac{7 d x}{2}\right)+69 B \sin \left(\frac{d x}{2}\right)\right)}{24 a d (\cos (c+d x)+1)}","\frac{(3 A-4 B) \sin ^3(c+d x)}{3 a d}-\frac{(3 A-4 B) \sin (c+d x)}{a d}+\frac{(A-B) \sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}+\frac{3 (A-B) \sin (c+d x) \cos (c+d x)}{2 a d}+\frac{3 x (A-B)}{2 a}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(36*(A - B)*d*x*Cos[(d*x)/2] + 36*(A - B)*d*x*Cos[c + (d*x)/2] - 60*A*Sin[(d*x)/2] + 69*B*Sin[(d*x)/2] - 12*A*Sin[c + (d*x)/2] + 21*B*Sin[c + (d*x)/2] - 9*A*Sin[c + (3*d*x)/2] + 18*B*Sin[c + (3*d*x)/2] - 9*A*Sin[2*c + (3*d*x)/2] + 18*B*Sin[2*c + (3*d*x)/2] + 3*A*Sin[2*c + (5*d*x)/2] - 2*B*Sin[2*c + (5*d*x)/2] + 3*A*Sin[3*c + (5*d*x)/2] - 2*B*Sin[3*c + (5*d*x)/2] + B*Sin[3*c + (7*d*x)/2] + B*Sin[4*c + (7*d*x)/2]))/(24*a*d*(1 + Cos[c + d*x]))","B",1
40,1,197,90,0.5049138,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{a+a \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[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(-4 d x (2 A-3 B) \cos \left(c+\frac{d x}{2}\right)-4 d x (2 A-3 B) \cos \left(\frac{d x}{2}\right)+4 A \sin \left(c+\frac{d x}{2}\right)+4 A \sin \left(c+\frac{3 d x}{2}\right)+4 A \sin \left(2 c+\frac{3 d x}{2}\right)+20 A \sin \left(\frac{d x}{2}\right)-4 B \sin \left(c+\frac{d x}{2}\right)-3 B \sin \left(c+\frac{3 d x}{2}\right)-3 B \sin \left(2 c+\frac{3 d x}{2}\right)+B \sin \left(2 c+\frac{5 d x}{2}\right)+B \sin \left(3 c+\frac{5 d x}{2}\right)-20 B \sin \left(\frac{d x}{2}\right)\right)}{8 a d (\cos (c+d x)+1)}","\frac{(A-B) \sin (c+d x)}{a d}+\frac{(A-B) \sin (c+d x)}{a d (\cos (c+d x)+1)}-\frac{x (A-B)}{a}+\frac{B \sin (c+d x) \cos (c+d x)}{2 a d}+\frac{B x}{2 a}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(-4*(2*A - 3*B)*d*x*Cos[(d*x)/2] - 4*(2*A - 3*B)*d*x*Cos[c + (d*x)/2] + 20*A*Sin[(d*x)/2] - 20*B*Sin[(d*x)/2] + 4*A*Sin[c + (d*x)/2] - 4*B*Sin[c + (d*x)/2] + 4*A*Sin[c + (3*d*x)/2] - 3*B*Sin[c + (3*d*x)/2] + 4*A*Sin[2*c + (3*d*x)/2] - 3*B*Sin[2*c + (3*d*x)/2] + B*Sin[2*c + (5*d*x)/2] + B*Sin[3*c + (5*d*x)/2]))/(8*a*d*(1 + Cos[c + d*x]))","B",1
41,1,126,54,0.281873,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{a+a \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[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 d x (A-B) \cos \left(c+\frac{d x}{2}\right)+2 d x (A-B) \cos \left(\frac{d x}{2}\right)-4 A \sin \left(\frac{d x}{2}\right)+B \sin \left(c+\frac{d x}{2}\right)+B \sin \left(c+\frac{3 d x}{2}\right)+B \sin \left(2 c+\frac{3 d x}{2}\right)+5 B \sin \left(\frac{d x}{2}\right)\right)}{2 a d (\cos (c+d x)+1)}","-\frac{(A-B) \sin (c+d x)}{a d (\cos (c+d x)+1)}+\frac{x (A-B)}{a}+\frac{B \sin (c+d x)}{a d}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(2*(A - B)*d*x*Cos[(d*x)/2] + 2*(A - B)*d*x*Cos[c + (d*x)/2] - 4*A*Sin[(d*x)/2] + 5*B*Sin[(d*x)/2] + B*Sin[c + (d*x)/2] + B*Sin[c + (3*d*x)/2] + B*Sin[2*c + (3*d*x)/2]))/(2*a*d*(1 + Cos[c + d*x]))","B",1
42,1,72,34,0.1492768,"\int \frac{A+B \cos (c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[(A + B*Cos[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 (A-B) \sin \left(\frac{d x}{2}\right)+B d x \cos \left(c+\frac{d x}{2}\right)+B d x \cos \left(\frac{d x}{2}\right)\right)}{a d (\cos (c+d x)+1)}","\frac{(A-B) \sin (c+d x)}{d (a \cos (c+d x)+a)}+\frac{B x}{a}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(B*d*x*Cos[(d*x)/2] + B*d*x*Cos[c + (d*x)/2] + 2*(A - B)*Sin[(d*x)/2]))/(a*d*(1 + Cos[c + d*x]))","B",1
43,1,109,44,0.2730413,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x])/(a + a*Cos[c + d*x]),x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left((B-A) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+A \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{A \tanh ^{-1}(\sin (c+d x))}{a d}-\frac{(A-B) \sin (c+d x)}{d (a \cos (c+d x)+a)}",1,"(2*Cos[(c + 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]]) + (-A + B)*Sec[c/2]*Sin[(d*x)/2]))/(a*d*(1 + Cos[c + d*x]))","B",1
44,1,201,69,1.3410303,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + a*Cos[c + d*x]),x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left((A-B) \sec \left(\frac{c}{2}\right) \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)}","\frac{(2 A-B) \tan (c+d x)}{a d}-\frac{(A-B) \tanh ^{-1}(\sin (c+d x))}{a d}-\frac{(A-B) \tan (c+d x)}{d (a \cos (c+d x)+a)}",1,"(2*Cos[(c + d*x)/2]*((A - B)*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]))","B",1
45,1,289,107,3.6360585,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + a*Cos[c + d*x]),x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(4 (B-A) \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 (A-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)}+(4 B-6 A) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \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 B \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 (A-B) \tan (c+d x)}{a d}+\frac{(3 A-2 B) \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{(3 A-2 B) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(A-B) \tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}",1,"(Cos[(c + d*x)/2]*(4*(-A + B)*Sec[c/2]*Sin[(d*x)/2] + Cos[(c + d*x)/2]*((-6*A + 4*B)*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*B*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 - 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])))))/(2*a*d*(1 + Cos[c + d*x]))","B",1
46,1,490,131,4.6055344,"\int \frac{(A+B \cos (c+d x)) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^4)/(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(6 (A+B) \sin \left(\frac{d x}{2}\right)+3 (13 A-9 B) \sin \left(\frac{3 d x}{2}\right)-24 A \sin \left(c-\frac{d x}{2}\right)-6 A \sin \left(c+\frac{d x}{2}\right)-24 A \sin \left(2 c+\frac{d x}{2}\right)+21 A \sin \left(c+\frac{3 d x}{2}\right)+9 A \sin \left(2 c+\frac{3 d x}{2}\right)-9 A \sin \left(3 c+\frac{3 d x}{2}\right)+7 A \sin \left(c+\frac{5 d x}{2}\right)+A \sin \left(2 c+\frac{5 d x}{2}\right)-3 A \sin \left(3 c+\frac{5 d x}{2}\right)-9 A \sin \left(4 c+\frac{5 d x}{2}\right)+16 A \sin \left(2 c+\frac{7 d x}{2}\right)+10 A \sin \left(3 c+\frac{7 d x}{2}\right)+6 A \sin \left(4 c+\frac{7 d x}{2}\right)+12 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)-9 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)-3 B \sin \left(c+\frac{5 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)+9 B \sin \left(4 c+\frac{5 d x}{2}\right)-12 B \sin \left(2 c+\frac{7 d x}{2}\right)-6 B \sin \left(3 c+\frac{7 d x}{2}\right)-6 B \sin \left(4 c+\frac{7 d x}{2}\right)\right)+144 (A-B) \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 A-3 B) \tan ^3(c+d x)}{3 a d}+\frac{(4 A-3 B) \tan (c+d x)}{a d}-\frac{3 (A-B) \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{3 (A-B) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{(A-B) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}",1,"(Cos[(c + d*x)/2]*(144*(A - 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]]) + Sec[c/2]*Sec[c]*Sec[c + d*x]^3*(6*(A + B)*Sin[(d*x)/2] + 3*(13*A - 9*B)*Sin[(3*d*x)/2] - 24*A*Sin[c - (d*x)/2] + 12*B*Sin[c - (d*x)/2] - 6*A*Sin[c + (d*x)/2] + 6*B*Sin[c + (d*x)/2] - 24*A*Sin[2*c + (d*x)/2] + 24*B*Sin[2*c + (d*x)/2] + 21*A*Sin[c + (3*d*x)/2] - 9*B*Sin[c + (3*d*x)/2] + 9*A*Sin[2*c + (3*d*x)/2] - 9*B*Sin[2*c + (3*d*x)/2] - 9*A*Sin[3*c + (3*d*x)/2] + 9*B*Sin[3*c + (3*d*x)/2] + 7*A*Sin[c + (5*d*x)/2] - 3*B*Sin[c + (5*d*x)/2] + A*Sin[2*c + (5*d*x)/2] + 3*B*Sin[2*c + (5*d*x)/2] - 3*A*Sin[3*c + (5*d*x)/2] + 3*B*Sin[3*c + (5*d*x)/2] - 9*A*Sin[4*c + (5*d*x)/2] + 9*B*Sin[4*c + (5*d*x)/2] + 16*A*Sin[2*c + (7*d*x)/2] - 12*B*Sin[2*c + (7*d*x)/2] + 10*A*Sin[3*c + (7*d*x)/2] - 6*B*Sin[3*c + (7*d*x)/2] + 6*A*Sin[4*c + (7*d*x)/2] - 6*B*Sin[4*c + (7*d*x)/2])))/(48*a*d*(1 + Cos[c + d*x]))","B",1
47,1,369,170,0.7026774,"\int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^4*(A + B*Cos[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(36 d x (7 A-10 B) \cos \left(c+\frac{d x}{2}\right)+36 d x (7 A-10 B) \cos \left(\frac{d x}{2}\right)+147 A \sin \left(c+\frac{d x}{2}\right)-239 A \sin \left(c+\frac{3 d x}{2}\right)-63 A \sin \left(2 c+\frac{3 d x}{2}\right)-15 A \sin \left(2 c+\frac{5 d x}{2}\right)-15 A \sin \left(3 c+\frac{5 d x}{2}\right)+3 A \sin \left(3 c+\frac{7 d x}{2}\right)+3 A \sin \left(4 c+\frac{7 d x}{2}\right)+84 A d x \cos \left(c+\frac{3 d x}{2}\right)+84 A d x \cos \left(2 c+\frac{3 d x}{2}\right)-381 A \sin \left(\frac{d x}{2}\right)-156 B \sin \left(c+\frac{d x}{2}\right)+342 B \sin \left(c+\frac{3 d x}{2}\right)+118 B \sin \left(2 c+\frac{3 d x}{2}\right)+30 B \sin \left(2 c+\frac{5 d x}{2}\right)+30 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)+B \sin \left(4 c+\frac{9 d x}{2}\right)+B \sin \left(5 c+\frac{9 d x}{2}\right)-120 B d x \cos \left(c+\frac{3 d x}{2}\right)-120 B d x \cos \left(2 c+\frac{3 d x}{2}\right)+516 B \sin \left(\frac{d x}{2}\right)\right)}{48 a^2 d (\cos (c+d x)+1)^2}","\frac{4 (2 A-3 B) \sin ^3(c+d x)}{3 a^2 d}-\frac{4 (2 A-3 B) \sin (c+d x)}{a^2 d}+\frac{(7 A-10 B) \sin (c+d x) \cos ^3(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{(7 A-10 B) \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{x (7 A-10 B)}{2 a^2}+\frac{(A-B) \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*A - 10*B)*d*x*Cos[(d*x)/2] + 36*(7*A - 10*B)*d*x*Cos[c + (d*x)/2] + 84*A*d*x*Cos[c + (3*d*x)/2] - 120*B*d*x*Cos[c + (3*d*x)/2] + 84*A*d*x*Cos[2*c + (3*d*x)/2] - 120*B*d*x*Cos[2*c + (3*d*x)/2] - 381*A*Sin[(d*x)/2] + 516*B*Sin[(d*x)/2] + 147*A*Sin[c + (d*x)/2] - 156*B*Sin[c + (d*x)/2] - 239*A*Sin[c + (3*d*x)/2] + 342*B*Sin[c + (3*d*x)/2] - 63*A*Sin[2*c + (3*d*x)/2] + 118*B*Sin[2*c + (3*d*x)/2] - 15*A*Sin[2*c + (5*d*x)/2] + 30*B*Sin[2*c + (5*d*x)/2] - 15*A*Sin[3*c + (5*d*x)/2] + 30*B*Sin[3*c + (5*d*x)/2] + 3*A*Sin[3*c + (7*d*x)/2] - 3*B*Sin[3*c + (7*d*x)/2] + 3*A*Sin[4*c + (7*d*x)/2] - 3*B*Sin[4*c + (7*d*x)/2] + B*Sin[4*c + (9*d*x)/2] + B*Sin[5*c + (9*d*x)/2]))/(48*a^2*d*(1 + Cos[c + d*x])^2)","B",1
48,1,315,147,0.8440844,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[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(-36 d x (4 A-7 B) \cos \left(c+\frac{d x}{2}\right)-36 d x (4 A-7 B) \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)\right)}{48 a^2 d (\cos (c+d x)+1)^2}","\frac{2 (5 A-8 B) \sin (c+d x)}{3 a^2 d}+\frac{(5 A-8 B) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(4 A-7 B) \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{x (4 A-7 B)}{2 a^2}+\frac{(A-B) \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*A - 7*B)*d*x*Cos[(d*x)/2] - 36*(4*A - 7*B)*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] - 48*A*d*x*Cos[2*c + (3*d*x)/2] + 84*B*d*x*Cos[2*c + (3*d*x)/2] + 264*A*Sin[(d*x)/2] - 381*B*Sin[(d*x)/2] - 120*A*Sin[c + (d*x)/2] + 147*B*Sin[c + (d*x)/2] + 164*A*Sin[c + (3*d*x)/2] - 239*B*Sin[c + (3*d*x)/2] + 36*A*Sin[2*c + (3*d*x)/2] - 63*B*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] + 12*A*Sin[3*c + (5*d*x)/2] - 15*B*Sin[3*c + (5*d*x)/2] + 3*B*Sin[3*c + (7*d*x)/2] + 3*B*Sin[4*c + (7*d*x)/2]))/(48*a^2*d*(1 + Cos[c + d*x])^2)","B",1
49,1,137,99,0.7653573,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(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 (A-2 B)+B \sin (c+d x))+(A-B) \tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)+(A-B) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)-2 (5 A-8 B) \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{(A-4 B) \sin (c+d x)}{3 a^2 d}-\frac{(A-2 B) \sin (c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{x (A-2 B)}{a^2}+\frac{(A-B) \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]*((A - B)*Sec[c/2]*Sin[(d*x)/2] - 2*(5*A - 8*B)*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + 6*Cos[(c + d*x)/2]^3*((A - 2*B)*d*x + B*Sin[c + d*x]) + (A - B)*Cos[(c + d*x)/2]*Tan[c/2]))/(3*a^2*d*(1 + Cos[c + d*x])^2)","A",1
50,1,153,70,0.3725931,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(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 A \sin \left(c+\frac{d x}{2}\right)+4 A \sin \left(c+\frac{3 d x}{2}\right)+6 A \sin \left(\frac{d x}{2}\right)+12 B \sin \left(c+\frac{d x}{2}\right)-10 B \sin \left(c+\frac{3 d x}{2}\right)+9 B d x \cos \left(c+\frac{d x}{2}\right)+3 B d x \cos \left(c+\frac{3 d x}{2}\right)+3 B d x \cos \left(2 c+\frac{3 d x}{2}\right)-18 B \sin \left(\frac{d x}{2}\right)+9 B d x \cos \left(\frac{d x}{2}\right)\right)}{24 a^2 d}","\frac{(2 A-5 B) \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{B x}{a^2}-\frac{(A-B) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^3*(9*B*d*x*Cos[(d*x)/2] + 9*B*d*x*Cos[c + (d*x)/2] + 3*B*d*x*Cos[c + (3*d*x)/2] + 3*B*d*x*Cos[2*c + (3*d*x)/2] + 6*A*Sin[(d*x)/2] - 18*B*Sin[(d*x)/2] - 6*A*Sin[c + (d*x)/2] + 12*B*Sin[c + (d*x)/2] + 4*A*Sin[c + (3*d*x)/2] - 10*B*Sin[c + (3*d*x)/2]))/(24*a^2*d)","B",1
51,1,76,65,0.193836,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[(A + B*Cos[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((A+2 B) \sin \left(c+\frac{3 d x}{2}\right)+3 (A+B) \sin \left(\frac{d x}{2}\right)-3 B \sin \left(c+\frac{d x}{2}\right)\right)}{3 a^2 d (\cos (c+d x)+1)^2}","\frac{(A+2 B) \sin (c+d x)}{3 d \left(a^2 \cos (c+d x)+a^2\right)}+\frac{(A-B) \sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(3*(A + B)*Sin[(d*x)/2] - 3*B*Sin[c + (d*x)/2] + (A + 2*B)*Sin[c + (3*d*x)/2]))/(3*a^2*d*(1 + Cos[c + d*x])^2)","A",1
52,1,170,79,0.5407488,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x])*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-B) \tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)+(A-B) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+2 (4 A-B) \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{(4 A-B) \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) \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 - B)*Sec[c/2]*Sin[(d*x)/2] + 2*(4*A - B)*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + (A - B)*Cos[(c + d*x)/2]*Tan[c/2]))/(3*a^2*d*(1 + Cos[c + d*x])^2)","B",1
53,1,264,107,1.8528919,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x])*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((A-B) \tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)+(A-B) \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 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)+2 (7 A-4 B) \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 A-2 B) \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) \tan (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(2*Cos[(c + d*x)/2]*((A - B)*Sec[c/2]*Sin[(d*x)/2] + 2*(7*A - 4*B)*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)*Cos[(c + d*x)/2]*Tan[c/2]))/(3*a^2*d*(1 + Cos[c + d*x])^2)","B",1
54,1,496,152,3.4297784,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^2,x]","-\frac{96 (7 A-4 B) \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(-14 (A-B) \sin \left(\frac{d x}{2}\right)+(97 A-64 B) \sin \left(\frac{3 d x}{2}\right)-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)+84 B \sin \left(c-\frac{d x}{2}\right)-42 B \sin \left(c+\frac{d x}{2}\right)+56 B \sin \left(2 c+\frac{d x}{2}\right)+6 B \sin \left(c+\frac{3 d x}{2}\right)-34 B \sin \left(2 c+\frac{3 d x}{2}\right)+36 B \sin \left(3 c+\frac{3 d x}{2}\right)-48 B \sin \left(c+\frac{5 d x}{2}\right)-6 B \sin \left(2 c+\frac{5 d x}{2}\right)-30 B \sin \left(3 c+\frac{5 d x}{2}\right)+12 B \sin \left(4 c+\frac{5 d x}{2}\right)-20 B \sin \left(2 c+\frac{7 d x}{2}\right)-6 B \sin \left(3 c+\frac{7 d x}{2}\right)-14 B \sin \left(4 c+\frac{7 d x}{2}\right)\right)}{48 a^2 d (\cos (c+d x)+1)^2}","-\frac{2 (8 A-5 B) \tan (c+d x)}{3 a^2 d}+\frac{(7 A-4 B) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac{(7 A-4 B) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{(8 A-5 B) \tan (c+d x) \sec (c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(A-B) \tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"-1/48*(96*(7*A - 4*B)*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*(A - B)*Sin[(d*x)/2] + (97*A - 64*B)*Sin[(3*d*x)/2] - 126*A*Sin[c - (d*x)/2] + 84*B*Sin[c - (d*x)/2] + 42*A*Sin[c + (d*x)/2] - 42*B*Sin[c + (d*x)/2] - 98*A*Sin[2*c + (d*x)/2] + 56*B*Sin[2*c + (d*x)/2] - 3*A*Sin[c + (3*d*x)/2] + 6*B*Sin[c + (3*d*x)/2] + 37*A*Sin[2*c + (3*d*x)/2] - 34*B*Sin[2*c + (3*d*x)/2] - 63*A*Sin[3*c + (3*d*x)/2] + 36*B*Sin[3*c + (3*d*x)/2] + 75*A*Sin[c + (5*d*x)/2] - 48*B*Sin[c + (5*d*x)/2] + 15*A*Sin[2*c + (5*d*x)/2] - 6*B*Sin[2*c + (5*d*x)/2] + 39*A*Sin[3*c + (5*d*x)/2] - 30*B*Sin[3*c + (5*d*x)/2] - 21*A*Sin[4*c + (5*d*x)/2] + 12*B*Sin[4*c + (5*d*x)/2] + 32*A*Sin[2*c + (7*d*x)/2] - 20*B*Sin[2*c + (7*d*x)/2] + 12*A*Sin[3*c + (7*d*x)/2] - 6*B*Sin[3*c + (7*d*x)/2] + 20*A*Sin[4*c + (7*d*x)/2] - 14*B*Sin[4*c + (7*d*x)/2]))/(a^2*d*(1 + Cos[c + d*x])^2)","B",1
55,1,609,179,5.5995661,"\int \frac{(A+B \cos (c+d x)) \sec ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^4)/(a + a*Cos[c + d*x])^2,x]","\frac{192 (10 A-7 B) \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((45 B-6 A) \sin \left(\frac{d x}{2}\right)+(310 A-201 B) \sin \left(\frac{3 d x}{2}\right)-306 A \sin \left(c-\frac{d x}{2}\right)+42 A \sin \left(c+\frac{d x}{2}\right)-270 A \sin \left(2 c+\frac{d x}{2}\right)+50 A \sin \left(c+\frac{3 d x}{2}\right)+90 A \sin \left(2 c+\frac{3 d x}{2}\right)-170 A \sin \left(3 c+\frac{3 d x}{2}\right)+198 A \sin \left(c+\frac{5 d x}{2}\right)+42 A \sin \left(2 c+\frac{5 d x}{2}\right)+66 A \sin \left(3 c+\frac{5 d x}{2}\right)-90 A \sin \left(4 c+\frac{5 d x}{2}\right)+114 A \sin \left(2 c+\frac{7 d x}{2}\right)+36 A \sin \left(3 c+\frac{7 d x}{2}\right)+48 A \sin \left(4 c+\frac{7 d x}{2}\right)-30 A \sin \left(5 c+\frac{7 d x}{2}\right)+48 A \sin \left(3 c+\frac{9 d x}{2}\right)+22 A \sin \left(4 c+\frac{9 d x}{2}\right)+26 A \sin \left(5 c+\frac{9 d x}{2}\right)+195 B \sin \left(c-\frac{d x}{2}\right)-51 B \sin \left(c+\frac{d x}{2}\right)+189 B \sin \left(2 c+\frac{d x}{2}\right)-B \sin \left(c+\frac{3 d x}{2}\right)-81 B \sin \left(2 c+\frac{3 d x}{2}\right)+119 B \sin \left(3 c+\frac{3 d x}{2}\right)-129 B \sin \left(c+\frac{5 d x}{2}\right)-9 B \sin \left(2 c+\frac{5 d x}{2}\right)-57 B \sin \left(3 c+\frac{5 d x}{2}\right)+63 B \sin \left(4 c+\frac{5 d x}{2}\right)-75 B \sin \left(2 c+\frac{7 d x}{2}\right)-15 B \sin \left(3 c+\frac{7 d x}{2}\right)-39 B \sin \left(4 c+\frac{7 d x}{2}\right)+21 B \sin \left(5 c+\frac{7 d x}{2}\right)-32 B \sin \left(3 c+\frac{9 d x}{2}\right)-12 B \sin \left(4 c+\frac{9 d x}{2}\right)-20 B \sin \left(5 c+\frac{9 d x}{2}\right)\right)}{96 a^2 d (\cos (c+d x)+1)^2}","\frac{4 (3 A-2 B) \tan ^3(c+d x)}{3 a^2 d}+\frac{4 (3 A-2 B) \tan (c+d x)}{a^2 d}-\frac{(10 A-7 B) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}-\frac{(10 A-7 B) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{(10 A-7 B) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{(A-B) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(192*(10*A - 7*B)*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*((-6*A + 45*B)*Sin[(d*x)/2] + (310*A - 201*B)*Sin[(3*d*x)/2] - 306*A*Sin[c - (d*x)/2] + 195*B*Sin[c - (d*x)/2] + 42*A*Sin[c + (d*x)/2] - 51*B*Sin[c + (d*x)/2] - 270*A*Sin[2*c + (d*x)/2] + 189*B*Sin[2*c + (d*x)/2] + 50*A*Sin[c + (3*d*x)/2] - B*Sin[c + (3*d*x)/2] + 90*A*Sin[2*c + (3*d*x)/2] - 81*B*Sin[2*c + (3*d*x)/2] - 170*A*Sin[3*c + (3*d*x)/2] + 119*B*Sin[3*c + (3*d*x)/2] + 198*A*Sin[c + (5*d*x)/2] - 129*B*Sin[c + (5*d*x)/2] + 42*A*Sin[2*c + (5*d*x)/2] - 9*B*Sin[2*c + (5*d*x)/2] + 66*A*Sin[3*c + (5*d*x)/2] - 57*B*Sin[3*c + (5*d*x)/2] - 90*A*Sin[4*c + (5*d*x)/2] + 63*B*Sin[4*c + (5*d*x)/2] + 114*A*Sin[2*c + (7*d*x)/2] - 75*B*Sin[2*c + (7*d*x)/2] + 36*A*Sin[3*c + (7*d*x)/2] - 15*B*Sin[3*c + (7*d*x)/2] + 48*A*Sin[4*c + (7*d*x)/2] - 39*B*Sin[4*c + (7*d*x)/2] - 30*A*Sin[5*c + (7*d*x)/2] + 21*B*Sin[5*c + (7*d*x)/2] + 48*A*Sin[3*c + (9*d*x)/2] - 32*B*Sin[3*c + (9*d*x)/2] + 22*A*Sin[4*c + (9*d*x)/2] - 12*B*Sin[4*c + (9*d*x)/2] + 26*A*Sin[5*c + (9*d*x)/2] - 20*B*Sin[5*c + (9*d*x)/2]))/(96*a^2*d*(1 + Cos[c + d*x])^2)","B",1
56,1,491,218,1.0859601,"\int \frac{\cos ^5(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^5*(A + B*Cos[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(600 d x (13 A-23 B) \cos \left(c+\frac{d x}{2}\right)+600 d x (13 A-23 B) \cos \left(\frac{d x}{2}\right)+7560 A \sin \left(c+\frac{d x}{2}\right)-9230 A \sin \left(c+\frac{3 d x}{2}\right)+930 A \sin \left(2 c+\frac{3 d x}{2}\right)-2782 A \sin \left(2 c+\frac{5 d x}{2}\right)-750 A \sin \left(3 c+\frac{5 d x}{2}\right)-105 A \sin \left(3 c+\frac{7 d x}{2}\right)-105 A \sin \left(4 c+\frac{7 d x}{2}\right)+15 A \sin \left(4 c+\frac{9 d x}{2}\right)+15 A \sin \left(5 c+\frac{9 d x}{2}\right)+3900 A d x \cos \left(c+\frac{3 d x}{2}\right)+3900 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+780 A d x \cos \left(2 c+\frac{5 d x}{2}\right)+780 A d x \cos \left(3 c+\frac{5 d x}{2}\right)-12760 A \sin \left(\frac{d x}{2}\right)-11110 B \sin \left(c+\frac{d x}{2}\right)+15380 B \sin \left(c+\frac{3 d x}{2}\right)-380 B \sin \left(2 c+\frac{3 d x}{2}\right)+4777 B \sin \left(2 c+\frac{5 d x}{2}\right)+1625 B \sin \left(3 c+\frac{5 d x}{2}\right)+230 B \sin \left(3 c+\frac{7 d x}{2}\right)+230 B \sin \left(4 c+\frac{7 d x}{2}\right)-20 B \sin \left(4 c+\frac{9 d x}{2}\right)-20 B \sin \left(5 c+\frac{9 d x}{2}\right)+5 B \sin \left(5 c+\frac{11 d x}{2}\right)+5 B \sin \left(6 c+\frac{11 d x}{2}\right)-6900 B d x \cos \left(c+\frac{3 d x}{2}\right)-6900 B d x \cos \left(2 c+\frac{3 d x}{2}\right)-1380 B d x \cos \left(2 c+\frac{5 d x}{2}\right)-1380 B d x \cos \left(3 c+\frac{5 d x}{2}\right)+20410 B \sin \left(\frac{d x}{2}\right)\right)}{480 a^3 d (\cos (c+d x)+1)^3}","\frac{4 (19 A-34 B) \sin ^3(c+d x)}{15 a^3 d}-\frac{4 (19 A-34 B) \sin (c+d x)}{5 a^3 d}+\frac{(13 A-23 B) \sin (c+d x) \cos ^3(c+d x)}{3 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(13 A-23 B) \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac{x (13 A-23 B)}{2 a^3}+\frac{(A-B) \sin (c+d x) \cos ^5(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(8 A-13 B) \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*(13*A - 23*B)*d*x*Cos[(d*x)/2] + 600*(13*A - 23*B)*d*x*Cos[c + (d*x)/2] + 3900*A*d*x*Cos[c + (3*d*x)/2] - 6900*B*d*x*Cos[c + (3*d*x)/2] + 3900*A*d*x*Cos[2*c + (3*d*x)/2] - 6900*B*d*x*Cos[2*c + (3*d*x)/2] + 780*A*d*x*Cos[2*c + (5*d*x)/2] - 1380*B*d*x*Cos[2*c + (5*d*x)/2] + 780*A*d*x*Cos[3*c + (5*d*x)/2] - 1380*B*d*x*Cos[3*c + (5*d*x)/2] - 12760*A*Sin[(d*x)/2] + 20410*B*Sin[(d*x)/2] + 7560*A*Sin[c + (d*x)/2] - 11110*B*Sin[c + (d*x)/2] - 9230*A*Sin[c + (3*d*x)/2] + 15380*B*Sin[c + (3*d*x)/2] + 930*A*Sin[2*c + (3*d*x)/2] - 380*B*Sin[2*c + (3*d*x)/2] - 2782*A*Sin[2*c + (5*d*x)/2] + 4777*B*Sin[2*c + (5*d*x)/2] - 750*A*Sin[3*c + (5*d*x)/2] + 1625*B*Sin[3*c + (5*d*x)/2] - 105*A*Sin[3*c + (7*d*x)/2] + 230*B*Sin[3*c + (7*d*x)/2] - 105*A*Sin[4*c + (7*d*x)/2] + 230*B*Sin[4*c + (7*d*x)/2] + 15*A*Sin[4*c + (9*d*x)/2] - 20*B*Sin[4*c + (9*d*x)/2] + 15*A*Sin[5*c + (9*d*x)/2] - 20*B*Sin[5*c + (9*d*x)/2] + 5*B*Sin[5*c + (11*d*x)/2] + 5*B*Sin[6*c + (11*d*x)/2]))/(480*a^3*d*(1 + Cos[c + d*x])^3)","B",1
57,1,435,193,0.9234167,"\int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^4*(A + B*Cos[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(-600 d x (6 A-13 B) \cos \left(c+\frac{d x}{2}\right)-600 d x (6 A-13 B) \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)\right)}{480 a^3 d (\cos (c+d x)+1)^3}","\frac{8 (9 A-19 B) \sin (c+d x)}{15 a^3 d}+\frac{4 (9 A-19 B) \sin (c+d x) \cos ^2(c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(6 A-13 B) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{x (6 A-13 B)}{2 a^3}+\frac{(A-B) \sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(6 A-11 B) \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*A - 13*B)*d*x*Cos[(d*x)/2] - 600*(6*A - 13*B)*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] - 1800*A*d*x*Cos[2*c + (3*d*x)/2] + 3900*B*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] - 360*A*d*x*Cos[3*c + (5*d*x)/2] + 780*B*d*x*Cos[3*c + (5*d*x)/2] + 7020*A*Sin[(d*x)/2] - 12760*B*Sin[(d*x)/2] - 4500*A*Sin[c + (d*x)/2] + 7560*B*Sin[c + (d*x)/2] + 4860*A*Sin[c + (3*d*x)/2] - 9230*B*Sin[c + (3*d*x)/2] - 900*A*Sin[2*c + (3*d*x)/2] + 930*B*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] + 300*A*Sin[3*c + (5*d*x)/2] - 750*B*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] + 60*A*Sin[4*c + (7*d*x)/2] - 105*B*Sin[4*c + (7*d*x)/2] + 15*B*Sin[4*c + (9*d*x)/2] + 15*B*Sin[5*c + (9*d*x)/2]))/(480*a^3*d*(1 + Cos[c + d*x])^3)","B",1
58,1,361,147,0.9631275,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[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(300 d x (A-3 B) \cos \left(c+\frac{d x}{2}\right)+300 d x (A-3 B) \cos \left(\frac{d x}{2}\right)+540 A \sin \left(c+\frac{d x}{2}\right)-460 A \sin \left(c+\frac{3 d x}{2}\right)+180 A \sin \left(2 c+\frac{3 d x}{2}\right)-128 A \sin \left(2 c+\frac{5 d x}{2}\right)+150 A d x \cos \left(c+\frac{3 d x}{2}\right)+150 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+30 A d x \cos \left(2 c+\frac{5 d x}{2}\right)+30 A d x \cos \left(3 c+\frac{5 d x}{2}\right)-740 A \sin \left(\frac{d x}{2}\right)-1125 B \sin \left(c+\frac{d x}{2}\right)+1215 B \sin \left(c+\frac{3 d x}{2}\right)-225 B \sin \left(2 c+\frac{3 d x}{2}\right)+363 B \sin \left(2 c+\frac{5 d x}{2}\right)+75 B \sin \left(3 c+\frac{5 d x}{2}\right)+15 B \sin \left(3 c+\frac{7 d x}{2}\right)+15 B \sin \left(4 c+\frac{7 d x}{2}\right)-450 B d x \cos \left(c+\frac{3 d x}{2}\right)-450 B d x \cos \left(2 c+\frac{3 d x}{2}\right)-90 B d x \cos \left(2 c+\frac{5 d x}{2}\right)-90 B d x \cos \left(3 c+\frac{5 d x}{2}\right)+1755 B \sin \left(\frac{d x}{2}\right)\right)}{120 a^3 d (\cos (c+d x)+1)^3}","-\frac{(7 A-27 B) \sin (c+d x)}{15 a^3 d}-\frac{(A-3 B) \sin (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{x (A-3 B)}{a^3}+\frac{(A-B) \sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(4 A-9 B) \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*(A - 3*B)*d*x*Cos[(d*x)/2] + 300*(A - 3*B)*d*x*Cos[c + (d*x)/2] + 150*A*d*x*Cos[c + (3*d*x)/2] - 450*B*d*x*Cos[c + (3*d*x)/2] + 150*A*d*x*Cos[2*c + (3*d*x)/2] - 450*B*d*x*Cos[2*c + (3*d*x)/2] + 30*A*d*x*Cos[2*c + (5*d*x)/2] - 90*B*d*x*Cos[2*c + (5*d*x)/2] + 30*A*d*x*Cos[3*c + (5*d*x)/2] - 90*B*d*x*Cos[3*c + (5*d*x)/2] - 740*A*Sin[(d*x)/2] + 1755*B*Sin[(d*x)/2] + 540*A*Sin[c + (d*x)/2] - 1125*B*Sin[c + (d*x)/2] - 460*A*Sin[c + (3*d*x)/2] + 1215*B*Sin[c + (3*d*x)/2] + 180*A*Sin[2*c + (3*d*x)/2] - 225*B*Sin[2*c + (3*d*x)/2] - 128*A*Sin[2*c + (5*d*x)/2] + 363*B*Sin[2*c + (5*d*x)/2] + 75*B*Sin[3*c + (5*d*x)/2] + 15*B*Sin[3*c + (7*d*x)/2] + 15*B*Sin[4*c + (7*d*x)/2]))/(120*a^3*d*(1 + Cos[c + d*x])^3)","B",1
59,1,241,116,0.6242238,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(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 A \sin \left(c+\frac{d x}{2}\right)+40 A \sin \left(c+\frac{3 d x}{2}\right)-30 A \sin \left(2 c+\frac{3 d x}{2}\right)+14 A \sin \left(2 c+\frac{5 d x}{2}\right)+80 A \sin \left(\frac{d x}{2}\right)+270 B \sin \left(c+\frac{d x}{2}\right)-230 B \sin \left(c+\frac{3 d x}{2}\right)+90 B \sin \left(2 c+\frac{3 d x}{2}\right)-64 B \sin \left(2 c+\frac{5 d x}{2}\right)+150 B d x \cos \left(c+\frac{d x}{2}\right)+75 B d x \cos \left(c+\frac{3 d x}{2}\right)+75 B d x \cos \left(2 c+\frac{3 d x}{2}\right)+15 B d x \cos \left(2 c+\frac{5 d x}{2}\right)+15 B d x \cos \left(3 c+\frac{5 d x}{2}\right)-370 B \sin \left(\frac{d x}{2}\right)+150 B d x \cos \left(\frac{d x}{2}\right)\right)}{480 a^3 d}","\frac{(4 A-29 B) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{B x}{a^3}+\frac{(A-B) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(2 A-7 B) \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*B*d*x*Cos[(d*x)/2] + 150*B*d*x*Cos[c + (d*x)/2] + 75*B*d*x*Cos[c + (3*d*x)/2] + 75*B*d*x*Cos[2*c + (3*d*x)/2] + 15*B*d*x*Cos[2*c + (5*d*x)/2] + 15*B*d*x*Cos[3*c + (5*d*x)/2] + 80*A*Sin[(d*x)/2] - 370*B*Sin[(d*x)/2] - 60*A*Sin[c + (d*x)/2] + 270*B*Sin[c + (d*x)/2] + 40*A*Sin[c + (3*d*x)/2] - 230*B*Sin[c + (3*d*x)/2] - 30*A*Sin[2*c + (3*d*x)/2] + 90*B*Sin[2*c + (3*d*x)/2] + 14*A*Sin[2*c + (5*d*x)/2] - 64*B*Sin[2*c + (5*d*x)/2]))/(480*a^3*d)","B",1
60,1,135,102,0.3634537,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[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 (A+2 B) \sin \left(c+\frac{d x}{2}\right)+5 (3 A+8 B) \sin \left(\frac{d x}{2}\right)+15 A \sin \left(c+\frac{3 d x}{2}\right)+3 A \sin \left(2 c+\frac{5 d x}{2}\right)+20 B \sin \left(c+\frac{3 d x}{2}\right)-15 B \sin \left(2 c+\frac{3 d x}{2}\right)+7 B \sin \left(2 c+\frac{5 d x}{2}\right)\right)}{30 a^3 d (\cos (c+d x)+1)^3}","\frac{(3 A+7 B) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(3 A-8 B) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(5*(3*A + 8*B)*Sin[(d*x)/2] - 15*(A + 2*B)*Sin[c + (d*x)/2] + 15*A*Sin[c + (3*d*x)/2] + 20*B*Sin[c + (3*d*x)/2] - 15*B*Sin[2*c + (3*d*x)/2] + 3*A*Sin[2*c + (5*d*x)/2] + 7*B*Sin[2*c + (5*d*x)/2]))/(30*a^3*d*(1 + Cos[c + d*x])^3)","A",1
61,1,96,102,0.2932639,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[(A + B*Cos[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 A+3 B) \left(5 \sin \left(c+\frac{3 d x}{2}\right)+\sin \left(2 c+\frac{5 d x}{2}\right)\right)+5 (4 A+3 B) \sin \left(\frac{d x}{2}\right)-15 B \sin \left(c+\frac{d x}{2}\right)\right)}{30 a^3 d (\cos (c+d x)+1)^3}","\frac{(2 A+3 B) \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(2 A+3 B) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}+\frac{(A-B) \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)*Sin[(d*x)/2] - 15*B*Sin[c + (d*x)/2] + (2*A + 3*B)*(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
62,1,197,117,1.0391605,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[((A + B*Cos[c + d*x])*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(-5 (29 A-4 B) \sin \left(\frac{d x}{2}\right)+75 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)\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{2 (11 A-B) \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) \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B) \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 - 4*B)*Sin[(d*x)/2] + 75*A*Sin[c + (d*x)/2] - 95*A*Sin[c + (3*d*x)/2] + 10*B*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]))/(30*a^3*d*(1 + Cos[c + d*x])^3)","A",1
63,1,482,145,3.2788587,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^3,x]","\frac{960 (3 A-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)+\sec \left(\frac{c}{2}\right) \sec (c) \cos \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(-5 (51 A-32 B) \sin \left(\frac{d x}{2}\right)+(567 A-167 B) \sin \left(\frac{3 d x}{2}\right)-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)+170 B \sin \left(c-\frac{d x}{2}\right)-170 B \sin \left(c+\frac{d x}{2}\right)+160 B \sin \left(2 c+\frac{d x}{2}\right)+75 B \sin \left(c+\frac{3 d x}{2}\right)-167 B \sin \left(2 c+\frac{3 d x}{2}\right)+75 B \sin \left(3 c+\frac{3 d x}{2}\right)-95 B \sin \left(c+\frac{5 d x}{2}\right)+15 B \sin \left(2 c+\frac{5 d x}{2}\right)-95 B \sin \left(3 c+\frac{5 d x}{2}\right)+15 B \sin \left(4 c+\frac{5 d x}{2}\right)-22 B \sin \left(2 c+\frac{7 d x}{2}\right)-22 B \sin \left(4 c+\frac{7 d x}{2}\right)\right)}{120 a^3 d (\cos (c+d x)+1)^3}","\frac{2 (36 A-11 B) \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) \tan (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B) \tan (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"(960*(3*A - 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]*Sec[c]*Sec[c + d*x]*(-5*(51*A - 32*B)*Sin[(d*x)/2] + (567*A - 167*B)*Sin[(3*d*x)/2] - 600*A*Sin[c - (d*x)/2] + 170*B*Sin[c - (d*x)/2] + 375*A*Sin[c + (d*x)/2] - 170*B*Sin[c + (d*x)/2] - 480*A*Sin[2*c + (d*x)/2] + 160*B*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] - 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] + 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] - 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] + 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]))/(120*a^3*d*(1 + Cos[c + d*x])^3)","B",1
64,1,610,196,5.2235319,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^3,x]","-\frac{1920 (13 A-6 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)+\sec \left(\frac{c}{2}\right) \sec (c) \cos \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left((870 B-1235 A) \sin \left(\frac{d x}{2}\right)+5 (761 A-366 B) \sin \left(\frac{3 d x}{2}\right)-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)+2094 B \sin \left(c-\frac{d x}{2}\right)-1314 B \sin \left(c+\frac{d x}{2}\right)+1650 B \sin \left(2 c+\frac{d x}{2}\right)+450 B \sin \left(c+\frac{3 d x}{2}\right)-1230 B \sin \left(2 c+\frac{3 d x}{2}\right)+1050 B \sin \left(3 c+\frac{3 d x}{2}\right)-1278 B \sin \left(c+\frac{5 d x}{2}\right)+90 B \sin \left(2 c+\frac{5 d x}{2}\right)-918 B \sin \left(3 c+\frac{5 d x}{2}\right)+450 B \sin \left(4 c+\frac{5 d x}{2}\right)-630 B \sin \left(2 c+\frac{7 d x}{2}\right)-60 B \sin \left(3 c+\frac{7 d x}{2}\right)-480 B \sin \left(4 c+\frac{7 d x}{2}\right)+90 B \sin \left(5 c+\frac{7 d x}{2}\right)-144 B \sin \left(3 c+\frac{9 d x}{2}\right)-30 B \sin \left(4 c+\frac{9 d x}{2}\right)-114 B \sin \left(5 c+\frac{9 d x}{2}\right)\right)}{480 a^3 d (\cos (c+d x)+1)^3}","-\frac{8 (19 A-9 B) \tan (c+d x)}{15 a^3 d}+\frac{(13 A-6 B) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{(13 A-6 B) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{4 (19 A-9 B) \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) \tan (c+d x) \sec (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B) \tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"-1/480*(1920*(13*A - 6*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]*Sec[c]*Sec[c + d*x]^2*((-1235*A + 870*B)*Sin[(d*x)/2] + 5*(761*A - 366*B)*Sin[(3*d*x)/2] - 4329*A*Sin[c - (d*x)/2] + 2094*B*Sin[c - (d*x)/2] + 1989*A*Sin[c + (d*x)/2] - 1314*B*Sin[c + (d*x)/2] - 3575*A*Sin[2*c + (d*x)/2] + 1650*B*Sin[2*c + (d*x)/2] - 475*A*Sin[c + (3*d*x)/2] + 450*B*Sin[c + (3*d*x)/2] + 2005*A*Sin[2*c + (3*d*x)/2] - 1230*B*Sin[2*c + (3*d*x)/2] - 2275*A*Sin[3*c + (3*d*x)/2] + 1050*B*Sin[3*c + (3*d*x)/2] + 2673*A*Sin[c + (5*d*x)/2] - 1278*B*Sin[c + (5*d*x)/2] + 105*A*Sin[2*c + (5*d*x)/2] + 90*B*Sin[2*c + (5*d*x)/2] + 1593*A*Sin[3*c + (5*d*x)/2] - 918*B*Sin[3*c + (5*d*x)/2] - 975*A*Sin[4*c + (5*d*x)/2] + 450*B*Sin[4*c + (5*d*x)/2] + 1325*A*Sin[2*c + (7*d*x)/2] - 630*B*Sin[2*c + (7*d*x)/2] + 255*A*Sin[3*c + (7*d*x)/2] - 60*B*Sin[3*c + (7*d*x)/2] + 875*A*Sin[4*c + (7*d*x)/2] - 480*B*Sin[4*c + (7*d*x)/2] - 195*A*Sin[5*c + (7*d*x)/2] + 90*B*Sin[5*c + (7*d*x)/2] + 304*A*Sin[3*c + (9*d*x)/2] - 144*B*Sin[3*c + (9*d*x)/2] + 90*A*Sin[4*c + (9*d*x)/2] - 30*B*Sin[4*c + (9*d*x)/2] + 214*A*Sin[5*c + (9*d*x)/2] - 114*B*Sin[5*c + (9*d*x)/2]))/(a^3*d*(1 + Cos[c + d*x])^3)","B",1
65,1,555,229,1.4236617,"\int \frac{\cos ^5(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^5*(A + B*Cos[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(-14700 d x (8 A-21 B) \cos \left(c+\frac{d x}{2}\right)-14700 d x (8 A-21 B) \cos \left(\frac{d x}{2}\right)-184520 A \sin \left(c+\frac{d x}{2}\right)+184464 A \sin \left(c+\frac{3 d x}{2}\right)-72240 A \sin \left(2 c+\frac{3 d x}{2}\right)+77168 A \sin \left(2 c+\frac{5 d x}{2}\right)-8400 A \sin \left(3 c+\frac{5 d x}{2}\right)+15164 A \sin \left(3 c+\frac{7 d x}{2}\right)+2940 A \sin \left(4 c+\frac{7 d x}{2}\right)+420 A \sin \left(4 c+\frac{9 d x}{2}\right)+420 A \sin \left(5 c+\frac{9 d x}{2}\right)-70560 A d x \cos \left(c+\frac{3 d x}{2}\right)-70560 A d x \cos \left(2 c+\frac{3 d x}{2}\right)-23520 A d x \cos \left(2 c+\frac{5 d x}{2}\right)-23520 A d x \cos \left(3 c+\frac{5 d x}{2}\right)-3360 A d x \cos \left(3 c+\frac{7 d x}{2}\right)-3360 A d x \cos \left(4 c+\frac{7 d x}{2}\right)+243320 A \sin \left(\frac{d x}{2}\right)+386190 B \sin \left(c+\frac{d x}{2}\right)-422478 B \sin \left(c+\frac{3 d x}{2}\right)+132930 B \sin \left(2 c+\frac{3 d x}{2}\right)-181461 B \sin \left(2 c+\frac{5 d x}{2}\right)+3675 B \sin \left(3 c+\frac{5 d x}{2}\right)-36003 B \sin \left(3 c+\frac{7 d x}{2}\right)-9555 B \sin \left(4 c+\frac{7 d x}{2}\right)-945 B \sin \left(4 c+\frac{9 d x}{2}\right)-945 B \sin \left(5 c+\frac{9 d x}{2}\right)+105 B \sin \left(5 c+\frac{11 d x}{2}\right)+105 B \sin \left(6 c+\frac{11 d x}{2}\right)+185220 B d x \cos \left(c+\frac{3 d x}{2}\right)+185220 B d x \cos \left(2 c+\frac{3 d x}{2}\right)+61740 B d x \cos \left(2 c+\frac{5 d x}{2}\right)+61740 B d x \cos \left(3 c+\frac{5 d x}{2}\right)+8820 B d x \cos \left(3 c+\frac{7 d x}{2}\right)+8820 B d x \cos \left(4 c+\frac{7 d x}{2}\right)-539490 B \sin \left(\frac{d x}{2}\right)\right)}{6720 a^4 d (\cos (c+d x)+1)^4}","\frac{8 (83 A-216 B) \sin (c+d x)}{105 a^4 d}+\frac{(52 A-129 B) \sin (c+d x) \cos ^3(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{4 (83 A-216 B) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(8 A-21 B) \sin (c+d x) \cos (c+d x)}{2 a^4 d}-\frac{x (8 A-21 B)}{2 a^4}+\frac{(A-B) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(A-2 B) \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*(8*A - 21*B)*d*x*Cos[(d*x)/2] - 14700*(8*A - 21*B)*d*x*Cos[c + (d*x)/2] - 70560*A*d*x*Cos[c + (3*d*x)/2] + 185220*B*d*x*Cos[c + (3*d*x)/2] - 70560*A*d*x*Cos[2*c + (3*d*x)/2] + 185220*B*d*x*Cos[2*c + (3*d*x)/2] - 23520*A*d*x*Cos[2*c + (5*d*x)/2] + 61740*B*d*x*Cos[2*c + (5*d*x)/2] - 23520*A*d*x*Cos[3*c + (5*d*x)/2] + 61740*B*d*x*Cos[3*c + (5*d*x)/2] - 3360*A*d*x*Cos[3*c + (7*d*x)/2] + 8820*B*d*x*Cos[3*c + (7*d*x)/2] - 3360*A*d*x*Cos[4*c + (7*d*x)/2] + 8820*B*d*x*Cos[4*c + (7*d*x)/2] + 243320*A*Sin[(d*x)/2] - 539490*B*Sin[(d*x)/2] - 184520*A*Sin[c + (d*x)/2] + 386190*B*Sin[c + (d*x)/2] + 184464*A*Sin[c + (3*d*x)/2] - 422478*B*Sin[c + (3*d*x)/2] - 72240*A*Sin[2*c + (3*d*x)/2] + 132930*B*Sin[2*c + (3*d*x)/2] + 77168*A*Sin[2*c + (5*d*x)/2] - 181461*B*Sin[2*c + (5*d*x)/2] - 8400*A*Sin[3*c + (5*d*x)/2] + 3675*B*Sin[3*c + (5*d*x)/2] + 15164*A*Sin[3*c + (7*d*x)/2] - 36003*B*Sin[3*c + (7*d*x)/2] + 2940*A*Sin[4*c + (7*d*x)/2] - 9555*B*Sin[4*c + (7*d*x)/2] + 420*A*Sin[4*c + (9*d*x)/2] - 945*B*Sin[4*c + (9*d*x)/2] + 420*A*Sin[5*c + (9*d*x)/2] - 945*B*Sin[5*c + (9*d*x)/2] + 105*B*Sin[5*c + (11*d*x)/2] + 105*B*Sin[6*c + (11*d*x)/2]))/(6720*a^4*d*(1 + Cos[c + d*x])^4)","B",1
66,1,481,185,0.9779571,"\int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^4*(A + B*Cos[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(7350 d x (A-4 B) \cos \left(c+\frac{d x}{2}\right)+7350 d x (A-4 B) \cos \left(\frac{d x}{2}\right)+16520 A \sin \left(c+\frac{d x}{2}\right)-14280 A \sin \left(c+\frac{3 d x}{2}\right)+7560 A \sin \left(2 c+\frac{3 d x}{2}\right)-5600 A \sin \left(2 c+\frac{5 d x}{2}\right)+1680 A \sin \left(3 c+\frac{5 d x}{2}\right)-1040 A \sin \left(3 c+\frac{7 d x}{2}\right)+4410 A d x \cos \left(c+\frac{3 d x}{2}\right)+4410 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+1470 A d x \cos \left(2 c+\frac{5 d x}{2}\right)+1470 A d x \cos \left(3 c+\frac{5 d x}{2}\right)+210 A d x \cos \left(3 c+\frac{7 d x}{2}\right)+210 A d x \cos \left(4 c+\frac{7 d x}{2}\right)-19880 A \sin \left(\frac{d x}{2}\right)-46130 B \sin \left(c+\frac{d x}{2}\right)+46116 B \sin \left(c+\frac{3 d x}{2}\right)-18060 B \sin \left(2 c+\frac{3 d x}{2}\right)+19292 B \sin \left(2 c+\frac{5 d x}{2}\right)-2100 B \sin \left(3 c+\frac{5 d x}{2}\right)+3791 B \sin \left(3 c+\frac{7 d x}{2}\right)+735 B \sin \left(4 c+\frac{7 d x}{2}\right)+105 B \sin \left(4 c+\frac{9 d x}{2}\right)+105 B \sin \left(5 c+\frac{9 d x}{2}\right)-17640 B d x \cos \left(c+\frac{3 d x}{2}\right)-17640 B d x \cos \left(2 c+\frac{3 d x}{2}\right)-5880 B d x \cos \left(2 c+\frac{5 d x}{2}\right)-5880 B d x \cos \left(3 c+\frac{5 d x}{2}\right)-840 B d x \cos \left(3 c+\frac{7 d x}{2}\right)-840 B d x \cos \left(4 c+\frac{7 d x}{2}\right)+60830 B \sin \left(\frac{d x}{2}\right)\right)}{1680 a^4 d (\cos (c+d x)+1)^4}","-\frac{(55 A-244 B) \sin (c+d x)}{105 a^4 d}+\frac{(25 A-88 B) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(A-4 B) \sin (c+d x)}{a^4 d (\cos (c+d x)+1)}+\frac{x (A-4 B)}{a^4}+\frac{(A-B) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(5 A-12 B) \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*(A - 4*B)*d*x*Cos[(d*x)/2] + 7350*(A - 4*B)*d*x*Cos[c + (d*x)/2] + 4410*A*d*x*Cos[c + (3*d*x)/2] - 17640*B*d*x*Cos[c + (3*d*x)/2] + 4410*A*d*x*Cos[2*c + (3*d*x)/2] - 17640*B*d*x*Cos[2*c + (3*d*x)/2] + 1470*A*d*x*Cos[2*c + (5*d*x)/2] - 5880*B*d*x*Cos[2*c + (5*d*x)/2] + 1470*A*d*x*Cos[3*c + (5*d*x)/2] - 5880*B*d*x*Cos[3*c + (5*d*x)/2] + 210*A*d*x*Cos[3*c + (7*d*x)/2] - 840*B*d*x*Cos[3*c + (7*d*x)/2] + 210*A*d*x*Cos[4*c + (7*d*x)/2] - 840*B*d*x*Cos[4*c + (7*d*x)/2] - 19880*A*Sin[(d*x)/2] + 60830*B*Sin[(d*x)/2] + 16520*A*Sin[c + (d*x)/2] - 46130*B*Sin[c + (d*x)/2] - 14280*A*Sin[c + (3*d*x)/2] + 46116*B*Sin[c + (3*d*x)/2] + 7560*A*Sin[2*c + (3*d*x)/2] - 18060*B*Sin[2*c + (3*d*x)/2] - 5600*A*Sin[2*c + (5*d*x)/2] + 19292*B*Sin[2*c + (5*d*x)/2] + 1680*A*Sin[3*c + (5*d*x)/2] - 2100*B*Sin[3*c + (5*d*x)/2] - 1040*A*Sin[3*c + (7*d*x)/2] + 3791*B*Sin[3*c + (7*d*x)/2] + 735*B*Sin[4*c + (7*d*x)/2] + 105*B*Sin[4*c + (9*d*x)/2] + 105*B*Sin[5*c + (9*d*x)/2]))/(1680*a^4*d*(1 + Cos[c + d*x])^4)","B",1
67,1,329,154,0.8291378,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/(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(-1260 A \sin \left(c+\frac{d x}{2}\right)+882 A \sin \left(c+\frac{3 d x}{2}\right)-630 A \sin \left(2 c+\frac{3 d x}{2}\right)+294 A \sin \left(2 c+\frac{5 d x}{2}\right)-210 A \sin \left(3 c+\frac{5 d x}{2}\right)+72 A \sin \left(3 c+\frac{7 d x}{2}\right)+1260 A \sin \left(\frac{d x}{2}\right)+8260 B \sin \left(c+\frac{d x}{2}\right)-7140 B \sin \left(c+\frac{3 d x}{2}\right)+3780 B \sin \left(2 c+\frac{3 d x}{2}\right)-2800 B \sin \left(2 c+\frac{5 d x}{2}\right)+840 B \sin \left(3 c+\frac{5 d x}{2}\right)-520 B \sin \left(3 c+\frac{7 d x}{2}\right)+3675 B d x \cos \left(c+\frac{d x}{2}\right)+2205 B d x \cos \left(c+\frac{3 d x}{2}\right)+2205 B d x \cos \left(2 c+\frac{3 d x}{2}\right)+735 B d x \cos \left(2 c+\frac{5 d x}{2}\right)+735 B d x \cos \left(3 c+\frac{5 d x}{2}\right)+105 B d x \cos \left(3 c+\frac{7 d x}{2}\right)+105 B d x \cos \left(4 c+\frac{7 d x}{2}\right)-9940 B \sin \left(\frac{d x}{2}\right)+3675 B d x \cos \left(\frac{d x}{2}\right)\right)}{13440 a^4 d}","\frac{(12 A-215 B) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(6 A-55 B) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{B x}{a^4}+\frac{(A-B) \sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(3 A-10 B) \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*B*d*x*Cos[(d*x)/2] + 3675*B*d*x*Cos[c + (d*x)/2] + 2205*B*d*x*Cos[c + (3*d*x)/2] + 2205*B*d*x*Cos[2*c + (3*d*x)/2] + 735*B*d*x*Cos[2*c + (5*d*x)/2] + 735*B*d*x*Cos[3*c + (5*d*x)/2] + 105*B*d*x*Cos[3*c + (7*d*x)/2] + 105*B*d*x*Cos[4*c + (7*d*x)/2] + 1260*A*Sin[(d*x)/2] - 9940*B*Sin[(d*x)/2] - 1260*A*Sin[c + (d*x)/2] + 8260*B*Sin[c + (d*x)/2] + 882*A*Sin[c + (3*d*x)/2] - 7140*B*Sin[c + (3*d*x)/2] - 630*A*Sin[2*c + (3*d*x)/2] + 3780*B*Sin[2*c + (3*d*x)/2] + 294*A*Sin[2*c + (5*d*x)/2] - 2800*B*Sin[2*c + (5*d*x)/2] - 210*A*Sin[3*c + (5*d*x)/2] + 840*B*Sin[3*c + (5*d*x)/2] + 72*A*Sin[3*c + (7*d*x)/2] - 520*B*Sin[3*c + (7*d*x)/2]))/(13440*a^4*d)","B",1
68,1,193,136,0.5003838,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[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(-35 (5 A+18 B) \sin \left(c+\frac{d x}{2}\right)+70 (4 A+9 B) \sin \left(\frac{d x}{2}\right)+168 A \sin \left(c+\frac{3 d x}{2}\right)-105 A \sin \left(2 c+\frac{3 d x}{2}\right)+91 A \sin \left(2 c+\frac{5 d x}{2}\right)+13 A \sin \left(3 c+\frac{7 d x}{2}\right)+441 B \sin \left(c+\frac{3 d x}{2}\right)-315 B \sin \left(2 c+\frac{3 d x}{2}\right)+147 B \sin \left(2 c+\frac{5 d x}{2}\right)-105 B \sin \left(3 c+\frac{5 d x}{2}\right)+36 B \sin \left(3 c+\frac{7 d x}{2}\right)\right)}{420 a^4 d (\cos (c+d x)+1)^4}","\frac{(13 A+36 B) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{2 (A+27 B) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{(A-B) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{(A-8 B) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(70*(4*A + 9*B)*Sin[(d*x)/2] - 35*(5*A + 18*B)*Sin[c + (d*x)/2] + 168*A*Sin[c + (3*d*x)/2] + 441*B*Sin[c + (3*d*x)/2] - 105*A*Sin[2*c + (3*d*x)/2] - 315*B*Sin[2*c + (3*d*x)/2] + 91*A*Sin[2*c + (5*d*x)/2] + 147*B*Sin[2*c + (5*d*x)/2] - 105*B*Sin[3*c + (5*d*x)/2] + 13*A*Sin[3*c + (7*d*x)/2] + 36*B*Sin[3*c + (7*d*x)/2]))/(420*a^4*d*(1 + Cos[c + d*x])^4)","A",1
69,1,163,138,0.4264628,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[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(-35 (4 A+5 B) \sin \left(c+\frac{d x}{2}\right)+140 (A+2 B) \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)\right)}{420 a^4 d (\cos (c+d x)+1)^4}","\frac{(8 A+13 B) \sin (c+d x)}{105 d \left(a^4 \cos (c+d x)+a^4\right)}+\frac{(8 A+13 B) \sin (c+d x)}{105 d \left(a^2 \cos (c+d x)+a^2\right)^2}+\frac{(4 A-11 B) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A-B) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(140*(A + 2*B)*Sin[(d*x)/2] - 35*(4*A + 5*B)*Sin[c + (d*x)/2] + 168*A*Sin[c + (3*d*x)/2] + 168*B*Sin[c + (3*d*x)/2] - 105*B*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] + 8*A*Sin[3*c + (7*d*x)/2] + 13*B*Sin[3*c + (7*d*x)/2]))/(420*a^4*d*(1 + Cos[c + d*x])^4)","A",1
70,1,109,138,0.37264,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^4} \, dx","Integrate[(A + B*Cos[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((3 A+4 B) \left(21 \sin \left(c+\frac{3 d x}{2}\right)+7 \sin \left(2 c+\frac{5 d x}{2}\right)+\sin \left(3 c+\frac{7 d x}{2}\right)\right)+35 (3 A+2 B) \sin \left(\frac{d x}{2}\right)-70 B \sin \left(c+\frac{d x}{2}\right)\right)}{210 a^4 d (\cos (c+d x)+1)^4}","\frac{2 (3 A+4 B) \sin (c+d x)}{105 d \left(a^4 \cos (c+d x)+a^4\right)}+\frac{2 (3 A+4 B) \sin (c+d x)}{105 d \left(a^2 \cos (c+d x)+a^2\right)^2}+\frac{(3 A+4 B) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac{(A-B) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(35*(3*A + 2*B)*Sin[(d*x)/2] - 70*B*Sin[c + (d*x)/2] + (3*A + 4*B)*(21*Sin[c + (3*d*x)/2] + 7*Sin[2*c + (5*d*x)/2] + Sin[3*c + (7*d*x)/2])))/(210*a^4*d*(1 + Cos[c + d*x])^4)","A",1
71,1,239,147,1.5942633,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(a+a \cos (c+d x))^4} \, dx","Integrate[((A + B*Cos[c + d*x])*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 (49 A-3 B) \sin \left(\frac{d x}{2}\right)+2170 A \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)\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{2 (80 A-3 B) \sin (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(55 A-6 B) \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) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A-B) \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 - 3*B)*Sin[(d*x)/2] + 2170*A*Sin[c + (d*x)/2] - 2625*A*Sin[c + (3*d*x)/2] + 126*B*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] + 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]))/(420*a^4*d*(1 + Cos[c + d*x])^4)","A",1
72,1,595,175,5.7261517,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^4,x]","\frac{26880 (4 A-B) \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)+\sec \left(\frac{c}{2}\right) \sec (c) \cos \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(-245 (44 A-17 B) \sin \left(\frac{d x}{2}\right)+7 (2684 A-635 B) \sin \left(\frac{3 d x}{2}\right)-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)+4795 B \sin \left(c-\frac{d x}{2}\right)-4795 B \sin \left(c+\frac{d x}{2}\right)+4165 B \sin \left(2 c+\frac{d x}{2}\right)+2275 B \sin \left(c+\frac{3 d x}{2}\right)-4445 B \sin \left(2 c+\frac{3 d x}{2}\right)+2275 B \sin \left(3 c+\frac{3 d x}{2}\right)-2785 B \sin \left(c+\frac{5 d x}{2}\right)+735 B \sin \left(2 c+\frac{5 d x}{2}\right)-2785 B \sin \left(3 c+\frac{5 d x}{2}\right)+735 B \sin \left(4 c+\frac{5 d x}{2}\right)-1015 B \sin \left(2 c+\frac{7 d x}{2}\right)+105 B \sin \left(3 c+\frac{7 d x}{2}\right)-1015 B \sin \left(4 c+\frac{7 d x}{2}\right)+105 B \sin \left(5 c+\frac{7 d x}{2}\right)-160 B \sin \left(3 c+\frac{9 d x}{2}\right)-160 B \sin \left(5 c+\frac{9 d x}{2}\right)\right)}{1680 a^4 d (\cos (c+d x)+1)^4}","\frac{8 (83 A-20 B) \tan (c+d x)}{105 a^4 d}-\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{(88 A-25 B) \tan (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(12 A-5 B) \tan (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A-B) \tan (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"(26880*(4*A - B)*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]*Sec[c]*Sec[c + d*x]*(-245*(44*A - 17*B)*Sin[(d*x)/2] + 7*(2684*A - 635*B)*Sin[(3*d*x)/2] - 20524*A*Sin[c - (d*x)/2] + 4795*B*Sin[c - (d*x)/2] + 14644*A*Sin[c + (d*x)/2] - 4795*B*Sin[c + (d*x)/2] - 16660*A*Sin[2*c + (d*x)/2] + 4165*B*Sin[2*c + (d*x)/2] - 4690*A*Sin[c + (3*d*x)/2] + 2275*B*Sin[c + (3*d*x)/2] + 14378*A*Sin[2*c + (3*d*x)/2] - 4445*B*Sin[2*c + (3*d*x)/2] - 9100*A*Sin[3*c + (3*d*x)/2] + 2275*B*Sin[3*c + (3*d*x)/2] + 11668*A*Sin[c + (5*d*x)/2] - 2785*B*Sin[c + (5*d*x)/2] - 630*A*Sin[2*c + (5*d*x)/2] + 735*B*Sin[2*c + (5*d*x)/2] + 9358*A*Sin[3*c + (5*d*x)/2] - 2785*B*Sin[3*c + (5*d*x)/2] - 2940*A*Sin[4*c + (5*d*x)/2] + 735*B*Sin[4*c + (5*d*x)/2] + 4228*A*Sin[2*c + (7*d*x)/2] - 1015*B*Sin[2*c + (7*d*x)/2] + 315*A*Sin[3*c + (7*d*x)/2] + 105*B*Sin[3*c + (7*d*x)/2] + 3493*A*Sin[4*c + (7*d*x)/2] - 1015*B*Sin[4*c + (7*d*x)/2] - 420*A*Sin[5*c + (7*d*x)/2] + 105*B*Sin[5*c + (7*d*x)/2] + 664*A*Sin[3*c + (9*d*x)/2] - 160*B*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] - 160*B*Sin[5*c + (9*d*x)/2]))/(1680*a^4*d*(1 + Cos[c + d*x])^4)","B",1
73,1,798,232,6.5085146,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^4,x]","-\frac{8 (21 A-8 B) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^4}+\frac{8 (21 A-8 B) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (c+d x) a+a)^4}+\frac{\sec \left(\frac{c}{2}\right) \sec (c) \sec ^2(c+d x) \left(73206 A \sin \left(\frac{d x}{2}\right)-38668 B \sin \left(\frac{d x}{2}\right)-166668 A \sin \left(\frac{3 d x}{2}\right)+64384 B \sin \left(\frac{3 d x}{2}\right)+183162 A \sin \left(c-\frac{d x}{2}\right)-70896 B \sin \left(c-\frac{d x}{2}\right)-100842 A \sin \left(c+\frac{d x}{2}\right)+50316 B \sin \left(c+\frac{d x}{2}\right)+155526 A \sin \left(2 c+\frac{d x}{2}\right)-59248 B \sin \left(2 c+\frac{d x}{2}\right)+37380 A \sin \left(c+\frac{3 d x}{2}\right)-22820 B \sin \left(c+\frac{3 d x}{2}\right)-101148 A \sin \left(2 c+\frac{3 d x}{2}\right)+48004 B \sin \left(2 c+\frac{3 d x}{2}\right)+102900 A \sin \left(3 c+\frac{3 d x}{2}\right)-39200 B \sin \left(3 c+\frac{3 d x}{2}\right)-119364 A \sin \left(c+\frac{5 d x}{2}\right)+46032 B \sin \left(c+\frac{5 d x}{2}\right)+8820 A \sin \left(2 c+\frac{5 d x}{2}\right)-8750 B \sin \left(2 c+\frac{5 d x}{2}\right)-78204 A \sin \left(3 c+\frac{5 d x}{2}\right)+35742 B \sin \left(3 c+\frac{5 d x}{2}\right)+49980 A \sin \left(4 c+\frac{5 d x}{2}\right)-19040 B \sin \left(4 c+\frac{5 d x}{2}\right)-64053 A \sin \left(2 c+\frac{7 d x}{2}\right)+24664 B \sin \left(2 c+\frac{7 d x}{2}\right)-3885 A \sin \left(3 c+\frac{7 d x}{2}\right)-1050 B \sin \left(3 c+\frac{7 d x}{2}\right)-44733 A \sin \left(4 c+\frac{7 d x}{2}\right)+19834 B \sin \left(4 c+\frac{7 d x}{2}\right)+15435 A \sin \left(5 c+\frac{7 d x}{2}\right)-5880 B \sin \left(5 c+\frac{7 d x}{2}\right)-21987 A \sin \left(3 c+\frac{9 d x}{2}\right)+8456 B \sin \left(3 c+\frac{9 d x}{2}\right)-3675 A \sin \left(4 c+\frac{9 d x}{2}\right)+630 B \sin \left(4 c+\frac{9 d x}{2}\right)-16107 A \sin \left(5 c+\frac{9 d x}{2}\right)+6986 B \sin \left(5 c+\frac{9 d x}{2}\right)+2205 A \sin \left(6 c+\frac{9 d x}{2}\right)-840 B \sin \left(6 c+\frac{9 d x}{2}\right)-3456 A \sin \left(4 c+\frac{11 d x}{2}\right)+1328 B \sin \left(4 c+\frac{11 d x}{2}\right)-840 A \sin \left(5 c+\frac{11 d x}{2}\right)+210 B \sin \left(5 c+\frac{11 d x}{2}\right)-2616 A \sin \left(6 c+\frac{11 d x}{2}\right)+1118 B \sin \left(6 c+\frac{11 d x}{2}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{6720 d (\cos (c+d x) a+a)^4}","-\frac{8 (216 A-83 B) \tan (c+d x)}{105 a^4 d}+\frac{(21 A-8 B) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{(21 A-8 B) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{4 (216 A-83 B) \tan (c+d x) \sec (c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(129 A-52 B) \tan (c+d x) \sec (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{(2 A-B) \tan (c+d x) \sec (c+d x)}{5 a d (a \cos (c+d x)+a)^3}-\frac{(A-B) \tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"(-8*(21*A - 8*B)*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 - 8*B)*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] - 38668*B*Sin[(d*x)/2] - 166668*A*Sin[(3*d*x)/2] + 64384*B*Sin[(3*d*x)/2] + 183162*A*Sin[c - (d*x)/2] - 70896*B*Sin[c - (d*x)/2] - 100842*A*Sin[c + (d*x)/2] + 50316*B*Sin[c + (d*x)/2] + 155526*A*Sin[2*c + (d*x)/2] - 59248*B*Sin[2*c + (d*x)/2] + 37380*A*Sin[c + (3*d*x)/2] - 22820*B*Sin[c + (3*d*x)/2] - 101148*A*Sin[2*c + (3*d*x)/2] + 48004*B*Sin[2*c + (3*d*x)/2] + 102900*A*Sin[3*c + (3*d*x)/2] - 39200*B*Sin[3*c + (3*d*x)/2] - 119364*A*Sin[c + (5*d*x)/2] + 46032*B*Sin[c + (5*d*x)/2] + 8820*A*Sin[2*c + (5*d*x)/2] - 8750*B*Sin[2*c + (5*d*x)/2] - 78204*A*Sin[3*c + (5*d*x)/2] + 35742*B*Sin[3*c + (5*d*x)/2] + 49980*A*Sin[4*c + (5*d*x)/2] - 19040*B*Sin[4*c + (5*d*x)/2] - 64053*A*Sin[2*c + (7*d*x)/2] + 24664*B*Sin[2*c + (7*d*x)/2] - 3885*A*Sin[3*c + (7*d*x)/2] - 1050*B*Sin[3*c + (7*d*x)/2] - 44733*A*Sin[4*c + (7*d*x)/2] + 19834*B*Sin[4*c + (7*d*x)/2] + 15435*A*Sin[5*c + (7*d*x)/2] - 5880*B*Sin[5*c + (7*d*x)/2] - 21987*A*Sin[3*c + (9*d*x)/2] + 8456*B*Sin[3*c + (9*d*x)/2] - 3675*A*Sin[4*c + (9*d*x)/2] + 630*B*Sin[4*c + (9*d*x)/2] - 16107*A*Sin[5*c + (9*d*x)/2] + 6986*B*Sin[5*c + (9*d*x)/2] + 2205*A*Sin[6*c + (9*d*x)/2] - 840*B*Sin[6*c + (9*d*x)/2] - 3456*A*Sin[4*c + (11*d*x)/2] + 1328*B*Sin[4*c + (11*d*x)/2] - 840*A*Sin[5*c + (11*d*x)/2] + 210*B*Sin[5*c + (11*d*x)/2] - 2616*A*Sin[6*c + (11*d*x)/2] + 1118*B*Sin[6*c + (11*d*x)/2]))/(6720*d*(a + a*Cos[c + d*x])^4)","B",1
74,1,103,187,0.7045645,"\int \cos ^3(c+d x) \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (94 (9 A+8 B) \cos (c+d x)+4 (54 A+83 B) \cos (2 (c+d x))+90 A \cos (3 (c+d x))+1368 A+80 B \cos (3 (c+d x))+35 B \cos (4 (c+d x))+1321 B)}{1260 d}","\frac{2 a (9 A+8 B) \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{4 (9 A+8 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 a d}-\frac{8 (9 A+8 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{4 a (9 A+8 B) \sin (c+d x)}{45 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a B \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(1368*A + 1321*B + 94*(9*A + 8*B)*Cos[c + d*x] + 4*(54*A + 83*B)*Cos[2*(c + d*x)] + 90*A*Cos[3*(c + d*x)] + 80*B*Cos[3*(c + d*x)] + 35*B*Cos[4*(c + d*x)])*Tan[(c + d*x)/2])/(1260*d)","A",1
75,1,80,144,0.3700539,"\int \cos ^2(c+d x) \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((112 A+141 B) \cos (c+d x)+6 (7 A+6 B) \cos (2 (c+d x))+266 A+15 B \cos (3 (c+d x))+228 B)}{210 d}","\frac{2 (7 A+6 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 a d}-\frac{4 (7 A+6 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 a (7 A+6 B) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a B \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(266*A + 228*B + (112*A + 141*B)*Cos[c + d*x] + 6*(7*A + 6*B)*Cos[2*(c + d*x)] + 15*B*Cos[3*(c + d*x)])*Tan[(c + d*x)/2])/(210*d)","A",1
76,1,64,101,0.2114562,"\int \cos (c+d x) \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]*Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (2 (5 A+4 B) \cos (c+d x)+20 A+3 B \cos (2 (c+d x))+19 B)}{15 d}","\frac{2 (5 A-2 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 a (5 A+7 B) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(20*A + 19*B + 2*(5*A + 4*B)*Cos[c + d*x] + 3*B*Cos[2*(c + d*x)])*Tan[(c + d*x)/2])/(15*d)","A",1
77,1,46,62,0.086744,"\int \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (3 A+B \cos (c+d x)+2 B)}{3 d}","\frac{2 a (3 A+B) \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 B \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}",1,"(2*Sqrt[a*(1 + Cos[c + d*x])]*(3*A + 2*B + B*Cos[c + d*x])*Tan[(c + d*x)/2])/(3*d)","A",1
78,1,66,66,0.0991567,"\int \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec (c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(\sqrt{2} A \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 B \sin \left(\frac{1}{2} (c+d x)\right)\right)}{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 B \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*B*Sin[(c + d*x)/2]))/d","A",1
79,1,85,68,0.2218504,"\int \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*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 A \sin \left(\frac{1}{2} (c+d x)\right)\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 \tan (c+d x)}{d \sqrt{a \cos (c+d x)+a}}",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*Sin[(c + d*x)/2]))/(2*d)","A",1
80,1,101,117,0.8829483,"\int \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{\sqrt{a (\cos (c+d x)+1)} \left(6 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) ((3 A+4 B) \cos (c+d x)+2 A)+3 \sqrt{2} (3 A+4 B) \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)}{24 d}","\frac{a (3 A+4 B) \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (3 A+4 B) \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) \sec (c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*(3*Sqrt[2]*(3*A + 4*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Sec[(c + d*x)/2] + 6*(2*A + (3*A + 4*B)*Cos[c + d*x])*Sec[c + d*x]^2*Tan[(c + d*x)/2]))/(24*d)","A",1
81,1,129,160,2.0141057,"\int \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{\sec ^3(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(\tan \left(\frac{1}{2} (c+d x)\right) (4 (5 A+6 B) \cos (c+d x)+3 (5 A+6 B) \cos (2 (c+d x))+31 A+18 B)+3 \sqrt{2} (5 A+6 B) \cos ^3(c+d x) \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+6 B) \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (5 A+6 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a (5 A+6 B) \tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{a A \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[c + d*x]^3*(3*Sqrt[2]*(5*A + 6*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^3*Sec[(c + d*x)/2] + (31*A + 18*B + 4*(5*A + 6*B)*Cos[c + d*x] + 3*(5*A + 6*B)*Cos[2*(c + d*x)])*Tan[(c + d*x)/2]))/(48*d)","A",1
82,1,125,234,1.0725016,"\int \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((35156 A+34734 B) \cos (c+d x)+8 (1507 A+1743 B) \cos (2 (c+d x))+3740 A \cos (3 (c+d x))+770 A \cos (4 (c+d x))+59158 A+4935 B \cos (3 (c+d x))+1470 B \cos (4 (c+d x))+315 B \cos (5 (c+d x))+55482 B)}{27720 d}","\frac{2 a^2 (11 A+12 B) \sin (c+d x) \cos ^4(c+d x)}{99 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (187 A+168 B) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (187 A+168 B) \sin (c+d x)}{495 d \sqrt{a \cos (c+d x)+a}}+\frac{4 (187 A+168 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 d}-\frac{8 a (187 A+168 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3465 d}+\frac{2 a B \sin (c+d x) \cos ^4(c+d x) \sqrt{a \cos (c+d x)+a}}{11 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(59158*A + 55482*B + (35156*A + 34734*B)*Cos[c + d*x] + 8*(1507*A + 1743*B)*Cos[2*(c + d*x)] + 3740*A*Cos[3*(c + d*x)] + 4935*B*Cos[3*(c + d*x)] + 770*A*Cos[4*(c + d*x)] + 1470*B*Cos[4*(c + d*x)] + 315*B*Cos[5*(c + d*x)])*Tan[(c + d*x)/2])/(27720*d)","A",1
83,1,103,189,0.5990528,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (2 (759 A+799 B) \cos (c+d x)+(468 A+548 B) \cos (2 (c+d x))+90 A \cos (3 (c+d x))+2964 A+170 B \cos (3 (c+d x))+35 B \cos (4 (c+d x))+2689 B)}{1260 d}","\frac{2 a^2 (9 A+10 B) \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (39 A+34 B) \sin (c+d x)}{45 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (39 A+34 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}-\frac{4 a (39 A+34 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 a B \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{9 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(2964*A + 2689*B + 2*(759*A + 799*B)*Cos[c + d*x] + (468*A + 548*B)*Cos[2*(c + d*x)] + 90*A*Cos[3*(c + d*x)] + 170*B*Cos[3*(c + d*x)] + 35*B*Cos[4*(c + d*x)])*Tan[(c + d*x)/2])/(1260*d)","A",1
84,1,81,138,0.3996239,"\int \cos (c+d x) (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((252 A+253 B) \cos (c+d x)+6 (7 A+13 B) \cos (2 (c+d x))+546 A+15 B \cos (3 (c+d x))+494 B)}{210 d}","\frac{8 a^2 (21 A+19 B) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 (7 A-2 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 a (21 A+19 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 B \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*A + 494*B + (252*A + 253*B)*Cos[c + d*x] + 6*(7*A + 13*B)*Cos[2*(c + d*x)] + 15*B*Cos[3*(c + d*x)])*Tan[(c + d*x)/2])/(210*d)","A",1
85,1,65,101,0.20227,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (2 (5 A+9 B) \cos (c+d x)+50 A+3 B \cos (2 (c+d x))+39 B)}{15 d}","\frac{8 a^2 (5 A+3 B) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a (5 A+3 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(50*A + 39*B + 2*(5*A + 9*B)*Cos[c + d*x] + 3*B*Cos[2*(c + d*x)])*Tan[(c + d*x)/2])/(15*d)","A",1
86,1,85,105,0.2179727,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x],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) (3 A+B \cos (c+d x)+5 B)+3 \sqrt{2} A \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{3 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 (3 A+4 B) \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a B \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}",1,"(a*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*A + 5*B + B*Cos[c + d*x])*Sin[(c + d*x)/2]))/(3*d)","A",1
87,1,98,103,0.3417202,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*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) (A+2 B \cos (c+d x))+\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)}{2 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 (A-2 B) \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}+\frac{a A \tan (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]*(Sqrt[2]*(3*A + 2*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x] + 2*(A + 2*B*Cos[c + d*x])*Sin[(c + d*x)/2]))/(2*d)","A",1
88,1,109,119,0.5825059,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*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)+\sqrt{2} (7 A+12 B) \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) \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) \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{a A \tan (c+d x) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{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)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^2 + 2*(2*A + (7*A + 4*B)*Cos[c + d*x])*Sin[(c + d*x)/2]))/(8*d)","A",1
89,1,132,164,1.0103036,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*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) (4 (11 A+6 B) \cos (c+d x)+(33 A+42 B) \cos (2 (c+d x))+7 (7 A+6 B))+3 \sqrt{2} (11 A+14 B) \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) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 (11 A+14 B) \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (7 A+6 B) \tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{a A \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{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)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^3 + (7*(7*A + 6*B) + 4*(11*A + 6*B)*Cos[c + d*x] + (33*A + 42*B)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
90,1,151,209,1.602852,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*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) \cos (c+d x)+4 (75 A+88 B) \cos (2 (c+d x))+225 A \cos (3 (c+d x))+492 A+264 B \cos (3 (c+d x))+352 B)+6 \sqrt{2} (75 A+88 B) \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) \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) \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (9 A+8 B) \tan (c+d x) \sec ^2(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (75 A+88 B) \tan (c+d x) \sec (c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a A \tan (c+d x) \sec ^3(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]^4*(6*Sqrt[2]*(75*A + 88*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^4 + (492*A + 352*B + (1155*A + 1048*B)*Cos[c + d*x] + 4*(75*A + 88*B)*Cos[2*(c + d*x)] + 225*A*Cos[3*(c + d*x)] + 264*B*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(768*d)","A",1
91,1,127,237,1.132566,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((68552 A+69890 B) \cos (c+d x)+16 (1397 A+1625 B) \cos (2 (c+d x))+5720 A \cos (3 (c+d x))+770 A \cos (4 (c+d x))+124366 A+8675 B \cos (3 (c+d x))+2240 B \cos (4 (c+d x))+315 B \cos (5 (c+d x))+114640 B)}{27720 d}","\frac{2 a^3 (209 A+194 B) \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (803 A+710 B) \sin (c+d x)}{495 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (11 A+14 B) \sin (c+d x) \cos ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{99 d}-\frac{4 a^2 (803 A+710 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3465 d}+\frac{2 a (803 A+710 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{1155 d}+\frac{2 a B \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(124366*A + 114640*B + (68552*A + 69890*B)*Cos[c + d*x] + 16*(1397*A + 1625*B)*Cos[2*(c + d*x)] + 5720*A*Cos[3*(c + d*x)] + 8675*B*Cos[3*(c + d*x)] + 770*A*Cos[4*(c + d*x)] + 2240*B*Cos[4*(c + d*x)] + 315*B*Cos[5*(c + d*x)])*Tan[(c + d*x)/2])/(27720*d)","A",1
92,1,105,175,0.7533743,"\int \cos (c+d x) (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((3030 A+3116 B) \cos (c+d x)+8 (90 A+127 B) \cos (2 (c+d x))+90 A \cos (3 (c+d x))+6240 A+260 B \cos (3 (c+d x))+35 B \cos (4 (c+d x))+5653 B)}{1260 d}","\frac{64 a^3 (15 A+13 B) \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (15 A+13 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 (9 A-2 B) \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{63 d}+\frac{2 a (15 A+13 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}+\frac{2 B \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*A + 5653*B + (3030*A + 3116*B)*Cos[c + d*x] + 8*(90*A + 127*B)*Cos[2*(c + d*x)] + 90*A*Cos[3*(c + d*x)] + 260*B*Cos[3*(c + d*x)] + 35*B*Cos[4*(c + d*x)])*Tan[(c + d*x)/2])/(1260*d)","A",1
93,1,83,138,0.3564535,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((392 A+505 B) \cos (c+d x)+6 (7 A+20 B) \cos (2 (c+d x))+1246 A+15 B \cos (3 (c+d x))+1040 B)}{210 d}","\frac{64 a^3 (7 A+5 B) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (7 A+5 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}+\frac{2 a (7 A+5 B) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(1246*A + 1040*B + (392*A + 505*B)*Cos[c + d*x] + 6*(7*A + 20*B)*Cos[2*(c + d*x)] + 15*B*Cos[3*(c + d*x)])*Tan[(c + d*x)/2])/(210*d)","A",1
94,1,104,142,0.4229961,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*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(\sin \left(\frac{1}{2} (c+d x)\right) (2 (5 A+14 B) \cos (c+d x)+80 A+3 B \cos (2 (c+d x))+89 B)+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^{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 (35 A+32 B) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (5 A+8 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 a B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(15*Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + (80*A + 89*B + 2*(5*A + 14*B)*Cos[c + d*x] + 3*B*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(15*d)","A",1
95,1,120,144,0.5628885,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*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) (2 (3 A+8 B) \cos (c+d x)+3 A+B \cos (2 (c+d x))+B)+3 \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)}{6 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 (3 A+14 B) \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (3 A-2 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{a A \tan (c+d x) (a \cos (c+d x)+a)^{3/2}}{d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]*(3*Sqrt[2]*(5*A + 2*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x] + 2*(3*A + B + 2*(3*A + 8*B)*Cos[c + d*x] + B*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(6*d)","A",1
96,1,126,156,0.6751397,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*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(2 \sin \left(\frac{1}{2} (c+d x)\right) ((11 A+4 B) \cos (c+d x)+2 (A+2 B \cos (2 (c+d x))+2 B))+\sqrt{2} (19 A+20 B) \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^{5/2} (19 A+20 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^3 (9 A-4 B) \sin (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (7 A+4 B) \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{a A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^{3/2}}{2 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^2*(Sqrt[2]*(19*A + 20*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^2 + 2*((11*A + 4*B)*Cos[c + d*x] + 2*(A + 2*B + 2*B*Cos[2*(c + d*x)]))*Sin[(c + d*x)/2]))/(8*d)","A",1
97,1,131,164,1.1128303,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*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) \cos (c+d x)+(75 A+66 B) \cos (2 (c+d x))+91 A+66 B)+3 \sqrt{2} (25 A+38 B) \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) \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) \tan (c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (3 A+2 B) \tan (c+d x) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{a A \tan (c+d x) \sec ^2(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*(3*Sqrt[2]*(25*A + 38*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^3 + (91*A + 66*B + 4*(17*A + 6*B)*Cos[c + d*x] + (75*A + 66*B)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
98,1,152,209,1.7985626,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*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) \cos (c+d x)+(652 A+544 B) \cos (2 (c+d x))+489 A \cos (3 (c+d x))+844 A+600 B \cos (3 (c+d x))+544 B)+6 \sqrt{2} (163 A+200 B) \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) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^3 (163 A+200 B) \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (95 A+104 B) \tan (c+d x) \sec (c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (11 A+8 B) \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{24 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^{3/2}}{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)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^4 + (844*A + 544*B + (2203*A + 2056*B)*Cos[c + d*x] + (652*A + 544*B)*Cos[2*(c + d*x)] + 489*A*Cos[3*(c + d*x)] + 600*B*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(768*d)","A",1
99,1,176,254,2.2492009,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^6(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*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) (36 (781 A+650 B) \cos (c+d x)+4 (6509 A+6730 B) \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)+60 \sqrt{2} (283 A+326 B) \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) \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) \tan (c+d x)}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (157 A+170 B) \tan (c+d x) \sec ^2(c+d x)}{240 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (283 A+326 B) \tan (c+d x) \sec (c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (13 A+10 B) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{40 d}+\frac{a A \tan (c+d x) \sec ^4(c+d x) (a \cos (c+d x)+a)^{3/2}}{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)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^5 + (24863*A + 22030*B + 36*(781*A + 650*B)*Cos[c + d*x] + 4*(6509*A + 6730*B)*Cos[2*(c + d*x)] + 5660*A*Cos[3*(c + d*x)] + 6520*B*Cos[3*(c + d*x)] + 4245*A*Cos[4*(c + d*x)] + 4890*B*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]))/(15360*d)","A",1
100,1,111,202,0.7219646,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/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) ((169 B-28 A) \cos (c+d x)+6 (7 A-B) \cos (2 (c+d x))+406 A+15 B \cos (3 (c+d x))-178 B)-420 (A-B) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{210 d \sqrt{a (\cos (c+d x)+1)}}","\frac{2 (7 A-B) \sin (c+d x) \cos ^2(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}-\frac{2 (7 A-31 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 a d}+\frac{4 (49 A-37 B) \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B) \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 B \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)*ArcTanh[Sin[(c + d*x)/2]] + 2*(406*A - 178*B + (-28*A + 169*B)*Cos[c + d*x] + 6*(7*A - B)*Cos[2*(c + d*x)] + 15*B*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(210*d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
101,1,94,159,0.3654453,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(15 (A-B) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\sin \left(\frac{1}{2} (c+d x)\right) (2 (5 A-B) \cos (c+d x)-10 A+3 B \cos (2 (c+d x))+29 B)\right)}{15 d \sqrt{a (\cos (c+d x)+1)}}","\frac{2 (5 A-B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 a d}-\frac{4 (5 A-7 B) \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (A-B) \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 B \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)*ArcTanh[Sin[(c + d*x)/2]] + (-10*A + 29*B + 2*(5*A - B)*Cos[c + d*x] + 3*B*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(15*d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
102,1,78,118,0.1725333,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(-3 (A-B) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 A \sin \left(\frac{1}{2} (c+d x)\right)-4 B \sin ^3\left(\frac{1}{2} (c+d x)\right)\right)}{3 d \sqrt{a (\cos (c+d x)+1)}}","\frac{2 (3 A-2 B) \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (A-B) \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 B \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 a d}",1,"(2*Cos[(c + d*x)/2]*(-3*(A - B)*ArcTanh[Sin[(c + d*x)/2]] + 6*A*Sin[(c + d*x)/2] - 4*B*Sin[(c + d*x)/2]^3))/(3*d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
103,1,60,78,0.0743117,"\int \frac{A+B \cos (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left((A-B) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 B \sin \left(\frac{1}{2} (c+d x)\right)\right)}{d \sqrt{a (\cos (c+d x)+1)}}","\frac{\sqrt{2} (A-B) \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 B \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}",1,"(2*Cos[(c + d*x)/2]*((A - B)*ArcTanh[Sin[(c + d*x)/2]] + 2*B*Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
104,1,72,91,0.0842676,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x])/Sqrt[a + a*Cos[c + d*x]],x]","-\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left((A-B) \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)\right)}{d \sqrt{a (\cos (c+d x)+1)}}","\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{\sqrt{2} (A-B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(-2*((A - B)*ArcTanh[Sin[(c + d*x)/2]] - Sqrt[2]*A*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]])*Cos[(c + d*x)/2])/(d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
105,1,95,119,0.3634194,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x])*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) \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{(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{\sqrt{2} (A-B) \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}}",1,"(Cos[(c + d*x)/2]*(2*(A - B)*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
106,1,114,165,0.8449913,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x])*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) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\sqrt{2} (7 A-4 B) \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{(A-4 B) \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{(7 A-4 B) \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) \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 (c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*(-8*(A - B)*ArcTanh[Sin[(c + d*x)/2]] + Sqrt[2]*(7*A - 4*B)*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
107,1,167,261,1.1659425,"\int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^4*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{105 (15 A-19 B) \cos ^5\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\frac{1}{2} \sin \left(\frac{1}{2} (c+d x)\right) \cos ^3\left(\frac{1}{2} (c+d x)\right) (6 (273 A-277 B) \cos (c+d x)+(256 B-84 A) \cos (2 (c+d x))+42 A \cos (3 (c+d x))+1974 A-18 B \cos (3 (c+d x))+15 B \cos (4 (c+d x))-2161 B)}{105 d \left(\sin ^2\left(\frac{1}{2} (c+d x)\right)-1\right) (a (\cos (c+d x)+1))^{3/2}}","-\frac{(15 A-19 B) \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{(273 A-397 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{210 a^2 d}+\frac{(A-B) \sin (c+d x) \cos ^4(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(7 A-11 B) \sin (c+d x) \cos ^3(c+d x)}{14 a d \sqrt{a \cos (c+d x)+a}}+\frac{(63 A-67 B) \sin (c+d x) \cos ^2(c+d x)}{70 a d \sqrt{a \cos (c+d x)+a}}+\frac{(651 A-799 B) \sin (c+d x)}{105 a d \sqrt{a \cos (c+d x)+a}}",1,"(105*(15*A - 19*B)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^5 - (Cos[(c + d*x)/2]^3*(1974*A - 2161*B + 6*(273*A - 277*B)*Cos[c + d*x] + (-84*A + 256*B)*Cos[2*(c + d*x)] + 42*A*Cos[3*(c + d*x)] - 18*B*Cos[3*(c + d*x)] + 15*B*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
108,1,142,216,0.9431812,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/(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 (20 A-39 B) \cos (c+d x)+(6 B-10 A) \cos (2 (c+d x))+85 A-3 B \cos (3 (c+d x))-141 B)-15 (11 A-15 B) \cos ^5\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{15 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) \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{(35 A-39 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{30 a^2 d}+\frac{(A-B) \sin (c+d x) \cos ^3(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(5 A-9 B) \sin (c+d x) \cos ^2(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}-\frac{(65 A-93 B) \sin (c+d x)}{15 a d \sqrt{a \cos (c+d x)+a}}",1,"(-15*(11*A - 15*B)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^5 + Cos[(c + d*x)/2]^3*(85*A - 141*B + 3*(20*A - 39*B)*Cos[c + d*x] + (-10*A + 6*B)*Cos[2*(c + d*x)] - 3*B*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
109,1,97,171,0.7666326,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(3/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) (12 (A-B) \cos (c+d x)+15 A+2 B \cos (2 (c+d x))-17 B)-3 (7 A-11 B) \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{6 a d \sqrt{a (\cos (c+d x)+1)}}","-\frac{(7 A-11 B) \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 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{6 a^2 d}+\frac{(A-B) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(9 A-13 B) \sin (c+d x)}{3 a d \sqrt{a \cos (c+d x)+a}}",1,"(-3*(7*A - 11*B)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2] + (15*A - 17*B + 12*(A - B)*Cos[c + d*x] + 2*B*Cos[2*(c + d*x)])*Tan[(c + d*x)/2])/(6*a*d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
110,1,104,118,0.4242971,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(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) (A-4 B \cos (c+d x)-5 B)-(3 A-7 B) \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 A-7 B) \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) \sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{2 B \sin (c+d x)}{a d \sqrt{a \cos (c+d x)+a}}",1,"(-((3*A - 7*B)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^5) + Cos[(c + d*x)/2]^3*(A - 5*B - 4*B*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
111,1,63,87,0.2023341,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(3/2),x]","\frac{\frac{1}{2} (A-B) \sin (c+d x)+(A+3 B) \cos ^3\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d (a (\cos (c+d x)+1))^{3/2}}","\frac{(A+3 B) \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) \sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"((A + 3*B)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 + ((A - B)*Sin[c + d*x])/2)/(d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
112,1,131,127,0.710046,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x])/(a + a*Cos[c + d*x])^(3/2),x]","\frac{(A-B) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^3\left(\frac{1}{2} (c+d x)\right)+(5 A-B) \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) \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) \sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"((5*A - B)*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)*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
113,1,141,170,1.1645596,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*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) \left(2 (9 A-5 B) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\frac{2 \sin \left(\frac{1}{2} (c+d x)\right) (-2 A \sec (c+d x)-3 A+B)+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)}{\sin ^2\left(\frac{1}{2} (c+d x)\right)-1}\right)}{2 d (a (\cos (c+d x)+1))^{3/2}}","-\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{(9 A-5 B) \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-B) \tan (c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B) \tan (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]^3*(2*(9*A - 5*B)*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 - 2*A*Sec[c + d*x])*Sin[(c + d*x)/2])/(-1 + Sin[(c + d*x)/2]^2)))/(2*d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
114,1,205,221,1.628731,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*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) \sec ^2(c+d x) \left(4 \sin \left(\frac{1}{2} (c+d x)\right) ((6 A-8 B) \cos (c+d x)+(7 A-6 B) \cos (2 (c+d x))+3 (A-2 B))+4 (13 A-9 B) \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)-2 \sqrt{2} (19 A-12 B) \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)\right)}{16 d \left(\sin ^2\left(\frac{1}{2} (c+d x)\right)-1\right) (a (\cos (c+d x)+1))^{3/2}}","\frac{(19 A-12 B) \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) \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) \tan (c+d x)}{4 a d \sqrt{a \cos (c+d x)+a}}+\frac{(2 A-B) \tan (c+d x) \sec (c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B) \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*Sec[c + d*x]^2*(4*(13*A - 9*B)*ArcTanh[Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Cos[(3*(c + d*x))/2])^2 - 2*Sqrt[2]*(19*A - 12*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Cos[(3*(c + d*x))/2])^2 + 4*(3*(A - 2*B) + (6*A - 8*B)*Cos[c + d*x] + (7*A - 6*B)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(16*d*(a*(1 + Cos[c + d*x]))^(3/2)*(-1 + Sin[(c + d*x)/2]^2))","A",1
115,1,139,261,1.6431755,"\int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^4*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) (-5 (479 A-887 B) \cos (c+d x)+(832 B-400 A) \cos (2 (c+d x))+40 A \cos (3 (c+d x))-1895 A-40 B \cos (3 (c+d x))+12 B \cos (4 (c+d x))+3491 B)+30 (163 A-283 B) \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{(163 A-283 B) \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{(475 A-787 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{240 a^3 d}-\frac{(85 A-157 B) \sin (c+d x) \cos ^2(c+d x)}{80 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(985 A-1729 B) \sin (c+d x)}{120 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(A-B) \sin (c+d x) \cos ^4(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(13 A-21 B) \sin (c+d x) \cos ^3(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"(30*(163*A - 283*B)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 + (-1895*A + 3491*B - 5*(479*A - 887*B)*Cos[c + d*x] + (-400*A + 832*B)*Cos[2*(c + d*x)] + 40*A*Cos[3*(c + d*x)] - 40*B*Cos[3*(c + d*x)] + 12*B*Cos[4*(c + d*x)])*Tan[(c + d*x)/2])/(240*a*d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
116,1,117,216,1.1099275,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) ((255 A-479 B) \cos (c+d x)+16 (3 A-5 B) \cos (2 (c+d x))+195 A+8 B \cos (3 (c+d x))-379 B)-6 (75 A-163 B) \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{(75 A-163 B) \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{(39 A-95 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{48 a^3 d}+\frac{(93 A-197 B) \sin (c+d x)}{24 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(A-B) \sin (c+d x) \cos ^3(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(9 A-17 B) \sin (c+d x) \cos ^2(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"(-6*(75*A - 163*B)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 + (195*A - 379*B + (255*A - 479*B)*Cos[c + d*x] + 16*(3*A - 5*B)*Cos[2*(c + d*x)] + 8*B*Cos[3*(c + d*x)])*Tan[(c + d*x)/2])/(48*a*d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
117,1,100,169,0.7792409,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) ((85 B-13 A) \cos (c+d x)-9 A+16 B \cos (2 (c+d x))+65 B)+2 (19 A-75 B) \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{(19 A-75 B) \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 B) \sin (c+d x)}{4 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(A-B) \sin (c+d x) \cos ^2(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{(5 A-13 B) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"(2*(19*A - 75*B)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 + (-9*A + 65*B + (-13*A + 85*B)*Cos[c + d*x] + 16*B*Cos[2*(c + d*x)])*Tan[(c + d*x)/2])/(16*a*d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
118,1,87,126,0.6190551,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) ((5 A-13 B) \cos (c+d x)+A-9 B)+2 (5 A+19 B) \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) \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-13 B) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(2*(5*A + 19*B)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 + (A - 9*B + (5*A - 13*B)*Cos[c + d*x])*Tan[(c + d*x)/2])/(16*a*d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
119,1,80,126,0.5167611,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\sin (c+d x) ((3 A+5 B) \cos (c+d x)+7 A+B)+4 (3 A+5 B) \cos ^5\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{16 d (a (\cos (c+d x)+1))^{5/2}}","\frac{(3 A+5 B) \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) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B) \sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(4*(3*A + 5*B)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^5 + (7*A + B + (3*A + 5*B)*Cos[c + d*x])*Sin[c + d*x])/(16*d*(a*(1 + Cos[c + d*x]))^(5/2))","A",1
120,1,126,164,1.6524512,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x])/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) ((3 B-11 A) \cos (c+d x)-15 A+7 B)-2 (43 A-3 B) \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-3 B) \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) \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(-2*(43*A - 3*B)*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 + 7*B + (-11*A + 3*B)*Cos[c + d*x])*Tan[(c + d*x)/2])/(16*a*d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
121,1,142,207,3.4714946,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\tan (c+d x) (10 (11 A-3 B) \cos (c+d x)+(35 A-11 B) \cos (2 (c+d x))+67 A-11 B)+8 (115 A-43 B) \cos ^5\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-128 \sqrt{2} (5 A-2 B) \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)}{32 d (a (\cos (c+d x)+1))^{5/2}}","-\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{(115 A-43 B) \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{(35 A-11 B) \tan (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(15 A-7 B) \tan (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \tan (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(8*(115*A - 43*B)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^5 - 128*Sqrt[2]*(5*A - 2*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^5 + (67*A - 11*B + 10*(11*A - 3*B)*Cos[c + d*x] + (35*A - 11*B)*Cos[2*(c + d*x)])*Tan[c + d*x])/(32*d*(a*(1 + Cos[c + d*x]))^(5/2))","A",1
122,1,178,264,6.2021726,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + a*Cos[c + d*x])^(5/2),x]","\frac{-8 (219 A-115 B) \cos ^3\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+32 \sqrt{2} (39 A-20 B) \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)+\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) ((169 B-269 A) \cos (c+d x)+(110 B-190 A) \cos (2 (c+d x))-63 A \cos (3 (c+d x))-158 A+35 B \cos (3 (c+d x))+110 B)}{64 a d (a (\cos (c+d x)+1))^{3/2}}","\frac{(39 A-20 B) \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) \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{7 (9 A-5 B) \tan (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(31 A-15 B) \tan (c+d x) \sec (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{(19 A-11 B) \tan (c+d x) \sec (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \tan (c+d x) \sec (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(-8*(219*A - 115*B)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 + 32*Sqrt[2]*(39*A - 20*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 + (-158*A + 110*B + (-269*A + 169*B)*Cos[c + d*x] + (-190*A + 110*B)*Cos[2*(c + d*x)] - 63*A*Cos[3*(c + d*x)] + 35*B*Cos[3*(c + d*x)])*Sec[c + d*x]^2*Tan[(c + d*x)/2])/(64*a*d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
123,1,914,159,6.3510541,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(-\frac{(9 A+7 B) \cot (c)}{15 d}+\frac{23 (A+B) \cos (d x) \sin (c)}{84 d}+\frac{(18 A+19 B) \cos (2 d x) \sin (2 c)}{180 d}+\frac{(A+B) \cos (3 d x) \sin (3 c)}{28 d}+\frac{B \cos (4 d x) \sin (4 c)}{72 d}+\frac{23 (A+B) \cos (c) \sin (d x)}{84 d}+\frac{(18 A+19 B) \cos (2 c) \sin (2 d x)}{180 d}+\frac{(A+B) \cos (3 c) \sin (3 d x)}{28 d}+\frac{B \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 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{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}}\right)","\frac{10 a (A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (9 A+7 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a (A+B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 a (9 A+7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 a (A+B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a B \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*B)*Cot[c])/d + (23*(A + B)*Cos[d*x]*Sin[c])/(84*d) + ((18*A + 19*B)*Cos[2*d*x]*Sin[2*c])/(180*d) + ((A + B)*Cos[3*d*x]*Sin[3*c])/(28*d) + (B*Cos[4*d*x]*Sin[4*c])/(72*d) + (23*(A + B)*Cos[c]*Sin[d*x])/(84*d) + ((18*A + 19*B)*Cos[2*c]*Sin[2*d*x])/(180*d) + ((A + B)*Cos[3*c]*Sin[3*d*x])/(28*d) + (B*Cos[4*c]*Sin[4*d*x])/(72*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]) - (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))","C",0
124,1,872,132,6.2743298,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(-\frac{3 (A+B) \cot (c)}{5 d}+\frac{(28 A+23 B) \cos (d x) \sin (c)}{84 d}+\frac{(A+B) \cos (2 d x) \sin (2 c)}{10 d}+\frac{B \cos (3 d x) \sin (3 c)}{28 d}+\frac{(28 A+23 B) \cos (c) \sin (d x)}{84 d}+\frac{(A+B) \cos (2 c) \sin (2 d x)}{10 d}+\frac{B \cos (3 c) \sin (3 d x)}{28 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{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}}\right)","\frac{2 a (7 A+5 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{6 a (A+B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (A+B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a (7 A+5 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a B \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*((-3*(A + B)*Cot[c])/(5*d) + ((28*A + 23*B)*Cos[d*x]*Sin[c])/(84*d) + ((A + B)*Cos[2*d*x]*Sin[2*c])/(10*d) + (B*Cos[3*d*x]*Sin[3*c])/(28*d) + ((28*A + 23*B)*Cos[c]*Sin[d*x])/(84*d) + ((A + B)*Cos[2*c]*Sin[2*d*x])/(10*d) + (B*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*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]) - (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",0
125,1,830,101,6.2586259,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(-\frac{(5 A+3 B) \cot (c)}{5 d}+\frac{(A+B) \cos (d x) \sin (c)}{3 d}+\frac{B \cos (2 d x) \sin (2 c)}{10 d}+\frac{(A+B) \cos (c) \sin (d x)}{3 d}+\frac{B \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 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{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}}\right)","\frac{2 a (A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (5 A+3 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (A+B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a B \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*B)*Cot[c])/d + ((A + B)*Cos[d*x]*Sin[c])/(3*d) + (B*Cos[2*d*x]*Sin[2*c])/(10*d) + ((A + B)*Cos[c]*Sin[d*x])/(3*d) + (B*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]]]])/(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]) - (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))","C",0
126,1,784,70,6.2730214,"\int \frac{(a+a \cos (c+d x)) (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","a \left(-\frac{A \csc (c) (\cos (c+d x)+1) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \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)}{2 d}-\frac{A \csc (c) (\cos (c+d x)+1) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \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{B \csc (c) (\cos (c+d x)+1) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \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)}{2 d}-\frac{B \csc (c) (\cos (c+d x)+1) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \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)} (\cos (c+d x)+1) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \left(-\frac{(A+B) \cot (c)}{d}+\frac{B \sin (c) \cos (d x)}{3 d}+\frac{B \cos (c) \sin (d x)}{3 d}\right)\right)","\frac{2 a (3 A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (A+B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a B \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*(-(((A + B)*Cot[c])/d) + (B*Cos[d*x]*Sin[c])/(3*d) + (B*Cos[c]*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]]]])/(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]) - (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",0
127,1,783,66,6.3120876,"\int \frac{(a+a \cos (c+d x)) (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","a \left(\frac{A \csc (c) (\cos (c+d x)+1) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \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)}{2 d}-\frac{A \csc (c) (\cos (c+d x)+1) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \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{B \csc (c) (\cos (c+d x)+1) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \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)}{2 d}-\frac{B \csc (c) (\cos (c+d x)+1) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \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)} (\cos (c+d x)+1) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{A \sec (c) \sin (d x) \sec (c+d x)}{d}-\frac{\csc (c) \sec (c) (-2 A+B \cos (2 c)+B)}{2 d}\right)\right)","\frac{2 a (A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 a (A-B) 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)}}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-1/2*((-2*A + B + B*Cos[2*c])*Csc[c]*Sec[c])/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]) + (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",0
128,1,813,95,6.3606514,"\int \frac{(a+a \cos (c+d x)) (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]))/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{(A+B) \csc (c) \sec (c)}{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{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}}\right)","\frac{2 a (A+3 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A+B) 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*(((A + B)*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]) + (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",0
129,1,865,132,6.4277486,"\int \frac{(a+a \cos (c+d x)) (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]))/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)) \sec (c+d x)}{15 d}+\frac{(3 A+5 B) \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{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}}\right)","\frac{2 a (A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (3 A+5 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (A+B) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a (3 A+5 B) \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)}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(((3*A + 5*B)*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*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]) + (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",0
130,1,944,194,6.2978302,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^2 \left(-\frac{(9 A+8 B) \cot (c)}{15 d}+\frac{(51 A+46 B) \cos (d x) \sin (c)}{168 d}+\frac{(36 A+37 B) \cos (2 d x) \sin (2 c)}{360 d}+\frac{(A+2 B) \cos (3 d x) \sin (3 c)}{56 d}+\frac{B \cos (4 d x) \sin (4 c)}{144 d}+\frac{(51 A+46 B) \cos (c) \sin (d x)}{168 d}+\frac{(36 A+37 B) \cos (2 c) \sin (2 d x)}{360 d}+\frac{(A+2 B) \cos (3 c) \sin (3 d x)}{56 d}+\frac{B \cos (4 c) \sin (4 d x)}{144 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{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{4 a^2 (6 A+5 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (9 A+8 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 (9 A+11 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{4 a^2 (9 A+8 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{4 a^2 (6 A+5 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{9 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)*Cot[c])/d + ((51*A + 46*B)*Cos[d*x]*Sin[c])/(168*d) + ((36*A + 37*B)*Cos[2*d*x]*Sin[2*c])/(360*d) + ((A + 2*B)*Cos[3*d*x]*Sin[3*c])/(56*d) + (B*Cos[4*d*x]*Sin[4*c])/(144*d) + ((51*A + 46*B)*Cos[c]*Sin[d*x])/(168*d) + ((36*A + 37*B)*Cos[2*c]*Sin[2*d*x])/(360*d) + ((A + 2*B)*Cos[3*c]*Sin[3*d*x])/(56*d) + (B*Cos[4*c]*Sin[4*d*x])/(144*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]) - (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)","C",0
131,1,898,161,6.2657509,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^2 \left(-\frac{(4 A+3 B) \cot (c)}{5 d}+\frac{(56 A+51 B) \cos (d x) \sin (c)}{168 d}+\frac{(A+2 B) \cos (2 d x) \sin (2 c)}{20 d}+\frac{B \cos (3 d x) \sin (3 c)}{56 d}+\frac{(56 A+51 B) \cos (c) \sin (d x)}{168 d}+\frac{(A+2 B) \cos (2 c) \sin (2 d x)}{20 d}+\frac{B \cos (3 c) \sin (3 d x)}{56 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{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{4 a^2 (7 A+6 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (4 A+3 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (7 A+9 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{4 a^2 (7 A+6 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{7 d}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(-1/5*((4*A + 3*B)*Cot[c])/d + ((56*A + 51*B)*Cos[d*x]*Sin[c])/(168*d) + ((A + 2*B)*Cos[2*d*x]*Sin[2*c])/(20*d) + (B*Cos[3*d*x]*Sin[3*c])/(56*d) + ((56*A + 51*B)*Cos[c]*Sin[d*x])/(168*d) + ((A + 2*B)*Cos[2*c]*Sin[2*d*x])/(20*d) + (B*Cos[3*c]*Sin[3*d*x])/(56*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]) - (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)","C",0
132,1,852,126,6.3016143,"\int \frac{(a+a \cos (c+d x))^2 (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^2 \left(-\frac{(5 A+4 B) \cot (c)}{5 d}+\frac{(A+2 B) \cos (d x) \sin (c)}{6 d}+\frac{B \cos (2 d x) \sin (2 c)}{20 d}+\frac{(A+2 B) \cos (c) \sin (d x)}{6 d}+\frac{B \cos (2 c) \sin (2 d x)}{20 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{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{4 a^2 (2 A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^2 (5 A+4 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (5 A+7 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)}{5 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)*Cot[c])/d + ((A + 2*B)*Cos[d*x]*Sin[c])/(6*d) + (B*Cos[2*d*x]*Sin[2*c])/(20*d) + ((A + 2*B)*Cos[c]*Sin[d*x])/(6*d) + (B*Cos[2*c]*Sin[2*d*x])/(20*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]) - (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)","C",0
133,1,623,118,6.3703471,"\int \frac{(a+a \cos (c+d x))^2 (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/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{B \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)}{2 d}-\frac{2 B \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) (A \cos (2 c)-A+2 B \cos (2 c)+2 B)}{4 d}+\frac{A \sec (c) \sin (d x) \sec (c+d x)}{2 d}+\frac{B \sin (c) \cos (d x)}{6 d}+\frac{B \cos (c) \sin (d x)}{6 d}\right)","\frac{4 a^2 (3 A+2 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a^2 (3 A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{d \sqrt{\cos (c+d x)}}+\frac{4 a^2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(-1/4*((-A + 2*B + A*Cos[2*c] + 2*B*Cos[2*c])*Csc[c]*Sec[c])/d + (B*Cos[d*x]*Sin[c])/(6*d) + (B*Cos[c]*Sin[d*x])/(6*d) + (A*Sec[c]*Sec[c + d*x]*Sin[d*x])/(2*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]) - (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
134,1,624,120,6.4481858,"\int \frac{(a+a \cos (c+d x))^2 (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{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)}{2 d}-\frac{2 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{B \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) (A \sin (c)+6 A \sin (d x)+3 B \sin (d x))}{6 d}-\frac{\csc (c) \sec (c) (-4 A+B \cos (2 c)-B)}{4 d}+\frac{A \sec (c) \sin (d x) \sec ^2(c+d x)}{6 d}\right)","\frac{4 a^2 (2 A+3 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^2 (5 A+3 B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}-\frac{4 a^2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{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 + B*Cos[2*c])*Csc[c]*Sec[c])/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]) + (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",0
135,1,883,159,6.532552,"\int \frac{(a+a \cos (c+d x))^2 (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/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)) \sec (c+d x)}{30 d}+\frac{(4 A+5 B) \csc (c) \sec (c)}{5 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{4 a^2 (A+2 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^2 (4 A+5 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (7 A+5 B) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (4 A+5 B) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{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*(((4*A + 5*B)*Csc[c]*Sec[c])/(5*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]))/(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]) + (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
136,1,925,194,6.627652,"\int \frac{(a+a \cos (c+d x))^2 (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/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)) \sec ^2(c+d x)}{210 d}+\frac{\sec (c) (30 A \sin (c)+35 B \sin (c)+63 A \sin (d x)+84 B \sin (d x)) \sec (c+d x)}{105 d}+\frac{(3 A+4 B) \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{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{4 a^2 (6 A+7 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (3 A+4 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a^2 (6 A+7 B) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (9 A+7 B) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2 (3 A+4 B) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{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)*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]))/(210*d) + (Sec[c]*Sec[c + d*x]*(30*A*Sin[c] + 35*B*Sin[c] + 63*A*Sin[d*x] + 84*B*Sin[d*x]))/(105*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]) + (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",0
137,1,990,237,6.3225792,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^3 \left(-\frac{(17 A+15 B) \cot (c)}{30 d}+\frac{(2134 A+1953 B) \cos (d x) \sin (c)}{7392 d}+\frac{(73 A+75 B) \cos (2 d x) \sin (2 c)}{720 d}+\frac{3 (44 A+63 B) \cos (3 d x) \sin (3 c)}{4928 d}+\frac{(A+3 B) \cos (4 d x) \sin (4 c)}{288 d}+\frac{B \cos (5 d x) \sin (5 c)}{704 d}+\frac{(2134 A+1953 B) \cos (c) \sin (d x)}{7392 d}+\frac{(73 A+75 B) \cos (2 c) \sin (2 d x)}{720 d}+\frac{3 (44 A+63 B) \cos (3 c) \sin (3 d x)}{4928 d}+\frac{(A+3 B) \cos (4 c) \sin (4 d x)}{288 d}+\frac{B \cos (5 c) \sin (5 d x)}{704 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{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{4 a^3 (121 A+105 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (17 A+15 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{20 a^3 (22 A+21 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{693 d}+\frac{4 a^3 (17 A+15 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 (11 A+15 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{99 d}+\frac{4 a^3 (121 A+105 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 a B \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])^3*Sec[c/2 + (d*x)/2]^6*(-1/30*((17*A + 15*B)*Cot[c])/d + ((2134*A + 1953*B)*Cos[d*x]*Sin[c])/(7392*d) + ((73*A + 75*B)*Cos[2*d*x]*Sin[2*c])/(720*d) + (3*(44*A + 63*B)*Cos[3*d*x]*Sin[3*c])/(4928*d) + ((A + 3*B)*Cos[4*d*x]*Sin[4*c])/(288*d) + (B*Cos[5*d*x]*Sin[5*c])/(704*d) + ((2134*A + 1953*B)*Cos[c]*Sin[d*x])/(7392*d) + ((73*A + 75*B)*Cos[2*c]*Sin[2*d*x])/(720*d) + (3*(44*A + 63*B)*Cos[3*c]*Sin[3*d*x])/(4928*d) + ((A + 3*B)*Cos[4*c]*Sin[4*d*x])/(288*d) + (B*Cos[5*c]*Sin[5*d*x])/(704*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]) - (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)","C",0
138,1,944,204,6.2967634,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]),x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^3 \left(-\frac{(21 A+17 B) \cot (c)}{30 d}+\frac{(107 A+97 B) \cos (d x) \sin (c)}{336 d}+\frac{(54 A+73 B) \cos (2 d x) \sin (2 c)}{720 d}+\frac{(A+3 B) \cos (3 d x) \sin (3 c)}{112 d}+\frac{B \cos (4 d x) \sin (4 c)}{288 d}+\frac{(107 A+97 B) \cos (c) \sin (d x)}{336 d}+\frac{(54 A+73 B) \cos (2 c) \sin (2 d x)}{720 d}+\frac{(A+3 B) \cos (3 c) \sin (3 d x)}{112 d}+\frac{B \cos (4 c) \sin (4 d x)}{288 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{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{4 a^3 (13 A+11 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (21 A+17 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^3 (24 A+23 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{2 (9 A+13 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{63 d}+\frac{4 a^3 (13 A+11 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a B \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])^3*Sec[c/2 + (d*x)/2]^6*(-1/30*((21*A + 17*B)*Cot[c])/d + ((107*A + 97*B)*Cos[d*x]*Sin[c])/(336*d) + ((54*A + 73*B)*Cos[2*d*x]*Sin[2*c])/(720*d) + ((A + 3*B)*Cos[3*d*x]*Sin[3*c])/(112*d) + (B*Cos[4*d*x]*Sin[4*c])/(288*d) + ((107*A + 97*B)*Cos[c]*Sin[d*x])/(336*d) + ((54*A + 73*B)*Cos[2*c]*Sin[2*d*x])/(720*d) + ((A + 3*B)*Cos[3*c]*Sin[3*d*x])/(112*d) + (B*Cos[4*c]*Sin[4*d*x])/(288*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]) - (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",0
139,1,898,171,6.3577038,"\int \frac{(a+a \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^3 \left(-\frac{(9 A+7 B) \cot (c)}{10 d}+\frac{(84 A+107 B) \cos (d x) \sin (c)}{336 d}+\frac{(A+3 B) \cos (2 d x) \sin (2 c)}{40 d}+\frac{B \cos (3 d x) \sin (3 c)}{112 d}+\frac{(84 A+107 B) \cos (c) \sin (d x)}{336 d}+\frac{(A+3 B) \cos (2 c) \sin (2 d x)}{40 d}+\frac{B \cos (3 c) \sin (3 d x)}{112 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{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{4 a^3 (21 A+13 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (9 A+7 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a^3 (42 A+41 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{2 (7 A+11 B) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{35 d}+\frac{2 a B \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])^3*Sec[c/2 + (d*x)/2]^6*(-1/10*((9*A + 7*B)*Cot[c])/d + ((84*A + 107*B)*Cos[d*x]*Sin[c])/(336*d) + ((A + 3*B)*Cos[2*d*x]*Sin[2*c])/(40*d) + (B*Cos[3*d*x]*Sin[3*c])/(112*d) + ((84*A + 107*B)*Cos[c]*Sin[d*x])/(336*d) + ((A + 3*B)*Cos[2*c]*Sin[2*d*x])/(40*d) + (B*Cos[3*c]*Sin[3*d*x])/(112*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]) - (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)","C",0
140,1,888,169,6.457577,"\int \frac{(a+a \cos (c+d x))^3 (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/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+18 B \cos (2 c)) \csc (c) \sec (c)}{40 d}+\frac{A \sec (c+d x) \sin (d x) \sec (c)}{4 d}+\frac{(A+3 B) \cos (d x) \sin (c)}{12 d}+\frac{B \cos (2 d x) \sin (2 c)}{40 d}+\frac{(A+3 B) \cos (c) \sin (d x)}{12 d}+\frac{B \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 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{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{4 a^3 (5 A+3 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^3 (5 A+9 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{4 a^3 (5 A-6 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}-\frac{2 (5 A-B) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{5 d}+\frac{2 a 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])^3*Sec[c/2 + (d*x)/2]^6*(-1/40*((5*A + 18*B + 15*A*Cos[2*c] + 18*B*Cos[2*c])*Csc[c]*Sec[c])/d + ((A + 3*B)*Cos[d*x]*Sin[c])/(12*d) + (B*Cos[2*d*x]*Sin[2*c])/(40*d) + ((A + 3*B)*Cos[c]*Sin[d*x])/(12*d) + (A*Sec[c]*Sec[c + d*x]*Sin[d*x])/(4*d) + (B*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]) - (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]) - (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)","C",0
141,1,879,161,6.5355461,"\int \frac{(a+a \cos (c+d x))^3 (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/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{(\cos (2 c) A-5 A+B+3 B \cos (2 c)) \csc (c) \sec (c)}{8 d}+\frac{B \cos (d x) \sin (c)}{12 d}+\frac{B \cos (c) \sin (d x)}{12 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{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{20 a^3 (A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{4 a^3 (4 A+B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 (7 A+3 B) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 a 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])^3*Sec[c/2 + (d*x)/2]^6*(-1/8*((-5*A + B + A*Cos[2*c] + 3*B*Cos[2*c])*Csc[c]*Sec[c])/d + (B*Cos[d*x]*Sin[c])/(12*d) + (B*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)) - (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]) + (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)","C",0
142,1,890,171,6.6181124,"\int \frac{(a+a \cos (c+d x))^3 (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/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)) \sec (c+d x)}{60 d}-\frac{(-36 A-25 B+5 B \cos (2 c)) \csc (c) \sec (c)}{40 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{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{4 a^3 (3 A+5 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 (9 A+5 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (9 A+5 B) \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 (21 A+20 B) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 a 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])^3*Sec[c/2 + (d*x)/2]^6*(-1/40*((-36*A - 25*B + 5*B*Cos[2*c])*Csc[c]*Sec[c])/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]))/(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]) + (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",0
143,1,925,204,6.653968,"\int \frac{(a+a \cos (c+d x))^3 (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/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)) \sec ^2(c+d x)}{420 d}+\frac{\sec (c) (130 A \sin (c)+105 B \sin (c)+294 A \sin (d x)+378 B \sin (d x)) \sec (c+d x)}{420 d}+\frac{(7 A+9 B) \csc (c) \sec (c)}{10 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{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{4 a^3 (13 A+21 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (7 A+9 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a^3 (41 A+42 B) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (11 A+7 B) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (7 A+9 B) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a 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])^3*Sec[c/2 + (d*x)/2]^6*(((7*A + 9*B)*Csc[c]*Sec[c])/(10*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]))/(420*d) + (Sec[c]*Sec[c + d*x]*(130*A*Sin[c] + 105*B*Sin[c] + 294*A*Sin[d*x] + 378*B*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]) + (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",0
144,1,967,237,6.7120216,"\int \frac{(a+a \cos (c+d x))^3 (A+B \cos (c+d x))}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/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)) \sec ^3(c+d x)}{1260 d}+\frac{\sec (c) (238 A \sin (c)+189 B \sin (c)+330 A \sin (d x)+390 B \sin (d x)) \sec ^2(c+d x)}{1260 d}+\frac{\sec (c) (55 A \sin (c)+65 B \sin (c)+119 A \sin (d x)+147 B \sin (d x)) \sec (c+d x)}{210 d}+\frac{(17 A+21 B) \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{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{4 a^3 (11 A+13 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (17 A+21 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^3 (11 A+13 B) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (23 A+24 B) \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (13 A+9 B) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (17 A+21 B) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 a 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])^3*Sec[c/2 + (d*x)/2]^6*(((17*A + 21*B)*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]*(55*A*Sin[c] + 65*B*Sin[c] + 119*A*Sin[d*x] + 147*B*Sin[d*x]))/(210*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]))/(1260*d) + (Sec[c]*Sec[c + d*x]^2*(238*A*Sin[c] + 189*B*Sin[c] + 330*A*Sin[d*x] + 390*B*Sin[d*x]))/(1260*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]) + (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)","C",0
145,1,1182,156,6.6227528,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{a+a \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]))/(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{\sqrt{\cos (c+d x)} \left(\frac{2 (10 \cos (c) A+5 A-5 B-16 B \cos (c)) \csc (c)}{5 d}+\frac{4 (A-B) \cos (d x) \sin (c)}{3 d}+\frac{2 B \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)\right)}{d}+\frac{4 (A-B) \cos (c) \sin (d x)}{3 d}+\frac{2 B \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{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{5 (A-B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 (5 A-7 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(5 A-7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}+\frac{5 (A-B) \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)*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]) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*((2*(5*A - 5*B + 10*A*Cos[c] - 16*B*Cos[c])*Csc[c])/(5*d) + (4*(A - B)*Cos[d*x]*Sin[c])/(3*d) + (2*B*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]))/d + (4*(A - B)*Cos[c]*Sin[d*x])/(3*d) + (2*B*Cos[2*c]*Sin[2*d*x])/(5*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",0
146,1,1129,123,6.555312,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{a+a \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/(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{\sqrt{\cos (c+d x)} \left(-\frac{2 (A-B) (2 \cos (c)+1) \csc (c)}{d}+\frac{4 B \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)\right)}{d}+\frac{4 B \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 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{(3 A-5 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}-\frac{(3 A-5 B) \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]) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*((-2*(A - B)*(1 + 2*Cos[c])*Csc[c])/d + (4*B*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]))/d + (4*B*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*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",0
147,1,1098,85,6.4547927,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{a+a \cos (c+d x)} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*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 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{\sqrt{\cos (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)-B \sin \left(\frac{d x}{2}\right)\right)}{d}-\frac{2 (-A+B+2 B \cos (c)) \csc (c)}{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{(A-B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-3 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \cos (c+d x)+a)}",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]) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*((-2*(-A + B + 2*B*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]))/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",1
148,1,1094,83,6.4854026,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))} \, dx","Integrate[(A + B*Cos[c + d*x])/(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{\sqrt{\cos (c+d x)} \left(-\frac{2 (A-B) \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)\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{(A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-B) \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]) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*((-2*(A - B)*Csc[c])/d - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - B*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",1
149,1,1130,119,6.7079639,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))} \, dx","Integrate[(A + B*Cos[c + d*x])/(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{\sqrt{\cos (c+d x)} \left(\frac{(\cos (c) A+2 A-B \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)\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{(A-B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(3 A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(3 A-B) \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}-\frac{(A-B) \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]) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*(((2*A + A*Cos[c] - B*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]))/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",1
150,1,1167,153,7.0894571,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))} \, dx","Integrate[(A + B*Cos[c + d*x])/(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{\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{(A-B) (\cos (c)+2) \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)\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{(5 A-3 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-B) \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{(5 A-3 B) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{3 (A-B) \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]) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*(-(((A - B)*(2 + 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]))/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",1
151,1,1262,203,6.8593398,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x]))/(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{28 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)}{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)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (A-B) \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)-4 B \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{4 (20 \cos (c) A+15 A-20 B-36 B \cos (c)) \csc (c)}{5 d}+\frac{8 (A-2 B) \cos (d x) \sin (c)}{3 d}+\frac{4 B \cos (2 d x) \sin (2 c)}{5 d}+\frac{8 (A-2 B) \cos (c) \sin (d x)}{3 d}+\frac{4 B \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{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)}{d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}","\frac{5 (2 A-3 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{7 (5 A-8 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}+\frac{(2 A-3 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{7 (5 A-8 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a^2 d}+\frac{5 (2 A-3 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{3 d (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 + (((28*I)/5)*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 - (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]]]])/(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*(15*A - 20*B + 20*A*Cos[c] - 36*B*Cos[c])*Csc[c])/(5*d) + (8*(A - 2*B)*Cos[d*x]*Sin[c])/(3*d) + (4*B*Cos[2*d*x]*Sin[2*c])/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(3*A*Sin[(d*x)/2] - 4*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]))/(3*d) + (8*(A - 2*B)*Cos[c]*Sin[d*x])/(3*d) + (4*B*Cos[2*c]*Sin[2*d*x])/(5*d) - (2*(A - B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])^2","C",0
152,1,1218,166,6.7583137,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]))/(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{\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)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{2 (A-B) \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)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{4 (2 \cos (c) A+2 A-3 B-4 B \cos (c)) \csc (c)}{d}+\frac{8 B \cos (d x) \sin (c)}{3 d}+\frac{8 B \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{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{5 (A-2 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(4 A-7 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{(4 A-7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{5 (A-2 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{5}{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 + (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]) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*((-4*(2*A - 3*B + 2*A*Cos[c] - 4*B*Cos[c])*Csc[c])/d + (8*B*Cos[d*x]*Sin[c])/(3*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(2*A*Sin[(d*x)/2] - 3*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]))/(3*d) + (8*B*Cos[c]*Sin[d*x])/(3*d) + (2*(A - B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])^2","C",0
153,1,1184,136,6.6572773,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*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{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{\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)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (A-B) \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)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{4 (-A+2 B+2 B \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{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{(2 A-5 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(A-4 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{(2 A-5 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d (\cos (c+d x)+1)}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{3}{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 - (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]) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*((-4*(-A + 2*B + 2*B*Cos[c])*Csc[c])/d + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - 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]))/(3*d) - (2*(A - B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])^2","C",1
154,1,815,121,6.5202033,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{(a+a \cos (c+d x))^2} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/(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{\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)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{2 (A-B) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 B \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{4 B \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{(A+2 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d (\cos (c+d x)+1)}+\frac{(A-B) \sin (c+d x) \sqrt{\cos (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*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]) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*((4*B*Csc[c])/d + (4*B*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] - B*Sin[(d*x)/2]))/(3*d) + (2*(A - B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])^2","C",1
155,1,815,121,6.5484591,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2} \, dx","Integrate[(A + B*Cos[c + d*x])/(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{\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)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (A-B) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 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{4 A \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{(2 A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{A \sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d (\cos (c+d x)+1)}-\frac{(A-B) \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 - (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]) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*((-4*A*Csc[c])/d - (4*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] - B*Sin[(d*x)/2]))/(3*d) - (2*(A - B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])^2","C",1
156,1,1217,168,6.8117246,"\int \frac{A+B \cos (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])/(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)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{2 (A-B) \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{(5 A-2 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\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{(5 A-2 B) \sin (c+d x)}{3 a^2 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1)}-\frac{(A-B) \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]) + (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 + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2]))/(3*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(2*A*Sin[(d*x)/2] - B*Sin[(d*x)/2]))/d + (8*A*Sec[c]*Sec[c + d*x]*Sin[d*x])/d + (2*(A - B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])^2","C",1
157,1,1258,201,7.4266416,"\int \frac{A+B \cos (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])/(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{\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)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (A-B) \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)\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)) \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{5 (2 A-B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(7 A-4 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(7 A-4 B) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x) (\cos (c+d x)+1)}+\frac{5 (2 A-B) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(7 A-4 B) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(A-B) \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 - (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]) + (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])*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]))/d - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] - B*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)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])^2","C",0
158,1,1346,254,7.1566301,"\int \frac{\cos ^{\frac{9}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(9/2)*(A + B*Cos[c + d*x]))/(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{231 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{\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)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{2 (A-B) \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(19 A \sin \left(\frac{d x}{2}\right)-24 B \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 (19 A-24 B) \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)-99 B \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 (60 \cos (c) A+59 A-99 B-132 B \cos (c)) \csc (c)}{5 d}+\frac{16 (A-3 B) \cos (d x) \sin (c)}{3 d}+\frac{8 B \cos (2 d x) \sin (2 c)}{5 d}+\frac{16 (A-3 B) \cos (c) \sin (d x)}{3 d}+\frac{8 B \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{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{42 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{(11 A-21 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{7 (17 A-33 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{3 (11 A-21 B) \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 (17 A-33 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 a^3 d}+\frac{(11 A-21 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a^3 d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(7 A-12 B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{15 a d (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 + (((231*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 - (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]) + (42*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]) + (Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*((4*(59*A - 99*B + 60*A*Cos[c] - 132*B*Cos[c])*Csc[c])/(5*d) + (16*(A - 3*B)*Cos[d*x]*Sin[c])/(3*d) + (8*B*Cos[2*d*x]*Sin[2*c])/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(59*A*Sin[(d*x)/2] - 99*B*Sin[(d*x)/2]))/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(19*A*Sin[(d*x)/2] - 24*B*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]))/(5*d) + (16*(A - 3*B)*Cos[c]*Sin[d*x])/(3*d) + (8*B*Cos[2*c]*Sin[2*d*x])/(5*d) - (4*(19*A - 24*B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (2*(A - B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",0
159,1,1306,219,6.9876697,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x]))/(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{\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)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{2 (A-B) \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)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 (14 A-19 B) \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)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (20 \cos (c) A+29 A-59 B-60 B \cos (c)) \csc (c)}{5 d}+\frac{16 B \cos (d x) \sin (c)}{3 d}+\frac{16 B \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{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{(13 A-33 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{7 (7 A-17 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{7 (7 A-17 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(13 A-33 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 a^3 d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(A-2 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 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 + (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]) + (Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*((-4*(29*A - 59*B + 20*A*Cos[c] - 60*B*Cos[c])*Csc[c])/(5*d) + (16*B*Cos[d*x]*Sin[c])/(3*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(29*A*Sin[(d*x)/2] - 59*B*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]))/(15*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2]))/(5*d) + (16*B*Cos[c]*Sin[d*x])/(3*d) + (4*(14*A - 19*B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (2*(A - B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",0
160,1,1273,188,6.8960237,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*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{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{\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)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{2 (A-B) \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)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 (9 A-14 B) \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)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (-9 A+29 B+20 B \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)}{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{(3 A-13 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(9 A-49 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(3 A-13 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(3 A-8 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 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 - (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]) + (Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*((-4*(-9*A + 29*B + 20*B*Cos[c])*Csc[c])/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(9*A*Sin[(d*x)/2] - 29*B*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]))/(15*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2]))/(5*d) - (4*(9*A - 14*B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (2*(A - B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",1
161,1,1265,180,6.8166907,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/(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{\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)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{2 (A-B) \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)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 (4 A-9 B) \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)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 (A+9 B) \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{(A+3 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(A+9 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(A+9 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{(A-6 B) \sin (c+d x) \sqrt{\cos (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 - (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]) + (Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*((4*(A + 9*B)*Csc[c])/(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]))/(15*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + 9*B*Sin[(d*x)/2]))/(5*d) + (4*(4*A - 9*B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (2*(A - B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",1
162,1,1264,178,6.7167603,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/(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{\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)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{2 (A-B) \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)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 (A+4 B) \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)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (A-B) \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{(A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A+4 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}+\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}",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 - (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]) + (Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*((-4*(A - B)*Csc[c])/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - B*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]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + 4*B*Sin[(d*x)/2]))/(15*d) + (4*(A + 4*B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (2*(A - B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",1
163,1,1265,182,6.8009713,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3} \, dx","Integrate[(A + B*Cos[c + d*x])/(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{\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)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{2 (A-B) \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)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 (6 A-B) \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)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (9 A+B) \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{(3 A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(9 A+B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(9 A+B) \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) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B) \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 - (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]) + (Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*((-4*(9*A + B)*Csc[c])/(5*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2]))/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(6*A*Sin[(d*x)/2] - B*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]))/(5*d) - (4*(6*A - B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (2*(A - B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",1
164,1,1305,221,7.112874,"\int \frac{A+B \cos (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])/(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{\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)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{2 (A-B) \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)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 (11 A-6 B) \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)\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)) \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{(13 A-3 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(49 A-9 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(49 A-9 B) \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{(13 A-3 B) \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) \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}-\frac{(A-B) \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 + (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]) + (Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*((2*(20*A + 29*A*Cos[c] - 9*B*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]))/(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]))/(15*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2]))/(5*d) + (16*A*Sec[c]*Sec[c + d*x]*Sin[d*x])/d + (4*(11*A - 6*B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (2*(A - B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",1
165,1,1346,254,7.8087579,"\int \frac{A+B \cos (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])/(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{\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)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{2 (A-B) \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)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 (16 A-11 B) \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)\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)) \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{(33 A-13 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{7 (17 A-7 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{7 (17 A-7 B) \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) \sin (c+d x)}{6 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{7 (17 A-7 B) \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\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}-\frac{(A-B) \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}",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 - (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]) + (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])*Csc[c/2]*Sec[c/2]*Sec[c])/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(59*A*Sin[(d*x)/2] - 29*B*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]))/(15*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*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)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (2*(A - B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",0
166,1,135,221,1.0303185,"\int \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(15 \sqrt{2} (8 A+7 B) \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 (40 A+53 B) \cos (c+d x)+4 (8 A+7 B) \cos (2 (c+d x))+152 A+12 B \cos (3 (c+d x))+133 B)\right)}{384 d}","\frac{a (8 A+7 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{5 a (8 A+7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{5 \sqrt{a} (8 A+7 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{5 a (8 A+7 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \cos ^{\frac{7}{2}}(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]*(15*Sqrt[2]*(8*A + 7*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(152*A + 133*B + 2*(40*A + 53*B)*Cos[c + d*x] + 4*(8*A + 7*B)*Cos[2*(c + d*x)] + 12*B*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(384*d)","A",1
167,1,118,176,0.5464358,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[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} (6 A+5 B) \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 (6 A+5 B) \cos (c+d x)+18 A+4 B \cos (2 (c+d x))+19 B)\right)}{48 d}","\frac{a (6 A+5 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (6 A+5 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a (6 A+5 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \cos ^{\frac{5}{2}}(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]*(6*A + 5*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(18*A + 19*B + 2*(6*A + 5*B)*Cos[c + d*x] + 4*B*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
168,1,100,131,0.2991966,"\int \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]*(A + B*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} (4 A+3 B) \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 A+2 B \cos (c+d x)+3 B)\right)}{8 d}","\frac{\sqrt{a} (4 A+3 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a (4 A+3 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(Sqrt[2]*(4*A + 3*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(4*A + 3*B + 2*B*Cos[c + d*x])*Sin[(c + d*x)/2]))/(8*d)","A",1
169,1,83,78,0.1594567,"\int \frac{\sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/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} (2 A+B) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 B \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)}\right)}{2 d}","\frac{\sqrt{a} (2 A+B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{a B \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(Sqrt[2]*(2*A + B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*B*Sqrt[Cos[c + d*x]]*Sin[(c + d*x)/2]))/(2*d)","A",1
170,1,86,76,0.1698189,"\int \frac{\sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/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 A \sin \left(\frac{1}{2} (c+d x)\right)+\sqrt{2} B \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}\right)}{d \sqrt{\cos (c+d x)}}","\frac{2 a A \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 \sqrt{a} B \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]*B*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*A*Sin[(c + d*x)/2]))/(d*Sqrt[Cos[c + d*x]])","A",1
171,1,57,85,0.151635,"\int \frac{\sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((2 A+3 B) \cos (c+d x)+A)}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 a (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 A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"(2*Sqrt[a*(1 + Cos[c + d*x])]*(A + (2*A + 3*B)*Cos[c + d*x])*Tan[(c + d*x)/2])/(3*d*Cos[c + d*x]^(3/2))","A",1
172,1,78,130,0.2528949,"\int \frac{\sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((4 A+5 B) \cos (c+d x)+(4 A+5 B) \cos (2 (c+d x))+7 A+5 B)}{15 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 a (4 A+5 B) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a (4 A+5 B) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"(2*Sqrt[a*(1 + Cos[c + d*x])]*(7*A + 5*B + (4*A + 5*B)*Cos[c + d*x] + (4*A + 5*B)*Cos[2*(c + d*x)])*Tan[(c + d*x)/2])/(15*d*Cos[c + d*x]^(5/2))","A",1
173,1,102,175,0.3960614,"\int \frac{\sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(9/2),x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (9 (6 A+7 B) \cos (c+d x)+2 (6 A+7 B) \cos (2 (c+d x))+12 A \cos (3 (c+d x))+27 A+14 B \cos (3 (c+d x))+14 B)}{105 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{8 a (6 A+7 B) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a (6 A+7 B) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a (6 A+7 B) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"(2*Sqrt[a*(1 + Cos[c + d*x])]*(27*A + 14*B + 9*(6*A + 7*B)*Cos[c + d*x] + 2*(6*A + 7*B)*Cos[2*(c + d*x)] + 12*A*Cos[3*(c + d*x)] + 14*B*Cos[3*(c + d*x)])*Tan[(c + d*x)/2])/(105*d*Cos[c + d*x]^(7/2))","A",1
174,1,136,227,1.1218139,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)*(A + B*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} (88 A+75 B) \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 (88 A+93 B) \cos (c+d x)+4 (8 A+15 B) \cos (2 (c+d x))+296 A+12 B \cos (3 (c+d x))+285 B)\right)}{384 d}","\frac{a^{3/2} (88 A+75 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^2 (8 A+9 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (88 A+75 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (88 A+75 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \cos ^{\frac{5}{2}}(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]*(3*Sqrt[2]*(88*A + 75*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(296*A + 285*B + 2*(88*A + 93*B)*Cos[c + d*x] + 4*(8*A + 15*B)*Cos[2*(c + d*x)] + 12*B*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(384*d)","A",1
175,1,119,180,0.6854844,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)*(A + B*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} (14 A+11 B) \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 (6 A+11 B) \cos (c+d x)+42 A+4 B \cos (2 (c+d x))+37 B)\right)}{48 d}","\frac{a^{3/2} (14 A+11 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 (6 A+7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (14 A+11 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(3*Sqrt[2]*(14*A + 11*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(42*A + 37*B + 2*(6*A + 11*B)*Cos[c + d*x] + 4*B*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
176,1,101,133,0.3838484,"\int \frac{(a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/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(\sqrt{2} (12 A+7 B) \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 A+2 B \cos (c+d x)+7 B)\right)}{8 d}","\frac{a^{3/2} (12 A+7 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a^2 (4 A+5 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(Sqrt[2]*(12*A + 7*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(4*A + 7*B + 2*B*Cos[c + d*x])*Sin[(c + d*x)/2]))/(8*d)","A",1
177,1,107,126,0.3310243,"\int \frac{(a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/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} (2 A+3 B) \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) (2 A+B \cos (c+d x))\right)}{2 d \sqrt{\cos (c+d x)}}","\frac{a^{3/2} (2 A+3 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a^2 (2 A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}+\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]*(Sqrt[2]*(2*A + 3*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*(2*A + B*Cos[c + d*x])*Sin[(c + d*x)/2]))/(2*d*Sqrt[Cos[c + d*x]])","A",1
178,1,106,125,0.3855898,"\int \frac{(a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/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(2 \sin \left(\frac{1}{2} (c+d x)\right) ((5 A+3 B) \cos (c+d x)+A)+3 \sqrt{2} B \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^{3/2} B \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+3 B) \sin (c+d x)}{3 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}}{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]*B*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(3/2) + 2*(A + (5*A + 3*B)*Cos[c + d*x])*Sin[(c + d*x)/2]))/(3*d*Cos[c + d*x]^(3/2))","A",1
179,1,80,134,0.3272863,"\int \frac{(a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} (2 (9 A+5 B) \cos (c+d x)+(18 A+25 B) \cos (2 (c+d x))+24 A+25 B)}{15 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 a^2 (6 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^2 (18 A+25 B) \sin (c+d x)}{15 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{5}{2}}(c+d x)}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(24*A + 25*B + 2*(9*A + 5*B)*Cos[c + d*x] + (18*A + 25*B)*Cos[2*(c + d*x)])*Tan[(c + d*x)/2])/(15*d*Cos[c + d*x]^(5/2))","A",1
180,1,102,181,0.5384456,"\int \frac{(a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/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)} (3 (78 A+77 B) \cos (c+d x)+(52 A+63 B) \cos (2 (c+d x))+52 A \cos (3 (c+d x))+82 A+63 B \cos (3 (c+d x))+63 B)}{105 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 a^2 (52 A+63 B) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (8 A+7 B) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (52 A+63 B) \sin (c+d x)}{105 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}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(82*A + 63*B + 3*(78*A + 77*B)*Cos[c + d*x] + (52*A + 63*B)*Cos[2*(c + d*x)] + 52*A*Cos[3*(c + d*x)] + 63*B*Cos[3*(c + d*x)])*Tan[(c + d*x)/2])/(105*d*Cos[c + d*x]^(7/2))","A",1
181,1,124,228,0.6914335,"\int \frac{(a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/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)} ((374 A+324 B) \cos (c+d x)+11 (34 A+39 B) \cos (2 (c+d x))+68 A \cos (3 (c+d x))+68 A \cos (4 (c+d x))+376 A+78 B \cos (3 (c+d x))+78 B \cos (4 (c+d x))+351 B)}{315 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{8 a^2 (34 A+39 B) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (34 A+39 B) \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (10 A+9 B) \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (34 A+39 B) \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}}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(376*A + 351*B + (374*A + 324*B)*Cos[c + d*x] + 11*(34*A + 39*B)*Cos[2*(c + d*x)] + 68*A*Cos[3*(c + d*x)] + 78*B*Cos[3*(c + d*x)] + 68*A*Cos[4*(c + d*x)] + 78*B*Cos[4*(c + d*x)])*Tan[(c + d*x)/2])/(315*d*Cos[c + d*x]^(9/2))","A",1
182,1,159,274,1.9783979,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)*(A + B*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(15 \sqrt{2} (326 A+283 B) \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)} ((3620 A+3874 B) \cos (c+d x)+4 (230 A+331 B) \cos (2 (c+d x))+120 A \cos (3 (c+d x))+5810 A+348 B \cos (3 (c+d x))+48 B \cos (4 (c+d x))+5521 B)\right)}{3840 d}","\frac{a^{5/2} (326 A+283 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{128 d}+\frac{a^3 (170 A+157 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{240 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (326 A+283 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{192 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (326 A+283 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{128 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (10 A+13 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{40 d}+\frac{a B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(15*Sqrt[2]*(326*A + 283*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(5810*A + 5521*B + (3620*A + 3874*B)*Cos[c + d*x] + 4*(230*A + 331*B)*Cos[2*(c + d*x)] + 120*A*Cos[3*(c + d*x)] + 348*B*Cos[3*(c + d*x)] + 48*B*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]))/(3840*d)","A",1
183,1,137,227,1.2467964,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)*(A + B*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} (200 A+163 B) \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)} ((272 A+362 B) \cos (c+d x)+4 (8 A+23 B) \cos (2 (c+d x))+632 A+12 B \cos (3 (c+d x))+581 B)\right)}{384 d}","\frac{a^{5/2} (200 A+163 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{a^3 (104 A+95 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (200 A+163 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (8 A+11 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{24 d}+\frac{a B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{4 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(3*Sqrt[2]*(200*A + 163*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(632*A + 581*B + (272*A + 362*B)*Cos[c + d*x] + 4*(8*A + 23*B)*Cos[2*(c + d*x)] + 12*B*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(384*d)","A",1
184,1,121,180,0.777665,"\int \frac{(a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/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} (38 A+25 B) \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 (6 A+17 B) \cos (c+d x)+66 A+4 B \cos (2 (c+d x))+79 B)\right)}{48 d}","\frac{a^{5/2} (38 A+25 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^3 (54 A+49 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (2 A+3 B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{4 d}+\frac{a B \sin (c+d x) \sqrt{\cos (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]*(3*Sqrt[2]*(38*A + 25*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(66*A + 79*B + 2*(6*A + 17*B)*Cos[c + d*x] + 4*B*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
185,1,126,178,0.6855867,"\int \frac{(a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/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(\sqrt{2} (20 A+19 B) \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) ((4 A+11 B) \cos (c+d x)+8 A+B \cos (2 (c+d x))+B)\right)}{8 d \sqrt{\cos (c+d x)}}","\frac{a^{5/2} (20 A+19 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}-\frac{a^3 (4 A-9 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (4 A-B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}+\frac{2 a A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{d \sqrt{\cos (c+d x)}}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(Sqrt[2]*(20*A + 19*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*(8*A + B + (4*A + 11*B)*Cos[c + d*x] + B*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(8*d*Sqrt[Cos[c + d*x]])","A",1
186,1,130,173,0.7134273,"\int \frac{(a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/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(3 \sqrt{2} (2 A+5 B) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{3}{2}}(c+d x)+\sin \left(\frac{1}{2} (c+d x)\right) (4 (8 A+3 B) \cos (c+d x)+4 A+3 B \cos (2 (c+d x))+3 B)\right)}{6 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{a^{5/2} (2 A+5 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a^3 (14 A+3 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (2 A+B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{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)}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(3*Sqrt[2]*(2*A + 5*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(3/2) + (4*A + 3*B + 4*(8*A + 3*B)*Cos[c + d*x] + 3*B*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(6*d*Cos[c + d*x]^(3/2))","A",1
187,1,130,172,0.7825466,"\int \frac{(a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/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) (2 (14 A+5 B) \cos (c+d x)+(43 A+40 B) \cos (2 (c+d x))+49 A+40 B)+30 \sqrt{2} B \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^{5/2} B \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+35 B) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (8 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 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(30*Sqrt[2]*B*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(5/2) + 2*(49*A + 40*B + 2*(14*A + 5*B)*Cos[c + d*x] + (43*A + 40*B)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(30*d*Cos[c + d*x]^(5/2))","A",1
188,1,104,181,0.6305729,"\int \frac{(a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(9/2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} ((930 A+987 B) \cos (c+d x)+2 (115 A+98 B) \cos (2 (c+d x))+230 A \cos (3 (c+d x))+290 A+301 B \cos (3 (c+d x))+196 B)}{210 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 a^3 (10 A+11 B) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (230 A+301 B) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (10 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 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(290*A + 196*B + (930*A + 987*B)*Cos[c + d*x] + 2*(115*A + 98*B)*Cos[2*(c + d*x)] + 230*A*Cos[3*(c + d*x)] + 301*B*Cos[3*(c + d*x)])*Tan[(c + d*x)/2])/(210*d*Cos[c + d*x]^(7/2))","A",1
189,1,126,228,0.8576499,"\int \frac{(a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/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)} ((1396 A+1215 B) \cos (c+d x)+2 (803 A+870 B) \cos (2 (c+d x))+292 A \cos (3 (c+d x))+292 A \cos (4 (c+d x))+1454 A+345 B \cos (3 (c+d x))+345 B \cos (4 (c+d x))+1395 B)}{630 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{2 a^3 (292 A+345 B) \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (124 A+135 B) \sin (c+d x)}{315 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a^3 (292 A+345 B) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (4 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 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(1454*A + 1395*B + (1396*A + 1215*B)*Cos[c + d*x] + 2*(803*A + 870*B)*Cos[2*(c + d*x)] + 292*A*Cos[3*(c + d*x)] + 345*B*Cos[3*(c + d*x)] + 292*A*Cos[4*(c + d*x)] + 345*B*Cos[4*(c + d*x)])*Tan[(c + d*x)/2])/(630*d*Cos[c + d*x]^(9/2))","A",1
190,1,147,275,0.9861905,"\int \frac{(a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/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)} ((25070 A+24827 B) \cos (c+d x)+(9230 A+9284 B) \cos (2 (c+d x))+9230 A \cos (3 (c+d x))+1420 A \cos (4 (c+d x))+1420 A \cos (5 (c+d x))+9070 A+10439 B \cos (3 (c+d x))+1606 B \cos (4 (c+d x))+1606 B \cos (5 (c+d x))+7678 B)}{6930 d \cos ^{\frac{11}{2}}(c+d x)}","\frac{8 a^3 (710 A+803 B) \sin (c+d x)}{3465 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (710 A+803 B) \sin (c+d x)}{1155 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (194 A+209 B) \sin (c+d x)}{693 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a^3 (710 A+803 B) \sin (c+d x)}{3465 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (14 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 A \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d \cos ^{\frac{11}{2}}(c+d x)}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(9070*A + 7678*B + (25070*A + 24827*B)*Cos[c + d*x] + (9230*A + 9284*B)*Cos[2*(c + d*x)] + 9230*A*Cos[3*(c + d*x)] + 10439*B*Cos[3*(c + d*x)] + 1420*A*Cos[4*(c + d*x)] + 1606*B*Cos[4*(c + d*x)] + 1420*A*Cos[5*(c + d*x)] + 1606*B*Cos[5*(c + d*x)])*Tan[(c + d*x)/2])/(6930*d*Cos[c + d*x]^(11/2))","A",1
191,1,348,190,1.9707163,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/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+2 B \cos (c+d x)-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) \log \left(1+e^{i (c+d x)}\right)+i (4 A-7 B) \sinh ^{-1}\left(e^{i (c+d x)}\right)-4 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)-4 A d x+7 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)+7 B d x\right)}{d \sqrt{1+e^{2 i (c+d x)}}}\right)}{8 \sqrt{a (\cos (c+d x)+1)}}","-\frac{(4 A-7 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) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (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 \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))]*(-4*A*d*x + 7*B*d*x + I*(4*A - 7*B)*ArcSinh[E^(I*(c + d*x))] - (8*I)*Sqrt[2]*(A - B)*Log[1 + E^(I*(c + d*x))] - (4*I)*A*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] + (7*I)*B*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))]]))/(d*Sqrt[1 + E^((2*I)*(c + d*x))]) + (4*Sqrt[Cos[c + d*x]]*(4*A - B + 2*B*Cos[c + d*x])*Sin[(c + d*x)/2])/d))/(8*Sqrt[a*(1 + Cos[c + d*x])])","C",1
192,1,222,141,1.2748679,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(\frac{4 B \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)}}{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((2 A-B) \sinh ^{-1}\left(e^{i (c+d x)}\right)+2 \sqrt{2} (A-B) \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+(B-2 A) \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{d \sqrt{1+e^{2 i (c+d x)}}}\right)}{2 \sqrt{a (\cos (c+d x)+1)}}","\frac{(2 A-B) \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) \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 \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*(((-I)*Sqrt[2]*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*((2*A - B)*ArcSinh[E^(I*(c + d*x))] + 2*Sqrt[2]*(A - B)*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] + (-2*A + B)*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))/(d*Sqrt[1 + E^((2*I)*(c + d*x))]) + (4*B*Sqrt[Cos[c + d*x]]*Sin[(c + d*x)/2])/d))/(2*Sqrt[a*(1 + Cos[c + d*x])])","C",1
193,1,82,100,0.1459964,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]),x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left((A-B) \tan ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)+\sqrt{2} B \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{d \sqrt{a (\cos (c+d x)+1)}}","\frac{\sqrt{2} (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{2 B \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(2*(Sqrt[2]*B*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + (A - B)*ArcTan[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]])*Cos[(c + d*x)/2])/(d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
194,1,203,99,1.6364557,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]),x]","\frac{2 \sin \left(\frac{1}{2} (c+d x)\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(10 B \cos (c+d x)-(A-B) \left(\frac{1}{2} \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)-\frac{5}{4} (4 \cos (c+d x)+\cos (2 (c+d x))+1) \csc ^4\left(\frac{1}{2} (c+d x)\right) \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)\right)}{5 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)}}","\frac{2 A \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (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}",1,"(2*Cos[(c + d*x)/2]*Sin[(c + d*x)/2]*(10*B*Cos[c + d*x] - (A - B)*((-5*(1 + 4*Cos[c + d*x] + Cos[2*(c + d*x)])*Csc[(c + d*x)/2]^4*(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]]))/4 + (Hypergeometric2F1[2, 5/2, 7/2, -(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]*Sin[c + d*x]*Tan[c + d*x])/2)))/(5*d*Cos[c + d*x]^(3/2)*Sqrt[a*(1 + Cos[c + d*x])])","C",0
195,1,627,142,6.8149504,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]),x]","\frac{2 (A-B) \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{2 (A-3 B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (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{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*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)*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
196,1,1728,187,7.8173576,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]),x]","-\frac{2 (A-B) \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{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{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 (13 A-5 B) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (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{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)) + (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)*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",0
197,1,362,197,2.2021223,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*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 \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 B \cos (c+d x)+3 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(i \sqrt{2} (5 A-9 B) \log \left(1+e^{i (c+d x)}\right)-2 i (2 A-3 B) \sinh ^{-1}\left(e^{i (c+d x)}\right)+4 i A \log \left(1+\sqrt{1+e^{2 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)+4 A d x-6 i B \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)+9 i \sqrt{2} B \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)-6 B d x\right)}{d \sqrt{1+e^{2 i (c+d x)}}}\right)}{2 (a (\cos (c+d x)+1))^{3/2}}","\frac{(2 A-3 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(5 A-9 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-3 B) \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*A*d*x - 6*B*d*x - (2*I)*(2*A - 3*B)*ArcSinh[E^(I*(c + d*x))] + I*Sqrt[2]*(5*A - 9*B)*Log[1 + E^(I*(c + d*x))] + (4*I)*A*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] - (6*I)*B*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))]] + (9*I)*Sqrt[2]*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))]) + (2*Sqrt[Cos[c + d*x]]*(-A + 3*B + 2*B*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
198,1,226,145,1.9071861,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*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-B) \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} (A-5 B) \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+4 B \sinh ^{-1}\left(e^{i (c+d x)}\right)-4 B \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{(A-5 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)}{2 \sqrt{2} a^{3/2} d}+\frac{2 B \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) \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*B*ArcSinh[E^(I*(c + d*x))] - Sqrt[2]*(A - 5*B)*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - 4*B*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))/(Sqrt[2]*d*Sqrt[1 + E^((2*I)*(c + d*x))]) + ((A - B)*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
199,1,212,107,1.154935,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)),x]","\frac{\frac{1}{2} i (A-B) e^{-\frac{1}{2} i (c+d x)} \left(-1+e^{i (c+d x)}\right) \sqrt{1+e^{2 i (c+d x)}} \sqrt{\cos (c+d x)} \cos \left(\frac{1}{2} (c+d x)\right)+i (3 A+B) e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \cos ^3\left(\frac{1}{2} (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)}{d \sqrt{1+e^{2 i (c+d x)}} (a (\cos (c+d x)+1))^{3/2}}","\frac{(3 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)}{2 \sqrt{2} a^{3/2} d}-\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(I*(3*A + B)*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))])]*Cos[(c + d*x)/2]^3 + ((I/2)*(A - B)*(-1 + E^(I*(c + d*x)))*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[(c + d*x)/2]*Sqrt[Cos[c + d*x]])/E^((I/2)*(c + d*x)))/(d*Sqrt[1 + E^((2*I)*(c + d*x))]*(a*(1 + Cos[c + d*x]))^(3/2))","C",1
200,1,423,156,3.8694283,"\int \frac{A+B \cos (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])/(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) \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) \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) \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) \sqrt{\cos (c+d x)}}{\sin \left(\frac{1}{2} (c+d x)\right)-1}-\frac{20 (A-B) \sqrt{\cos (c+d x)}}{\sin \left(\frac{1}{2} (c+d x)\right)+1}+30 (A-B) \tan ^{-1}\left(\frac{1-2 \sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)-30 (A-B) \tan ^{-1}\left(\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)+1}{\sqrt{\cos (c+d x)}}\right)\right)}{10 d (a (\cos (c+d x)+1))^{3/2}}","-\frac{(7 A-3 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A-B) \sin (c+d x)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(A-B) \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)*ArcTan[(1 - 2*Sin[(c + d*x)/2])/Sqrt[Cos[c + d*x]]] - 30*(A - B)*ArcTan[(1 + 2*Sin[(c + d*x)/2])/Sqrt[Cos[c + d*x]]] - (20*(A - B)*Sqrt[Cos[c + d*x]])/(-1 + Sin[(c + d*x)/2]) - (20*(A - B)*Sqrt[Cos[c + d*x]])/(1 + Sin[(c + d*x)/2]) + (5*(A - B)*(-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)*(1 + 2*Sin[(c + d*x)/2]))/(Sqrt[Cos[c + d*x]]*(-1 + Sin[(c + d*x)/2])) + ((A + 3*B)*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
201,1,1054,203,6.8061292,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)),x]","-\frac{(A-B) \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) \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{(A-B) \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) \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) \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) \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) \sin (c+d x)}{6 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(A-B) \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) \sin (c+d x)}{6 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"-1/6*((A - B)*Cos[c/2 + (d*x)/2]^3*(1 - 2*Sin[c/2 + (d*x)/2]))/(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)*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)*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)*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)*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
202,1,376,246,3.5334691,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*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(\sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) ((55 B-15 A) \cos (c+d x)-11 A+8 B \cos (2 (c+d x))+43 B)+\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} (43 A-115 B) \log \left(1+e^{i (c+d x)}\right)-16 i (2 A-5 B) \sinh ^{-1}\left(e^{i (c+d x)}\right)+32 i A \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)-43 i \sqrt{2} A \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)+32 A d x-80 i B \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)+115 i \sqrt{2} B \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)-80 B d x\right)}{\sqrt{1+e^{2 i (c+d x)}}}\right)}{8 d (a (\cos (c+d x)+1))^{5/2}}","\frac{(2 A-5 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(43 A-115 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)}{16 \sqrt{2} a^{5/2} d}-\frac{(11 A-35 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a^2 d \sqrt{a \cos (c+d x)+a}}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(7 A-15 B) \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*A*d*x - 80*B*d*x - (16*I)*(2*A - 5*B)*ArcSinh[E^(I*(c + d*x))] + I*Sqrt[2]*(43*A - 115*B)*Log[1 + E^(I*(c + d*x))] + (32*I)*A*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] - (80*I)*B*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] - (43*I)*Sqrt[2]*A*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]] + (115*I)*Sqrt[2]*B*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]]*(-11*A + 43*B + (-15*A + 55*B)*Cos[c + d*x] + 8*B*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
203,1,246,194,2.173402,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*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(\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) \cos (c+d x)+3 A-11 B)-\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-43 B) \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+32 B \sinh ^{-1}\left(e^{i (c+d x)}\right)-32 B \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-43 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)}{16 \sqrt{2} a^{5/2} d}+\frac{2 B \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) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{(3 A-11 B) \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*B*ArcSinh[E^(I*(c + d*x))] - Sqrt[2]*(3*A - 43*B)*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - 32*B*ArcTanh[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 + (7*A - 15*B)*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
204,1,198,154,1.5117797,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*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) ((A+7 B) \cos (c+d x)+5 A+3 B)+\frac{i (5 A+3 B) 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 A+3 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)}{16 \sqrt{2} a^{5/2} d}+\frac{(A+7 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B) \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*(5*A + 3*B)*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]]*(5*A + 3*B + (A + 7*B)*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
205,1,200,156,1.4725872,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(A + B*Cos[c + d*x])/(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) \cos (c+d x)+13 A-5 B)+\frac{i (19 A+5 B) 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) \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) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \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)*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 + (9*A - B)*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
206,1,217,203,2.770903,"\int \frac{A+B \cos (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])/(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) \cos (c+d x)+(49 A-9 B) \cos (2 (c+d x))+113 A-9 B)}{4 \sqrt{\cos (c+d x)}}-\frac{i (75 A-19 B) 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) \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) \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) \sin (c+d x)}{16 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \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)*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 + 2*(85*A - 13*B)*Cos[c + d*x] + (49*A - 9*B)*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
207,1,239,250,3.5839803,"\int \frac{A+B \cos (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])/(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) \cos (c+d x)+2 (503 A-255 B) \cos (2 (c+d x))+299 A \cos (3 (c+d x))+878 A-147 B \cos (3 (c+d x))-510 B)}{8 \cos ^{\frac{3}{2}}(c+d x)}+\frac{3 i (163 A-75 B) 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) \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) \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) \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) \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) \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)*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 + (1537*A - 825*B)*Cos[c + d*x] + 2*(503*A - 255*B)*Cos[2*(c + d*x)] + 299*A*Cos[3*(c + d*x)] - 147*B*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
208,1,396,293,5.8166697,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx","Integrate[(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(7/2),x]","\frac{\cos ^7\left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{4} \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) ((3172 B-724 A) \cos (c+d x)+(1099 B-247 A) \cos (2 (c+d x))-541 A+96 B \cos (3 (c+d x))+2233 B)+\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(i \sqrt{2} (177 A-637 B) \log \left(1+e^{i (c+d x)}\right)-64 i (2 A-7 B) \sinh ^{-1}\left(e^{i (c+d x)}\right)+128 i A \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)-177 i \sqrt{2} A \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)+128 A d x-448 i B \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)+637 i \sqrt{2} B \log \left(\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)}+1\right)-448 B d x\right)}{\sqrt{1+e^{2 i (c+d x)}}}\right)}{48 d (a (\cos (c+d x)+1))^{7/2}}","\frac{(2 A-7 B) \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{7/2} d}-\frac{(177 A-637 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)}{64 \sqrt{2} a^{7/2} d}-\frac{7 (7 A-27 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 a^3 d \sqrt{a \cos (c+d x)+a}}+\frac{(79 A-259 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}+\frac{(3 A-7 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{5/2}}",1,"(Cos[(c + d*x)/2]^7*((3*Sqrt[2]*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(128*A*d*x - 448*B*d*x - (64*I)*(2*A - 7*B)*ArcSinh[E^(I*(c + d*x))] + I*Sqrt[2]*(177*A - 637*B)*Log[1 + E^(I*(c + d*x))] + (128*I)*A*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] - (448*I)*B*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]] - (177*I)*Sqrt[2]*A*Log[1 - E^(I*(c + d*x)) + Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]] + (637*I)*Sqrt[2]*B*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]]*(-541*A + 2233*B + (-724*A + 3172*B)*Cos[c + d*x] + (-247*A + 1099*B)*Cos[2*(c + d*x)] + 96*B*Cos[3*(c + d*x)])*Sec[(c + d*x)/2]^5*Tan[(c + d*x)/2])/4))/(48*d*(a*(1 + Cos[c + d*x]))^(7/2))","C",1
209,1,266,241,3.2352085,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(7/2),x]","\frac{\cos ^7\left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{4} \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) (4 (25 A-181 B) \cos (c+d x)+(67 A-247 B) \cos (2 (c+d x))+97 A-541 B)-\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(-\sqrt{2} (5 A-177 B) \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+128 B \sinh ^{-1}\left(e^{i (c+d x)}\right)-128 B \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{\sqrt{1+e^{2 i (c+d x)}}}\right)}{48 d (a (\cos (c+d x)+1))^{7/2}}","\frac{(5 A-177 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)}{64 \sqrt{2} a^{7/2} d}+\frac{2 B \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{7/2} d}+\frac{(5 A-49 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{64 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}+\frac{(5 A-17 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{48 a d (a \cos (c+d x)+a)^{5/2}}",1,"(Cos[(c + d*x)/2]^7*(((-3*I)*Sqrt[2]*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(128*B*ArcSinh[E^(I*(c + d*x))] - Sqrt[2]*(5*A - 177*B)*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - 128*B*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))/Sqrt[1 + E^((2*I)*(c + d*x))] + (Sqrt[Cos[c + d*x]]*(97*A - 541*B + 4*(25*A - 181*B)*Cos[c + d*x] + (67*A - 247*B)*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^5*Tan[(c + d*x)/2])/4))/(48*d*(a*(1 + Cos[c + d*x]))^(7/2))","C",1
210,1,217,201,2.35719,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(7/2),x]","\frac{\cos ^7\left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{8} \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) (20 (7 A+5 B) \cos (c+d x)+(17 A+67 B) \cos (2 (c+d x))+59 A+97 B)+\frac{3 i (7 A+5 B) 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)}{24 d (a (\cos (c+d x)+1))^{7/2}}","\frac{(7 A+5 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)}{64 \sqrt{2} a^{7/2} d}+\frac{(17 A+67 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}+\frac{(A-13 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{48 a d (a \cos (c+d x)+a)^{5/2}}",1,"(Cos[(c + d*x)/2]^7*(((3*I)*(7*A + 5*B)*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]]*(59*A + 97*B + 20*(7*A + 5*B)*Cos[c + d*x] + (17*A + 67*B)*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^5*Tan[(c + d*x)/2])/8))/(24*d*(a*(1 + Cos[c + d*x]))^(7/2))","C",1
211,1,215,201,2.1440047,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{(a+a \cos (c+d x))^{7/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^(7/2),x]","\frac{\cos ^7\left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{8} \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) (4 (A+35 B) \cos (c+d x)+(17 B-5 A) \cos (2 (c+d x))+73 A+59 B)+\frac{3 i (13 A+7 B) 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)}{24 d (a (\cos (c+d x)+1))^{7/2}}","\frac{(13 A+7 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)}{64 \sqrt{2} a^{7/2} d}-\frac{(5 A-17 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{(A+3 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{5/2}}+\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}",1,"(Cos[(c + d*x)/2]^7*(((3*I)*(13*A + 7*B)*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]]*(73*A + 59*B + 4*(A + 35*B)*Cos[c + d*x] + (-5*A + 17*B)*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^5*Tan[(c + d*x)/2])/8))/(24*d*(a*(1 + Cos[c + d*x]))^(7/2))","C",1
212,1,216,203,2.1550288,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{7/2}} \, dx","Integrate[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(7/2)),x]","\frac{\cos ^7\left(\frac{1}{2} (c+d x)\right) \left(-\frac{1}{8} \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) ((532 A-4 B) \cos (c+d x)+(103 A+5 B) \cos (2 (c+d x))+493 A-73 B)+\frac{3 i (63 A+13 B) 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)}{24 d (a (\cos (c+d x)+1))^{7/2}}","\frac{(63 A+13 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)}{64 \sqrt{2} a^{7/2} d}-\frac{(103 A+5 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(5 A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{5/2}}-\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}",1,"(Cos[(c + d*x)/2]^7*(((3*I)*(63*A + 13*B)*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]]*(493*A - 73*B + (532*A - 4*B)*Cos[c + d*x] + (103*A + 5*B)*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^5*Tan[(c + d*x)/2])/8))/(24*d*(a*(1 + Cos[c + d*x]))^(7/2))","C",1
213,1,240,250,2.9249779,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}} \, dx","Integrate[(A + B*Cos[c + d*x])/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(7/2)),x]","\frac{\cos ^7\left(\frac{1}{2} (c+d x)\right) \left(\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) (9 (941 A-121 B) \cos (c+d x)+4 (937 A-133 B) \cos (2 (c+d x))+691 A \cos (3 (c+d x))+5284 A-103 B \cos (3 (c+d x))-532 B)}{16 \sqrt{\cos (c+d x)}}-\frac{9 i (121 A-21 B) 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)}{24 d (a (\cos (c+d x)+1))^{7/2}}","-\frac{3 (121 A-21 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)}{64 \sqrt{2} a^{7/2} d}+\frac{(691 A-103 B) \sin (c+d x)}{192 a^3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(199 A-43 B) \sin (c+d x)}{192 a^2 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{(19 A-7 B) \sin (c+d x)}{48 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac{(A-B) \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{7/2}}",1,"(Cos[(c + d*x)/2]^7*(((-9*I)*(121*A - 21*B)*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))] + ((5284*A - 532*B + 9*(941*A - 121*B)*Cos[c + d*x] + 4*(937*A - 133*B)*Cos[2*(c + d*x)] + 691*A*Cos[3*(c + d*x)] - 103*B*Cos[3*(c + d*x)])*Sec[(c + d*x)/2]^5*Tan[(c + d*x)/2])/(16*Sqrt[Cos[c + d*x]])))/(24*d*(a*(1 + Cos[c + d*x]))^(7/2))","C",1
214,1,262,297,5.4046874,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}} \, dx","Integrate[(A + B*Cos[c + d*x])/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(7/2)),x]","\frac{\cos ^7\left(\frac{1}{2} (c+d x)\right) \left(-\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) (4 (9415 A-3579 B) \cos (c+d x)+8 (3069 A-1145 B) \cos (2 (c+d x))+10164 A \cos (3 (c+d x))+1887 A \cos (4 (c+d x))+21641 A-3748 B \cos (3 (c+d x))-691 B \cos (4 (c+d x))-8469 B)}{32 \cos ^{\frac{3}{2}}(c+d x)}+\frac{3 i (1015 A-363 B) 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)}{24 d (a (\cos (c+d x)+1))^{7/2}}","\frac{(1015 A-363 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)}{64 \sqrt{2} a^{7/2} d}+\frac{(579 A-199 B) \sin (c+d x)}{192 a^3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{(1887 A-691 B) \sin (c+d x)}{192 a^3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{(109 A-41 B) \sin (c+d x)}{64 a^2 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(23 A-11 B) \sin (c+d x)}{48 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}-\frac{(A-B) \sin (c+d x)}{6 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}",1,"(Cos[(c + d*x)/2]^7*(((3*I)*(1015*A - 363*B)*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))] - ((21641*A - 8469*B + 4*(9415*A - 3579*B)*Cos[c + d*x] + 8*(3069*A - 1145*B)*Cos[2*(c + d*x)] + 10164*A*Cos[3*(c + d*x)] - 3748*B*Cos[3*(c + d*x)] + 1887*A*Cos[4*(c + d*x)] - 691*B*Cos[4*(c + d*x)])*Sec[(c + d*x)/2]^5*Tan[(c + d*x)/2])/(32*Cos[c + d*x]^(3/2))))/(24*d*(a*(1 + Cos[c + d*x]))^(7/2))","C",1
215,1,91,105,0.2234938,"\int \cos ^2(c+d x) (a+b \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","\frac{-32 (a B+A b) \sin ^3(c+d x)+96 (a B+A b) \sin (c+d x)+24 (a A+b B) \sin (2 (c+d x))+48 a A c+48 a A d x+3 b B \sin (4 (c+d x))+36 b B c+36 b B d x}{96 d}","-\frac{(a B+A b) \sin ^3(c+d x)}{3 d}+\frac{(a B+A b) \sin (c+d x)}{d}+\frac{(4 a A+3 b B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x (4 a A+3 b B)+\frac{b B \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"(48*a*A*c + 36*b*B*c + 48*a*A*d*x + 36*b*B*d*x + 96*(A*b + a*B)*Sin[c + d*x] - 32*(A*b + a*B)*Sin[c + d*x]^3 + 24*(a*A + b*B)*Sin[2*(c + d*x)] + 3*b*B*Sin[4*(c + d*x)])/(96*d)","A",1
216,1,75,84,0.1621525,"\int \cos (c+d x) (a+b \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","\frac{3 (4 a A+3 b B) \sin (c+d x)+3 (a B+A b) \sin (2 (c+d x))+6 a B c+6 a B d x+6 A b c+6 A b d x+b B \sin (3 (c+d x))}{12 d}","\frac{(3 a A+2 b B) \sin (c+d x)}{3 d}+\frac{(a B+A b) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (a B+A b)+\frac{b B \sin (c+d x) \cos ^2(c+d x)}{3 d}",1,"(6*A*b*c + 6*a*B*c + 6*A*b*d*x + 6*a*B*d*x + 3*(4*a*A + 3*b*B)*Sin[c + d*x] + 3*(A*b + a*B)*Sin[2*(c + d*x)] + b*B*Sin[3*(c + d*x)])/(12*d)","A",1
217,1,51,52,0.0854084,"\int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","\frac{4 (a B+A b) \sin (c+d x)+4 a A d x+b B \sin (2 (c+d x))+2 b B c+2 b B d x}{4 d}","\frac{(a B+A b) \sin (c+d x)}{d}+\frac{1}{2} x (2 a A+b B)+\frac{b B \sin (c+d x) \cos (c+d x)}{2 d}",1,"(2*b*B*c + 4*a*A*d*x + 2*b*B*d*x + 4*(A*b + a*B)*Sin[c + d*x] + b*B*Sin[2*(c + d*x)])/(4*d)","A",1
218,1,46,35,0.0271344,"\int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+a B x+A b x+\frac{b B \sin (c) \cos (d x)}{d}+\frac{b B \cos (c) \sin (d x)}{d}","x (a B+A b)+\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b B \sin (c+d x)}{d}",1,"A*b*x + a*B*x + (a*A*ArcTanh[Sin[c + d*x]])/d + (b*B*Cos[d*x]*Sin[c])/d + (b*B*Cos[c]*Sin[d*x])/d","A",1
219,1,43,35,0.0135706,"\int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{a A \tan (c+d x)}{d}+\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A b \tanh ^{-1}(\sin (c+d x))}{d}+b B x","\frac{(a B+A b) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A \tan (c+d x)}{d}+b B x",1,"b*B*x + (A*b*ArcTanh[Sin[c + d*x]])/d + (a*B*ArcTanh[Sin[c + d*x]])/d + (a*A*Tan[c + d*x])/d","A",1
220,1,75,61,0.020418,"\int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x])*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 b \tan (c+d x)}{d}+\frac{b B \tanh ^{-1}(\sin (c+d x))}{d}","\frac{(a B+A b) \tan (c+d x)}{d}+\frac{(a A+2 b B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a A \tan (c+d x) \sec (c+d x)}{2 d}",1,"(a*A*ArcTanh[Sin[c + d*x]])/(2*d) + (b*B*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
221,1,67,93,0.2819416,"\int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{3 (a B+A b) \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+6 b B\right)}{6 d}","\frac{(2 a A+3 b B) \tan (c+d x)}{3 d}+\frac{(a B+A b) \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)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(6*a*A + 6*b*B + 3*(A*b + a*B)*Sec[c + d*x] + 2*a*A*Tan[c + d*x]^2))/(6*d)","A",1
222,1,85,114,0.6196771,"\int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^5,x]","\frac{3 (3 a A+4 b B) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x) \left(8 (a B+A b) (\cos (2 (c+d x))+2) \sec (c+d x)+6 a A \sec ^2(c+d x)+9 a A+12 b B\right)}{24 d}","\frac{(a B+A b) \tan ^3(c+d x)}{3 d}+\frac{(a B+A b) \tan (c+d x)}{d}+\frac{(3 a A+4 b B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(3 a A+4 b B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"(3*(3*a*A + 4*b*B)*ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*(9*a*A + 12*b*B + 8*(A*b + a*B)*(2 + Cos[2*(c + d*x)])*Sec[c + d*x] + 6*a*A*Sec[c + d*x]^2)*Tan[c + d*x])/(24*d)","A",1
223,1,146,189,0.4833166,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]","\frac{60 (c+d x) \left(4 a^2 A+6 a b B+3 A b^2\right)+60 \left(6 a^2 B+12 a A b+5 b^2 B\right) \sin (c+d x)+120 \left(a^2 A+2 a b B+A b^2\right) \sin (2 (c+d x))+10 \left(4 a^2 B+8 a A b+5 b^2 B\right) \sin (3 (c+d x))+15 b (2 a B+A b) \sin (4 (c+d x))+6 b^2 B \sin (5 (c+d x))}{480 d}","\frac{\left(4 a^2 A+6 a b B+3 A b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(4 a^2 A+6 a b B+3 A b^2\right)-\frac{\left(5 a (a B+2 A b)+4 b^2 B\right) \sin ^3(c+d x)}{15 d}+\frac{\left(5 a (a B+2 A b)+4 b^2 B\right) \sin (c+d x)}{5 d}+\frac{b (6 a B+5 A b) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{b B \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))}{5 d}",1,"(60*(4*a^2*A + 3*A*b^2 + 6*a*b*B)*(c + d*x) + 60*(12*a*A*b + 6*a^2*B + 5*b^2*B)*Sin[c + d*x] + 120*(a^2*A + A*b^2 + 2*a*b*B)*Sin[2*(c + d*x)] + 10*(8*a*A*b + 4*a^2*B + 5*b^2*B)*Sin[3*(c + d*x)] + 15*b*(A*b + 2*a*B)*Sin[4*(c + d*x)] + 6*b^2*B*Sin[5*(c + d*x)])/(480*d)","A",1
224,1,118,170,0.4552008,"\int \cos (c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]","\frac{12 (c+d x) \left(4 a^2 B+8 a A b+3 b^2 B\right)+24 \left(4 a^2 A+6 a b B+3 A b^2\right) \sin (c+d x)+24 \left(a^2 B+2 a A b+b^2 B\right) \sin (2 (c+d x))+8 b (2 a B+A b) \sin (3 (c+d x))+3 b^2 B \sin (4 (c+d x))}{96 d}","\frac{\left(-2 a^2 B+8 a A b+9 b^2 B\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(4 a^2 B+8 a A b+3 b^2 B\right)+\frac{\left(a^3 (-B)+4 a^2 A b+8 a b^2 B+4 A b^3\right) \sin (c+d x)}{6 b d}+\frac{(4 A b-a B) \sin (c+d x) (a+b \cos (c+d x))^2}{12 b d}+\frac{B \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d}",1,"(12*(8*a*A*b + 4*a^2*B + 3*b^2*B)*(c + d*x) + 24*(4*a^2*A + 3*A*b^2 + 6*a*b*B)*Sin[c + d*x] + 24*(2*a*A*b + a^2*B + b^2*B)*Sin[2*(c + d*x)] + 8*b*(A*b + 2*a*B)*Sin[3*(c + d*x)] + 3*b^2*B*Sin[4*(c + d*x)])/(96*d)","A",1
225,1,90,107,0.2254323,"\int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]","\frac{6 (c+d x) \left(2 a^2 A+2 a b B+A b^2\right)+3 \left(4 a^2 B+8 a A b+3 b^2 B\right) \sin (c+d x)+3 b (2 a B+A b) \sin (2 (c+d x))+b^2 B \sin (3 (c+d x))}{12 d}","\frac{2 \left(a^2 B+3 a A b+b^2 B\right) \sin (c+d x)}{3 d}+\frac{1}{2} x \left(2 a^2 A+2 a b B+A b^2\right)+\frac{b (2 a B+3 A b) \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,"(6*(2*a^2*A + A*b^2 + 2*a*b*B)*(c + d*x) + 3*(8*a*A*b + 4*a^2*B + 3*b^2*B)*Sin[c + d*x] + 3*b*(A*b + 2*a*B)*Sin[2*(c + d*x)] + b^2*B*Sin[3*(c + d*x)])/(12*d)","A",1
226,1,120,86,0.2333833,"\int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{2 (c+d x) \left(2 a^2 B+4 a A b+b^2 B\right)-4 a^2 A \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 \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 B+A b) \sin (c+d x)+b^2 B \sin (2 (c+d x))}{4 d}","\frac{1}{2} x \left(2 a^2 B+4 a A b+b^2 B\right)+\frac{a^2 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b (3 a B+2 A b) \sin (c+d x)}{2 d}+\frac{b B \sin (c+d x) (a+b \cos (c+d x))}{2 d}",1,"(2*(4*a*A*b + 2*a^2*B + b^2*B)*(c + d*x) - 4*a^2*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*a^2*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 4*b*(A*b + 2*a*B)*Sin[c + d*x] + b^2*B*Sin[2*(c + d*x)])/(4*d)","A",1
227,1,109,60,0.4965214,"\int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{a^2 A \tan (c+d x)+b (c+d x) (2 a B+A b)-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)+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)+b^2 B \sin (c+d x)}{d}","\frac{a^2 A \tan (c+d x)}{d}+\frac{a (a B+2 A b) \tanh ^{-1}(\sin (c+d x))}{d}+b x (2 a B+A b)+\frac{b^2 B \sin (c+d x)}{d}",1,"(b*(A*b + 2*a*B)*(c + d*x) - a*(2*A*b + a*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + a*(2*A*b + a*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + b^2*B*Sin[c + d*x] + a^2*A*Tan[c + d*x])/d","A",1
228,1,67,80,0.2745278,"\int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{\left(a^2 A+4 a b B+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))+a \tan (c+d x) (a A \sec (c+d x)+2 a B+4 A b)+2 b^2 B d x}{2 d}","\frac{\left(a^2 A+4 a b B+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 A \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a (a B+2 A b) \tan (c+d x)}{d}+b^2 B x",1,"(2*b^2*B*d*x + (a^2*A + 2*A*b^2 + 4*a*b*B)*ArcTanh[Sin[c + d*x]] + a*(4*A*b + 2*a*B + a*A*Sec[c + d*x])*Tan[c + d*x])/(2*d)","A",1
229,1,92,116,0.4722982,"\int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{3 \left(a^2 B+2 a A b+2 b^2 B\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(2 \left(a^2 A \tan ^2(c+d x)+3 a^2 A+6 a b B+3 A b^2\right)+3 a (a B+2 A b) \sec (c+d x)\right)}{6 d}","\frac{\left(2 a^2 A+6 a b B+3 A b^2\right) \tan (c+d x)}{3 d}+\frac{\left(a^2 B+2 a A b+2 b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 A \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a (a B+2 A b) \tan (c+d x) \sec (c+d x)}{2 d}",1,"(3*(2*a*A*b + a^2*B + 2*b^2*B)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(3*a*(2*A*b + a*B)*Sec[c + d*x] + 2*(3*a^2*A + 3*A*b^2 + 6*a*b*B + a^2*A*Tan[c + d*x]^2)))/(6*d)","A",1
230,1,120,156,0.7237583,"\int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^5,x]","\frac{3 \left(3 a^2 A+8 a b B+4 A b^2\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(3 \left(3 a^2 A+8 a b B+4 A b^2\right) \sec (c+d x)+24 \left(a^2 B+2 a A b+b^2 B\right)+6 a^2 A \sec ^3(c+d x)+8 a (a B+2 A b) \tan ^2(c+d x)\right)}{24 d}","\frac{\left(2 a^2 B+4 a A b+3 b^2 B\right) \tan (c+d x)}{3 d}+\frac{\left(3 a^2 A+8 a b B+4 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(3 a^2 A+8 a b B+4 A b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^2 A \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{a (a B+2 A b) \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"(3*(3*a^2*A + 4*A*b^2 + 8*a*b*B)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(24*(2*a*A*b + a^2*B + b^2*B) + 3*(3*a^2*A + 4*A*b^2 + 8*a*b*B)*Sec[c + d*x] + 6*a^2*A*Sec[c + d*x]^3 + 8*a*(2*A*b + a*B)*Tan[c + d*x]^2))/(24*d)","A",1
231,1,289,269,0.6854497,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]),x]","\frac{480 a^3 A c+480 a^3 A d x+80 a^3 B \sin (3 (c+d x))+240 a^2 A b \sin (3 (c+d x))+90 a^2 b B \sin (4 (c+d x))+1080 a^2 b B c+1080 a^2 b B d x+120 \left(6 a^3 B+18 a^2 A b+15 a b^2 B+5 A b^3\right) \sin (c+d x)+15 \left(16 a^3 A+48 a^2 b B+48 a A b^2+15 b^3 B\right) \sin (2 (c+d x))+90 a A b^2 \sin (4 (c+d x))+1080 a A b^2 c+1080 a A b^2 d x+300 a b^2 B \sin (3 (c+d x))+36 a b^2 B \sin (5 (c+d x))+100 A b^3 \sin (3 (c+d x))+12 A b^3 \sin (5 (c+d x))+45 b^3 B \sin (4 (c+d x))+5 b^3 B \sin (6 (c+d x))+300 b^3 B c+300 b^3 B d x}{960 d}","\frac{b \left(14 a^2 B+18 a A b+5 b^2 B\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}-\frac{\left(5 a^3 B+15 a^2 A b+12 a b^2 B+4 A b^3\right) \sin ^3(c+d x)}{15 d}+\frac{\left(5 a^3 B+15 a^2 A b+12 a b^2 B+4 A b^3\right) \sin (c+d x)}{5 d}+\frac{\left(8 a^3 A+18 a^2 b B+18 a A b^2+5 b^3 B\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left(8 a^3 A+18 a^2 b B+18 a A b^2+5 b^3 B\right)+\frac{b^2 (4 a B+3 A b) \sin (c+d x) \cos ^4(c+d x)}{15 d}+\frac{b B \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^2}{6 d}",1,"(480*a^3*A*c + 1080*a*A*b^2*c + 1080*a^2*b*B*c + 300*b^3*B*c + 480*a^3*A*d*x + 1080*a*A*b^2*d*x + 1080*a^2*b*B*d*x + 300*b^3*B*d*x + 120*(18*a^2*A*b + 5*A*b^3 + 6*a^3*B + 15*a*b^2*B)*Sin[c + d*x] + 15*(16*a^3*A + 48*a*A*b^2 + 48*a^2*b*B + 15*b^3*B)*Sin[2*(c + d*x)] + 240*a^2*A*b*Sin[3*(c + d*x)] + 100*A*b^3*Sin[3*(c + d*x)] + 80*a^3*B*Sin[3*(c + d*x)] + 300*a*b^2*B*Sin[3*(c + d*x)] + 90*a*A*b^2*Sin[4*(c + d*x)] + 90*a^2*b*B*Sin[4*(c + d*x)] + 45*b^3*B*Sin[4*(c + d*x)] + 12*A*b^3*Sin[5*(c + d*x)] + 36*a*b^2*B*Sin[5*(c + d*x)] + 5*b^3*B*Sin[6*(c + d*x)])/(960*d)","A",1
232,1,176,243,0.7184291,"\int \cos (c+d x) (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]),x]","\frac{10 b \left(12 a^2 B+12 a A b+5 b^2 B\right) \sin (3 (c+d x))+60 (c+d x) \left(4 a^3 B+12 a^2 A b+9 a b^2 B+3 A b^3\right)+60 \left(8 a^3 A+18 a^2 b B+18 a A b^2+5 b^3 B\right) \sin (c+d x)+120 \left(a^3 B+3 a^2 A b+3 a b^2 B+A b^3\right) \sin (2 (c+d x))+15 b^2 (3 a B+A b) \sin (4 (c+d x))+6 b^3 B \sin (5 (c+d x))}{480 d}","\frac{\left(-3 a^2 B+15 a A b+16 b^2 B\right) \sin (c+d x) (a+b \cos (c+d x))^2}{60 b d}+\frac{\left(-6 a^3 B+30 a^2 A b+71 a b^2 B+45 A b^3\right) \sin (c+d x) \cos (c+d x)}{120 d}+\frac{1}{8} x \left(4 a^3 B+12 a^2 A b+9 a b^2 B+3 A b^3\right)+\frac{\left(-3 a^4 B+15 a^3 A b+52 a^2 b^2 B+60 a A b^3+16 b^4 B\right) \sin (c+d x)}{30 b d}+\frac{(5 A b-a B) \sin (c+d x) (a+b \cos (c+d x))^3}{20 b d}+\frac{B \sin (c+d x) (a+b \cos (c+d x))^4}{5 b d}",1,"(60*(12*a^2*A*b + 3*A*b^3 + 4*a^3*B + 9*a*b^2*B)*(c + d*x) + 60*(8*a^3*A + 18*a*A*b^2 + 18*a^2*b*B + 5*b^3*B)*Sin[c + d*x] + 120*(3*a^2*A*b + A*b^3 + a^3*B + 3*a*b^2*B)*Sin[2*(c + d*x)] + 10*b*(12*a*A*b + 12*a^2*B + 5*b^2*B)*Sin[3*(c + d*x)] + 15*b^2*(A*b + 3*a*B)*Sin[4*(c + d*x)] + 6*b^3*B*Sin[5*(c + d*x)])/(480*d)","A",1
233,1,140,171,0.417114,"\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]),x]","\frac{24 b \left(3 a^2 B+3 a A b+b^2 B\right) \sin (2 (c+d x))+12 (c+d x) \left(8 a^3 A+12 a^2 b B+12 a A b^2+3 b^3 B\right)+24 \left(4 a^3 B+12 a^2 A b+9 a b^2 B+3 A b^3\right) \sin (c+d x)+8 b^2 (3 a B+A b) \sin (3 (c+d x))+3 b^3 B \sin (4 (c+d x))}{96 d}","\frac{b \left(6 a^2 B+20 a A b+9 b^2 B\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{\left(3 a^3 B+16 a^2 A b+12 a b^2 B+4 A b^3\right) \sin (c+d x)}{6 d}+\frac{1}{8} x \left(8 a^3 A+12 a^2 b B+12 a A b^2+3 b^3 B\right)+\frac{(3 a B+4 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{B \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}",1,"(12*(8*a^3*A + 12*a*A*b^2 + 12*a^2*b*B + 3*b^3*B)*(c + d*x) + 24*(12*a^2*A*b + 3*A*b^3 + 4*a^3*B + 9*a*b^2*B)*Sin[c + d*x] + 24*b*(3*a*A*b + 3*a^2*B + b^2*B)*Sin[2*(c + d*x)] + 8*b^2*(A*b + 3*a*B)*Sin[3*(c + d*x)] + 3*b^3*B*Sin[4*(c + d*x)])/(96*d)","A",1
234,1,159,137,0.395875,"\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{-12 a^3 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 a^3 A \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 B+4 a A b+b^2 B\right) \sin (c+d x)+6 (c+d x) \left(2 a^3 B+6 a^2 A b+3 a b^2 B+A b^3\right)+3 b^2 (3 a B+A b) \sin (2 (c+d x))+b^3 B \sin (3 (c+d x))}{12 d}","\frac{a^3 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b \left(8 a^2 B+9 a A b+2 b^2 B\right) \sin (c+d x)}{3 d}+\frac{1}{2} x \left(2 a^3 B+6 a^2 A b+3 a b^2 B+A b^3\right)+\frac{b^2 (5 a B+3 A b) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{b B \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}",1,"(6*(6*a^2*A*b + A*b^3 + 2*a^3*B + 3*a*b^2*B)*(c + d*x) - 12*a^3*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*a^3*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 9*b*(4*a*A*b + 4*a^2*B + b^2*B)*Sin[c + d*x] + 3*b^2*(A*b + 3*a*B)*Sin[2*(c + d*x)] + b^3*B*Sin[3*(c + d*x)])/(12*d)","A",1
235,1,217,131,0.6800132,"\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{\frac{4 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{4 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)}+2 b (c+d x) \left(6 a^2 B+6 a A b+b^2 B\right)-4 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)+4 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)+4 b^2 (3 a B+A b) \sin (c+d x)+b^3 B \sin (2 (c+d x))}{4 d}","-\frac{b \left(2 a^2 A-3 a b B-A b^2\right) \sin (c+d x)}{d}+\frac{1}{2} b x \left(6 a^2 B+6 a A b+b^2 B\right)+\frac{a^2 (a B+3 A b) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^2 (2 a A-b B) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a A \tan (c+d x) (a+b \cos (c+d x))^2}{d}",1,"(2*b*(6*a*A*b + 6*a^2*B + b^2*B)*(c + d*x) - 4*a^2*(3*A*b + a*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*a^2*(3*A*b + a*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (4*a^3*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (4*a^3*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 4*b^2*(A*b + 3*a*B)*Sin[c + d*x] + b^3*B*Sin[2*(c + d*x)])/(4*d)","A",1
236,1,277,124,2.103004,"\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*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 a \left(a^2 A+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+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 (c+d x) (3 a B+A b)+4 b^3 B \sin (c+d x)}{4 d}","\frac{a \left(a^2 A+6 a b B+6 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 (a B+2 A b) \tan (c+d x)}{d}-\frac{b^2 (a A-2 b B) \sin (c+d x)}{2 d}+b^2 x (3 a B+A b)+\frac{a A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}",1,"(4*b^2*(A*b + 3*a*B)*(c + d*x) - 2*a*(a^2*A + 6*A*b^2 + 6*a*b*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*a*(a^2*A + 6*A*b^2 + 6*a*b*B)*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^3*B*Sin[c + d*x])/(4*d)","B",1
237,1,108,145,0.5934183,"\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{2 a^3 A \tan ^3(c+d x)+3 a \tan (c+d x) \left(2 a^2 A+a (a B+3 A b) \sec (c+d x)+6 a b B+6 A b^2\right)+3 \left(a^3 B+3 a^2 A b+6 a b^2 B+2 A b^3\right) \tanh ^{-1}(\sin (c+d x))+6 b^3 B d x}{6 d}","\frac{a \left(2 a^2 A+9 a b B+8 A b^2\right) \tan (c+d x)}{3 d}+\frac{a^2 (3 a B+5 A b) \tan (c+d x) \sec (c+d x)}{6 d}+\frac{\left(a^3 B+3 a^2 A b+6 a b^2 B+2 A b^3\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}+b^3 B x",1,"(6*b^3*B*d*x + 3*(3*a^2*A*b + 2*A*b^3 + a^3*B + 6*a*b^2*B)*ArcTanh[Sin[c + d*x]] + 3*a*(2*a^2*A + 6*A*b^2 + 6*a*b*B + a*(3*A*b + a*B)*Sec[c + d*x])*Tan[c + d*x] + 2*a^3*A*Tan[c + d*x]^3)/(6*d)","A",1
238,1,140,188,0.8416937,"\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^5,x]","\frac{3 \left(3 a^3 A+12 a^2 b B+12 a A b^2+8 b^3 B\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(6 a^3 A \sec ^3(c+d x)+9 a \left(a^2 A+4 a b B+4 A b^2\right) \sec (c+d x)+8 a^2 (a B+3 A b) \tan ^2(c+d x)+24 \left(a^3 B+3 a^2 A b+3 a b^2 B+A b^3\right)\right)}{24 d}","\frac{a \left(3 a^2 A+12 a b B+10 A b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^2 (2 a B+3 A b) \tan (c+d x) \sec ^2(c+d x)}{6 d}+\frac{\left(2 a^3 B+6 a^2 A b+9 a b^2 B+3 A b^3\right) \tan (c+d x)}{3 d}+\frac{\left(3 a^3 A+12 a^2 b B+12 a A b^2+8 b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{4 d}",1,"(3*(3*a^3*A + 12*a*A*b^2 + 12*a^2*b*B + 8*b^3*B)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(24*(3*a^2*A*b + A*b^3 + a^3*B + 3*a*b^2*B) + 9*a*(a^2*A + 4*A*b^2 + 4*a*b*B)*Sec[c + d*x] + 6*a^3*A*Sec[c + d*x]^3 + 8*a^2*(3*A*b + a*B)*Tan[c + d*x]^2))/(24*d)","A",1
239,1,181,236,3.2603782,"\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^6(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^6,x]","\frac{15 \left(3 a^3 B+9 a^2 A b+12 a b^2 B+4 A b^3\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 \left(2 a^2 A+3 a b B+3 A b^2\right) \tan ^2(c+d x)+15 \left(a^3 A+3 a^2 b B+3 a A b^2+b^3 B\right)\right)+15 \left(3 a^3 B+9 a^2 A b+12 a b^2 B+4 A b^3\right) \sec (c+d x)\right)}{120 d}","\frac{a \left(4 a^2 A+15 a b B+12 A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{15 d}+\frac{a^2 (5 a B+7 A b) \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac{\left(8 a^3 A+30 a^2 b B+30 a A b^2+15 b^3 B\right) \tan (c+d x)}{15 d}+\frac{\left(3 a^3 B+9 a^2 A b+12 a b^2 B+4 A b^3\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(3 a^3 B+9 a^2 A b+12 a b^2 B+4 A b^3\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}",1,"(15*(9*a^2*A*b + 4*A*b^3 + 3*a^3*B + 12*a*b^2*B)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*(9*a^2*A*b + 4*A*b^3 + 3*a^3*B + 12*a*b^2*B)*Sec[c + d*x] + 30*a^2*(3*A*b + a*B)*Sec[c + d*x]^3 + 8*(15*(a^3*A + 3*a*A*b^2 + 3*a^2*b*B + b^3*B) + 5*a*(2*a^2*A + 3*A*b^2 + 3*a*b*B)*Tan[c + d*x]^2 + 3*a^3*A*Tan[c + d*x]^4)))/(120*d)","A",1
240,1,408,366,0.8784751,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x]),x]","\frac{3360 a^4 A c+3360 a^4 A d x+560 a^4 B \sin (3 (c+d x))+2240 a^3 A b \sin (3 (c+d x))+840 a^3 b B \sin (4 (c+d x))+10080 a^3 b B c+10080 a^3 b B d x+1260 a^2 A b^2 \sin (4 (c+d x))+15120 a^2 A b^2 c+15120 a^2 A b^2 d x+4200 a^2 b^2 B \sin (3 (c+d x))+504 a^2 b^2 B \sin (5 (c+d x))+105 \left(48 a^4 B+192 a^3 A b+240 a^2 b^2 B+160 a A b^3+35 b^4 B\right) \sin (c+d x)+105 \left(16 a^4 A+64 a^3 b B+96 a^2 A b^2+60 a b^3 B+15 A b^4\right) \sin (2 (c+d x))+2800 a A b^3 \sin (3 (c+d x))+336 a A b^3 \sin (5 (c+d x))+1260 a b^3 B \sin (4 (c+d x))+140 a b^3 B \sin (6 (c+d x))+8400 a b^3 B c+8400 a b^3 B d x+315 A b^4 \sin (4 (c+d x))+35 A b^4 \sin (6 (c+d x))+2100 A b^4 c+2100 A b^4 d x+735 b^4 B \sin (3 (c+d x))+147 b^4 B \sin (5 (c+d x))+15 b^4 B \sin (7 (c+d x))}{6720 d}","\frac{b^2 \left(31 a^2 B+49 a A b+18 b^2 B\right) \sin (c+d x) \cos ^4(c+d x)}{105 d}+\frac{b \left(104 a^3 B+224 a^2 A b+140 a b^2 B+35 A b^3\right) \sin (c+d x) \cos ^3(c+d x)}{168 d}-\frac{\left(35 a^4 B+140 a^3 A b+168 a^2 b^2 B+112 a A b^3+24 b^4 B\right) \sin ^3(c+d x)}{105 d}+\frac{\left(35 a^4 B+140 a^3 A b+168 a^2 b^2 B+112 a A b^3+24 b^4 B\right) \sin (c+d x)}{35 d}+\frac{\left(8 a^4 A+24 a^3 b B+36 a^2 A b^2+20 a b^3 B+5 A b^4\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left(8 a^4 A+24 a^3 b B+36 a^2 A b^2+20 a b^3 B+5 A b^4\right)+\frac{b (10 a B+7 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^2}{42 d}+\frac{b B \sin (c+d x) \cos ^3(c+d x) (a+b \cos (c+d x))^3}{7 d}",1,"(3360*a^4*A*c + 15120*a^2*A*b^2*c + 2100*A*b^4*c + 10080*a^3*b*B*c + 8400*a*b^3*B*c + 3360*a^4*A*d*x + 15120*a^2*A*b^2*d*x + 2100*A*b^4*d*x + 10080*a^3*b*B*d*x + 8400*a*b^3*B*d*x + 105*(192*a^3*A*b + 160*a*A*b^3 + 48*a^4*B + 240*a^2*b^2*B + 35*b^4*B)*Sin[c + d*x] + 105*(16*a^4*A + 96*a^2*A*b^2 + 15*A*b^4 + 64*a^3*b*B + 60*a*b^3*B)*Sin[2*(c + d*x)] + 2240*a^3*A*b*Sin[3*(c + d*x)] + 2800*a*A*b^3*Sin[3*(c + d*x)] + 560*a^4*B*Sin[3*(c + d*x)] + 4200*a^2*b^2*B*Sin[3*(c + d*x)] + 735*b^4*B*Sin[3*(c + d*x)] + 1260*a^2*A*b^2*Sin[4*(c + d*x)] + 315*A*b^4*Sin[4*(c + d*x)] + 840*a^3*b*B*Sin[4*(c + d*x)] + 1260*a*b^3*B*Sin[4*(c + d*x)] + 336*a*A*b^3*Sin[5*(c + d*x)] + 504*a^2*b^2*B*Sin[5*(c + d*x)] + 147*b^4*B*Sin[5*(c + d*x)] + 35*A*b^4*Sin[6*(c + d*x)] + 140*a*b^3*B*Sin[6*(c + d*x)] + 15*b^4*B*Sin[7*(c + d*x)])/(6720*d)","A",1
241,1,333,325,1.1608634,"\int \cos (c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x]),x]","\frac{480 a^4 B c+480 a^4 B d x+1920 a^3 A b c+1920 a^3 A b d x+320 a^3 b B \sin (3 (c+d x))+480 a^2 A b^2 \sin (3 (c+d x))+180 a^2 b^2 B \sin (4 (c+d x))+2160 a^2 b^2 B c+2160 a^2 b^2 B d x+120 \left(8 a^4 A+24 a^3 b B+36 a^2 A b^2+20 a b^3 B+5 A b^4\right) \sin (c+d x)+15 \left(16 a^4 B+64 a^3 A b+96 a^2 b^2 B+64 a A b^3+15 b^4 B\right) \sin (2 (c+d x))+120 a A b^3 \sin (4 (c+d x))+1440 a A b^3 c+1440 a A b^3 d x+400 a b^3 B \sin (3 (c+d x))+48 a b^3 B \sin (5 (c+d x))+100 A b^4 \sin (3 (c+d x))+12 A b^4 \sin (5 (c+d x))+45 b^4 B \sin (4 (c+d x))+5 b^4 B \sin (6 (c+d x))+300 b^4 B c+300 b^4 B d x}{960 d}","\frac{\left(-4 a^2 B+24 a A b+25 b^2 B\right) \sin (c+d x) (a+b \cos (c+d x))^3}{120 b d}+\frac{\left(-4 a^3 B+24 a^2 A b+53 a b^2 B+32 A b^3\right) \sin (c+d x) (a+b \cos (c+d x))^2}{120 b d}+\frac{\left(-8 a^4 B+48 a^3 A b+178 a^2 b^2 B+232 a A b^3+75 b^4 B\right) \sin (c+d x) \cos (c+d x)}{240 d}+\frac{1}{16} x \left(8 a^4 B+32 a^3 A b+36 a^2 b^2 B+24 a A b^3+5 b^4 B\right)+\frac{\left(-4 a^5 B+24 a^4 A b+121 a^3 b^2 B+224 a^2 A b^3+128 a b^4 B+32 A b^5\right) \sin (c+d x)}{60 b d}+\frac{(6 A b-a B) \sin (c+d x) (a+b \cos (c+d x))^4}{30 b d}+\frac{B \sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}",1,"(1920*a^3*A*b*c + 1440*a*A*b^3*c + 480*a^4*B*c + 2160*a^2*b^2*B*c + 300*b^4*B*c + 1920*a^3*A*b*d*x + 1440*a*A*b^3*d*x + 480*a^4*B*d*x + 2160*a^2*b^2*B*d*x + 300*b^4*B*d*x + 120*(8*a^4*A + 36*a^2*A*b^2 + 5*A*b^4 + 24*a^3*b*B + 20*a*b^3*B)*Sin[c + d*x] + 15*(64*a^3*A*b + 64*a*A*b^3 + 16*a^4*B + 96*a^2*b^2*B + 15*b^4*B)*Sin[2*(c + d*x)] + 480*a^2*A*b^2*Sin[3*(c + d*x)] + 100*A*b^4*Sin[3*(c + d*x)] + 320*a^3*b*B*Sin[3*(c + d*x)] + 400*a*b^3*B*Sin[3*(c + d*x)] + 120*a*A*b^3*Sin[4*(c + d*x)] + 180*a^2*b^2*B*Sin[4*(c + d*x)] + 45*b^4*B*Sin[4*(c + d*x)] + 12*A*b^4*Sin[5*(c + d*x)] + 48*a*b^3*B*Sin[5*(c + d*x)] + 5*b^4*B*Sin[6*(c + d*x)])/(960*d)","A",1
242,1,263,241,0.6555904,"\int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x]),x]","\frac{480 a^4 A c+480 a^4 A d x+960 a^3 b B c+960 a^3 b B d x+1440 a^2 A b^2 c+1440 a^2 A b^2 d x+240 a^2 b^2 B \sin (3 (c+d x))+120 b \left(4 a^3 B+6 a^2 A b+4 a b^2 B+A b^3\right) \sin (2 (c+d x))+60 \left(8 a^4 B+32 a^3 A b+36 a^2 b^2 B+24 a A b^3+5 b^4 B\right) \sin (c+d x)+160 a A b^3 \sin (3 (c+d x))+60 a b^3 B \sin (4 (c+d x))+720 a b^3 B c+720 a b^3 B d x+15 A b^4 \sin (4 (c+d x))+180 A b^4 c+180 A b^4 d x+50 b^4 B \sin (3 (c+d x))+6 b^4 B \sin (5 (c+d x))}{480 d}","\frac{\left(12 a^2 B+35 a A b+16 b^2 B\right) \sin (c+d x) (a+b \cos (c+d x))^2}{60 d}+\frac{b \left(24 a^3 B+130 a^2 A b+116 a b^2 B+45 A b^3\right) \sin (c+d x) \cos (c+d x)}{120 d}+\frac{\left(12 a^4 B+95 a^3 A b+112 a^2 b^2 B+80 a A b^3+16 b^4 B\right) \sin (c+d x)}{30 d}+\frac{1}{8} x \left(8 a^4 A+16 a^3 b B+24 a^2 A b^2+12 a b^3 B+3 A b^4\right)+\frac{(4 a B+5 A b) \sin (c+d x) (a+b \cos (c+d x))^3}{20 d}+\frac{B \sin (c+d x) (a+b \cos (c+d x))^4}{5 d}",1,"(480*a^4*A*c + 1440*a^2*A*b^2*c + 180*A*b^4*c + 960*a^3*b*B*c + 720*a*b^3*B*c + 480*a^4*A*d*x + 1440*a^2*A*b^2*d*x + 180*A*b^4*d*x + 960*a^3*b*B*d*x + 720*a*b^3*B*d*x + 60*(32*a^3*A*b + 24*a*A*b^3 + 8*a^4*B + 36*a^2*b^2*B + 5*b^4*B)*Sin[c + d*x] + 120*b*(6*a^2*A*b + A*b^3 + 4*a^3*B + 4*a*b^2*B)*Sin[2*(c + d*x)] + 160*a*A*b^3*Sin[3*(c + d*x)] + 240*a^2*b^2*B*Sin[3*(c + d*x)] + 50*b^4*B*Sin[3*(c + d*x)] + 15*A*b^4*Sin[4*(c + d*x)] + 60*a*b^3*B*Sin[4*(c + d*x)] + 6*b^4*B*Sin[5*(c + d*x)])/(480*d)","A",1
243,1,210,200,0.6104161,"\int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{-96 a^4 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+96 a^4 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+24 b^2 \left(6 a^2 B+4 a A b+b^2 B\right) \sin (2 (c+d x))+24 b \left(16 a^3 B+24 a^2 A b+12 a b^2 B+3 A b^3\right) \sin (c+d x)+12 (c+d x) \left(8 a^4 B+32 a^3 A b+24 a^2 b^2 B+16 a A b^3+3 b^4 B\right)+8 b^3 (4 a B+A b) \sin (3 (c+d x))+3 b^4 B \sin (4 (c+d x))}{96 d}","\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \left(26 a^2 B+32 a A b+9 b^2 B\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{b \left(19 a^3 B+34 a^2 A b+16 a b^2 B+4 A b^3\right) \sin (c+d x)}{6 d}+\frac{1}{8} x \left(8 a^4 B+32 a^3 A b+24 a^2 b^2 B+16 a A b^3+3 b^4 B\right)+\frac{b (7 a B+4 A b) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{b B \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}",1,"(12*(32*a^3*A*b + 16*a*A*b^3 + 8*a^4*B + 24*a^2*b^2*B + 3*b^4*B)*(c + d*x) - 96*a^4*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 96*a^4*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 24*b*(24*a^2*A*b + 3*A*b^3 + 16*a^3*B + 12*a*b^2*B)*Sin[c + d*x] + 24*b^2*(4*a*A*b + 6*a^2*B + b^2*B)*Sin[2*(c + d*x)] + 8*b^3*(A*b + 4*a*B)*Sin[3*(c + d*x)] + 3*b^4*B*Sin[4*(c + d*x)])/(96*d)","A",1
244,1,257,195,1.0737927,"\int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{\frac{12 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{12 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)}-12 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)+12 a^3 (a B+4 A b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+3 b^2 \left(24 a^2 B+16 a A b+3 b^2 B\right) \sin (c+d x)+6 b (c+d x) \left(8 a^3 B+12 a^2 A b+4 a b^2 B+A b^3\right)+3 b^3 (4 a B+A b) \sin (2 (c+d x))+b^4 B \sin (3 (c+d x))}{12 d}","\frac{a^3 (a B+4 A b) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^2 \left(6 a^2 A-8 a b B-3 A b^2\right) \sin (c+d x) \cos (c+d x)}{6 d}-\frac{b \left(6 a^3 A-17 a^2 b B-12 a A b^2-2 b^3 B\right) \sin (c+d x)}{3 d}+\frac{1}{2} b x \left(8 a^3 B+12 a^2 A b+4 a b^2 B+A b^3\right)-\frac{b (3 a A-b B) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}+\frac{a A \tan (c+d x) (a+b \cos (c+d x))^3}{d}",1,"(6*b*(12*a^2*A*b + A*b^3 + 8*a^3*B + 4*a*b^2*B)*(c + d*x) - 12*a^3*(4*A*b + a*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*a^3*(4*A*b + a*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (12*a^4*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (12*a^4*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 3*b^2*(16*a*A*b + 24*a^2*B + 3*b^2*B)*Sin[c + d*x] + 3*b^3*(A*b + 4*a*B)*Sin[2*(c + d*x)] + b^4*B*Sin[3*(c + d*x)])/(12*d)","A",1
245,1,310,209,1.8469802,"\int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{\frac{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{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{4 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{4 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)}+2 b^2 (c+d x) \left(12 a^2 B+8 a A b+b^2 B\right)-2 a^2 \left(a^2 A+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)+2 a^2 \left(a^2 A+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)+4 b^3 (4 a B+A b) \sin (c+d x)+b^4 B \sin (2 (c+d x))}{4 d}","\frac{a^2 \left(a^2 A+8 a b B+12 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{b^2 \left(2 a^2 B+6 a A b-b^2 B\right) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} b^2 x \left(12 a^2 B+8 a A b+b^2 B\right)-\frac{b \left(4 a^3 B+13 a^2 A b-8 a b^2 B-2 A b^3\right) \sin (c+d x)}{2 d}+\frac{a (2 a B+5 A b) \tan (c+d x) (a+b \cos (c+d x))^2}{2 d}+\frac{a A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^3}{2 d}",1,"(2*b^2*(8*a*A*b + 12*a^2*B + b^2*B)*(c + d*x) - 2*a^2*(a^2*A + 12*A*b^2 + 8*a*b*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*a^2*(a^2*A + 12*A*b^2 + 8*a*b*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a^4*A)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (4*a^3*(4*A*b + a*B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (a^4*A)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*a^3*(4*A*b + a*B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 4*b^3*(A*b + 4*a*B)*Sin[c + d*x] + b^4*B*Sin[2*(c + d*x)])/(4*d)","A",1
246,1,415,198,5.9537749,"\int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*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+3 B)+12 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^3 (a (A+3 B)+12 A b)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{8 a^2 \left(a^2 A+6 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)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{8 a^2 \left(a^2 A+6 a b B+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 a \left(a^3 B+4 a^2 A b+12 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 a \left(a^3 B+4 a^2 A b+12 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)+12 b^3 (c+d x) (4 a B+A b)+12 b^4 B \sin (c+d x)}{12 d}","-\frac{b^2 \left(3 a^2 B+8 a A b-6 b^2 B\right) \sin (c+d x)}{6 d}+\frac{a^2 \left(2 a^2 A+9 a b B+9 A b^2\right) \tan (c+d x)}{3 d}+\frac{a \left(a^3 B+4 a^2 A b+12 a b^2 B+8 A b^3\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+b^3 x (4 a B+A b)+\frac{a (a B+2 A b) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}",1,"(12*b^3*(A*b + 4*a*B)*(c + d*x) - 6*a*(4*a^2*A*b + 8*A*b^3 + a^3*B + 12*a*b^2*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*a*(4*a^2*A*b + 8*A*b^3 + a^3*B + 12*a*b^2*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a^3*(12*A*b + a*(A + 3*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 + (8*a^2*(a^2*A + 9*A*b^2 + 6*a*b*B)*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*(12*A*b + a*(A + 3*B)))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (8*a^2*(a^2*A + 9*A*b^2 + 6*a*b*B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 12*b^4*B*Sin[c + d*x])/(12*d)","B",1
247,1,160,216,1.0824326,"\int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^5,x]","\frac{8 a^3 (a B+4 A b) \tan ^3(c+d x)+3 a \tan (c+d x) \left(2 a^3 A \sec ^3(c+d x)+a \left(3 a^2 A+16 a b B+24 A b^2\right) \sec (c+d x)+8 \left(a^3 B+4 a^2 A b+6 a b^2 B+4 A b^3\right)\right)+3 \left(3 a^4 A+16 a^3 b B+24 a^2 A b^2+32 a b^3 B+8 A b^4\right) \tanh ^{-1}(\sin (c+d x))+24 b^4 B d x}{24 d}","\frac{a^2 \left(9 a^2 A+32 a b B+26 A b^2\right) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{a \left(4 a^3 B+16 a^2 A b+34 a b^2 B+19 A b^3\right) \tan (c+d x)}{6 d}+\frac{\left(3 a^4 A+16 a^3 b B+24 a^2 A b^2+32 a b^3 B+8 A b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (4 a B+7 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}+b^4 B x",1,"(24*b^4*B*d*x + 3*(3*a^4*A + 24*a^2*A*b^2 + 8*A*b^4 + 16*a^3*b*B + 32*a*b^3*B)*ArcTanh[Sin[c + d*x]] + 3*a*(8*(4*a^2*A*b + 4*A*b^3 + a^3*B + 6*a*b^2*B) + a*(3*a^2*A + 24*A*b^2 + 16*a*b*B)*Sec[c + d*x] + 2*a^3*A*Sec[c + d*x]^3)*Tan[c + d*x] + 8*a^3*(4*A*b + a*B)*Tan[c + d*x]^3)/(24*d)","A",1
248,1,198,267,4.2485243,"\int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^6(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^6,x]","\frac{15 \left(3 a^4 B+12 a^3 A b+24 a^2 b^2 B+16 a A b^3+8 b^4 B\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(24 a^4 A \tan ^4(c+d x)+30 a^3 (a B+4 A b) \sec ^3(c+d x)+80 a^2 \left(a^2 A+2 a b B+3 A b^2\right) \tan ^2(c+d x)+15 a \left(3 a^3 B+12 a^2 A b+24 a b^2 B+16 A b^3\right) \sec (c+d x)+120 \left(a^4 A+4 a^3 b B+6 a^2 A b^2+4 a b^3 B+A b^4\right)\right)}{120 d}","\frac{a^2 \left(8 a^2 A+25 a b B+18 A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{30 d}+\frac{a \left(15 a^3 B+60 a^2 A b+110 a b^2 B+56 A b^3\right) \tan (c+d x) \sec (c+d x)}{40 d}+\frac{\left(8 a^4 A+40 a^3 b B+60 a^2 A b^2+60 a b^3 B+15 A b^4\right) \tan (c+d x)}{15 d}+\frac{\left(3 a^4 B+12 a^3 A b+24 a^2 b^2 B+16 a A b^3+8 b^4 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (5 a B+8 A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{20 d}+\frac{a A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^3}{5 d}",1,"(15*(12*a^3*A*b + 16*a*A*b^3 + 3*a^4*B + 24*a^2*b^2*B + 8*b^4*B)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(120*(a^4*A + 6*a^2*A*b^2 + A*b^4 + 4*a^3*b*B + 4*a*b^3*B) + 15*a*(12*a^2*A*b + 16*A*b^3 + 3*a^3*B + 24*a*b^2*B)*Sec[c + d*x] + 30*a^3*(4*A*b + a*B)*Sec[c + d*x]^3 + 80*a^2*(a^2*A + 3*A*b^2 + 2*a*b*B)*Tan[c + d*x]^2 + 24*a^4*A*Tan[c + d*x]^4))/(120*d)","A",1
249,1,244,324,2.7735348,"\int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^7(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x])*Sec[c + d*x]^7,x]","\frac{15 \left(5 a^4 A+24 a^3 b B+36 a^2 A b^2+32 a b^3 B+8 A b^4\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(40 a^4 A \sec ^5(c+d x)+48 a^3 (a B+4 A b) \tan ^4(c+d x)+10 a^2 \left(5 a^2 A+24 a b B+36 A b^2\right) \sec ^3(c+d x)+160 a \left(a^3 B+4 a^2 A b+3 a b^2 B+2 A b^3\right) \tan ^2(c+d x)+15 \left(5 a^4 A+24 a^3 b B+36 a^2 A b^2+32 a b^3 B+8 A b^4\right) \sec (c+d x)+240 \left(a^4 B+4 a^3 A b+6 a^2 b^2 B+4 a A b^3+b^4 B\right)\right)}{240 d}","\frac{a^2 \left(25 a^2 A+72 a b B+48 A b^2\right) \tan (c+d x) \sec ^3(c+d x)}{120 d}+\frac{a \left(4 a^3 B+16 a^2 A b+27 a b^2 B+13 A b^3\right) \tan (c+d x) \sec ^2(c+d x)}{15 d}+\frac{\left(8 a^4 B+32 a^3 A b+60 a^2 b^2 B+40 a A b^3+15 b^4 B\right) \tan (c+d x)}{15 d}+\frac{\left(5 a^4 A+24 a^3 b B+36 a^2 A b^2+32 a b^3 B+8 A b^4\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\left(5 a^4 A+24 a^3 b B+36 a^2 A b^2+32 a b^3 B+8 A b^4\right) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{a (2 a B+3 A b) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{10 d}+\frac{a A \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{6 d}",1,"(15*(5*a^4*A + 36*a^2*A*b^2 + 8*A*b^4 + 24*a^3*b*B + 32*a*b^3*B)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(240*(4*a^3*A*b + 4*a*A*b^3 + a^4*B + 6*a^2*b^2*B + b^4*B) + 15*(5*a^4*A + 36*a^2*A*b^2 + 8*A*b^4 + 24*a^3*b*B + 32*a*b^3*B)*Sec[c + d*x] + 10*a^2*(5*a^2*A + 36*A*b^2 + 24*a*b*B)*Sec[c + d*x]^3 + 40*a^4*A*Sec[c + d*x]^5 + 160*a*(4*a^2*A*b + 2*A*b^3 + a^3*B + 3*a*b^2*B)*Tan[c + d*x]^2 + 48*a^3*(4*A*b + a*B)*Tan[c + d*x]^4))/(240*d)","A",1
250,1,152,178,0.4680685,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),x]","\frac{6 \left(2 a^2+b^2\right) (c+d x) (A b-a B)+3 b \left(4 a^2 B-4 a A b+3 b^2 B\right) \sin (c+d x)-\frac{24 a^3 (a B-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}}+3 b^2 (A b-a B) \sin (2 (c+d x))+b^3 B \sin (3 (c+d x))}{12 b^4 d}","-\frac{2 a^3 (A b-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 \sqrt{a-b} \sqrt{a+b}}+\frac{x \left(2 a^2+b^2\right) (A b-a B)}{2 b^4}-\frac{\left(-3 a^2 B+3 a A b-2 b^2 B\right) \sin (c+d x)}{3 b^3 d}+\frac{(A b-a B) \sin (c+d x) \cos (c+d x)}{2 b^2 d}+\frac{B \sin (c+d x) \cos ^2(c+d x)}{3 b d}",1,"(6*(2*a^2 + b^2)*(A*b - a*B)*(c + d*x) - (24*a^3*(-(A*b) + a*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + 3*b*(-4*a*A*b + 4*a^2*B + 3*b^2*B)*Sin[c + d*x] + 3*b^2*(A*b - a*B)*Sin[2*(c + d*x)] + b^3*B*Sin[3*(c + d*x)])/(12*b^4*d)","A",1
251,1,121,134,0.3134591,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),x]","\frac{2 (c+d x) \left(2 a^2 B-2 a A b+b^2 B\right)+\frac{8 a^2 (a B-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}}+4 b (A b-a B) \sin (c+d x)+b^2 B \sin (2 (c+d x))}{4 b^3 d}","\frac{2 a^2 (A b-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 \sqrt{a-b} \sqrt{a+b}}-\frac{x \left(-2 a^2 B+2 a A b-b^2 B\right)}{2 b^3}+\frac{(A b-a B) \sin (c+d x)}{b^2 d}+\frac{B \sin (c+d x) \cos (c+d x)}{2 b d}",1,"(2*(-2*a*A*b + 2*a^2*B + b^2*B)*(c + d*x) + (8*a^2*(-(A*b) + a*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + 4*b*(A*b - a*B)*Sin[c + d*x] + b^2*B*Sin[2*(c + d*x)])/(4*b^3*d)","A",1
252,1,85,89,0.2097564,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),x]","\frac{-\frac{2 a (a B-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}}+(c+d x) (A b-a B)+b B \sin (c+d x)}{b^2 d}","-\frac{2 a (A b-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 \sqrt{a-b} \sqrt{a+b}}+\frac{x (A b-a B)}{b^2}+\frac{B \sin (c+d x)}{b d}",1,"((A*b - a*B)*(c + d*x) - (2*a*(-(A*b) + a*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + b*B*Sin[c + d*x])/(b^2*d)","A",1
253,1,68,67,0.1209364,"\int \frac{A+B \cos (c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x]),x]","\frac{\frac{2 (a B-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}}+B (c+d x)}{b d}","\frac{2 (A b-a B) \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{B x}{b}",1,"(B*(c + d*x) + (2*(-(A*b) + a*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2])/(b*d)","A",1
254,1,112,76,0.1630665,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x])/(a + b*Cos[c + d*x]),x]","\frac{\frac{2 (A b-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 \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{A \tanh ^{-1}(\sin (c+d x))}{a d}-\frac{2 (A b-a B) \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*(A*b - a*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + A*(-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
255,1,129,99,0.564558,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + b*Cos[c + d*x]),x]","\frac{-\frac{2 b (A b-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-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)+a A \tan (c+d x)}{a^2 d}","\frac{2 b (A b-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 \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*b*(A*b - a*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (A*b - 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*A*Tan[c + d*x])/(a^2*d)","A",1
256,1,300,143,1.7821115,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + b*Cos[c + d*x]),x]","\frac{\frac{8 b^2 (A b-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}}-2 \left(a^2 A-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 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^2 (A b-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 \sqrt{a-b} \sqrt{a+b}}-\frac{(A b-a B) \tan (c+d x)}{a^2 d}+\frac{\left(a^2 A-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^2*(A*b - a*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] - 2*(a^2*A + 2*A*b^2 - 2*a*b*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(a^2*A + 2*A*b^2 - 2*a*b*B)*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
257,1,422,187,2.2724498,"\int \frac{(A+B \cos (c+d x)) \sec ^4(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x])*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(2 a^2 A-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)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 a \left(2 a^2 A-3 a b B+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)}-6 \left(a^2+2 b^2\right) (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)+6 \left(a^2+2 b^2\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{24 b^3 (a B-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}}+\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}}{12 a^4 d}","\frac{2 b^3 (A b-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 \sqrt{a-b} \sqrt{a+b}}-\frac{(A b-a B) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{\left(a^2+2 b^2\right) (A b-a B) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{\left(2 a^2 A-3 a b B+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^3*(-(A*b) + a*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] - 6*(a^2 + 2*b^2)*(-(A*b) + a*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*(a^2 + 2*b^2)*(-(A*b) + a*B)*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*(2*a^2*A + 3*A*b^2 - 3*a*b*B)*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*(2*a^2*A + 3*A*b^2 - 3*a*b*B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(12*a^4*d)","B",1
258,1,184,263,1.0720536,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^2,x]","\frac{\frac{4 a^3 b (A b-a B) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}+2 (c+d x) \left(6 a^2 B-4 a A b+b^2 B\right)-\frac{8 a^2 \left(3 a^3 B-2 a^2 A b-4 a b^2 B+3 A b^3\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 (A b-2 a B) \sin (c+d x)+b^2 B \sin (2 (c+d x))}{4 b^4 d}","\frac{a (A b-a B) \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 B+2 a A b+b^2 B\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 B+4 a A b-b^2 B\right)}{2 b^4}+\frac{\left(-3 a^3 B+2 a^2 A b+2 a b^2 B-A b^3\right) \sin (c+d x)}{b^3 d \left(a^2-b^2\right)}+\frac{2 a^2 \left(-3 a^3 B+2 a^2 A b+4 a b^2 B-3 A b^3\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*A*b + 6*a^2*B + b^2*B)*(c + d*x) - (8*a^2*(-2*a^2*A*b + 3*A*b^3 + 3*a^3*B - 4*a*b^2*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + 4*b*(A*b - 2*a*B)*Sin[c + d*x] + (4*a^3*b*(A*b - a*B)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])) + b^2*B*Sin[2*(c + d*x)])/(4*b^4*d)","A",1
259,1,147,155,0.8464452,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^2,x]","\frac{\frac{a^2 b (a B-A b) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}+\frac{2 a \left(2 a^3 B-a^2 A b-3 a b^2 B+2 A b^3\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) (A b-2 a B)+b B \sin (c+d x)}{b^3 d}","-\frac{a^2 (A b-a B) \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 B+a^2 A b+3 a b^2 B-2 A b^3\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 (A b-2 a B)}{b^3}+\frac{B \sin (c+d x)}{b^2 d}",1,"((A*b - 2*a*B)*(c + d*x) + (2*a*(-(a^2*A*b) + 2*A*b^3 + 2*a^3*B - 3*a*b^2*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + b*B*Sin[c + d*x] + (a^2*b*(-(A*b) + a*B)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])))/(b^3*d)","A",1
260,1,119,122,0.5522657,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^2,x]","\frac{-\frac{2 \left(a B \left(a^2-2 b^2\right)+A b^3\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 (A b-a B) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}+B (c+d x)}{b^2 d}","-\frac{2 \left(a^3 B-2 a b^2 B+A b^3\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 (A b-a B) \sin (c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{B x}{b^2}",1,"(B*(c + d*x) - (2*(A*b^3 + a*(a^2 - 2*b^2)*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + (a*b*(A*b - a*B)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])))/(b^2*d)","A",1
261,1,97,100,0.3466997,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2,x]","\frac{\frac{2 (a A-b B) \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-A b) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}}{d}","\frac{2 (a A-b B) \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{(A b-a B) \sin (c+d x)}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"((2*(a*A - b*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + ((-(A*b) + a*B)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])))/d","A",1
262,1,191,133,0.6284984,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x])/(a + b*Cos[c + d*x])^2,x]","\frac{\cos (c+d x) (A \sec (c+d x)+B) \left(\frac{2 \left(a^3 B-2 a^2 A b+A b^3\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 (A b-a B) \sin (c+d x)}{(a-b) (a+b) (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 (A+B \cos (c+d x))}","\frac{b (A b-a B) \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}-\frac{2 \left(a^3 (-B)+2 a^2 A b-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,"(Cos[c + d*x]*(B + A*Sec[c + d*x])*((2*(-2*a^2*A*b + A*b^3 + a^3*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) - 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*(A*b - a*B)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x]))))/(a^2*d*(A + B*Cos[c + d*x]))","A",1
263,1,240,189,1.9469236,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^2,x]","\frac{-\frac{2 b \left(2 a^3 B-3 a^2 A b-a b^2 B+2 A b^3\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 B-A b) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}+a A \tan (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)+2 A b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 A 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 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^3 d}+\frac{\left(a^2 A+a b B-2 A b^2\right) \tan (c+d x)}{a^2 d \left(a^2-b^2\right)}+\frac{b (A b-a B) \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 B+3 a^2 A b+a b^2 B-2 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^3 d (a-b)^{3/2} (a+b)^{3/2}}",1,"((-2*b*(-3*a^2*A*b + 2*A*b^3 + 2*a^3*B - a*b^2*B)*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]] - a*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 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]] + (a*b^2*(-(A*b) + a*B)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])) + a*A*Tan[c + d*x])/(a^3*d)","A",1
264,1,438,270,6.26826,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^2,x]","\frac{A b^4 \sin (c+d x)-a b^3 B \sin (c+d x)}{a^3 d (a-b) (a+b) (a+b \cos (c+d x))}+\frac{a B \sin \left(\frac{1}{2} (c+d x)\right)-2 A b \sin \left(\frac{1}{2} (c+d x)\right)}{a^3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{a B \sin \left(\frac{1}{2} (c+d x)\right)-2 A b \sin \left(\frac{1}{2} (c+d x)\right)}{a^3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{A}{4 a^2 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{A}{4 a^2 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 (-A)+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 a^4 d}+\frac{\left(a^2 A-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)}{2 a^4 d}-\frac{2 b^2 \left(3 a^3 B-4 a^2 A b-2 a b^2 B+3 A b^3\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}}","\frac{\left(a^2 A+2 a b B-3 A b^2\right) \tan (c+d x) \sec (c+d x)}{2 a^2 d \left(a^2-b^2\right)}+\frac{b (A b-a B) \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-4 a b B+6 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{\left(a^3 (-B)+2 a^2 A b+2 a b^2 B-3 A b^3\right) \tan (c+d x)}{a^3 d \left(a^2-b^2\right)}-\frac{2 b^2 \left(-3 a^3 B+4 a^2 A b+2 a b^2 B-3 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^4 d (a-b)^{3/2} (a+b)^{3/2}}",1,"(-2*b^2*(-4*a^2*A*b + 3*A*b^3 + 3*a^3*B - 2*a*b^2*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(a^4*(a^2 - b^2)*Sqrt[-a^2 + b^2]*d) + ((-(a^2*A) - 6*A*b^2 + 4*a*b*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(2*a^4*d) + ((a^2*A + 6*A*b^2 - 4*a*b*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(2*a^4*d) + A/(4*a^2*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) - A/(4*a^2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (-2*A*b*Sin[(c + d*x)/2] + a*B*Sin[(c + d*x)/2])/(a^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (-2*A*b*Sin[(c + d*x)/2] + a*B*Sin[(c + d*x)/2])/(a^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (A*b^4*Sin[c + d*x] - a*b^3*B*Sin[c + d*x])/(a^3*(a - b)*(a + b)*d*(a + b*Cos[c + d*x]))","A",1
265,1,734,398,3.6049243,"\int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^4*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{16 a^2 \left(12 a^5 B-6 a^4 A b-29 a^3 b^2 B+15 a^2 A b^3+20 a b^4 B-12 A b^5\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 B c+96 a^8 B d x-48 a^7 A b c-48 a^7 A b d x-96 a^7 b B \sin (c+d x)+48 a^6 A b^2 \sin (c+d x)-72 a^6 b^2 B \sin (2 (c+d x))-136 a^6 b^2 B c-136 a^6 b^2 B d x+36 a^5 A b^3 \sin (2 (c+d x))+72 a^5 A b^3 c+72 a^5 A b^3 d x+160 a^5 b^3 B \sin (c+d x)-8 a^5 b^3 B \sin (3 (c+d x))-84 a^4 A b^4 \sin (c+d x)+4 a^4 A b^4 \sin (3 (c+d x))+130 a^4 b^4 B \sin (2 (c+d x))+a^4 b^4 B \sin (4 (c+d x))-12 a^4 b^4 B c-12 a^4 b^4 B d x-64 a^3 A b^5 \sin (2 (c+d x))-32 a^3 b^5 B \sin (c+d x)+16 a^3 b^5 B \sin (3 (c+d x))+8 a^2 A b^6 \sin (c+d x)-8 a^2 A b^6 \sin (3 (c+d x))+16 a b \left(a^2-b^2\right)^2 (c+d x) \left(12 a^2 B-6 a A b+b^2 B\right) \cos (c+d x)+4 \left(b^3-a^2 b\right)^2 (c+d x) \left(12 a^2 B-6 a A b+b^2 B\right) \cos (2 (c+d x))-48 a^2 b^6 B \sin (2 (c+d x))-2 a^2 b^6 B \sin (4 (c+d x))+48 a^2 b^6 B c+48 a^2 b^6 B d x+16 a A b^7 \sin (2 (c+d x))-24 a A b^7 c-24 a A b^7 d x-8 a b^7 B \sin (c+d x)-8 a b^7 B \sin (3 (c+d x))+4 A b^8 \sin (c+d x)+4 A b^8 \sin (3 (c+d x))+2 b^8 B \sin (2 (c+d x))+b^8 B \sin (4 (c+d x))+4 b^8 B c+4 b^8 B d x}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}}{16 b^5 d}","\frac{a (A b-a B) \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 B+6 a A b-b^2 B\right)}{2 b^5}+\frac{a \left(-4 a^3 B+2 a^2 A b+7 a b^2 B-5 A b^3\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 B+3 a^3 A b+10 a^2 b^2 B-6 a A b^3-b^4 B\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 B+6 a^4 A b+21 a^3 b^2 B-11 a^2 A b^3-6 a b^4 B+2 A b^5\right) \sin (c+d x)}{2 b^4 d \left(a^2-b^2\right)^2}+\frac{a^2 \left(-12 a^5 B+6 a^4 A b+29 a^3 b^2 B-15 a^2 A b^3-20 a b^4 B+12 A b^5\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*A*b + 15*a^2*A*b^3 - 12*A*b^5 + 12*a^5*B - 29*a^3*b^2*B + 20*a*b^4*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (-48*a^7*A*b*c + 72*a^5*A*b^3*c - 24*a*A*b^7*c + 96*a^8*B*c - 136*a^6*b^2*B*c - 12*a^4*b^4*B*c + 48*a^2*b^6*B*c + 4*b^8*B*c - 48*a^7*A*b*d*x + 72*a^5*A*b^3*d*x - 24*a*A*b^7*d*x + 96*a^8*B*d*x - 136*a^6*b^2*B*d*x - 12*a^4*b^4*B*d*x + 48*a^2*b^6*B*d*x + 4*b^8*B*d*x + 16*a*b*(a^2 - b^2)^2*(-6*a*A*b + 12*a^2*B + b^2*B)*(c + d*x)*Cos[c + d*x] + 4*(-(a^2*b) + b^3)^2*(-6*a*A*b + 12*a^2*B + b^2*B)*(c + d*x)*Cos[2*(c + d*x)] + 48*a^6*A*b^2*Sin[c + d*x] - 84*a^4*A*b^4*Sin[c + d*x] + 8*a^2*A*b^6*Sin[c + d*x] + 4*A*b^8*Sin[c + d*x] - 96*a^7*b*B*Sin[c + d*x] + 160*a^5*b^3*B*Sin[c + d*x] - 32*a^3*b^5*B*Sin[c + d*x] - 8*a*b^7*B*Sin[c + d*x] + 36*a^5*A*b^3*Sin[2*(c + d*x)] - 64*a^3*A*b^5*Sin[2*(c + d*x)] + 16*a*A*b^7*Sin[2*(c + d*x)] - 72*a^6*b^2*B*Sin[2*(c + d*x)] + 130*a^4*b^4*B*Sin[2*(c + d*x)] - 48*a^2*b^6*B*Sin[2*(c + d*x)] + 2*b^8*B*Sin[2*(c + d*x)] + 4*a^4*A*b^4*Sin[3*(c + d*x)] - 8*a^2*A*b^6*Sin[3*(c + d*x)] + 4*A*b^8*Sin[3*(c + d*x)] - 8*a^5*b^3*B*Sin[3*(c + d*x)] + 16*a^3*b^5*B*Sin[3*(c + d*x)] - 8*a*b^7*B*Sin[3*(c + d*x)] + a^4*b^4*B*Sin[4*(c + d*x)] - 2*a^2*b^6*B*Sin[4*(c + d*x)] + b^8*B*Sin[4*(c + d*x)])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2))/(16*b^5*d)","A",1
266,1,232,280,2.1617804,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{a^3 b (A b-a B) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}+\frac{a^2 b \left(5 a^3 B-3 a^2 A b-8 a b^2 B+6 A b^3\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}-\frac{2 a \left(6 a^5 B-2 a^4 A b-15 a^3 b^2 B+5 a^2 A b^3+12 a b^4 B-6 A b^5\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) (A b-3 a B)+2 b B \sin (c+d x)}{2 b^4 d}","\frac{a (A b-a B) \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 B+a A b+2 b^2 B\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right)}-\frac{a^2 \left(-3 a^3 B+a^2 A b+6 a b^2 B-4 A b^3\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 B+2 a^4 A b+15 a^3 b^2 B-5 a^2 A b^3-12 a b^4 B+6 A b^5\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 (A b-3 a B)}{b^4}",1,"(2*(A*b - 3*a*B)*(c + d*x) - (2*a*(-2*a^4*A*b + 5*a^2*A*b^3 - 6*A*b^5 + 6*a^5*B - 15*a^3*b^2*B + 12*a*b^4*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + 2*b*B*Sin[c + d*x] + (a^3*b*(A*b - a*B)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^2) + (a^2*b*(-3*a^2*A*b + 6*A*b^3 + 5*a^3*B - 8*a*b^2*B)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])))/(2*b^4*d)","A",1
267,1,204,211,1.3594442,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{a^2 b (a B-A b) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}+\frac{a b \left(-3 a^3 B+a^2 A b+6 a b^2 B-4 A b^3\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}+\frac{2 \left(2 a^5 B-5 a^3 b^2 B-a^2 A b^3+6 a b^4 B-2 A b^5\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 (c+d x)}{2 b^3 d}","-\frac{a^2 (A b-a B) \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 B+a^2 A b+6 a b^2 B-4 A b^3\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 B+5 a^3 b^2 B+a^2 A b^3-6 a b^4 B+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)}{b^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{B x}{b^3}",1,"(2*B*(c + d*x) + (2*(-(a^2*A*b^3) - 2*A*b^5 + 2*a^5*B - 5*a^3*b^2*B + 6*a*b^4*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (a^2*b*(-(A*b) + a*B)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^2) + (a*b*(a^2*A*b - 4*A*b^3 - 3*a^3*B + 6*a*b^2*B)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])))/(2*b^3*d)","A",1
268,1,172,180,0.8562229,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^3,x]","\frac{-\frac{2 \left(a^2 B-3 a A b+2 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{\left(a^3 B+a^2 A b-4 a b^2 B+2 A b^3\right) \sin (c+d x)}{b (a-b)^2 (a+b)^2 (a+b \cos (c+d x))}+\frac{a (A b-a B) \sin (c+d x)}{b (a-b) (a+b) (a+b \cos (c+d x))^2}}{2 d}","-\frac{\left(a^2 (-B)+3 a A b-2 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{a (A b-a B) \sin (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{\left(a^3 B+a^2 A b-4 a b^2 B+2 A b^3\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*A*b + a^2*B + 2*b^2*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (a*(A*b - a*B)*Sin[c + d*x])/((a - b)*b*(a + b)*(a + b*Cos[c + d*x])^2) + ((a^2*A*b + 2*A*b^3 + a^3*B - 4*a*b^2*B)*Sin[c + d*x])/((a - b)^2*b*(a + b)^2*(a + b*Cos[c + d*x])))/(2*d)","A",1
269,1,157,164,0.6670626,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{\left(a^2 B-3 a A b+2 b^2 B\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}-\frac{2 \left(2 a^2 A-3 a 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)}{\left(b^2-a^2\right)^{5/2}}+\frac{(a B-A b) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}}{2 d}","\frac{\left(2 a^2 A-3 a b B+A 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{\left(a^2 (-B)+3 a A b-2 b^2 B\right) \sin (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{(A b-a B) \sin (c+d x)}{2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}",1,"((-2*(2*a^2*A + A*b^2 - 3*a*b*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + ((-(A*b) + a*B)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^2) + ((-3*a*A*b + a^2*B + 2*b^2*B)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])))/(2*d)","A",1
270,1,269,214,1.333261,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x])/(a + b*Cos[c + d*x])^3,x]","\frac{\cos (c+d x) (A \sec (c+d x)+B) \left(\frac{a^2 b (A b-a B) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}+\frac{a b \left(-3 a^3 B+5 a^2 A b-2 A b^3\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}-\frac{2 \left(2 a^5 B-6 a^4 A b+a^3 b^2 B+5 a^2 A b^3-2 A b^5\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 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 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 (A+B \cos (c+d x))}","\frac{A \tanh ^{-1}(\sin (c+d x))}{a^3 d}+\frac{b (A b-a B) \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 B+5 a^2 A b-2 A b^3\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 B+6 a^4 A b-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 + A*Sec[c + d*x])*((-2*(-6*a^4*A*b + 5*a^2*A*b^3 - 2*A*b^5 + 2*a^5*B + a^3*b^2*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) - 2*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a^2*b*(A*b - a*B)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^2) + (a*b*(5*a^2*A*b - 2*A*b^3 - 3*a^3*B)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x]))))/(2*a^3*d*(A + B*Cos[c + d*x]))","A",1
271,1,352,299,5.9712076,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{a^2 b^2 (a B-A b) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}+\frac{a b^2 \left(5 a^3 B-7 a^2 A b-2 a b^2 B+4 A b^3\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}-\frac{2 b \left(-6 a^5 B+12 a^4 A b+5 a^3 b^2 B-15 a^2 A b^3-2 a b^4 B+6 A b^5\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 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)+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)+\frac{2 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{2 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)}}{2 a^4 d}","-\frac{(3 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac{b (A b-a B) \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 B+6 a^2 A b+a b^2 B-3 A b^3\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 A+5 a^3 b B-11 a^2 A b^2-2 a b^3 B+6 A b^4\right) \tan (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}+\frac{b \left(-6 a^5 B+12 a^4 A b+5 a^3 b^2 B-15 a^2 A b^3-2 a b^4 B+6 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^4 d (a-b)^{5/2} (a+b)^{5/2}}",1,"((-2*b*(12*a^4*A*b - 15*a^2*A*b^3 + 6*A*b^5 - 6*a^5*B + 5*a^3*b^2*B - 2*a*b^4*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + 2*(3*A*b - a*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(-3*A*b + a*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*a*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (2*a*A*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + (a^2*b^2*(-(A*b) + a*B)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^2) + (a*b^2*(-7*a^2*A*b + 4*A*b^3 + 5*a^3*B - 2*a*b^2*B)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])))/(2*a^4*d)","A",1
272,1,507,402,2.9562066,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^3,x]","\frac{-8 \left(a^2 A-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-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^2 \left(-12 a^5 B+20 a^4 A b+15 a^3 b^2 B-29 a^2 A b^3-6 a b^4 B+12 A b^5\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))-6 a^4 A b^3 \cos (3 (c+d x))-32 a^4 b^3 B+68 a^3 A b^4-11 a^3 b^4 B \cos (3 (c+d x))+21 a^2 A b^5 \cos (3 (c+d x))+18 a^2 b^5 B+2 a b \left(4 a^5 B-11 a^4 A b-16 a^3 b^2 B+32 a^2 A b^3+9 a b^4 B-18 A b^5\right) \cos (2 (c+d x))+\left(8 a^7 B-16 a^6 A b-10 a^5 b^2 B+14 a^4 A b^3-25 a^3 b^4 B+47 a^2 A b^5+18 a b^6 B-36 A b^7\right) \cos (c+d x)-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{b (A b-a B) \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-6 a b B+12 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}+\frac{b \left(-5 a^3 B+7 a^2 A b+2 a b^2 B-4 A b^3\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+6 a^3 b B-10 a^2 A b^2-3 a b^3 B+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^2 \left(-12 a^5 B+20 a^4 A b+15 a^3 b^2 B-29 a^2 A b^3-6 a b^4 B+12 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^5 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\left(-2 a^5 B+6 a^4 A b+11 a^3 b^2 B-21 a^2 A b^3-6 a b^4 B+12 A b^5\right) \tan (c+d x)}{2 a^4 d \left(a^2-b^2\right)^2}",1,"((16*b^2*(20*a^4*A*b - 29*a^2*A*b^3 + 12*A*b^5 - 12*a^5*B + 15*a^3*b^2*B - 6*a*b^4*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) - 8*(a^2*A + 12*A*b^2 - 6*a*b*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 8*(a^2*A + 12*A*b^2 - 6*a*b*B)*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 + (-16*a^6*A*b + 14*a^4*A*b^3 + 47*a^2*A*b^5 - 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)*Cos[c + d*x] + 2*a*b*(-11*a^4*A*b + 32*a^2*A*b^3 - 18*A*b^5 + 4*a^5*B - 16*a^3*b^2*B + 9*a*b^4*B)*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)])*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
273,1,1278,409,6.6455968,"\int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^4*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^4,x]","\frac{96 B (c+d x) a^{10}-24 A b (c+d x) a^9+288 b B (c+d x) \cos (c+d x) a^9-96 b B \sin (c+d x) a^9-144 b^2 B (c+d x) a^8-72 A b^2 (c+d x) \cos (c+d x) a^8+144 b^2 B (c+d x) \cos (2 (c+d x)) a^8+24 A b^2 \sin (c+d x) a^8-120 b^2 B \sin (2 (c+d x)) a^8+36 A b^3 (c+d x) a^7-792 b^3 B (c+d x) \cos (c+d x) a^7-36 A b^3 (c+d x) \cos (2 (c+d x)) a^7+24 b^3 B (c+d x) \cos (3 (c+d x)) a^7+228 b^3 B \sin (c+d x) a^7+30 A b^3 \sin (2 (c+d x)) a^7-44 b^3 B \sin (3 (c+d x)) a^7-144 b^4 B (c+d x) a^6+198 A b^4 (c+d x) \cos (c+d x) a^6-432 b^4 B (c+d x) \cos (2 (c+d x)) a^6-6 A b^4 (c+d x) \cos (3 (c+d x)) a^6-57 A b^4 \sin (c+d x) a^6+336 b^4 B \sin (2 (c+d x)) a^6+11 A b^4 \sin (3 (c+d x)) a^6-3 b^4 B \sin (4 (c+d x)) a^6+36 A b^5 (c+d x) a^5+648 b^5 B (c+d x) \cos (c+d x) a^5+108 A b^5 (c+d x) \cos (2 (c+d x)) a^5-72 b^5 B (c+d x) \cos (3 (c+d x)) a^5-135 b^5 B \sin (c+d x) a^5-90 A b^5 \sin (2 (c+d x)) a^5+125 b^5 B \sin (3 (c+d x)) a^5+336 b^6 B (c+d x) a^4-162 A b^6 (c+d x) \cos (c+d x) a^4+432 b^6 B (c+d x) \cos (2 (c+d x)) a^4+18 A b^6 (c+d x) \cos (3 (c+d x)) a^4+72 A b^6 \sin (c+d x) a^4-300 b^6 B \sin (2 (c+d x)) a^4-32 A b^6 \sin (3 (c+d x)) a^4+9 b^6 B \sin (4 (c+d x)) a^4-84 A b^7 (c+d x) a^3-72 b^7 B (c+d x) \cos (c+d x) a^3-108 A b^7 (c+d x) \cos (2 (c+d x)) a^3+72 b^7 B (c+d x) \cos (3 (c+d x)) a^3-90 b^7 B \sin (c+d x) a^3+120 A b^7 \sin (2 (c+d x)) a^3-114 b^7 B \sin (3 (c+d x)) a^3-144 b^8 B (c+d x) a^2+18 A b^8 (c+d x) \cos (c+d x) a^2-144 b^8 B (c+d x) \cos (2 (c+d x)) a^2-18 A b^8 (c+d x) \cos (3 (c+d x)) a^2+36 A b^8 \sin (c+d x) a^2+18 b^8 B \sin (2 (c+d x)) a^2+36 A b^8 \sin (3 (c+d x)) a^2-9 b^8 B \sin (4 (c+d x)) a^2+36 A b^9 (c+d x) a-72 b^9 B (c+d x) \cos (c+d x) a+36 A b^9 (c+d x) \cos (2 (c+d x)) a-24 b^9 B (c+d x) \cos (3 (c+d x)) a+18 b^9 B \sin (c+d x) a+18 b^9 B \sin (3 (c+d x)) a+18 A b^{10} (c+d x) \cos (c+d x)+6 A b^{10} (c+d x) \cos (3 (c+d x))+6 b^{10} B \sin (2 (c+d x))+3 b^{10} B \sin (4 (c+d x))}{24 b^5 \left(b^2-a^2\right)^3 d (a+b \cos (c+d x))^3}-\frac{a \left(8 B a^7-2 A b a^6-28 b^2 B a^5+7 A b^3 a^4+35 b^4 B a^3-8 A b^5 a^2-20 b^6 B a+8 A b^7\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{b^5 \left(a^2-b^2\right)^3 \sqrt{b^2-a^2} d}","\frac{a (A b-a B) \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{a \left(-4 a^3 B+a^2 A b+9 a b^2 B-6 A b^3\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{\left(-12 a^4 B+3 a^3 A b+23 a^2 b^2 B-8 a A b^3-6 b^4 B\right) \sin (c+d x)}{6 b^4 d \left(a^2-b^2\right)^2}-\frac{a^2 \left(-4 a^5 B+a^4 A b+11 a^3 b^2 B-2 a^2 A b^3-12 a b^4 B+6 A b^5\right) \sin (c+d x)}{2 b^4 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{a \left(-8 a^7 B+2 a^6 A b+28 a^5 b^2 B-7 a^4 A b^3-35 a^3 b^4 B+8 a^2 A b^5+20 a b^6 B-8 A b^7\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 (A b-4 a B)}{b^5}",1,"-((a*(-2*a^6*A*b + 7*a^4*A*b^3 - 8*a^2*A*b^5 + 8*A*b^7 + 8*a^7*B - 28*a^5*b^2*B + 35*a^3*b^4*B - 20*a*b^6*B)*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)) + (-24*a^9*A*b*(c + d*x) + 36*a^7*A*b^3*(c + d*x) + 36*a^5*A*b^5*(c + d*x) - 84*a^3*A*b^7*(c + d*x) + 36*a*A*b^9*(c + d*x) + 96*a^10*B*(c + d*x) - 144*a^8*b^2*B*(c + d*x) - 144*a^6*b^4*B*(c + d*x) + 336*a^4*b^6*B*(c + d*x) - 144*a^2*b^8*B*(c + d*x) - 72*a^8*A*b^2*(c + d*x)*Cos[c + d*x] + 198*a^6*A*b^4*(c + d*x)*Cos[c + d*x] - 162*a^4*A*b^6*(c + d*x)*Cos[c + d*x] + 18*a^2*A*b^8*(c + d*x)*Cos[c + d*x] + 18*A*b^10*(c + d*x)*Cos[c + d*x] + 288*a^9*b*B*(c + d*x)*Cos[c + d*x] - 792*a^7*b^3*B*(c + d*x)*Cos[c + d*x] + 648*a^5*b^5*B*(c + d*x)*Cos[c + d*x] - 72*a^3*b^7*B*(c + d*x)*Cos[c + d*x] - 72*a*b^9*B*(c + d*x)*Cos[c + d*x] - 36*a^7*A*b^3*(c + d*x)*Cos[2*(c + d*x)] + 108*a^5*A*b^5*(c + d*x)*Cos[2*(c + d*x)] - 108*a^3*A*b^7*(c + d*x)*Cos[2*(c + d*x)] + 36*a*A*b^9*(c + d*x)*Cos[2*(c + d*x)] + 144*a^8*b^2*B*(c + d*x)*Cos[2*(c + d*x)] - 432*a^6*b^4*B*(c + d*x)*Cos[2*(c + d*x)] + 432*a^4*b^6*B*(c + d*x)*Cos[2*(c + d*x)] - 144*a^2*b^8*B*(c + d*x)*Cos[2*(c + d*x)] - 6*a^6*A*b^4*(c + d*x)*Cos[3*(c + d*x)] + 18*a^4*A*b^6*(c + d*x)*Cos[3*(c + d*x)] - 18*a^2*A*b^8*(c + d*x)*Cos[3*(c + d*x)] + 6*A*b^10*(c + d*x)*Cos[3*(c + d*x)] + 24*a^7*b^3*B*(c + d*x)*Cos[3*(c + d*x)] - 72*a^5*b^5*B*(c + d*x)*Cos[3*(c + d*x)] + 72*a^3*b^7*B*(c + d*x)*Cos[3*(c + d*x)] - 24*a*b^9*B*(c + d*x)*Cos[3*(c + d*x)] + 24*a^8*A*b^2*Sin[c + d*x] - 57*a^6*A*b^4*Sin[c + d*x] + 72*a^4*A*b^6*Sin[c + d*x] + 36*a^2*A*b^8*Sin[c + d*x] - 96*a^9*b*B*Sin[c + d*x] + 228*a^7*b^3*B*Sin[c + d*x] - 135*a^5*b^5*B*Sin[c + d*x] - 90*a^3*b^7*B*Sin[c + d*x] + 18*a*b^9*B*Sin[c + d*x] + 30*a^7*A*b^3*Sin[2*(c + d*x)] - 90*a^5*A*b^5*Sin[2*(c + d*x)] + 120*a^3*A*b^7*Sin[2*(c + d*x)] - 120*a^8*b^2*B*Sin[2*(c + d*x)] + 336*a^6*b^4*B*Sin[2*(c + d*x)] - 300*a^4*b^6*B*Sin[2*(c + d*x)] + 18*a^2*b^8*B*Sin[2*(c + d*x)] + 6*b^10*B*Sin[2*(c + d*x)] + 11*a^6*A*b^4*Sin[3*(c + d*x)] - 32*a^4*A*b^6*Sin[3*(c + d*x)] + 36*a^2*A*b^8*Sin[3*(c + d*x)] - 44*a^7*b^3*B*Sin[3*(c + d*x)] + 125*a^5*b^5*B*Sin[3*(c + d*x)] - 114*a^3*b^7*B*Sin[3*(c + d*x)] + 18*a*b^9*B*Sin[3*(c + d*x)] - 3*a^6*b^4*B*Sin[4*(c + d*x)] + 9*a^4*b^6*B*Sin[4*(c + d*x)] - 9*a^2*b^8*B*Sin[4*(c + d*x)] + 3*b^10*B*Sin[4*(c + d*x)])/(24*b^5*(-a^2 + b^2)^3*d*(a + b*Cos[c + d*x])^3)","B",1
274,1,717,301,3.3053919,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^4,x]","\frac{\frac{24 a^9 B c+24 a^9 B d x-24 a^8 b B \sin (c+d x)-30 a^7 b^2 B \sin (2 (c+d x))-36 a^7 b^2 B c-36 a^7 b^2 B d x+57 a^6 b^3 B \sin (c+d x)-11 a^6 b^3 B \sin (3 (c+d x))+6 a^6 b^3 B c \cos (3 (c+d x))+6 a^6 b^3 B d x \cos (3 (c+d x))+18 a^5 A b^4 \sin (c+d x)+2 a^5 A b^4 \sin (3 (c+d x))+90 a^5 b^4 B \sin (2 (c+d x))-36 a^5 b^4 B c-36 a^5 b^4 B d x+6 a^4 A b^5 \sin (2 (c+d x))-72 a^4 b^5 B \sin (c+d x)+32 a^4 b^5 B \sin (3 (c+d x))-18 a^4 b^5 B c \cos (3 (c+d x))-18 a^4 b^5 B d x \cos (3 (c+d x))+39 a^3 A b^6 \sin (c+d x)-5 a^3 A b^6 \sin (3 (c+d x))-120 a^3 b^6 B \sin (2 (c+d x))+84 a^3 b^6 B c+84 a^3 b^6 B d x+54 a^2 A b^7 \sin (2 (c+d x))-36 a^2 b^7 B \sin (c+d x)-36 a^2 b^7 B \sin (3 (c+d x))+18 a^2 b^7 B c \cos (3 (c+d x))+18 a^2 b^7 B d x \cos (3 (c+d x))+36 a b^2 B \left(a^2-b^2\right)^3 (c+d x) \cos (2 (c+d x))+18 b B \left(a^2-b^2\right)^3 \left(4 a^2+b^2\right) (c+d x) \cos (c+d x)+18 a A b^8 \sin (c+d x)+18 a A b^8 \sin (3 (c+d x))-36 a b^8 B c-36 a b^8 B d x-6 b^9 B c \cos (3 (c+d x))-6 b^9 B d x \cos (3 (c+d x))}{\left(a^2-b^2\right)^3 (a+b \cos (c+d x))^3}-\frac{24 \left(2 a^7 B-7 a^5 b^2 B+8 a^3 b^4 B+3 a^2 A b^5-8 a b^6 B+2 A b^7\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{a (A b-a B) \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^2 \left(3 a^3 B-8 a b^2 B+5 A b^3\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(9 a^5 B-28 a^3 b^2 B+a^2 A b^3+34 a b^4 B-16 A b^5\right) \sin (c+d x)}{6 b^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{\left(2 a^7 B-7 a^5 b^2 B+8 a^3 b^4 B+3 a^2 A b^5-8 a b^6 B+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)}{b^4 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{B x}{b^4}",1,"((-24*(3*a^2*A*b^5 + 2*A*b^7 + 2*a^7*B - 7*a^5*b^2*B + 8*a^3*b^4*B - 8*a*b^6*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) + (24*a^9*B*c - 36*a^7*b^2*B*c - 36*a^5*b^4*B*c + 84*a^3*b^6*B*c - 36*a*b^8*B*c + 24*a^9*B*d*x - 36*a^7*b^2*B*d*x - 36*a^5*b^4*B*d*x + 84*a^3*b^6*B*d*x - 36*a*b^8*B*d*x + 18*b*(a^2 - b^2)^3*(4*a^2 + b^2)*B*(c + d*x)*Cos[c + d*x] + 36*a*b^2*(a^2 - b^2)^3*B*(c + d*x)*Cos[2*(c + d*x)] + 6*a^6*b^3*B*c*Cos[3*(c + d*x)] - 18*a^4*b^5*B*c*Cos[3*(c + d*x)] + 18*a^2*b^7*B*c*Cos[3*(c + d*x)] - 6*b^9*B*c*Cos[3*(c + d*x)] + 6*a^6*b^3*B*d*x*Cos[3*(c + d*x)] - 18*a^4*b^5*B*d*x*Cos[3*(c + d*x)] + 18*a^2*b^7*B*d*x*Cos[3*(c + d*x)] - 6*b^9*B*d*x*Cos[3*(c + d*x)] + 18*a^5*A*b^4*Sin[c + d*x] + 39*a^3*A*b^6*Sin[c + d*x] + 18*a*A*b^8*Sin[c + d*x] - 24*a^8*b*B*Sin[c + d*x] + 57*a^6*b^3*B*Sin[c + d*x] - 72*a^4*b^5*B*Sin[c + d*x] - 36*a^2*b^7*B*Sin[c + d*x] + 6*a^4*A*b^5*Sin[2*(c + d*x)] + 54*a^2*A*b^7*Sin[2*(c + d*x)] - 30*a^7*b^2*B*Sin[2*(c + d*x)] + 90*a^5*b^4*B*Sin[2*(c + d*x)] - 120*a^3*b^6*B*Sin[2*(c + d*x)] + 2*a^5*A*b^4*Sin[3*(c + d*x)] - 5*a^3*A*b^6*Sin[3*(c + d*x)] + 18*a*A*b^8*Sin[3*(c + d*x)] - 11*a^6*b^3*B*Sin[3*(c + d*x)] + 32*a^4*b^5*B*Sin[3*(c + d*x)] - 36*a^2*b^7*B*Sin[3*(c + d*x)])/((a^2 - b^2)^3*(a + b*Cos[c + d*x])^3))/(24*b^4*d)","B",1
275,1,251,274,1.3247775,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^4,x]","\frac{\frac{2 \sin (c+d x) \left(10 a^5 B-25 a^4 A b+17 a^3 b^2 B-14 a^2 A b^3+6 a \left(a^4 A+a^3 b B-9 a^2 A b^2+9 a b^3 B-2 A b^4\right) \cos (c+d x)+\left(2 a^5 B+a^4 A b-5 a^3 b^2 B-10 a^2 A b^3+18 a b^4 B-6 A b^5\right) \cos (2 (c+d x))+18 a b^4 B-6 A b^5\right)}{(a+b \cos (c+d x))^3}-\frac{24 \left(a^3 A-3 a^2 b B+4 a A b^2-2 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)}{\sqrt{b^2-a^2}}}{24 d \left(a^2-b^2\right)^3}","-\frac{a^2 (A b-a B) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{\left(a^3 A-3 a^2 b B+4 a A b^2-2 b^3 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac{a \left(-4 a^3 B+a^2 A b+9 a b^2 B-6 A b^3\right) \sin (c+d x)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\left(2 a^5 B+a^4 A b-5 a^3 b^2 B-10 a^2 A b^3+18 a b^4 B-6 A b^5\right) \sin (c+d x)}{6 b^2 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}",1,"((-24*(a^3*A + 4*a*A*b^2 - 3*a^2*b*B - 2*b^3*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (2*(-25*a^4*A*b - 14*a^2*A*b^3 - 6*A*b^5 + 10*a^5*B + 17*a^3*b^2*B + 18*a*b^4*B + 6*a*(a^4*A - 9*a^2*A*b^2 - 2*A*b^4 + a^3*b*B + 9*a*b^3*B)*Cos[c + d*x] + (a^4*A*b - 10*a^2*A*b^3 - 6*A*b^5 + 2*a^5*B - 5*a^3*b^2*B + 18*a*b^4*B)*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
276,1,252,263,1.1594551,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^4,x]","\frac{\frac{2 \sin (c+d x) \left(12 a^5 A-25 a^4 b B+22 a^3 A b^2-14 a^2 b^3 B+b \left(a^4 B+2 a^3 A b-10 a^2 b^2 B+13 a A b^3-6 b^4 B\right) \cos (2 (c+d x))+6 \left(a^5 B+2 a^4 A b-9 a^3 b^2 B+9 a^2 A b^3-2 a b^4 B-A b^5\right) \cos (c+d x)+11 a A b^4-6 b^5 B\right)}{(a+b \cos (c+d x))^3}-\frac{24 \left(a^3 B-4 a^2 A b+4 a b^2 B-A b^3\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 (A b-a B) \sin (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}-\frac{\left(a^3 (-B)+4 a^2 A b-4 a b^2 B+A b^3\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{\left(a^3 B+2 a^2 A b-6 a b^2 B+3 A b^3\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^4 B+2 a^3 A b-10 a^2 b^2 B+13 a A b^3-6 b^4 B\right) \sin (c+d x)}{6 b d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}",1,"((-24*(-4*a^2*A*b - A*b^3 + a^3*B + 4*a*b^2*B)*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 + 6*(2*a^4*A*b + 9*a^2*A*b^3 - A*b^5 + a^5*B - 9*a^3*b^2*B - 2*a*b^4*B)*Cos[c + d*x] + b*(2*a^3*A*b + 13*a*A*b^3 + a^4*B - 10*a^2*b^2*B - 6*b^4*B)*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
277,1,227,237,2.3707189,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x))^4} \, dx","Integrate[(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^4,x]","\frac{\frac{\left(2 a^2 B-5 a A b+3 b^2 B\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))^2}+\frac{\left(2 a^3 B-11 a^2 A b+13 a b^2 B-4 A b^3\right) \sin (c+d x)}{(a-b)^3 (a+b)^3 (a+b \cos (c+d x))}+\frac{6 \left(2 a^3 A-4 a^2 b B+3 a A b^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)^{7/2}}+\frac{2 (a B-A b) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^3}}{6 d}","-\frac{\left(-2 a^2 B+5 a A b-3 b^2 B\right) \sin (c+d x)}{6 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{(A b-a B) \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{\left(2 a^3 A-4 a^2 b B+3 a A b^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)}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left(-2 a^3 B+11 a^2 A b-13 a b^2 B+4 A b^3\right) \sin (c+d x)}{6 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}",1,"((6*(2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) + (2*(-(A*b) + a*B)*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^3) + ((-5*a*A*b + 2*a^2*B + 3*b^2*B)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])^2) + ((-11*a^2*A*b - 4*A*b^3 + 2*a^3*B + 13*a*b^2*B)*Sin[c + d*x])/((a - b)^3*(a + b)^3*(a + b*Cos[c + d*x])))/(6*d)","A",1
278,1,368,301,1.7331325,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(a+b \cos (c+d x))^4} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x])/(a + b*Cos[c + d*x])^4,x]","\frac{\cos (c+d x) (A \sec (c+d x)+B) \left(\frac{24 \left(2 a^7 B-8 a^6 A b+3 a^5 b^2 B+8 a^4 A b^3-7 a^2 A b^5+2 A b^7\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 b \sin (c+d x) \left(36 a^7 B-72 a^6 A b+a^5 b^2 B+38 a^4 A b^3+8 a^3 b^4 B-5 a^2 A b^5+6 a b \left(9 a^5 B-20 a^4 A b+a^3 b^2 B+15 a^2 A b^3-5 A b^5\right) \cos (c+d x)+b^2 \left(11 a^5 B-26 a^4 A b+4 a^3 b^2 B+17 a^2 A b^3-6 A b^5\right) \cos (2 (c+d x))-6 A b^7\right)}{\left(a^2-b^2\right)^3 (a+b \cos (c+d x))^3}-24 A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+24 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{24 a^4 d (A+B \cos (c+d x))}","\frac{A \tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac{b (A b-a B) \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{b \left(-5 a^3 B+8 a^2 A b-3 A b^3\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(-11 a^5 B+26 a^4 A b-4 a^3 b^2 B-17 a^2 A b^3+6 A b^5\right) \sin (c+d x)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{\left(-2 a^7 B+8 a^6 A b-3 a^5 b^2 B-8 a^4 A b^3+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}}",1,"(Cos[c + d*x]*(B + A*Sec[c + d*x])*((24*(-8*a^6*A*b + 8*a^4*A*b^3 - 7*a^2*A*b^5 + 2*A*b^7 + 2*a^7*B + 3*a^5*b^2*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) - 24*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 24*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - (2*a*b*(-72*a^6*A*b + 38*a^4*A*b^3 - 5*a^2*A*b^5 - 6*A*b^7 + 36*a^7*B + a^5*b^2*B + 8*a^3*b^4*B + 6*a*b*(-20*a^4*A*b + 15*a^2*A*b^3 - 5*A*b^5 + 9*a^5*B + a^3*b^2*B)*Cos[c + d*x] + b^2*(-26*a^4*A*b + 17*a^2*A*b^3 - 6*A*b^5 + 11*a^5*B + 4*a^3*b^2*B)*Cos[2*(c + d*x)])*Sin[c + d*x])/((a^2 - b^2)^3*(a + b*Cos[c + d*x])^3)))/(24*a^4*d*(A + B*Cos[c + d*x]))","A",1
279,1,549,420,3.2699248,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^4,x]","\frac{-\frac{48 b \left(8 a^7 B-20 a^6 A b-8 a^5 b^2 B+35 a^4 A b^3+7 a^3 b^4 B-28 a^2 A b^5-2 a b^6 B+8 A b^7\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 \tan (c+d x) \left(24 a^9 A-36 a^7 A b^2+6 a^6 A b^3 \cos (3 (c+d x))+120 a^6 b^3 B-246 a^5 A b^4+26 a^5 b^4 B \cos (3 (c+d x))-65 a^4 A b^5 \cos (3 (c+d x))-90 a^4 b^5 B+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 \left(6 a^6 A+20 a^5 b B-53 a^4 A b^2-15 a^3 b^3 B+57 a^2 A b^4+5 a b^5 B-20 A b^6\right) \cos (2 (c+d x))+b \left(72 a^8 A+144 a^7 b B-438 a^6 A b^2-50 a^5 b^3 B+305 a^4 A b^4-7 a^3 b^5 B+28 a^2 A b^6+18 a b^7 B-72 A b^8\right) \cos (c+d x)-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) \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) \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{(4 A b-a B) \tanh ^{-1}(\sin (c+d x))}{a^5 d}+\frac{b (A b-a B) \tan (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{b \left(-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right) \tan (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{b \left(-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right) \tan (c+d x)}{2 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{\left(6 a^6 A+26 a^5 b B-65 a^4 A b^2-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right) \tan (c+d x)}{6 a^4 d \left(a^2-b^2\right)^3}+\frac{b \left(-8 a^7 B+20 a^6 A b+8 a^5 b^2 B-35 a^4 A b^3-7 a^3 b^4 B+28 a^2 A b^5+2 a b^6 B-8 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^5 d (a-b)^{7/2} (a+b)^{7/2}}",1,"((-48*b*(-20*a^6*A*b + 35*a^4*A*b^3 - 28*a^2*A*b^5 + 8*A*b^7 + 8*a^7*B - 8*a^5*b^2*B + 7*a^3*b^4*B - 2*a*b^6*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) + 48*(4*A*b - a*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 48*(-4*A*b + a*B)*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 + b*(72*a^8*A - 438*a^6*A*b^2 + 305*a^4*A*b^4 + 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)*Cos[c + d*x] + 6*a*b^2*(6*a^6*A - 53*a^4*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)*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)])*Tan[c + d*x])/((a^2 - b^2)^3*(a + b*Cos[c + d*x])^3))/(48*a^5*d)","A",1
280,1,781,547,5.2883298,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^4,x]","\frac{-48 \left(a^2 A-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)+48 \left(a^2 A-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)+\frac{96 b^2 \left(20 a^7 B-40 a^6 A b-35 a^5 b^2 B+84 a^4 A b^3+28 a^3 b^4 B-69 a^2 A b^5-8 a b^6 B+20 A b^7\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 \tan (c+d x) \sec (c+d x) \left(24 a^{10} A+72 a^9 b B-324 a^8 A b^2+36 a^8 b^2 B \cos (3 (c+d x))-138 a^7 A b^3 \cos (3 (c+d x))+6 a^7 b^3 B \cos (4 (c+d x))-438 a^7 b^3 B-24 a^6 A b^4 \cos (4 (c+d x))+1116 a^6 A b^4-318 a^6 b^4 B \cos (3 (c+d x))+738 a^5 A b^5 \cos (3 (c+d x))-65 a^5 b^5 B \cos (4 (c+d x))+305 a^5 b^5 B+146 a^4 A b^6 \cos (4 (c+d x))-830 a^4 A b^6+342 a^4 b^6 B \cos (3 (c+d x))-840 a^3 A b^7 \cos (3 (c+d x))+68 a^3 b^7 B \cos (4 (c+d x))+28 a^3 b^7 B-167 a^2 A b^8 \cos (4 (c+d x))-61 a^2 A b^8-120 a^2 b^8 B \cos (3 (c+d x))+6 a \left(8 a^9 B-20 a^8 A b-6 a^7 b^2 B-9 a^6 A b^3-135 a^5 b^4 B+309 a^4 A b^5+163 a^3 b^6 B-400 a^2 A b^7-60 a b^8 B+150 A b^9\right) \cos (c+d x)+12 b \left(6 a^9 B-21 a^8 A b-36 a^7 b^2 B+85 a^6 A b^3+20 a^5 b^4 B-55 a^4 A b^5+8 a^3 b^6 B-19 a^2 A b^7-8 a b^8 B+20 A b^9\right) \cos (2 (c+d x))+300 a A b^9 \cos (3 (c+d x))-24 a b^9 B \cos (4 (c+d x))-72 a b^9 B+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}}{96 a^6 d}","\frac{b (A b-a B) \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-8 a b B+20 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}+\frac{b \left(-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\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{b \left(-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\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))}+\frac{\left(a^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-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{b^2 \left(-20 a^7 B+40 a^6 A b+35 a^5 b^2 B-84 a^4 A b^3-28 a^3 b^4 B+69 a^2 A b^5+8 a b^6 B-20 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^6 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left(-6 a^7 B+24 a^6 A b+65 a^5 b^2 B-146 a^4 A b^3-68 a^3 b^4 B+167 a^2 A b^5+24 a b^6 B-60 A b^7\right) \tan (c+d x)}{6 a^5 d \left(a^2-b^2\right)^3}",1,"((96*b^2*(-40*a^6*A*b + 84*a^4*A*b^3 - 69*a^2*A*b^5 + 20*A*b^7 + 20*a^7*B - 35*a^5*b^2*B + 28*a^3*b^4*B - 8*a*b^6*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) - 48*(a^2*A + 20*A*b^2 - 8*a*b*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 48*(a^2*A + 20*A*b^2 - 8*a*b*B)*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 + 72*a^9*b*B - 438*a^7*b^3*B + 305*a^5*b^5*B + 28*a^3*b^7*B - 72*a*b^9*B + 6*a*(-20*a^8*A*b - 9*a^6*A*b^3 + 309*a^4*A*b^5 - 400*a^2*A*b^7 + 150*A*b^9 + 8*a^9*B - 6*a^7*b^2*B - 135*a^5*b^4*B + 163*a^3*b^6*B - 60*a*b^8*B)*Cos[c + d*x] + 12*b*(-21*a^8*A*b + 85*a^6*A*b^3 - 55*a^4*A*b^5 - 19*a^2*A*b^7 + 20*A*b^9 + 6*a^9*B - 36*a^7*b^2*B + 20*a^5*b^4*B + 8*a^3*b^6*B - 8*a*b^8*B)*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)] + 36*a^8*b^2*B*Cos[3*(c + d*x)] - 318*a^6*b^4*B*Cos[3*(c + d*x)] + 342*a^4*b^6*B*Cos[3*(c + d*x)] - 120*a^2*b^8*B*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)] + 6*a^7*b^3*B*Cos[4*(c + d*x)] - 65*a^5*b^5*B*Cos[4*(c + d*x)] + 68*a^3*b^7*B*Cos[4*(c + d*x)] - 24*a*b^9*B*Cos[4*(c + d*x)])*Sec[c + d*x]*Tan[c + d*x])/((a^2 - b^2)^3*(a + b*Cos[c + d*x])^3))/(96*a^6*d)","A",1
281,1,28,28,0.0084386,"\int \frac{\cos ^3(c+d x) (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),x]","B \left(\frac{\sin (c+d x)}{d}-\frac{\sin ^3(c+d x)}{3 d}\right)","\frac{B \sin (c+d x)}{d}-\frac{B \sin ^3(c+d x)}{3 d}",1,"B*(Sin[c + d*x]/d - Sin[c + d*x]^3/(3*d))","A",1
282,1,24,27,0.0203614,"\int \frac{\cos ^2(c+d x) (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),x]","\frac{B (2 (c+d x)+\sin (2 (c+d x)))}{4 d}","\frac{B \sin (c+d x) \cos (c+d x)}{2 d}+\frac{B x}{2}",1,"(B*(2*(c + d*x) + Sin[2*(c + d*x)]))/(4*d)","A",1
283,1,23,11,0.0076745,"\int \frac{\cos (c+d x) (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),x]","B \left(\frac{\sin (c) \cos (d x)}{d}+\frac{\cos (c) \sin (d x)}{d}\right)","\frac{B \sin (c+d x)}{d}",1,"B*((Cos[d*x]*Sin[c])/d + (Cos[c]*Sin[d*x])/d)","B",1
284,1,3,3,0.0003764,"\int \frac{a B+b B \cos (c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x]),x]","B x","B x",1,"B*x","A",1
285,1,12,12,0.0029429,"\int \frac{(a B+b B \cos (c+d x)) \sec (c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x])/(a + b*Cos[c + d*x]),x]","\frac{B \tanh ^{-1}(\sin (c+d x))}{d}","\frac{B \tanh ^{-1}(\sin (c+d x))}{d}",1,"(B*ArcTanh[Sin[c + d*x]])/d","A",1
286,1,11,11,0.0046519,"\int \frac{(a B+b B \cos (c+d x)) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + b*Cos[c + d*x]),x]","\frac{B \tan (c+d x)}{d}","\frac{B \tan (c+d x)}{d}",1,"(B*Tan[c + d*x])/d","A",1
287,1,36,36,0.0077153,"\int \frac{(a B+b B \cos (c+d x)) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + b*Cos[c + d*x]),x]","B \left(\frac{\tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{\tan (c+d x) \sec (c+d x)}{2 d}\right)","\frac{B \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{B \tan (c+d x) \sec (c+d x)}{2 d}",1,"B*(ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d))","A",1
288,1,24,28,0.0399672,"\int \frac{(a B+b B \cos (c+d x)) \sec ^4(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x]^4)/(a + b*Cos[c + d*x]),x]","\frac{B \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}","\frac{B \tan ^3(c+d x)}{3 d}+\frac{B \tan (c+d x)}{d}",1,"(B*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d","A",1
289,1,98,114,0.2505679,"\int \frac{\cos ^3(c+d x) (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^2,x]","\frac{B \left(2 \left(2 a^2+b^2\right) (c+d x)+\frac{8 a^3 \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 \sin (c+d x)+b^2 \sin (2 (c+d x))\right)}{4 b^3 d}","-\frac{2 a^3 B \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{B x \left(2 a^2+b^2\right)}{2 b^3}-\frac{a B \sin (c+d x)}{b^2 d}+\frac{B \sin (c+d x) \cos (c+d x)}{2 b d}",1,"(B*(2*(2*a^2 + b^2)*(c + d*x) + (8*a^3*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^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
290,1,73,79,0.1382776,"\int \frac{\cos ^2(c+d x) (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^2,x]","\frac{B \left(-\frac{2 a^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}}-a (c+d x)+b \sin (c+d x)\right)}{b^2 d}","\frac{2 a^2 B \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 B x}{b^2}+\frac{B \sin (c+d x)}{b d}",1,"(B*(-(a*(c + d*x)) - (2*a^2*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + b*Sin[c + d*x]))/(b^2*d)","A",1
291,1,59,61,0.0796867,"\int \frac{\cos (c+d x) (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^2,x]","\frac{B \left(\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}}+c+d x\right)}{b d}","\frac{B x}{b}-\frac{2 a B \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}}",1,"(B*(c + d*x + (2*a*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2]))/(b*d)","A",1
292,1,49,50,0.0375725,"\int \frac{a B+b B \cos (c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^2,x]","-\frac{2 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{2 B \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}}",1,"(-2*B*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(Sqrt[-a^2 + b^2]*d)","A",1
293,1,103,70,0.0808015,"\int \frac{(a B+b B \cos (c+d x)) \sec (c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x])/(a + b*Cos[c + d*x])^2,x]","\frac{B \left(\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}}-\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 d}","\frac{B \tanh ^{-1}(\sin (c+d x))}{a d}-\frac{2 b B \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,"(B*((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*d)","A",1
294,1,116,88,0.3736304,"\int \frac{(a B+b B \cos (c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^2,x]","\frac{B \left(-\frac{2 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}}+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)\right)}{a^2 d}","\frac{2 b^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{b B \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{B \tan (c+d x)}{a d}",1,"(B*((-2*b^2*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*Tan[c + d*x]))/(a^2*d)","A",1
295,1,239,123,1.0532524,"\int \frac{(a B+b B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^2,x]","\frac{B \left(\frac{8 b^3 \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)-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)\right)}{4 a^3 d}","-\frac{2 b^3 B \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 \tan (c+d x)}{a^2 d}+\frac{B \left(a^2+2 b^2\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,"(B*((8*b^3*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]] - 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)","A",1
296,1,292,386,1.5722996,"\int \cos ^3(c+d x) \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{8 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b^2 \left(-4 a^3 B+6 a^2 A b+111 a b^2 B+75 A b^3\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(-16 a^4 B+24 a^3 A b-24 a^2 b^2 B+57 a A b^3+147 b^4 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)-b (a+b \cos (c+d x)) \left(-b \left(\left(-24 a^2 B+36 a A b+266 b^2 B\right) \sin (2 (c+d x))+5 b (2 (a B+9 A b) \sin (3 (c+d x))+7 b B \sin (4 (c+d x)))\right)-2 \left(32 a^3 B-48 a^2 A b+57 a b^2 B+345 A b^3\right) \sin (c+d x)\right)}{1260 b^4 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(-24 a^2 B+36 a A b-49 b^2 B\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^3 d}+\frac{2 \left(-16 a^3 B+24 a^2 A b-36 a b^2 B+75 A b^3\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^3 d}-\frac{2 \left(a^2-b^2\right) \left(-16 a^3 B+24 a^2 A b-36 a b^2 B+75 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)}{315 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-16 a^4 B+24 a^3 A b-24 a^2 b^2 B+57 a A b^3+147 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)}{315 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (3 A b-2 a B) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{21 b^2 d}+\frac{2 B \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*A*b + 75*A*b^3 - 4*a^3*B + 111*a*b^2*B)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (24*a^3*A*b + 57*a*A*b^3 - 16*a^4*B - 24*a^2*b^2*B + 147*b^4*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])*(-2*(-48*a^2*A*b + 345*A*b^3 + 32*a^3*B + 57*a*b^2*B)*Sin[c + d*x] - b*((36*a*A*b - 24*a^2*B + 266*b^2*B)*Sin[2*(c + d*x)] + 5*b*(2*(9*A*b + a*B)*Sin[3*(c + d*x)] + 7*b*B*Sin[4*(c + d*x)]))))/(1260*b^4*d*Sqrt[a + b*Cos[c + d*x]])","A",1
297,1,232,303,1.0102599,"\int \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{b (a+b \cos (c+d x)) \left(\left(-16 a^2 B+28 a A b+115 b^2 B\right) \sin (c+d x)+3 b (2 (a B+7 A b) \sin (2 (c+d x))+5 b B \sin (3 (c+d x)))\right)+4 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b^2 \left(2 a^2 B+49 a A b+25 b^2 B\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(8 a^3 B-14 a^2 A b+19 a b^2 B+63 A b^3\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 B+14 a A b-25 b^2 B\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 B+14 a A b-25 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)}{105 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-8 a^3 B+14 a^2 A b-19 a b^2 B-63 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)}{105 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (7 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b^2 d}+\frac{2 B \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*A*b + 2*a^2*B + 25*b^2*B)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (-14*a^2*A*b + 63*A*b^3 + 8*a^3*B + 19*a*b^2*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])*((28*a*A*b - 16*a^2*B + 115*b^2*B)*Sin[c + d*x] + 3*b*(2*(7*A*b + a*B)*Sin[2*(c + d*x)] + 5*b*B*Sin[3*(c + d*x)])))/(210*b^3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
298,1,179,231,0.8913963,"\int \cos (c+d x) \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(\left(-2 a^2 B+5 a A b+9 b^2 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)+b^2 (7 a B+5 A 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 B+5 A b+3 b B \cos (c+d x))}{15 b^2 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(a^2-b^2\right) (5 A b-2 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)}{15 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-2 a^2 B+5 a A b+9 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)}{15 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (5 A b-2 a B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b d}+\frac{2 B \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*A*b + 7*a*B)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (5*a*A*b - 2*a^2*B + 9*b^2*B)*((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*A*b + a*B + 3*b*B*Cos[c + d*x])*Sin[c + d*x])/(15*b^2*d*Sqrt[a + b*Cos[c + d*x]])","A",1
299,1,146,171,0.5941015,"\int \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{-2 B \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 B+3 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)+2 b B \sin (c+d x) (a+b \cos (c+d x))}{3 b d \sqrt{a+b \cos (c+d x)}}","-\frac{2 B \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 B+3 A 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 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}",1,"(2*(a + b)*(3*A*b + 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)*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + 2*b*B*(a + b*Cos[c + d*x])*Sin[c + d*x])/(3*b*d*Sqrt[a + b*Cos[c + d*x]])","A",1
300,1,107,178,2.4010137,"\int \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec (c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(A \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)+B (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 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)}{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 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,"(2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*((a + b)*B*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + A*(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
301,1,372,213,10.5162978,"\int \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{\frac{2 (4 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)}{\sqrt{a+b \cos (c+d x)}}+4 A \tan (c+d x) \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 b \sqrt{-\frac{1}{a+b}}}+\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)}}}{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 \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{d}-\frac{A \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*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*(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*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
302,1,420,292,4.2559295,"\int \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{\frac{2 \left(8 a^2 A+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 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 d}","\frac{\left(4 a^2 A+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+A b) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a d}+\frac{(4 a B+3 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 d \sqrt{a+b \cos (c+d x)}}-\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*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 - 3*A*b^2 + 4*a*b*B)*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
303,1,635,378,6.547444,"\int \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*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)+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 \left(-16 a^2 A b-6 a b^2 B+3 A b^3\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 B+8 a^2 A b-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{\left(16 a^2 A+6 a b B-3 A b^2\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 a^2 d}+\frac{\left(16 a^2 A+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(16 a^2 A+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 A b-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)*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)*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) + (A*Sec[c + d*x]^2*Tan[c + d*x])/3))/d","C",1
304,1,291,378,1.5337054,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{b (a+b \cos (c+d x)) \left(b \left(2 \left(6 a^2 B+144 a A b+133 b^2 B\right) \sin (2 (c+d x))+5 b (2 (10 a B+9 A b) \sin (3 (c+d x))+7 b B \sin (4 (c+d x)))\right)+\left(-32 a^3 B+72 a^2 A b+804 a b^2 B+690 A b^3\right) \sin (c+d x)\right)+8 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b^2 \left(2 a^3 B+153 a^2 A b+186 a b^2 B+75 A b^3\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(8 a^4 B-18 a^3 A b+33 a^2 b^2 B+246 a A b^3+147 b^4 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)}{1260 b^3 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(-8 a^2 B+18 a A b-49 b^2 B\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}-\frac{2 \left(-8 a^3 B+18 a^2 A b-39 a b^2 B-75 A b^3\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 B+18 a^2 A b-39 a b^2 B-75 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)}{315 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-8 a^4 B+18 a^3 A b-33 a^2 b^2 B-246 a A b^3-147 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)}{315 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (9 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac{2 B \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*A*b + 75*A*b^3 + 2*a^3*B + 186*a*b^2*B)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*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])*((72*a^2*A*b + 690*A*b^3 - 32*a^3*B + 804*a*b^2*B)*Sin[c + d*x] + b*(2*(144*a*A*b + 6*a^2*B + 133*b^2*B)*Sin[2*(c + d*x)] + 5*b*(2*(9*A*b + 10*a*B)*Sin[3*(c + d*x)] + 7*b*B*Sin[4*(c + d*x)]))))/(1260*b^3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
305,1,233,297,1.0715613,"\int \cos (c+d x) (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{b (a+b \cos (c+d x)) \left(\left(12 a^2 B+168 a A b+115 b^2 B\right) \sin (c+d x)+3 b (2 (8 a B+7 A b) \sin (2 (c+d x))+5 b B \sin (3 (c+d x)))\right)+4 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b^2 \left(51 a^2 B+84 a A b+25 b^2 B\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(-6 a^3 B+21 a^2 A b+82 a b^2 B+63 A b^3\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 B+21 a A b+25 b^2 B\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 B+21 a A b+25 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)}{105 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-6 a^3 B+21 a^2 A b+82 a b^2 B+63 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)}{105 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (7 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b d}+\frac{2 B \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*A*b + 51*a^2*B + 25*b^2*B)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (21*a^2*A*b + 63*A*b^3 - 6*a^3*B + 82*a*b^2*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])*((168*a*A*b + 12*a^2*B + 115*b^2*B)*Sin[c + d*x] + 3*b*(2*(7*A*b + 8*a*B)*Sin[2*(c + d*x)] + 5*b*B*Sin[3*(c + d*x)])))/(210*b^2*d*Sqrt[a + b*Cos[c + d*x]])","A",1
306,1,203,225,0.784418,"\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{2 \left(b \left(15 a^2 A+12 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)+\left(3 a^2 B+20 a A b+9 b^2 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)+b \sin (c+d x) (a+b \cos (c+d x)) (6 a B+5 A b+3 b B \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 B+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)}{15 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(3 a^2 B+20 a A b+9 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)}{15 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (3 a B+5 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{2 B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}",1,"(2*(b*(15*a^2*A + 5*A*b^2 + 12*a*b*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (20*a*A*b + 3*a^2*B + 9*b^2*B)*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*A*b + 6*a*B + 3*b*B*Cos[c + d*x])*Sin[c + d*x]))/(15*b*d*Sqrt[a + b*Cos[c + d*x]])","A",1
307,1,406,236,2.5834475,"\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{\frac{4 \left(3 a^2 B+6 a A b+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(6 a^2 A+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)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 i (4 a B+3 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 b B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{6 d}","\frac{2 \left(a^2 (-B)+3 a A b+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 d \sqrt{a+b \cos (c+d x)}}+\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 (4 a B+3 A 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 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}",1,"((4*(6*a*A*b + 3*a^2*B + b^2*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*(6*a^2*A + 3*A*b^2 + 4*a*b*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)*(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*b*B*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(6*d)","C",1
308,1,398,232,2.5362566,"\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{\frac{2 \left(4 a^2 B+5 a A b+2 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 (2 a B+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)}}+\frac{2 i (2 b B-a 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 b \sqrt{-\frac{1}{a+b}}}+4 a A \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}","\frac{\left(a^2 A+2 a b B+2 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)}{d \sqrt{a+b \cos (c+d x)}}-\frac{(a A-2 b 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}}}+\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{a A \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{d}",1,"((8*b*(A*b + 2*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*(5*a*A*b + 4*a^2*B + 2*b^2*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*A) + 2*b*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*a*A*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*d)","C",1
309,1,422,295,4.9528086,"\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{\frac{2 \left(8 a^2 A+20 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)}{\sqrt{a+b \cos (c+d x)}}+\frac{8 b (a A+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)}}+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)-\frac{2 i (4 a B+5 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}}}}{16 d}","\frac{\left(4 a^2 B+7 a A b+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+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) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{(4 a B+5 A 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 A \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d}",1,"((8*b*(a*A + 4*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 + A*b^2 + 20*a*b*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)*(5*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 + (5*A*b + 4*a*B)*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(16*d)","C",1
310,1,634,375,6.7108584,"\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*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)+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\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 A b-30 a b^2 B-3 A b^3\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 B+56 a^2 A b+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{\left(16 a^2 A+30 a b B+3 A b^2\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 a d}+\frac{\left(16 a^2 A+42 a b B+17 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 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(16 a^2 A+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 A b+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{(6 a B+7 A b) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{12 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}",1,"((2*(28*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*(56*a^2*A*b - 9*A*b^3 + 48*a^3*B + 6*a*b^2*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)*(-16*a^2*A*b - 3*A*b^3 - 30*a*b^2*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)))/(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) + (a*A*Sec[c + d*x]^2*Tan[c + d*x])/3))/d","C",1
311,1,357,462,2.1189081,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]),x]","\frac{b (a+b \cos (c+d x)) \left(b \left(5 b \left(\left(452 a^2 B+836 a A b+513 b^2 B\right) \sin (3 (c+d x))+7 b ((46 a B+22 A b) \sin (4 (c+d x))+9 b B \sin (5 (c+d x)))\right)+4 \left(30 a^3 B+1650 a^2 A b+3095 a b^2 B+1463 A b^3\right) \sin (2 (c+d x))\right)+\left(-320 a^4 B+880 a^3 A b+18660 a^2 b^2 B+32868 a A b^3+13050 b^4 B\right) \sin (c+d x)\right)+16 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b^2 \left(10 a^4 B+1705 a^3 A b+3315 a^2 b^2 B+2871 a A b^3+675 b^4 B\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(40 a^5 B-110 a^4 A b+255 a^3 b^2 B+3069 a^2 A b^3+3705 a b^4 B+1617 A b^5\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 B+22 a A b-81 b^2 B\right) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}-\frac{2 \left(-40 a^3 B+110 a^2 A b-335 a b^2 B-539 A b^3\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3465 b^2 d}-\frac{2 \left(-40 a^4 B+110 a^3 A b-285 a^2 b^2 B-1254 a A b^3-675 b^4 B\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 B+110 a^3 A b-285 a^2 b^2 B-1254 a A b^3-675 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)}{3465 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-40 a^5 B+110 a^4 A b-255 a^3 b^2 B-3069 a^2 A b^3-3705 a b^4 B-1617 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)}{3465 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (11 A b-4 a B) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac{2 B \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*A*b + 2871*a*A*b^3 + 10*a^4*B + 3315*a^2*b^2*B + 675*b^4*B)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*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])*((880*a^3*A*b + 32868*a*A*b^3 - 320*a^4*B + 18660*a^2*b^2*B + 13050*b^4*B)*Sin[c + d*x] + b*(4*(1650*a^2*A*b + 1463*A*b^3 + 30*a^3*B + 3095*a*b^2*B)*Sin[2*(c + d*x)] + 5*b*((836*a*A*b + 452*a^2*B + 513*b^2*B)*Sin[3*(c + d*x)] + 7*b*((22*A*b + 46*a*B)*Sin[4*(c + d*x)] + 9*b*B*Sin[5*(c + d*x)])))))/(27720*b^3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
312,1,291,372,1.6002631,"\int \cos (c+d x) (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]),x]","\frac{b (a+b \cos (c+d x)) \left(b \left(\left(300 a^2 B+540 a A b+266 b^2 B\right) \sin (2 (c+d x))+5 b (2 (19 a B+9 A b) \sin (3 (c+d x))+7 b B \sin (4 (c+d x)))\right)+2 \left(20 a^3 B+540 a^2 A b+747 a b^2 B+345 A b^3\right) \sin (c+d x)\right)+8 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b^2 \left(155 a^3 B+405 a^2 A b+261 a b^2 B+75 A b^3\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(-10 a^4 B+45 a^3 A b+279 a^2 b^2 B+435 a A b^3+147 b^4 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)}{1260 b^2 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(-10 a^2 B+45 a A b+49 b^2 B\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b d}+\frac{2 \left(-10 a^3 B+45 a^2 A b+114 a b^2 B+75 A b^3\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 B+45 a^2 A b+114 a b^2 B+75 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)}{315 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-10 a^4 B+45 a^3 A b+279 a^2 b^2 B+435 a A b^3+147 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)}{315 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (9 A b-2 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b d}+\frac{2 B \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*A*b + 75*A*b^3 + 155*a^3*B + 261*a*b^2*B)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (45*a^3*A*b + 435*a*A*b^3 - 10*a^4*B + 279*a^2*b^2*B + 147*b^4*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])*(2*(540*a^2*A*b + 345*A*b^3 + 20*a^3*B + 747*a*b^2*B)*Sin[c + d*x] + b*((540*a*A*b + 300*a^2*B + 266*b^2*B)*Sin[2*(c + d*x)] + 5*b*(2*(9*A*b + 19*a*B)*Sin[3*(c + d*x)] + 7*b*B*Sin[4*(c + d*x)]))))/(1260*b^2*d*Sqrt[a + b*Cos[c + d*x]])","A",1
313,1,254,288,1.0847764,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]),x]","\frac{b \sin (c+d x) (a+b \cos (c+d x)) \left(90 a^2 B+6 b (15 a B+7 A b) \cos (c+d x)+154 a A b+15 b^2 B \cos (2 (c+d x))+65 b^2 B\right)+2 b \left(105 a^3 A+135 a^2 b B+119 a A b^2+25 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)+2 \left(15 a^3 B+161 a^2 A b+145 a b^2 B+63 A b^3\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 B+56 a A b+25 b^2 B\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 B+56 a A b+25 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)}{105 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(15 a^3 B+161 a^2 A b+145 a b^2 B+63 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)}{105 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (5 a B+7 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d}+\frac{2 B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}",1,"(2*b*(105*a^3*A + 119*a*A*b^2 + 135*a^2*b*B + 25*b^3*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + 2*(161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*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*A*b + 90*a^2*B + 65*b^2*B + 6*b*(7*A*b + 15*a*B)*Cos[c + d*x] + 15*b^2*B*Cos[2*(c + d*x)])*Sin[c + d*x])/(105*b*d*Sqrt[a + b*Cos[c + d*x]])","A",1
314,1,453,292,2.911697,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{\frac{2 i \left(23 a^2 B+35 a A b+9 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(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 B+45 a^2 A b+17 a b^2 B+5 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 \left(30 a^3 A+23 a^2 b B+35 a A b^2+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)}}+4 b \sin (c+d x) \sqrt{a+b \cos (c+d x)} (11 a B+5 A b+3 b B \cos (c+d x))}{30 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(23 a^2 B+35 a A b+9 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)}{15 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \left(-8 a^3 B+10 a^2 A b+8 a b^2 B+5 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)}{15 d \sqrt{a+b \cos (c+d x)}}+\frac{2 b (8 a B+5 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}+\frac{2 b B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}",1,"((4*(45*a^2*A*b + 5*A*b^3 + 15*a^3*B + 17*a*b^2*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*(30*a^3*A + 35*a*A*b^2 + 23*a^2*b*B + 9*b^3*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)*(35*a*A*b + 23*a^2*B + 9*b^2*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]]*(5*A*b + 11*a*B + 3*b*B*Cos[c + d*x])*Sin[c + d*x])/(30*d)","C",1
315,1,442,296,3.9369121,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{4 \tan (c+d x) \sqrt{a+b \cos (c+d x)} \left(3 a^2 A+2 b^2 B \cos (c+d x)\right)+\frac{8 b \left(9 a^2 B+9 a A b+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(-3 a^2 A+14 a b B+6 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 \sqrt{-\frac{1}{a+b}}}+\frac{2 \left(12 a^3 B+27 a^2 A b+14 a b^2 B+6 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)}}}{12 d}","-\frac{\left(3 a^2 A-14 a b B-6 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)}{3 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(3 a^3 A+4 a^2 b B+12 a A b^2+2 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 d \sqrt{a+b \cos (c+d x)}}-\frac{b (3 a A-2 b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{a A \tan (c+d x) (a+b \cos (c+d x))^{3/2}}{d}",1,"((8*b*(9*a*A*b + 9*a^2*B + b^2*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*(27*a^2*A*b + 6*A*b^3 + 12*a^3*B + 14*a*b^2*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)*(-3*a^2*A + 6*A*b^2 + 14*a*b*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]]*(3*a^2*A + 2*b^2*B*Cos[c + d*x])*Tan[c + d*x])/(12*d)","C",1
316,1,451,315,5.8752151,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{\frac{8 b \left(a^2 A+12 a b B+4 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 \left(-4 a^2 B-9 a A b+8 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(8 a^3 A+36 a^2 b B+21 a A b^2+8 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)}}+4 a \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)} ((4 a B+9 A b) \cos (c+d x)+2 a A)}{16 d}","-\frac{\left(4 a^2 B+9 a A b-8 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)}{4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{a \left(4 a^2 A+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(4 a^3 B+11 a^2 A b+16 a b^2 B+8 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)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{a (4 a B+7 A b) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{a A \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d}",1,"((8*b*(a^2*A + 4*A*b^2 + 12*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^3*A + 21*a*A*b^2 + 36*a^2*b*B + 8*b^3*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)*(-9*a*A*b - 4*a^2*B + 8*b^2*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*a*Sqrt[a + b*Cos[c + d*x]]*(2*a*A + (9*A*b + 4*a*B)*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(16*d)","C",1
317,1,486,376,6.0520614,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{\frac{8 b \left(6 a^2 B+13 a A b+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(\left(8 a^2 A+27 a b B+\frac{33 A b^2}{2}\right) \sin (2 (c+d x))+8 a^2 A \tan (c+d x)+2 a (6 a B+13 A b) \sin (c+d x)\right)-\frac{2 i \left(16 a^2 A+54 a b B+33 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 \sqrt{-\frac{1}{a+b}}}+\frac{2 \left(48 a^3 B+104 a^2 A b+126 a b^2 B-3 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 d}","\frac{\left(16 a^2 A+54 a b B+33 A b^2\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d}-\frac{\left(16 a^2 A+54 a b B+33 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 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(16 a^3 A+66 a^2 b B+59 a A b^2+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 A b+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{a (2 a B+3 A b) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{a A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}",1,"((8*b*(13*a*A*b + 6*a^2*B + 24*b^2*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*(104*a^2*A*b - 3*A*b^3 + 48*a^3*B + 126*a*b^2*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)*(16*a^2*A + 33*A*b^2 + 54*a*b*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]]*Sec[c + d*x]^2*(2*a*(13*A*b + 6*a*B)*Sin[c + d*x] + (8*a^2*A + (33*A*b^2)/2 + 27*a*b*B)*Sin[2*(c + d*x)] + 8*a^2*A*Tan[c + d*x]))/(96*d)","C",1
318,1,729,465,6.7700458,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*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)+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)+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+416 a^2 b^2 B+236 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(-128 a^3 b B-284 a^2 A b^2-264 a b^3 B-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+832 a^3 b B+436 a^2 A b^2-24 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{\left(36 a^2 A+104 a b B+59 A b^2\right) \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{96 d}+\frac{\left(128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{192 a d}+\frac{\left(128 a^3 B+356 a^2 A b+472 a b^2 B+133 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 d \sqrt{a+b \cos (c+d x)}}-\frac{\left(128 a^3 B+284 a^2 A b+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(48 a^4 A+160 a^3 b B+120 a^2 A b^2+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{a (8 a B+11 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{a 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 + 236*a*A*b^3 + 416*a^2*b^2*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*(288*a^4*A + 436*a^2*A*b^2 - 45*A*b^4 + 832*a^3*b*B - 24*a*b^3*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)*(-284*a^2*A*b^2 - 15*A*b^4 - 128*a^3*b*B - 264*a*b^3*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)))/(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]))/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]))/(192*a) + (a^2*A*Sec[c + d*x]^3*Tan[c + d*x])/4))/d","C",0
319,1,230,320,1.0581542,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/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 B+6 b (7 A b-6 a B) \cos (c+d x)-56 a A b+15 b^2 B \cos (2 (c+d x))+65 b^2 B\right)+4 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b^2 \left(-12 a^2 B+14 a A b+25 b^2 B\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-\left(48 a^3 B-56 a^2 A b+44 a b^2 B-63 A b^3\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 \left(-24 a^2 B+28 a A b-25 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b^3 d}+\frac{2 \left(-48 a^3 B+56 a^2 A b-44 a b^2 B+63 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)}{105 b^4 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 \left(-48 a^4 B+56 a^3 A b-32 a^2 b^2 B+49 a A b^3-25 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)}{105 b^4 d \sqrt{a+b \cos (c+d x)}}+\frac{2 (7 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{35 b^2 d}+\frac{2 B \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*(14*a*A*b - 12*a^2*B + 25*b^2*B)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - (-56*a^2*A*b - 63*A*b^3 + 48*a^3*B + 44*a*b^2*B)*((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])*(-56*a*A*b + 48*a^2*B + 65*b^2*B + 6*b*(7*A*b - 6*a*B)*Cos[c + d*x] + 15*b^2*B*Cos[2*(c + d*x)])*Sin[c + d*x])/(210*b^4*d*Sqrt[a + b*Cos[c + d*x]])","A",1
320,1,180,246,0.9068556,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/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 B-10 a A b+9 b^2 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)+b^2 (2 a B+5 A 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 B+5 A b+3 b B \cos (c+d x))}{15 b^3 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(-8 a^2 B+10 a A b-9 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)}{15 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \left(-8 a^3 B+10 a^2 A b-7 a b^2 B+5 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)}{15 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 (5 A b-4 a B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^2 d}+\frac{2 B \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*A*b + 2*a*B)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (-10*a*A*b + 8*a^2*B + 9*b^2*B)*((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*A*b - 4*a*B + 3*b*B*Cos[c + d*x])*Sin[c + d*x])/(15*b^3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
321,1,154,183,0.691353,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \left(2 a^2 B-3 a A b+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)-2 (a+b) (2 a B-3 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)+2 b B \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 B+3 a A b-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 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 (3 A b-2 a B) \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 B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d}",1,"(-2*(a + b)*(-3*A*b + 2*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + 2*(-3*a*A*b + 2*a^2*B + b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + 2*b*B*(a + b*Cos[c + d*x])*Sin[c + d*x])/(3*b^2*d*Sqrt[a + b*Cos[c + d*x]])","A",1
322,1,93,130,3.2943385,"\int \frac{A+B \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left((A b-a B) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+B (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 (A b-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)}{b d \sqrt{a+b \cos (c+d x)}}+\frac{2 B \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)*B*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + (A*b - a*B)*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/(b*d*Sqrt[a + b*Cos[c + d*x]])","A",1
323,1,81,118,0.2025051,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x])*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(A \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+B 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 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 \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)]*(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
324,1,320,216,6.5252046,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\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)}}+4 A \tan (c+d x) \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 b \sqrt{-\frac{1}{a+b}}}}{4 a d}","-\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 \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{a d}+\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)}{d \sqrt{a+b \cos (c+d x)}}-\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*(-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*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*a*d)","C",1
325,1,420,299,6.0205517,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\frac{2 \left(8 a^2 A-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-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*(8*a^2*A + 9*A*b^2 - 12*a*b*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)*(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
326,1,304,387,1.8649888,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\frac{30 a^3 b (a B-A b) \sin (c+d x)}{b^2-a^2}+\frac{2 b^2 \left(12 a^3 B-10 a^2 A b+3 a b^2 B-5 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)}{(a-b) (a+b)}+\frac{2 \left(48 a^4 B-40 a^3 A b-24 a^2 b^2 B+25 a A b^3-9 b^4 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)}{(a-b) (a+b)}+2 b (5 A b-9 a B) \sin (c+d x) (a+b \cos (c+d x))+3 b^2 B \sin (2 (c+d x)) (a+b \cos (c+d x))}{15 b^4 d \sqrt{a+b \cos (c+d x)}}","\frac{2 a (A b-a B) \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 B+5 a A b+b^2 B\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 B+20 a^2 A b+9 a b^2 B-5 A b^3\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 B+40 a^2 A b-12 a b^2 B+5 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)}{15 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-48 a^4 B+40 a^3 A b+24 a^2 b^2 B-25 a A b^3+9 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)}{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*A*b - 5*A*b^3 + 12*a^3*B + 3*a*b^2*B)*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*A*b + 25*a*A*b^3 + 48*a^4*B - 24*a^2*b^2*B - 9*b^4*B)*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*(-(A*b) + a*B)*Sin[c + d*x])/(-a^2 + b^2) + 2*b*(5*A*b - 9*a*B)*(a + b*Cos[c + d*x])*Sin[c + d*x] + 3*b^2*B*(a + b*Cos[c + d*x])*Sin[2*(c + d*x)])/(15*b^4*d*Sqrt[a + b*Cos[c + d*x]])","A",1
327,1,189,262,1.5093798,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 \left(b \sin (c+d x) \left(\frac{a \left(-4 a^2 B+3 a A b+b^2 B\right)}{b^2-a^2}+b B \cos (c+d x)\right)+\frac{\sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left((a-b) \left(8 a^2 B-6 a A b+b^2 B\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(-8 a^3 B+6 a^2 A b+5 a b^2 B-3 A b^3\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 (A b-a B) \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 B+6 a A b-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 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-8 a^3 B+6 a^2 A b+5 a b^2 B-3 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)}{3 b^3 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 B \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*A*b - 3*A*b^3 - 8*a^3*B + 5*a*b^2*B)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + (a - b)*(-6*a*A*b + 8*a^2*B + b^2*B)*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/(a - b) + b*((a*(3*a*A*b - 4*a^2*B + b^2*B))/(-a^2 + b^2) + b*B*Cos[c + d*x])*Sin[c + d*x]))/(3*b^3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
328,1,170,204,0.8206838,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 \left(\left(a^2-b^2\right) (2 a B-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)-\left((a+b) \left(2 a^2 B-a A b-b^2 B\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 B-A b) \sin (c+d x)\right)}{b^2 d (a-b) (a+b) \sqrt{a+b \cos (c+d x)}}","\frac{2 a (A b-a B) \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 B+a A b+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)}{b^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 (A b-2 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)}{b^2 d \sqrt{a+b \cos (c+d x)}}",1,"(-2*(-((a + b)*(-(a*A*b) + 2*a^2*B - b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)]) + (a^2 - b^2)*(-(A*b) + 2*a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + a*b*(-(A*b) + a*B)*Sin[c + d*x]))/((a - b)*b^2*(a + b)*d*Sqrt[a + b*Cos[c + d*x]])","A",1
329,1,151,185,0.5646304,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 \left(B \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 B-A b) \sin (c+d x)-\left((a+b) (a B-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)\right)\right)}{b d (a-b) (a+b) \sqrt{a+b \cos (c+d x)}}","-\frac{2 (A b-a B) \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 (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)}{b 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}} 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)*(-(A*b) + a*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)]) + (a^2 - b^2)*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + b*(-(A*b) + a*B)*Sin[c + d*x]))/((a - b)*b*(a + b)*d*Sqrt[a + b*Cos[c + d*x]])","A",1
330,1,460,190,3.9796909,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x])/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\cos (c+d x) (A \sec (c+d x)+B) \left(\frac{4 b (A b-a B) \sin (c+d x)}{\left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{\frac{2 \left(2 a^2 A+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)}{\sqrt{a+b \cos (c+d x)}}+\frac{4 a (a B-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)}}-\frac{2 i (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 b \sqrt{-\frac{1}{a+b}}}}{(b-a) (a+b)}\right)}{2 a d (A+B \cos (c+d x))}","\frac{2 b (A b-a B) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 (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)}{a 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)}}",1,"(Cos[c + d*x]*(B + A*Sec[c + d*x])*(-(((4*a*(-(A*b) + 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*(2*a^2*A - 3*A*b^2 + a*b*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 - 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)]))/((-a + b)*(a + b))) + (4*b*(A*b - a*B)*Sin[c + d*x])/((a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]])))/(2*a*d*(A + B*Cos[c + d*x]))","C",1
331,1,482,303,5.7382944,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\frac{4 \tan (c+d x) \left(b \left(a^2 A+2 a b B-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{2 i \left(a^2 A+2 a b B-3 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)}{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 B-7 a^2 A b-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)}}-\frac{8 a b (a B-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)}}}{(a-b) (a+b)}}{4 a^2 d}","\frac{b \left(a^2 A+2 a b B-3 A b^2\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 A+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,"(((-8*a*b*(-(A*b) + 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*(-7*a^2*A*b + 9*A*b^3 + 4*a^3*B - 6*a*b^2*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^2*A - 3*A*b^2 + 2*a*b*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)]))/((a - b)*(a + b)) + (4*(a*A*(a^2 - b^2) + b*(a^2*A - 3*A*b^2 + 2*a*b*B)*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
332,1,678,398,6.9355847,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*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 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^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 \left(-4 a^3 b B+7 a^2 A b^2+12 a b^3 B-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-28 a^3 b B+29 a^2 A b^2+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-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 \left(-4 a^3 B+7 a^2 A b+12 a b^2 B-15 A b^3\right) \sin (c+d x)}{4 a^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{\left(-4 a^3 B+7 a^2 A b+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)*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)*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)*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^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
333,1,372,550,4.4341763,"\int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^4*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{b \left(\frac{10 a^4 (a B-A b) \sin (c+d x)}{a^2-b^2}-\frac{10 a^3 \left(11 a^3 B-8 a^2 A b-15 a b^2 B+12 A b^3\right) \sin (c+d x) (a+b \cos (c+d x))}{\left(a^2-b^2\right)^2}+2 (5 A b-14 a B) \sin (c+d x) (a+b \cos (c+d x))^2+3 b B \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 B+20 a^4 A b+44 a^3 b^2 B-35 a^2 A b^3+8 a b^4 B-5 A b^5\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-\left(128 a^6 B-80 a^5 A b-212 a^4 b^2 B+140 a^3 A b^3+55 a^2 b^4 B-40 a A b^5+9 b^6 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)}}{15 b^5 d (a+b \cos (c+d x))^{3/2}}","\frac{2 a (A b-a B) \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{2 a \left(-8 a^3 B+5 a^2 A b+12 a b^2 B-9 A b^3\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{2 \left(-48 a^4 B+30 a^3 A b+71 a^2 b^2 B-50 a A b^3-3 b^4 B\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(-64 a^5 B+40 a^4 A b+98 a^3 b^2 B-65 a^2 A b^3-14 a b^4 B+5 A b^5\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 \left(-128 a^5 B+80 a^4 A b+116 a^3 b^2 B-80 a^2 A b^3+17 a b^4 B-5 A b^5\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 B+80 a^5 A b+212 a^4 b^2 B-140 a^3 A b^3-55 a^2 b^4 B+40 a A b^5-9 b^6 B\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*A*b - 35*a^2*A*b^3 - 5*A*b^5 - 32*a^5*B + 44*a^3*b^2*B + 8*a*b^4*B)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - (-80*a^5*A*b + 140*a^3*A*b^3 - 40*a*A*b^5 + 128*a^6*B - 212*a^4*b^2*B + 55*a^2*b^4*B + 9*b^6*B)*((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^4*(-(A*b) + a*B)*Sin[c + d*x])/(a^2 - b^2) - (10*a^3*(-8*a^2*A*b + 12*A*b^3 + 11*a^3*B - 15*a*b^2*B)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2 + 2*(5*A*b - 14*a*B)*(a + b*Cos[c + d*x])^2*Sin[c + d*x] + 3*b*B*(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
334,1,334,413,2.8854204,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/(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 B+2 a^3 A b+7 a^2 b^2 B-6 a A b^3+b^4 B\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-\left(16 a^5 B-8 a^4 A b-28 a^3 b^2 B+15 a^2 A b^3+8 a b^4 B-3 A b^5\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 B-8 a^5 A b-25 a^4 b^2 B+16 a^3 A b^3+B \left(b^3-a^2 b\right)^2 \cos (2 (c+d x))+2 a b \left(10 a^4 B-5 a^3 A b-16 a^2 b^2 B+9 a A b^3+2 b^4 B\right) \cos (c+d x)+b^6 B\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 (A b-a B) \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 B+a A b+b^2 B\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 B+3 a^2 A b+10 a b^2 B-7 A b^3\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 B+8 a^3 A b+16 a^2 b^2 B-9 a A b^3+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)}{3 b^4 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(-16 a^5 B+8 a^4 A b+28 a^3 b^2 B-15 a^2 A b^3-8 a b^4 B+3 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)}{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*A*b - 6*a*A*b^3 - 4*a^4*B + 7*a^2*b^2*B + b^4*B)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - (-8*a^4*A*b + 15*a^2*A*b^3 - 3*A*b^5 + 16*a^5*B - 28*a^3*b^2*B + 8*a*b^4*B)*((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*A*b + 16*a^3*A*b^3 + 16*a^6*B - 25*a^4*b^2*B + b^6*B + 2*a*b*(-5*a^3*A*b + 9*a*A*b^3 + 10*a^4*B - 16*a^2*b^2*B + 2*b^4*B)*Cos[c + d*x] + (-(a^2*b) + b^3)^2*B*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
335,1,274,331,2.3298824,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(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 B+a^2 A b-6 a b^2 B+3 A b^3\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(8 a^4 B-2 a^3 A b-15 a^2 b^2 B+6 a A b^3+3 b^4 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{a b \sin (c+d x) \left(b \left(5 a^3 B-2 a^2 A b-9 a b^2 B+6 A b^3\right) \cos (c+d x)+a \left(4 a^3 B-a^2 A b-8 a b^2 B+5 A b^3\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 (A b-a B) \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 B+2 a^2 A b+9 a b^2 B-6 A b^3\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 B+2 a^2 A b+9 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)}{3 b^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(-8 a^4 B+2 a^3 A b+15 a^2 b^2 B-6 a A b^3-3 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 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*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (-2*a^3*A*b + 6*a*A*b^3 + 8*a^4*B - 15*a^2*b^2*B + 3*b^4*B)*((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*A*b) + 5*A*b^3 + 4*a^3*B - 8*a*b^2*B) + b*(-2*a^2*A*b + 6*A*b^3 + 5*a^3*B - 9*a*b^2*B)*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
336,1,224,307,2.0260786,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(\frac{b \sin (c+d x) \left(b \left(2 a^3 B+a^2 A b-6 a b^2 B+3 A b^3\right) \cos (c+d x)+a \left(a^3 B+2 a^2 A b-5 a b^2 B+2 A b^3\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 B+a^2 A b-6 a b^2 B+3 A b^3\right) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-(a-b) \left(2 a^2 B+a A b-3 b^2 B\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 (A b-a B) \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 B+a A b-3 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 b^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(2 a^3 B+a^2 A b-6 a b^2 B+3 A b^3\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 B+a^2 A b-6 a b^2 B+3 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)}{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*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - (a - b)*(a*A*b + 2*a^2*B - 3*b^2*B)*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/(a - b)^2) + (b*(a*(2*a^2*A*b + 2*A*b^3 + a^3*B - 5*a*b^2*B) + b*(a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*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
337,1,193,275,1.6476182,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(\frac{\sin (c+d x) \left(2 a^3 B+b \left(a^2 B-4 a A b+3 b^2 B\right) \cos (c+d x)-5 a^2 A b+2 a b^2 B+A b^3\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 B-4 a A b+3 b^2 B\right) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-(a-b) (a B-A 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 (-B)+4 a A b-3 b^2 B\right) \sin (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 (A b-a B) \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 (A b-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)}{3 b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2 (-B)+4 a A b-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)}{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*A*b + a^2*B + 3*b^2*B)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - (a - b)*(-(A*b) + a*B)*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/((a - b)^2*b)) + ((-5*a^2*A*b + A*b^3 + 2*a^3*B + 2*a*b^2*B + b*(-4*a*A*b + a^2*B + 3*b^2*B)*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
338,1,743,349,6.811976,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x])/(a + b*Cos[c + d*x])^(5/2),x]","\frac{\cos (c+d x) \sqrt{a+b \cos (c+d x)} (A \sec (c+d x)+B) \left(-\frac{2 \left(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 \left(4 a^3 b B \sin (c+d x)-7 a^2 A b^2 \sin (c+d x)+3 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 (A+B \cos (c+d x))}+\frac{\cos (c+d x) (A \sec (c+d x)+B) \left(-\frac{2 i \left(4 a^3 b B-7 a^2 A b^2+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(6 a^4 A+4 a^3 b B-19 a^2 A b^2+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)}}+\frac{2 \left(6 a^4 B-12 a^3 A b+2 a^2 b^2 B+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)}}\right)}{6 a^2 d (a-b)^2 (a+b)^2 (A+B \cos (c+d x))}","\frac{2 b (A b-a B) \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 (A b-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)}{3 a 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 b \left(-4 a^3 B+7 a^2 A b-3 A b^3\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 B+7 a^2 A b-3 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)}{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]*(B + A*Sec[c + d*x])*((2*(-12*a^3*A*b + 4*a*A*b^3 + 6*a^4*B + 2*a^2*b^2*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*(6*a^4*A - 19*a^2*A*b^2 + 9*A*b^4 + 4*a^3*b*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)*(-7*a^2*A*b^2 + 3*A*b^4 + 4*a^3*b*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))))/(6*a^2*(a - b)^2*(a + b)^2*d*(A + B*Cos[c + d*x])) + (Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*(B + A*Sec[c + d*x])*((-2*(-(A*b^2*Sin[c + d*x]) + a*b*B*Sin[c + d*x]))/(3*a*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) - (2*(-7*a^2*A*b^2*Sin[c + d*x] + 3*A*b^4*Sin[c + d*x] + 4*a^3*b*B*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/(d*(A + B*Cos[c + d*x]))","C",0
339,1,750,437,7.2630778,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(a + b*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{a+b \cos (c+d x)} \left(\frac{A \tan (c+d x)}{a^3}+\frac{2 \left(a b^2 B \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{2 \left(7 a^3 b^2 B \sin (c+d x)-10 a^2 A b^3 \sin (c+d x)-3 a b^4 B \sin (c+d x)+6 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}+\frac{\frac{2 \left(-24 a^4 b B+36 a^3 A b^2+8 a^2 b^3 B-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 i \left(-3 a^4 A b-14 a^3 b^2 B+26 a^2 A b^3+6 a b^4 B-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)}+\frac{2 \left(12 a^5 B-33 a^4 A b-38 a^3 b^2 B+86 a^2 A b^3+18 a b^4 B-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)}}}{12 a^3 d (b-a)^2 (a+b)^2}","-\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 \left(3 a^2 A+2 a b B-5 A b^2\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 A+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 \left(3 a^4 A+14 a^3 b B-26 a^2 A b^2-6 a b^3 B+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(3 a^4 A+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,"((2*(36*a^3*A*b^2 - 20*a*A*b^4 - 24*a^4*b*B + 8*a^2*b^3*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*(-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)*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)*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*(-(A*b^3*Sin[c + d*x]) + a*b^2*B*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) + (2*(-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]))/(3*a^3*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) + (A*Tan[c + d*x])/a^3))/d","C",0
340,1,820,532,7.951443,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^(5/2),x]","\frac{\frac{2 \left(12 A b a^5+144 b^2 B a^4-216 A b^3 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^6-132 b B a^5+195 A b^2 a^4+344 b^3 B a^3-566 A b^4 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+104 a^3 B b^3+33 a^4 A 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 b^3 B \sin (c+d x)-A b^4 \sin (c+d x)\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+10 a^3 B \sin (c+d x) b^3\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-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 \left(-12 a^3 B+27 a^2 A b+20 a b^2 B-35 A b^3\right) \sin (c+d x)}{12 a^3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{\left(-12 a^3 B+27 a^2 A b+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 \left(-12 a^5 B+33 a^4 A b+104 a^3 b^2 B-170 a^2 A b^3-60 a b^4 B+105 A b^5\right) \sin (c+d x)}{12 a^4 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{\left(-12 a^5 B+33 a^4 A b+104 a^3 b^2 B-170 a^2 A b^3-60 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)*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)*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)*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]))/(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]))/(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
341,1,58,58,0.0502458,"\int \frac{a B+b B \cos (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 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 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)}}",1,"(2*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])","A",1
342,1,59,59,0.0792661,"\int \frac{(a B+b B \cos (c+d x)) \sec (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x])/(a + b*Cos[c + d*x])^(3/2),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 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,"(2*B*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])","A",1
343,1,84,108,0.2138373,"\int \frac{a B+b B \cos (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(a*B + b*B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(5/2),x]","\frac{B \left(2 (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)-2 b \sin (c+d x)\right)}{d (a-b) (a+b) \sqrt{a+b \cos (c+d x)}}","\frac{2 B \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 b B \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}",1,"(B*(2*(a + b)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 2*b*Sin[c + d*x]))/((a - b)*(a + b)*d*Sqrt[a + b*Cos[c + d*x]])","A",1
344,1,403,179,4.8679198,"\int \frac{(a B+b B \cos (c+d x)) \sec (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x])/(a + b*Cos[c + d*x])^(5/2),x]","\frac{B \left(\frac{4 b^2 \sin (c+d x)}{\left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{\frac{2 \left(2 a^2-3 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{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)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \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}}}}{(b-a) (a+b)}\right)}{2 a d}","\frac{2 b^2 B \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 b B \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,"(B*(-(((-4*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*(2*a^2 - 3*b^2)*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)*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)]))/((-a + b)*(a + b))) + (4*b^2*Sin[c + d*x])/((a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]])))/(2*a*d)","C",1
345,1,125,170,1.2934141,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","\frac{300 (a B+A b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+84 (9 a A+7 b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} (7 (36 a A+43 b B) \cos (c+d x)+5 (18 (a B+A b) \cos (2 (c+d x))+78 a B+78 A b+7 b B \cos (3 (c+d x))))}{630 d}","\frac{10 (a B+A b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 (9 a A+7 b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (a B+A b) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 (9 a A+7 b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 (a B+A b) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}",1,"(84*(9*a*A + 7*b*B)*EllipticE[(c + d*x)/2, 2] + 300*(A*b + a*B)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*(36*a*A + 43*b*B)*Cos[c + d*x] + 5*(78*A*b + 78*a*B + 18*(A*b + a*B)*Cos[2*(c + d*x)] + 7*b*B*Cos[3*(c + d*x)]))*Sin[c + d*x])/(630*d)","A",1
346,1,103,140,0.8607765,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","\frac{10 (7 a A+5 b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+126 (a B+A b) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} (42 (a B+A b) \cos (c+d x)+70 a A+15 b B \cos (2 (c+d x))+65 b B)}{105 d}","\frac{2 (7 a A+5 b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{6 (a B+A b) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (a B+A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 (7 a A+5 b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}",1,"(126*(A*b + a*B)*EllipticE[(c + d*x)/2, 2] + 10*(7*a*A + 5*b*B)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(70*a*A + 65*b*B + 42*(A*b + a*B)*Cos[c + d*x] + 15*b*B*Cos[2*(c + d*x)])*Sin[c + d*x])/(105*d)","A",1
347,1,86,108,0.4143046,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x)) (A+B \cos (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]),x]","\frac{2 \left(5 (a B+A b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 (5 a A+3 b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} (5 a B+5 A b+3 b B \cos (c+d x))\right)}{15 d}","\frac{2 (a B+A b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (5 a A+3 b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (a B+A b) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(2*(3*(5*a*A + 3*b*B)*EllipticE[(c + d*x)/2, 2] + 5*(A*b + a*B)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(5*A*b + 5*a*B + 3*b*B*Cos[c + d*x])*Sin[c + d*x]))/(15*d)","A",1
348,1,67,75,0.2308676,"\int \frac{(a+b \cos (c+d x)) (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{2 \left((3 a A+b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 (a B+A b) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b B \sin (c+d x) \sqrt{\cos (c+d x)}\right)}{3 d}","\frac{2 (3 a A+b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (a B+A b) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(2*(3*(A*b + a*B)*EllipticE[(c + d*x)/2, 2] + (3*a*A + b*B)*EllipticF[(c + d*x)/2, 2] + b*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x]))/(3*d)","A",1
349,1,64,71,0.3554866,"\int \frac{(a+b \cos (c+d x)) (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{2 \left((a B+A b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+(b B-a A) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{a A \sin (c+d x)}{\sqrt{\cos (c+d x)}}\right)}{d}","\frac{2 (a B+A b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 (a A-b B) 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)}}",1,"(2*((-(a*A) + b*B)*EllipticE[(c + d*x)/2, 2] + (A*b + a*B)*EllipticF[(c + d*x)/2, 2] + (a*A*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/d","A",1
350,1,107,103,0.4767709,"\int \frac{(a+b \cos (c+d x)) (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{2 \left((a A+3 b B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-3 (a B+A b) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+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 (a A+3 b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 (a B+A b) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{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)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + (a*A + 3*b*B)*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
351,1,134,140,0.8318261,"\int \frac{(a+b \cos (c+d x)) (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","\frac{10 (a B+A b) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 (3 a A+5 b B) \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)+10 a B \sin (c+d x)+10 A b \sin (c+d x)+15 b B \sin (2 (c+d x))}{15 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 (a B+A b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 (3 a A+5 b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (a B+A b) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (3 a A+5 b B) \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)}",1,"(-6*(3*a*A + 5*b*B)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 10*(A*b + a*B)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 10*A*b*Sin[c + d*x] + 10*a*B*Sin[c + d*x] + 9*a*A*Sin[2*(c + d*x)] + 15*b*B*Sin[2*(c + d*x)] + 6*a*A*Tan[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",1
352,1,196,264,1.7690162,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]","\frac{1200 \left(11 a^2 B+22 a A b+9 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3696 \left(9 a^2 A+14 a b B+7 A b^2\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 A+86 a b B+43 A b^2\right) \cos (c+d x)+180 \left(11 a^2 B+22 a A b+16 b^2 B\right) \cos (2 (c+d x))+15 \left(572 a^2 B+1144 a A b+21 b^2 B \cos (4 (c+d x))+531 b^2 B\right)+770 b (2 a B+A b) \cos (3 (c+d x))\right)}{27720 d}","\frac{2 \left(9 a^2 A+14 a b B+7 A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(9 a^2 A+14 a b B+7 A b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 \left(11 a (a B+2 A b)+9 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 \left(11 a (a B+2 A b)+9 b^2 B\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{10 \left(11 a (a B+2 A b)+9 b^2 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 b (13 a B+11 A b) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))}{11 d}",1,"(3696*(9*a^2*A + 7*A*b^2 + 14*a*b*B)*EllipticE[(c + d*x)/2, 2] + 1200*(22*a*A*b + 11*a^2*B + 9*b^2*B)*EllipticF[(c + d*x)/2, 2] + 2*Sqrt[Cos[c + d*x]]*(154*(36*a^2*A + 43*A*b^2 + 86*a*b*B)*Cos[c + d*x] + 180*(22*a*A*b + 11*a^2*B + 16*b^2*B)*Cos[2*(c + d*x)] + 770*b*(A*b + 2*a*B)*Cos[3*(c + d*x)] + 15*(1144*a*A*b + 572*a^2*B + 531*b^2*B + 21*b^2*B*Cos[4*(c + d*x)]))*Sin[c + d*x])/(27720*d)","A",1
353,1,167,223,1.411574,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]","\frac{60 \left(7 a^2 A+10 a b B+5 A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+84 \left(9 a^2 B+18 a A b+7 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(7 \left(36 a^2 B+72 a A b+43 b^2 B\right) \cos (c+d x)+5 \left(84 a^2 A+18 b (2 a B+A b) \cos (2 (c+d x))+156 a b B+78 A b^2+7 b^2 B \cos (3 (c+d x))\right)\right)}{630 d}","\frac{2 \left(7 a^2 A+10 a b B+5 A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(7 a^2 A+10 a b B+5 A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 \left(9 a (a B+2 A b)+7 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(9 a (a B+2 A b)+7 b^2 B\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 b (11 a B+9 A b) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d}",1,"(84*(18*a*A*b + 9*a^2*B + 7*b^2*B)*EllipticE[(c + d*x)/2, 2] + 60*(7*a^2*A + 5*A*b^2 + 10*a*b*B)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*(72*a*A*b + 36*a^2*B + 43*b^2*B)*Cos[c + d*x] + 5*(84*a^2*A + 78*A*b^2 + 156*a*b*B + 18*b*(A*b + 2*a*B)*Cos[2*(c + d*x)] + 7*b^2*B*Cos[3*(c + d*x)]))*Sin[c + d*x])/(630*d)","A",1
354,1,139,182,1.127075,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]),x]","\frac{10 \left(7 a^2 B+14 a A b+5 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+42 \left(5 a^2 A+6 a b B+3 A b^2\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 B+28 a A b+3 b^2 B \cos (2 (c+d x))+13 b^2 B\right)+42 b (2 a B+A b) \cos (c+d x)\right)}{105 d}","\frac{2 \left(5 a^2 A+6 a b B+3 A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(7 a (a B+2 A b)+5 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(7 a (a B+2 A b)+5 b^2 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b (9 a B+7 A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 b B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}{7 d}",1,"(42*(5*a^2*A + 3*A*b^2 + 6*a*b*B)*EllipticE[(c + d*x)/2, 2] + 10*(14*a*A*b + 7*a^2*B + 5*b^2*B)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(42*b*(A*b + 2*a*B)*Cos[c + d*x] + 5*(28*a*A*b + 14*a^2*B + 13*b^2*B + 3*b^2*B*Cos[2*(c + d*x)]))*Sin[c + d*x])/(105*d)","A",1
355,1,106,140,0.6004717,"\int \frac{(a+b \cos (c+d x))^2 (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{2 \left(5 \left(3 a^2 A+2 a b B+A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 \left(5 a^2 B+10 a A b+3 b^2 B\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 B+5 A b+3 b B \cos (c+d x))\right)}{15 d}","\frac{2 \left(3 a^2 A+2 a b B+A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 \left(5 a (a B+2 A b)+3 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b (7 a B+5 A b) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 b B \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}{5 d}",1,"(2*(3*(10*a*A*b + 5*a^2*B + 3*b^2*B)*EllipticE[(c + d*x)/2, 2] + 5*(3*a^2*A + A*b^2 + 2*a*b*B)*EllipticF[(c + d*x)/2, 2] + b*Sqrt[Cos[c + d*x]]*(5*A*b + 10*a*B + 3*b*B*Cos[c + d*x])*Sin[c + d*x]))/(15*d)","A",1
356,1,102,121,0.6407826,"\int \frac{(a+b \cos (c+d x))^2 (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{2 \left(\left(3 a^2 B+6 a A b+b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\left(-3 a^2 A+6 a b B+3 A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{\sin (c+d x) \left(3 a^2 A+b^2 B \cos (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)}{3 d}","\frac{2 \left(3 a^2 B+6 a A b+b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(a^2 A-2 a b B-A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 b^2 B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(2*((-3*a^2*A + 3*A*b^2 + 6*a*b*B)*EllipticE[(c + d*x)/2, 2] + (6*a*A*b + 3*a^2*B + b^2*B)*EllipticF[(c + d*x)/2, 2] + ((3*a^2*A + b^2*B*Cos[c + d*x])*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(3*d)","A",1
357,1,105,126,1.1821324,"\int \frac{(a+b \cos (c+d x))^2 (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{2 \left(\left(a^2 A+6 a b B+3 A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-3 \left(a^2 B+2 a A b-b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{a \sin (c+d x) (3 (a B+2 A b) \cos (c+d x)+a A)}{\cos ^{\frac{3}{2}}(c+d x)}\right)}{3 d}","\frac{2 \left(a^2 A+6 a b B+3 A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(a^2 B+2 a A b-b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a (a B+2 A b) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(2*(-3*(2*a*A*b + a^2*B - b^2*B)*EllipticE[(c + d*x)/2, 2] + (a^2*A + 3*A*b^2 + 6*a*b*B)*EllipticF[(c + d*x)/2, 2] + (a*(a*A + 3*(2*A*b + a*B)*Cos[c + d*x])*Sin[c + d*x])/Cos[c + d*x]^(3/2)))/(3*d)","A",1
358,1,175,172,1.1106768,"\int \frac{(a+b \cos (c+d x))^2 (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","\frac{10 \left(a^2 B+2 a A b+3 b^2 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^2 A+10 a b B+5 A b^2\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)+10 a^2 B \sin (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 \left(a^2 B+2 a A b+3 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(3 a^2 A+10 a b B+5 A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(3 a^2 A+10 a b B+5 A b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a (a B+2 A b) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-6*(3*a^2*A + 5*A*b^2 + 10*a*b*B)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 10*(2*a*A*b + a^2*B + 3*b^2*B)*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)] + 6*a^2*A*Tan[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",1
359,1,235,305,1.9916362,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]),x]","\frac{240 \left(77 a^3 A+165 a^2 b B+165 a A b^2+45 b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3696 \left(9 a^3 B+27 a^2 A b+21 a b^2 B+7 A b^3\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 B+33 a A b+16 b^2 B\right) \cos (2 (c+d x))+154 \left(36 a^3 B+108 a^2 A b+129 a b^2 B+43 A b^3\right) \cos (c+d x)+15 \left(616 a^3 A+1716 a^2 b B+1716 a A b^2+21 b^3 B \cos (4 (c+d x))+531 b^3 B\right)+770 b^2 (3 a B+A b) \cos (3 (c+d x))\right)}{27720 d}","\frac{2 b \left(26 a^2 B+33 a A b+9 b^2 B\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{2 \left(77 a^3 A+165 a^2 b B+165 a A b^2+45 b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{2 \left(9 a^3 B+27 a^2 A b+21 a b^2 B+7 A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(9 a^3 B+27 a^2 A b+21 a b^2 B+7 A b^3\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 \left(77 a^3 A+165 a^2 b B+165 a A b^2+45 b^3 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 b^2 (15 a B+11 A b) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d}+\frac{2 b B \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*A*b + 7*A*b^3 + 9*a^3*B + 21*a*b^2*B)*EllipticE[(c + d*x)/2, 2] + 240*(77*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 45*b^3*B)*EllipticF[(c + d*x)/2, 2] + 2*Sqrt[Cos[c + d*x]]*(154*(108*a^2*A*b + 43*A*b^3 + 36*a^3*B + 129*a*b^2*B)*Cos[c + d*x] + 180*b*(33*a*A*b + 33*a^2*B + 16*b^2*B)*Cos[2*(c + d*x)] + 770*b^2*(A*b + 3*a*B)*Cos[3*(c + d*x)] + 15*(616*a^3*A + 1716*a*A*b^2 + 1716*a^2*b*B + 531*b^3*B + 21*b^3*B*Cos[4*(c + d*x)]))*Sin[c + d*x])/(27720*d)","A",1
360,1,197,255,1.2290588,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]),x]","\frac{60 \left(7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+84 \left(15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\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 B+108 a A b+43 b^2 B\right) \cos (c+d x)+5 \left(84 a^3 B+252 a^2 A b+18 b^2 (3 a B+A b) \cos (2 (c+d x))+234 a b^2 B+78 A b^3+7 b^3 B \cos (3 (c+d x))\right)\right)}{630 d}","\frac{2 b \left(22 a^2 B+27 a A b+7 b^2 B\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 \left(7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b^2 (13 a B+9 A b) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 b B \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*A + 27*a*A*b^2 + 27*a^2*b*B + 7*b^3*B)*EllipticE[(c + d*x)/2, 2] + 60*(21*a^2*A*b + 5*A*b^3 + 7*a^3*B + 15*a*b^2*B)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*b*(108*a*A*b + 108*a^2*B + 43*b^2*B)*Cos[c + d*x] + 5*(252*a^2*A*b + 78*A*b^3 + 84*a^3*B + 234*a*b^2*B + 18*b^2*(A*b + 3*a*B)*Cos[2*(c + d*x)] + 7*b^3*B*Cos[3*(c + d*x)]))*Sin[c + d*x])/(630*d)","A",1
361,1,158,205,1.3042005,"\int \frac{(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{b \sin (c+d x) \sqrt{\cos (c+d x)} \left(5 \left(42 a^2 B+42 a A b+3 b^2 B \cos (2 (c+d x))+13 b^2 B\right)+42 b (3 a B+A b) \cos (c+d x)\right)+10 \left(21 a^3 A+21 a^2 b B+21 a A b^2+5 b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+42 \left(5 a^3 B+15 a^2 A b+9 a b^2 B+3 A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{105 d}","\frac{2 b \left(18 a^2 B+21 a A b+5 b^2 B\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 \left(21 a^3 A+21 a^2 b B+21 a A b^2+5 b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(5 a^3 B+15 a^2 A b+9 a b^2 B+3 A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b^2 (11 a B+7 A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 b B \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}",1,"(42*(15*a^2*A*b + 3*A*b^3 + 5*a^3*B + 9*a*b^2*B)*EllipticE[(c + d*x)/2, 2] + 10*(21*a^3*A + 21*a*A*b^2 + 21*a^2*b*B + 5*b^3*B)*EllipticF[(c + d*x)/2, 2] + b*Sqrt[Cos[c + d*x]]*(42*b*(A*b + 3*a*B)*Cos[c + d*x] + 5*(42*a*A*b + 42*a^2*B + 13*b^2*B + 3*b^2*B*Cos[2*(c + d*x)]))*Sin[c + d*x])/(105*d)","A",1
362,1,150,202,1.153201,"\int \frac{(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{\frac{\sin (c+d x) \left(3 \left(10 a^3 A+b^3 B \cos (2 (c+d x))+b^3 B\right)+10 b^2 (3 a B+A b) \cos (c+d x)\right)}{\sqrt{\cos (c+d x)}}+10 \left(3 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\left(-30 a^3 A+90 a^2 b B+90 a A b^2+18 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}","-\frac{2 b \left(6 a^2 A-3 a b B-A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 \left(3 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(5 a^3 A-15 a^2 b B-15 a A b^2-3 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{2 b^2 (5 a A-b B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a A \sin (c+d x) (a+b \cos (c+d x))^2}{d \sqrt{\cos (c+d x)}}",1,"((-30*a^3*A + 90*a*A*b^2 + 90*a^2*b*B + 18*b^3*B)*EllipticE[(c + d*x)/2, 2] + 10*(9*a^2*A*b + A*b^3 + 3*a^3*B + 3*a*b^2*B)*EllipticF[(c + d*x)/2, 2] + ((10*b^2*(A*b + 3*a*B)*Cos[c + d*x] + 3*(10*a^3*A + b^3*B + b^3*B*Cos[2*(c + d*x)]))*Sin[c + d*x])/Sqrt[Cos[c + d*x]])/(15*d)","A",1
363,1,165,192,1.1139839,"\int \frac{(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{2 a^3 A \tan (c+d x)+6 a^3 B \sin (c+d x)+18 a^2 A b \sin (c+d x)+2 \left(a^3 A+9 a^2 b B+9 a A b^2+b^3 B\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b^3 B \sin (2 (c+d x))}{3 d \sqrt{\cos (c+d x)}}","\frac{2 a^2 (3 a B+7 A b) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 \left(a^3 A+9 a^2 b B+9 a A b^2+b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 b^2 (a A-b B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a A \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*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*(a^3*A + 9*a*A*b^2 + 9*a^2*b*B + b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 18*a^2*A*b*Sin[c + d*x] + 6*a^3*B*Sin[c + d*x] + b^3*B*Sin[2*(c + d*x)] + 2*a^3*A*Tan[c + d*x])/(3*d*Sqrt[Cos[c + d*x]])","A",1
364,1,176,204,2.2122335,"\int \frac{(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","\frac{6 a^3 A \tan (c+d x)+9 a \left(a^2 A+5 a b B+5 A b^2\right) \sin (2 (c+d x))+10 a^2 (a B+3 A b) \sin (c+d x)+10 \left(a^3 B+3 a^2 A b+9 a b^2 B+3 A b^3\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 A+15 a^2 b B+15 a A b^2-5 b^3 B\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 A+15 a b B+14 A b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 (5 a B+9 A b) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(a^3 B+3 a^2 A b+9 a b^2 B+3 A b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(3 a^3 A+15 a^2 b B+15 a A b^2-5 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a A \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*A + 15*a*A*b^2 + 15*a^2*b*B - 5*b^3*B)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 10*(3*a^2*A*b + 3*A*b^3 + a^3*B + 9*a*b^2*B)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 10*a^2*(3*A*b + a*B)*Sin[c + d*x] + 9*a*(a^2*A + 5*A*b^2 + 5*a*b*B)*Sin[2*(c + d*x)] + 6*a^3*A*Tan[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",1
365,1,260,182,2.4690796,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),x]","\frac{\frac{2 b^2 \left(5 a^2 B-5 a A b+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 \left(5 a^2 B-5 a A b+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 \sqrt{\sin ^2(c+d x)}}+4 b^2 \sin (c+d x) \sqrt{\cos (c+d x)} (-5 a B+5 A b+3 b B \cos (c+d x))+2 b^2 (4 a B+5 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)}{30 b^4 d}","-\frac{2 a^3 (A b-a B) \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) (A b-a B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d}-\frac{2 \left(-5 a^2 B+5 a A b-3 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^3 d}+\frac{2 (A b-a B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d}+\frac{2 B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 b d}",1,"((2*b^2*(-5*a*A*b + 5*a^2*B + 9*b^2*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + 2*b^2*(5*A*b + 4*a*B)*(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*A*b - 5*a*B + 3*b*B*Cos[c + d*x])*Sin[c + d*x] + (6*(-5*a*A*b + 5*a^2*B + 3*b^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*Sqrt[Sin[c + d*x]^2]))/(30*b^4*d)","A",1
366,1,207,137,1.4384513,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),x]","\frac{\frac{3 (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)}{a b^2 \sqrt{\sin ^2(c+d x)}}+\frac{(3 A b-a B) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+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)+2 B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d}","\frac{2 a^2 (A b-a B) \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 B+3 a A b-b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^3 d}+\frac{2 (A b-a B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d}",1,"(((3*A*b - a*B)*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)) + 2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x] + (3*(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])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/(3*b*d)","A",1
367,1,128,89,0.910495,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{a+b \cos (c+d x)} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),x]","\frac{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)-\frac{2 B \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 (A b-a B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}-\frac{2 a (A b-a B) \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 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(A*b*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)) - (2*B*(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
368,1,58,61,0.2083399,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx","Integrate[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])),x]","\frac{2 \left((A b-a B) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)+B (a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{b d (a+b)}","\frac{2 (A b-a B) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a+b)}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(2*((a + b)*B*EllipticF[(c + d*x)/2, 2] + (A*b - a*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(b*(a + b)*d)","A",1
369,1,206,86,2.4823038,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Integrate[(A + B*Cos[c + d*x])/(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{2 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)}{b}+\frac{4 A \sin (c+d x)}{\sqrt{\cos (c+d x)}}}{2 a d}","-\frac{2 (A b-a B) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a 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)}}",1,"((2*(-3*A*b + 2*a*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) - (2*a*A*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)))/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)","B",1
370,1,260,150,2.2886681,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Integrate[(A + B*Cos[c + d*x])/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])),x]","\frac{\frac{2 a \left(2 a^2 A-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 (A b-a B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{2 b (A b-a B) \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)}{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*(2*a^2*A + 9*A*b^2 - 9*a*b*B)*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
371,1,318,303,3.2415328,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]))/(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 B-A b)}{\left(a^2-b^2\right) (a+b \cos (c+d x))}+2 B\right)-\frac{\frac{8 \left(2 a^2 B-3 a A b+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)}{a+b}+\frac{2 \left(5 a^3 B-3 a^2 A b-8 a b^2 B+6 A b^3\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 B-3 a^2 A b-4 a b^2 B+2 A b^3\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 (A b-a B) \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 B+3 a A b+2 b^2 B\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 B+3 a^2 A b+4 a b^2 B-2 A b^3\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 B+3 a^2 A b+7 a b^2 B-5 A b^3\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 B+9 a^3 A b+16 a^2 b^2 B-12 a A b^3+2 b^4 B\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*B + (3*a^2*(-(A*b) + a*B))/((a^2 - b^2)*(a + b*Cos[c + d*x])))*Sin[c + d*x] - ((2*(-3*a^2*A*b + 6*A*b^3 + 5*a^3*B - 8*a*b^2*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(-3*a*A*b + 2*a^2*B + b^2*B)*((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*A*b + 2*A*b^3 + 5*a^3*B - 4*a*b^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^2*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b)))/(12*b^2*d)","A",1
372,1,280,224,2.7361578,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^2,x]","\frac{\frac{\frac{2 \left(a^2 B+a A b-2 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 \left(3 a^2 B-a A b-2 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^2 \sqrt{\sin ^2(c+d x)}}+\frac{8 (a B-A 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 B-A 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 B+a A b+2 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}+\frac{a (A b-a B) \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 B+a^2 A b+4 a b^2 B-2 A b^3\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 B+a^2 A b+5 a b^2 B-3 A b^3\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*(-(A*b) + a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*(a + b*Cos[c + d*x])) + ((2*(a*A*b + a^2*B - 2*b^2*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(-(A*b) + a*B)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (2*(-(a*A*b) + 3*a^2*B - 2*b^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^2*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b)))/(4*b*d)","A",1
373,1,260,198,2.3778875,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^2,x]","\frac{\frac{4 (a B-A 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 (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)}{a b^2 \sqrt{\sin ^2(c+d x)}}+\frac{2 (a B-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-4 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}}{(b-a) (a+b)}}{4 d}","\frac{\left(a^2 B+a A b-2 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}+\frac{(A b-a B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}-\frac{(A b-a B) \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 B+a^2 A b-3 a b^2 B+A b^3\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*(-(A*b) + a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*(a + b*Cos[c + d*x])) - ((2*(-(A*b) + a*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + ((4*a*A - 4*b*B)*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)))/b + (2*(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])/(a*b^2*Sqrt[Sin[c + d*x]^2]))/((-a + b)*(a + b)))/(4*d)","A",1
374,1,274,200,2.6767665,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2} \, dx","Integrate[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2),x]","\frac{\frac{4 b (A b-a 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 \left(4 a^2 A-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 (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)}{a b \sqrt{\sin ^2(c+d x)}}+\frac{4 a (a B-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}}{(a-b) (a+b)}}{4 a d}","-\frac{(A b-a B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}-\frac{(A b-a B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \left(a^2-b^2\right)}+\frac{b (A b-a B) \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 (-B)+3 a^2 A b-a b^2 B-A b^3\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*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*(a + b*Cos[c + d*x])) + ((2*(4*a^2*A - 3*A*b^2 - a*b*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (4*a*(-(A*b) + a*B)*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)))/b + (2*(-(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])/(a*b*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b)))/(4*a*d)","A",1
375,1,316,256,4.2189489,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2),x]","\frac{4 \sqrt{\cos (c+d x)} \left(\frac{b^2 (A b-a B) \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{8 a \left(a^2 A+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 \left(2 a^2 A+a b B-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)}}+\frac{2 \left(4 a^3 B-10 a^2 A b-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{(A b-a B) 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 A+a b B-3 A b^2\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 A+a b B-3 A b^2\right) \sin (c+d x)}{a^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}+\frac{b (A b-a B) \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 B+5 a^2 A b+a b^2 B-3 A b^3\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*A*b + 9*A*b^3 + 4*a^3*B - 3*a*b^2*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) - (8*a*(a^2*A - 2*A*b^2 + a*b*B)*((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*A - 3*A*b^2 + a*b*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]))/((-a + b)*(a + b))) + 4*Sqrt[Cos[c + d*x]]*((b^2*(A*b - a*B)*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
376,1,427,345,6.9188121,"\int \frac{A+B \cos (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])/(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 b^4 \sin (c+d x)-a b^3 B \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 B+12 a^2 A b^2+9 a b^3 B-15 A b^4\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 B+28 a^3 A b+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-30 a^3 b B+44 a^2 A b^2+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{\left(2 a^2 A+3 a b B-5 A b^2\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 (A b-a B) \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 A+3 a b B-5 A b^2\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 B+4 a^2 A b+3 a b^2 B-5 A b^3\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 B+7 a^2 A b+3 a b^2 B-5 A b^3\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 B+4 a^2 A b+3 a b^2 B-5 A b^3\right) \sin (c+d x)}{a^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}",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)*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)*(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)*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^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
377,1,390,367,5.058895,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{\frac{8 \left(a^3 B+a^2 A b-4 a b^2 B+2 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)}{a+b}+\frac{\left(5 a^4 B-a^3 A b-7 a^2 b^2 B-5 a A b^3+8 b^4 B\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 B-3 a^3 A b-29 a^2 b^2 B+9 a A b^3+8 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^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 B-3 a^2 A b-13 a b^2 B+9 A b^3\right) \cos (c+d x)+a \left(5 a^3 B-a^2 A b-11 a b^2 B+7 A b^3\right)\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}}{8 b^2 d}","\frac{a (A b-a B) \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 B+a^2 A b+11 a b^2 B-7 A b^3\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 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\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 B+3 a^4 A b+33 a^3 b^2 B-5 a^2 A b^3-24 a b^4 B+8 A b^5\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 B+3 a^4 A b+38 a^3 b^2 B-6 a^2 A b^3-35 a b^4 B+15 A b^5\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*A*b) + 7*A*b^3 + 5*a^3*B - 11*a*b^2*B) + b*(-3*a^2*A*b + 9*A*b^3 + 7*a^3*B - 13*a*b^2*B)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) + (((-(a^3*A*b) - 5*a*A*b^3 + 5*a^4*B - 7*a^2*b^2*B + 8*b^4*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(a^2*A*b + 2*A*b^3 + a^3*B - 4*a*b^2*B)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + ((-3*a^3*A*b + 9*a*A*b^3 + 15*a^4*B - 29*a^2*b^2*B + 8*b^4*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^2*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(8*b^2*d)","A",1
378,1,360,344,3.7299013,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/(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 B+a^2 A b-9 a b^2 B+5 A b^3\right) \cos (c+d x)+a \left(a^3 B+3 a^2 A b-7 a b^2 B+3 A b^3\right)\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{-\frac{8 \left(a^2 B-3 a A b+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)}{a+b}+\frac{\left(a^3 B-5 a^2 A b+5 a b^2 B-A b^3\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 B+a^2 A b-9 a b^2 B+5 A b^3\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 (A b-a B) \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 B+a^2 A b-9 a b^2 B+5 A b^3\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 B+a^2 A b-9 a b^2 B+5 A b^3\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 B+a^3 A b-5 a^2 b^2 B-7 a A b^3+8 b^4 B\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 B+a^4 A b-6 a^3 b^2 B-10 a^2 A b^3+15 a b^4 B-3 A b^5\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*A*b + 3*A*b^3 + a^3*B - 7*a*b^2*B) + b*(a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) - (((-5*a^2*A*b - A*b^3 + a^3*B + 5*a*b^2*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) - (8*(-3*a*A*b + a^2*B + 2*b^2*B)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + ((a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^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^2*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(8*b*d)","A",1
379,1,365,337,4.5503207,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{(a+b \cos (c+d x))^3} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*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 B-5 a^2 A b+5 a b^2 B-A b^3\right) \cos (c+d x)+a \left(3 a^3 B-7 a^2 A b+3 a b^2 B+A b^3\right)\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\frac{16 a \left(2 a^2 A-3 a b B+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(5 a^3 B-9 a^2 A b+a b^2 B+3 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^3 B-5 a^2 A b+5 a b^2 B-A b^3\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{(A b-a B) \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 B+3 a^2 A b-7 a b^2 B+3 A b^3\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 (-B)+5 a^2 A b-5 a b^2 B+A b^3\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 (-B)+5 a^2 A b-5 a b^2 B+A b^3\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 B+3 a^4 A b-10 a^3 b^2 B+10 a^2 A b^3-3 a b^4 B-A b^5\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*A*b + A*b^3 + 3*a^3*B + 3*a*b^2*B) + b*(-5*a^2*A*b - A*b^3 + a^3*B + 5*a*b^2*B)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) + ((2*(-9*a^2*A*b + 3*A*b^3 + 5*a^3*B + a*b^2*B)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (16*a*(2*a^2*A + A*b^2 - 3*a*b*B)*((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*A*b - A*b^3 + a^3*B + 5*a*b^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^2*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(16*a*d)","A",1
380,1,383,345,4.8867201,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3} \, dx","Integrate[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3),x]","\frac{\frac{\frac{8 a \left(2 a^3 B-4 a^2 A b+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{\left(5 a^3 B-9 a^2 A b+a b^2 B+3 A b^3\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 A-9 a^3 b B-19 a^2 A b^2+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}}{(a-b)^2 (a+b)^2}-\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} \left(b \left(5 a^3 B-9 a^2 A b+a b^2 B+3 A b^3\right) \cos (c+d x)+a \left(7 a^3 B-11 a^2 A b-a b^2 B+5 A b^3\right)\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}}{8 a^2 d}","\frac{b (A b-a B) \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 B+7 a^2 A b-3 a b^2 B-A b^3\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 B+9 a^2 A b-a b^2 B-3 A b^3\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 B+9 a^2 A b-a b^2 B-3 A b^3\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 B+15 a^4 A b-10 a^3 b^2 B-6 a^2 A b^3+a b^4 B+3 A b^5\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*A*b + 5*A*b^3 + 7*a^3*B - a*b^2*B) + b*(-9*a^2*A*b + 3*A*b^3 + 5*a^3*B + a*b^2*B)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^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)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*a*(-4*a^2*A*b + A*b^3 + 2*a^3*B + a*b^2*B)*((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*A*b + 3*A*b^3 + 5*a^3*B + a*b^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]))/((a - b)^2*(a + b)^2))/(8*a^2*d)","A",1
381,1,458,420,5.5073345,"\int \frac{A+B \cos (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])/(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(8 a^4 A+9 a^3 b B-29 a^2 A b^2-3 a b^3 B+15 A b^4\right) \sin (2 (c+d x))+2 a b \left(16 a^4 A+11 a^3 b B-47 a^2 A b^2-5 a b^3 B+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 a \left(2 a^4 A+4 a^3 b B-10 a^2 A b^2-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{\left(8 a^4 A+9 a^3 b B-29 a^2 A b^2-3 a b^3 B+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)}}+\frac{\left(-16 a^5 B+56 a^4 A b+19 a^3 b^2 B-95 a^2 A b^3-9 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}}{8 a^3 d}","\frac{b (A b-a B) \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 B+11 a^2 A b+a b^2 B-5 A b^3\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 B+11 a^2 A b+a b^2 B-5 A b^3\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 A+9 a^3 b B-29 a^2 A b^2-3 a b^3 B+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(8 a^4 A+9 a^3 b B-29 a^2 A b^2-3 a b^3 B+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(-15 a^5 B+35 a^4 A b+6 a^3 b^2 B-38 a^2 A b^3-3 a b^4 B+15 A b^5\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*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)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*a*(2*a^4*A - 10*a^2*A*b^2 + 5*A*b^4 + 4*a^3*b*B - a*b^3*B)*((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*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*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]))/((a - b)^2*(a + b)^2)) + (Sqrt[Cos[c + d*x]]*(2*a*b*(16*a^4*A - 47*a^2*A*b^2 + 25*A*b^4 + 11*a^3*b*B - 5*a*b^3*B)*Sin[c + d*x] + b^2*(8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B)*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
382,1,570,523,7.2749168,"\int \frac{A+B \cos (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])/(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 b^4 \sin (c+d x)-a b^3 B \sin (c+d x)}{2 a^3 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{-13 a^3 b^3 B \sin (c+d x)+17 a^2 A b^4 \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 \left(-24 a^5 b B+72 a^4 A b^2+87 a^3 b^3 B-195 a^2 A b^4-45 a b^5 B+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{\left(-48 a^6 B+160 a^5 A b+240 a^4 b^2 B-512 a^3 A b^3-120 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-168 a^5 b B+328 a^4 A b^2+285 a^3 b^3 B-641 a^2 A b^4-135 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{b (A b-a B) \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(-9 a^3 B+13 a^2 A b+3 a b^2 B-7 A b^3\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{\left(8 a^4 A+33 a^3 b B-61 a^2 A b^2-15 a b^3 B+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(8 a^4 A+33 a^3 b B-61 a^2 A b^2-15 a b^3 B+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)}+\frac{\left(-8 a^5 B+24 a^4 A b+29 a^3 b^2 B-65 a^2 A b^3-15 a b^4 B+35 A b^5\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{b \left(-35 a^5 B+63 a^4 A b+38 a^3 b^2 B-86 a^2 A b^3-15 a b^4 B+35 A b^5\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(-8 a^5 B+24 a^4 A b+29 a^3 b^2 B-65 a^2 A b^3-15 a b^4 B+35 A b^5\right) \sin (c+d x)}{4 a^4 d \left(a^2-b^2\right)^2 \sqrt{\cos (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 - 168*a^5*b*B + 285*a^3*b^3*B - 135*a*b^5*B)*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)*(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)*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])/(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])/(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",1
383,1,41,44,0.0385227,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),x]","\frac{B \left(6 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (2 (c+d x)) \sqrt{\cos (c+d x)}\right)}{5 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}",1,"(B*(6*EllipticE[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*Sin[2*(c + d*x)]))/(5*d)","A",1
384,1,37,44,0.0359513,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),x]","\frac{2 B \left(F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)}\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) \sqrt{\cos (c+d x)}}{3 d}",1,"(2*B*(EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*Sin[c + d*x]))/(3*d)","A",1
385,1,17,17,0.0153801,"\int \frac{\sqrt{\cos (c+d x)} (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x]),x]","\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}","\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*B*EllipticE[(c + d*x)/2, 2])/d","A",1
386,1,17,17,0.018278,"\int \frac{a B+b B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx","Integrate[(a*B + b*B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])),x]","\frac{2 B F\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}",1,"(2*B*EllipticF[(c + d*x)/2, 2])/d","A",1
387,1,40,40,0.053494,"\int \frac{a B+b B \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Integrate[(a*B + b*B*Cos[c + d*x])/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])),x]","B \left(\frac{2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}\right)","\frac{2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"B*((-2*EllipticE[(c + d*x)/2, 2])/d + (2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]))","A",1
388,1,37,44,0.0618733,"\int \frac{a B+b B \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Integrate[(a*B + b*B*Cos[c + d*x])/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])),x]","\frac{2 B \left(F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{\sin (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)}\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)}",1,"(2*B*(EllipticF[(c + d*x)/2, 2] + Sin[c + d*x]/Cos[c + d*x]^(3/2)))/(3*d)","A",1
389,1,159,116,1.5781052,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^2,x]","\frac{B \left(-\frac{6 \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{6 a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+4 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+4 \sin (c+d x) \sqrt{\cos (c+d x)}\right)}{6 b d}","-\frac{2 a^3 B \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 \left(3 a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^3 d}-\frac{2 a B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d}",1,"(B*(4*EllipticF[(c + d*x)/2, 2] - (6*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + 4*Sqrt[Cos[c + d*x]]*Sin[c + d*x] - (6*(-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
390,1,82,78,0.1104573,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^2,x]","-\frac{2 B \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)}{b^2 d \sqrt{\sin ^2(c+d x)}}","\frac{2 a^2 B \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 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(-2*B*(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])/(b^2*d*Sqrt[Sin[c + d*x]^2])","A",1
391,1,49,55,0.0563211,"\int \frac{\sqrt{\cos (c+d x)} (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^2,x]","\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 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}-\frac{2 a B \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a+b)}",1,"(B*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)))/(b*d)","A",1
392,1,30,30,0.0652668,"\int \frac{a B+b B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2} \, dx","Integrate[(a*B + b*B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2),x]","\frac{2 B \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{d (a+b)}","\frac{2 B \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{d (a+b)}",1,"(2*B*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a + b)*d)","A",1
393,1,196,80,2.6755167,"\int \frac{a B+b B \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx","Integrate[(a*B + b*B*Cos[c + d*x])/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2),x]","-\frac{B \left(\frac{2 \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{6 b \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{2 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)}{b}-\frac{4 \sin (c+d x)}{\sqrt{\cos (c+d x)}}\right)}{2 a d}","-\frac{2 b B \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,"-1/2*(B*((6*b*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (2*a*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)))/b - (4*Sin[c + d*x])/Sqrt[Cos[c + d*x]] + (2*(-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)","B",1
394,1,211,133,4.1151734,"\int \frac{a B+b B \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx","Integrate[(a*B + b*B*Cos[c + d*x])/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2),x]","\frac{B \left(\frac{2 \left(2 a^2+9 b^2\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(\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 \sin (c+d x) (a-3 b \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)}+8 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)\right)}{6 a^2 d}","\frac{2 b^2 B \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 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{2 b B \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,"(B*((2*(2*a^2 + 9*b^2)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + 8*a*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)) + (4*(a - 3*b*Cos[c + d*x])*Sin[c + d*x])/Cos[c + d*x]^(3/2) + (6*(-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
395,1,1224,560,6.3269947,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{(6 A b+a B) \sin (c+d x)}{12 b}+\frac{1}{6} B \sin (2 (c+d x))\right)}{d}-\frac{-\frac{4 a \left(B a^2-18 A b a-16 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(-24 A b^2-28 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(3 B a^2-6 A b a-16 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)}{48 b d}","\frac{\left(-3 a^2 B+6 a A b+16 b^2 B\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 B+6 a A b+16 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)}{24 a b^2 d}+\frac{\sqrt{a+b} \left(a^3 (-B)+2 a^2 A b-4 a b^2 B-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}} \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 B+6 A b+8 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)}{24 b^2 d}+\frac{(2 A b-a B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}+\frac{B \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*A*b + a^2*B - 16*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*(-24*A*b^2 - 28*a*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*(-6*a*A*b + 3*a^2*B - 16*b^2*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]])))/(b*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(((6*A*b + a*B)*Sin[c + d*x])/(12*b) + (B*Sin[2*(c + d*x)])/6))/d","C",0
396,1,1175,473,21.106361,"\int \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(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
397,1,408,385,11.360952,"\int \frac{\sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{\sqrt{\cos (c+d x)} \left(-4 (a (B-A)+A 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 A 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 B (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 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 a B \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \tan \left(\frac{1}{2} (c+d x)\right)-b B \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \tan \left(\frac{1}{2} (c+d x)\right)+b B \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 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{\sqrt{a+b} (a B+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}} \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{B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{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}",1,"(Sqrt[Cos[c + d*x]]*(2*(a + b)*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)] - 4*(A*b + a*(-A + B))*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*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*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)] + b*B*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sin[(3*(c + d*x))/2] + 2*a*B*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Tan[(c + d*x)/2] - b*B*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
398,1,273,351,12.7433785,"\int \frac{\sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{2 (a (A+B)+b (A-B)) \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 A \tan \left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))}{\sqrt{\cos (c+d x)}}-2 A (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 B \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} (A b-a (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 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 d}-\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}} \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)*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*(A - B) + a*(A + B))*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*B*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*A*(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
399,1,407,284,13.5013264,"\int \frac{\sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\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)+A b \sin (c+d x))}{3 a}+\frac{2}{3} A \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 B+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))\right)+2 a (a+b) (A+3 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 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 \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}","\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 (a-b) \sqrt{a+b} (A-3 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 d}+\frac{2 A \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)*(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*(a + b)*(A + 3*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)] - (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*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]*(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","A",0
400,1,1315,350,6.3714791,"\int \frac{\sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/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+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+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\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-5 a B+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(9 a^2 A+5 a b 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}} 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)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-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)*((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) + (2*A*Sec[c + d*x]^2*Tan[c + d*x])/5))/d","C",1
401,1,1408,433,6.4647052,"\int \frac{\sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(9/2),x]","\frac{-\frac{4 a \left(25 A a^4-14 b B a^3-17 A b^2 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+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^2-63 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+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^2-14 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 \left(25 a^2 A+7 a b B-4 A b^2\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 A-63 B)+2 a b (3 A-7 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) \sqrt{a+b} \left(63 a^3 B+19 a^2 A b-14 a b^2 B+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}} 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)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-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)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-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)*((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]))/(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]))/(105*a^3) + (2*A*Sec[c + d*x]^3*Tan[c + d*x])/7))/d","C",0
402,1,1284,670,6.4127116,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{\left(3 B a^2+56 A b a+42 b^2 B\right) \sin (c+d x)}{96 b}+\frac{1}{48} (8 A b+9 a B) \sin (2 (c+d x))+\frac{1}{16} b B \sin (3 (c+d x))\right)}{d}-\frac{-\frac{4 a \left(3 B a^3-136 A b a^2-228 b^2 B a-128 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(-144 B b^3-416 a A b^2-228 a^2 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(9 B a^3-24 A b a^2-156 b^2 B a-128 A b^3\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 B+8 a A b+12 b^2 B\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 B+24 a^2 A b+156 a b^2 B+128 A b^3\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 B-6 a^2 b (4 A+B)-4 a b^2 (28 A+39 B)-8 b^3 (16 A+9 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)}{192 b^2 d}-\frac{(a-b) \sqrt{a+b} \left(-9 a^3 B+24 a^2 A b+156 a b^2 B+128 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}} 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 B+8 a^3 A b-24 a^2 b^2 B-96 a A b^3-48 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}} \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 A b-3 a B) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{24 b d}+\frac{B \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*A*b - 128*A*b^3 + 3*a^3*B - 228*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*(-416*a*A*b^2 - 228*a^2*b*B - 144*b^3*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*(-24*a^2*A*b - 128*A*b^3 + 9*a^3*B - 156*a*b^2*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]])))/(b*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(((56*a*A*b + 3*a^2*B + 42*b^2*B)*Sin[c + d*x])/(96*b) + ((8*A*b + 9*a*B)*Sin[2*(c + d*x)])/48 + (b*B*Sin[3*(c + d*x)])/16))/d","C",0
403,1,1227,566,6.3209291,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{-\frac{4 a \left(17 B a^2+42 A b a+16 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(48 A a^2+52 b B a+24 A b^2\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 a^2+30 A b a+16 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)}{48 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{1}{12} (6 A b+7 a B) \sin (c+d x)+\frac{1}{6} b B \sin (2 (c+d x))\right)}{d}","\frac{\left(3 a^2 B+30 a A b+16 b^2 B\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 B+30 a A b+14 a b B+12 A b^2+16 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)}{24 b d}-\frac{(a-b) \sqrt{a+b} \left(3 a^2 B+30 a A b+16 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)}{24 a b d}-\frac{\sqrt{a+b} \left(a^3 (-B)+6 a^2 A b+12 a b^2 B+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}} \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 B+6 A b) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{12 d}+\frac{b B \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*A*b + 17*a^2*B + 16*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*(48*a^2*A + 24*A*b^2 + 52*a*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*(30*a*A*b + 3*a^2*B + 16*b^2*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]])))/(48*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(((6*A*b + 7*a*B)*Sin[c + d*x])/12 + (b*B*Sin[2*(c + d*x)])/6))/d","C",0
404,1,1198,472,6.3488303,"\int \frac{(a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{b B \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{2 d}+\frac{-\frac{4 a \left(8 A a^2+7 b B a+4 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(8 B a^2+16 A b a+4 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(4 A b^2+5 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)}{8 d}","-\frac{\sqrt{a+b} \left(3 a^2 B+12 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 d}+\frac{(5 a B+4 A 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 A+5 a B+4 A b+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)}{4 d}-\frac{(a-b) \sqrt{a+b} (5 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 d}+\frac{b B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}",1,"(b*B*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + ((-4*a*(8*a^2*A + 4*A*b^2 + 7*a*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]]) - 4*a*(16*a*A*b + 8*a^2*B + 4*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]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(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^2 + 5*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]])))/(8*d)","C",1
405,1,1196,449,6.3498409,"\int \frac{(a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{2 a A \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{\frac{4 a \left(-2 B a^2-2 A b 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(2 A a^2-4 b B a-2 A b^2\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 A b-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)}{2 d}","-\frac{(2 a A-b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} (2 a (A-B)-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)}{d}+\frac{(a-b) \sqrt{a+b} (2 a A-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)}{a d}-\frac{\sqrt{a+b} (3 a B+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}} \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 A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(2*a*A*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + ((4*a*(-2*a*A*b - 2*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*(2*a^2*A - 2*A*b^2 - 4*a*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*(2*a*A*b - b^2*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]])))/(2*d)","C",1
406,1,1236,419,6.364883,"\int \frac{(a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{-\frac{4 a \left(A a^2+3 b B 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(-3 B a^2-4 A b a+3 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(-4 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 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{2 \sqrt{a+b} \left(a^2 (A-3 B)-a (4 A b-6 b B)+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}} 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 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)}{3 a d}+\frac{2 a A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b 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}} \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*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]]) - 4*a*(-4*a*A*b - 3*a^2*B + 3*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]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(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^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*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
407,1,1314,353,6.4463526,"\int \frac{(a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/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+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+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+20 b B a^2+3 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(3 A b^3+20 a B b^2+9 a^2 A 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 A+20 a b B+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)}{15 a^2 d}+\frac{2 (5 a B+6 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-b) \sqrt{a+b} (9 a A-5 a B-3 A b+15 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 d}+\frac{2 a 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*(-3*a^2*A*b + 3*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 + 3*a*A*b^2 + 20*a^2*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*(9*a^2*A*b + 3*A*b^3 + 20*a*b^2*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) + (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*a*A*Sec[c + d*x]^2*Tan[c + d*x])/5))/d","C",1
408,1,1407,433,6.5436023,"\int \frac{(a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(9/2),x]","\frac{-\frac{4 a \left(25 A a^4+21 b B a^3-31 A b^2 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^3-21 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^2-63 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+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+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 \left(25 a^2 A+42 a b B+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(-\left(a^2 (25 A-63 B)\right)+3 a b (19 A-7 B)+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{2 (a-b) \sqrt{a+b} \left(63 a^3 B+82 a^2 A b+21 a b^2 B-6 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}} 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+8 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 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 - 31*a^2*A*b^2 + 6*A*b^4 + 21*a^3*b*B - 21*a*b^3*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*(-82*a^3*A*b + 6*a*A*b^3 - 63*a^4*B - 21*a^2*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]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(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)*((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]))/(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]))/(105*a^2) + (2*a*A*Sec[c + d*x]^3*Tan[c + d*x])/7))/d","C",0
409,1,1515,522,6.6299005,"\int \frac{(a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/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+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+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+246 b B \sin (c+d x) a^3+33 A b^2 \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+93 b^2 B a^3+31 A b^3 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+246 b B a^4+33 A b^2 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+246 a^3 B b^2+147 a^4 A 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 \left(49 a^2 A+72 a b B+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 \left(75 a^3 B+88 a^2 A b+9 a b^2 B-4 A b^3\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 (a-b) \sqrt{a+b} \left(-\left(a^3 (147 A-75 B)\right)+3 a^2 b (13 A-57 B)+6 a b^2 (A-3 B)+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-b) \sqrt{a+b} \left(147 a^4 A+246 a^3 b B+33 a^2 A b^2-18 a b^3 B+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 (9 a B+10 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a A \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{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)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-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)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-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)*((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]))/(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]))/(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]))/(315*a^3) + (2*a*A*Sec[c + d*x]^4*Tan[c + d*x])/9))/d","C",0
410,1,1353,779,6.5336751,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{1}{40} B \sin (4 (c+d x)) b^2+\frac{1}{160} (10 A b+21 a B) \sin (3 (c+d x)) b+\frac{1}{480} \left(93 B a^2+170 A b a+88 b^2 B\right) \sin (2 (c+d x))+\frac{\left(15 B a^3+590 A b a^2+898 b^2 B a+420 A b^3\right) \sin (c+d x)}{960 b}\right)}{d}-\frac{-\frac{4 a \left(15 B a^4-1330 A b a^3-3236 b^2 B a^2-3560 A b^3 a-1024 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(-1440 A b^4-4624 a B b^3-6440 a^2 A b^2-2292 a^3 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(45 B a^4-150 A b a^3-1692 b^2 B a^2-2840 A b^3 a-1024 b^4 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)}{3840 b d}","\frac{\left(-15 a^2 B+50 a A b+64 b^2 B\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 B+50 a^2 A b+172 a b^2 B+120 A b^3\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 B+150 a^3 A b+1692 a^2 b^2 B+2840 a A b^3+1024 b^4 B\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 B-30 a^3 b (5 A+B)-4 a^2 b^2 (295 A+423 B)-8 a b^3 (355 A+193 B)-16 b^4 (45 A+64 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)}{1920 b^2 d}-\frac{(a-b) \sqrt{a+b} \left(-45 a^4 B+150 a^3 A b+1692 a^2 b^2 B+2840 a A b^3+1024 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)}{1920 a b^2 d}+\frac{\sqrt{a+b} \left(-3 a^5 B+10 a^4 A b-40 a^3 b^2 B-240 a^2 A b^3-240 a b^4 B-96 A b^5\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 A b-3 a B) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{40 b d}+\frac{B \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*A*b - 3560*a*A*b^3 + 15*a^4*B - 3236*a^2*b^2*B - 1024*b^4*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*(-6440*a^2*A*b^2 - 1440*A*b^4 - 2292*a^3*b*B - 4624*a*b^3*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*(-150*a^3*A*b - 2840*a*A*b^3 + 45*a^4*B - 1692*a^2*b^2*B - 1024*b^4*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]])))/(b*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(((590*a^2*A*b + 420*A*b^3 + 15*a^3*B + 898*a*b^2*B)*Sin[c + d*x])/(960*b) + ((170*a*A*b + 93*a^2*B + 88*b^2*B)*Sin[2*(c + d*x)])/480 + (b*(10*A*b + 21*a*B)*Sin[3*(c + d*x)])/160 + (b^2*B*Sin[4*(c + d*x)])/40))/d","C",0
411,1,1287,664,6.4186098,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]),x]","\frac{-\frac{4 a \left(133 B a^3+472 A b a^2+356 b^2 B a+128 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(384 A a^3+644 b B a^2+608 A b^2 a+144 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(15 B a^3+264 A b a^2+284 b^2 B a+128 A b^3\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} B \sin (3 (c+d x)) b^2+\frac{1}{48} (8 A b+17 a B) \sin (2 (c+d x)) b+\frac{1}{96} \left(59 B a^2+104 A b a+42 b^2 B\right) \sin (c+d x)\right)}{d}","\frac{\left(5 a^2 B+24 a A b+12 b^2 B\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{32 d}+\frac{\left(15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\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 B+2 a^2 b (132 A+59 B)+4 a b^2 (52 A+71 B)+8 b^3 (16 A+9 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)}{192 b d}-\frac{(a-b) \sqrt{a+b} \left(15 a^3 B+264 a^2 A b+284 a b^2 B+128 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}} 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 B+40 a^3 A b+120 a^2 b^2 B+160 a A b^3+48 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}} \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 B+8 A b) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{24 d}+\frac{b B \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*A*b + 128*A*b^3 + 133*a^3*B + 356*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*(384*a^3*A + 608*a*A*b^2 + 644*a^2*b*B + 144*b^3*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*(264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*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]])))/(384*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*(((104*a*A*b + 59*a^2*B + 42*b^2*B)*Sin[c + d*x])/96 + (b*(8*A*b + 17*a*B)*Sin[2*(c + d*x)])/48 + (b^2*B*Sin[3*(c + d*x)])/16))/d","C",0
412,1,1251,564,6.5123349,"\int \frac{(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{-\frac{4 a \left(48 A a^3+59 b B a^2+66 A b^2 a+16 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(48 B a^3+144 A b a^2+76 b^2 B a+24 A b^3\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 B b^3+54 a A b^2+33 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)}{48 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{1}{6} B \sin (2 (c+d x)) b^2+\frac{1}{12} (6 A b+13 a B) \sin (c+d x) b\right)}{d}","\frac{\left(33 a^2 B+54 a A b+16 b^2 B\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 B)+a (54 A b+26 b B)+4 b^2 (3 A+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)}{24 d}-\frac{(a-b) \sqrt{a+b} \left(33 a^2 B+54 a A b+16 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)}{24 a d}-\frac{\sqrt{a+b} \left(5 a^3 B+30 a^2 A b+20 a b^2 B+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}} \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 B+2 A b) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{b B \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*A + 66*a*A*b^2 + 59*a^2*b*B + 16*b^3*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*(144*a^2*A*b + 24*A*b^3 + 48*a^3*B + 76*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]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(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*A*b^2 + 33*a^2*b*B + 16*b^3*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]])))/(48*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((b*(6*A*b + 13*a*B)*Sin[c + d*x])/12 + (b^2*B*Sin[2*(c + d*x)])/6))/d","C",0
413,1,1241,547,6.4904631,"\int \frac{(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{\frac{4 a \left(-8 B a^3-16 A b a^2-11 b^2 B a-4 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(8 A a^3-24 b B a^2-24 A b^2 a-4 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(-4 A b^3-9 a B b^2+8 a^2 A 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 A \tan (c+d x) a^2+\frac{1}{2} b^2 B \sin (c+d x)\right)}{d}","-\frac{\left(8 a^2 A-9 a b B-4 A b^2\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 (A-B)-3 a b (8 A+3 B)-2 b^2 (2 A+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)}{4 d}+\frac{(a-b) \sqrt{a+b} \left(8 a^2 A-9 a b B-4 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)}{4 a d}-\frac{\sqrt{a+b} \left(15 a^2 B+20 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 d}-\frac{b (4 a A-b B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}+\frac{2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \sqrt{\cos (c+d x)}}",1,"((4*a*(-16*a^2*A*b - 4*A*b^3 - 8*a^3*B - 11*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*(8*a^3*A - 24*a*A*b^2 - 24*a^2*b*B - 4*b^3*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*(8*a^2*A*b - 4*A*b^3 - 9*a*b^2*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) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((b^2*B*Sin[c + d*x])/2 + 2*a^2*A*Tan[c + d*x]))/d","C",0
414,1,1269,536,6.5111771,"\int \frac{(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{-\frac{4 a \left(2 A a^3+12 b B a^2+4 A b^2 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(-6 B a^3-14 A b a^2+18 b^2 B a+6 A b^3\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^3-14 a A b^2-6 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)}{6 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{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{\left(6 a^2 B+14 a A b-3 b^2 B\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 (A-3 B)+2 a b (7 A-9 B)-3 b^2 (6 A+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 d}+\frac{(a-b) \sqrt{a+b} \left(6 a^2 B+14 a A b-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)}{3 a d}+\frac{2 a (a B+2 A 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 B+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}} \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 A \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*A + 4*a*A*b^2 + 12*a^2*b*B + 3*b^3*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*(-14*a^2*A*b + 6*A*b^3 - 6*a^3*B + 18*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]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(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*A*b^2 - 6*a^2*b*B + 3*b^3*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]])))/(6*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((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",1
415,1,1319,493,6.5678699,"\int \frac{(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","\frac{\frac{4 a \left(-5 B a^3-8 A b 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+35 b B a^2+23 A b^2 a-15 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(23 A b^3+35 a B b^2+9 a^2 A 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 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+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 (a-b) \sqrt{a+b} \left(9 a^2 A+35 a b B+23 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 d}+\frac{2 \sqrt{a+b} \left(-\left(a^3 (9 A-5 B)\right)+a^2 b (17 A-35 B)-a b^2 (23 A-45 B)+15 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 (5 a B+8 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 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^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}} \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*(-8*a^2*A*b + 8*A*b^3 - 5*a^3*B - 10*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 + 23*a*A*b^2 + 35*a^2*b*B - 15*b^3*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*(9*a^2*A*b + 23*A*b^3 + 35*a*b^2*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]])))/(15*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 + (2*a^2*A*Sec[c + d*x]^2*Tan[c + d*x])/5))/d","C",0
416,1,1409,434,6.6339536,"\int \frac{(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(9/2),x]","\frac{-\frac{4 a \left(25 A a^4+56 b B a^3-10 A b^2 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^3-161 b^2 B a^2-15 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(-15 A b^4-161 a B b^3-145 a^2 A 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+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+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 \left(25 a^2 A+77 a b B+45 A b^2\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 A-63 B)-8 a b (15 A-7 B)+15 b^2 (A-7 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 d}+\frac{2 (a-b) \sqrt{a+b} \left(63 a^3 B+145 a^2 A b+161 a b^2 B+15 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}} 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 B+10 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 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 - 10*a^2*A*b^2 - 15*A*b^4 + 56*a^3*b*B - 56*a*b^3*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*(-145*a^3*A*b - 15*a*A*b^3 - 63*a^4*B - 161*a^2*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]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(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)*((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]))/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]))/(105*a) + (2*a^2*A*Sec[c + d*x]^3*Tan[c + d*x])/7))/d","C",0
417,1,1517,522,6.7291398,"\int \frac{(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/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+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+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+435 b B \sin (c+d x) a^3+279 A b^2 \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+30 b^2 B a^3+124 A b^3 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+435 b B a^4+279 A b^2 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+435 a^3 B b^2+147 a^4 A 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 A+135 a b B+75 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 \left(75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\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 A-25 B)-6 a^2 b (19 A-60 B)+15 a b^2 (11 A-3 B)+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(147 a^4 A+435 a^3 b B+279 a^2 A b^2+45 a b^3 B-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 (3 a B+4 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 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*(-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)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-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)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-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)*((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]))/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]))/(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]))/(315*a^2) + (2*a^2*A*Sec[c + d*x]^4*Tan[c + d*x])/9))/d","C",0
418,1,1640,622,6.8476474,"\int \frac{(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(13/2),x]","\frac{-\frac{4 a \left(675 A a^6+1254 b B a^5-390 A b^2 a^4-1364 b^3 B a^3-245 A b^4 a^2+110 b^5 B a-40 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(-1617 B a^6-3705 A b a^5-3069 b^2 B a^4-255 A b^3 a^3+110 b^4 B a^2-40 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(-40 A b^6+110 a B b^5-255 a^2 A b^4-3069 a^3 B b^3-3705 a^4 A b^2-1617 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)}{3465 a^3 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (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 ^5(c+d x)+\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+209 b B \sin (c+d x) a+113 A b^2 \sin (c+d x)\right) \sec ^4(c+d x)+\frac{2 \left(539 B \sin (c+d x) a^3+1145 A b \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 ^3(c+d x)}{3465 a}+\frac{2 \left(675 A \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+55 b^3 B \sin (c+d x) a-20 A b^4 \sin (c+d x)\right) \sec ^2(c+d x)}{3465 a^2}+\frac{2 \left(1617 B \sin (c+d x) a^5+3705 A b \sin (c+d x) a^4+3069 b^2 B \sin (c+d x) a^3+255 A b^3 \sin (c+d x) a^2-110 b^4 B \sin (c+d x) a+40 A b^5 \sin (c+d x)\right) \sec (c+d x)}{3465 a^3}\right)}{d}","\frac{2 \left(81 a^2 A+209 a b B+113 A b^2\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 B+1145 a^2 A b+825 a b^2 B+15 A b^3\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 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\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 A-539 B)-6 a^3 b (505 A-209 B)+15 a^2 b^2 (19 A-121 B)+10 a b^3 (3 A-11 B)+40 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)}{3465 a^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(1617 a^5 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 a b^4 B+40 A b^5\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 B+14 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 a A \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*A - 390*a^4*A*b^2 - 245*a^2*A*b^4 - 40*A*b^6 + 1254*a^5*b*B - 1364*a^3*b^3*B + 110*a*b^5*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*(-3705*a^5*A*b - 255*a^3*A*b^3 - 40*a*A*b^5 - 1617*a^6*B - 3069*a^4*b^2*B + 110*a^2*b^4*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*(-3705*a^4*A*b^2 - 255*a^2*A*b^4 - 40*A*b^6 - 1617*a^5*b*B - 3069*a^3*b^3*B + 110*a*b^5*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]])))/(3465*a^3*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^5*(23*a*A*b*Sin[c + d*x] + 11*a^2*B*Sin[c + d*x]))/99 + (2*Sec[c + d*x]^4*(81*a^2*A*Sin[c + d*x] + 113*A*b^2*Sin[c + d*x] + 209*a*b*B*Sin[c + d*x]))/693 + (2*Sec[c + d*x]^3*(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]))/(3465*a) + (2*Sec[c + d*x]^2*(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]))/(3465*a^2) + (2*Sec[c + d*x]*(3705*a^4*A*b*Sin[c + d*x] + 255*a^2*A*b^3*Sin[c + d*x] + 40*A*b^5*Sin[c + d*x] + 1617*a^5*B*Sin[c + d*x] + 3069*a^3*b^2*B*Sin[c + d*x] - 110*a*b^4*B*Sin[c + d*x]))/(3465*a^3) + (2*a^2*A*Sec[c + d*x]^5*Tan[c + d*x])/11))/d","C",0
419,1,1236,418,19.4517879,"\int \frac{(a+b \cos (c+d x))^{5/2} \left(\frac{3 b B}{2 a}+B \cos (c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*((3*b*B)/(2*a) + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\sec (c+d x) \left(2 B \sin (c+d x) a^2+7 b^2 B \sin (c+d x)\right)+a b B \sec (c+d x) \tan (c+d x)\right)}{d}-\frac{B \left(-\frac{4 a \left(-5 b a^3-3 b^3 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(2 a^4+b^2 a^2-3 b^4\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 a^3+6 b^3 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)\right)}{2 a d}","-\frac{B (a-3 b) \sqrt{a+b} \left(2 a^2-a b+3 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 d}+\frac{2 B (a-b) \sqrt{a+b} \left(a^2+3 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 d}-\frac{b B \sqrt{a+b} \left(\frac{3 b^2}{a}+5 a\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)}{d}+\frac{b B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{d \cos ^{\frac{3}{2}}(c+d x)}",1,"-1/2*(B*((-4*a*(-5*a^3*b - 3*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*(2*a^4 + a^2*b^2 - 3*b^4)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-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^3*b + 6*a*b^3)*((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]]*(Sec[c + d*x]*(2*a^2*B*Sin[c + d*x] + 7*b^2*B*Sin[c + d*x]) + a*b*B*Sec[c + d*x]*Tan[c + d*x]))/d","C",1
420,1,1175,479,12.4332747,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/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 b d}+\frac{-\frac{4 a (4 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)}}-16 a 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-3 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 b d}","\frac{\sqrt{a+b} \left(-3 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^3 d}+\frac{(4 A b-3 a B) \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 B+4 A b+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)}{4 b^2 d}-\frac{(a-b) \sqrt{a+b} (4 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)}{4 a b^2 d}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 b d}",1,"(B*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*b*d) + ((-4*a*(4*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]]) - 16*a*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 - 3*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*b*d)","C",1
421,1,4017,427,17.355279,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Sqrt[a + b*Cos[c + d*x]],x]","\text{Result too large to show}","-\frac{\sqrt{a+b} (2 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}} \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{B \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a+b \cos (c+d x)}}+\frac{a B \sin (c+d x)}{b d \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{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)}{b d}-\frac{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 b d}",1,"((1 + Cos[c + d*x])^(3/2)*((A*Sqrt[Cos[c + d*x]])/Sqrt[a + b*Cos[c + d*x]] + (B*Cos[c + d*x]^(3/2))/Sqrt[a + b*Cos[c + d*x]])*Sec[(c + d*x)/2]^2*((2*I)*(a - b)*B*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)*(A*b - a*B)*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)*A*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*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))] + b*Sqrt[(a - b)/(a + b)]*B*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)]*B*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Tan[(c + d*x)/2] - b*Sqrt[(a - b)/(a + b)]*B*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)*B*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)*(A*b - a*B)*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)*A*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*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))] + b*Sqrt[(a - b)/(a + b)]*B*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)]*B*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Tan[(c + d*x)/2] - b*Sqrt[(a - b)/(a + b)]*B*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)*B*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)*(A*b - a*B)*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)*A*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*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))] + b*Sqrt[(a - b)/(a + b)]*B*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)]*B*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Tan[(c + d*x)/2] - b*Sqrt[(a - b)/(a + b)]*B*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)*B*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)*(A*b - a*B)*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)*A*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*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))] + b*Sqrt[(a - b)/(a + b)]*B*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)]*B*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Tan[(c + d*x)/2] - b*Sqrt[(a - b)/(a + b)]*B*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)]*B*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)]*B*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]^2 - (b*Sqrt[(a - b)/(a + b)]*B*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]^2)/2 + (I*(a - b)*B*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)*(A*b - a*B)*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)*A*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*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]))] + (b*Sqrt[(a - b)/(a + b)]*B*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)]*B*((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)]*B*((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)]*B*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)]*(A*b - a*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]*Sqrt[1 + ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a - 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])/Sqrt[1 + ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + (4*A*b*Sqrt[(a - b)/(a + 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)]*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)])))/(4*b*Sqrt[(a - b)/(a + b)]*Sqrt[a + b*Cos[c + d*x]])))","C",0
422,1,144,228,1.4805261,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*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((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)}{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 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 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}} \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]))]*((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)]))/(d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2])","A",1
423,1,299,230,13.0005085,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{2 \left(A \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 (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)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+A \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))+2 A (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 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}",1,"(2*(A*(a + b*Cos[c + d*x])*Sin[c + d*x] - (2*Sqrt[2]*(Cos[(c + d*x)/2]^2)^(3/2)*(2*A*(a + 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*(A + 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]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + A*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
424,1,416,290,15.7268756,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(5/2)*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 B \sin (c+d x)-2 A b \sin (c+d x))}{3 a^2}+\frac{2 A \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 A b-3 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 (a (A+3 B)-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)-2 (a+b) (3 a B-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)}} 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 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} (a (A-3 B)+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,"(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*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))*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*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*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","A",0
425,1,1319,363,6.4109341,"\int \frac{A+B \cos (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])/(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^2-10 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^2-10 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^3-10 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\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} \left(9 a^2 A-10 a b 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}} 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 A-5 B)-2 a b (A+5 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 - 5*a^3*B - 10*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 + 8*a*A*b^2 - 10*a^2*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*(9*a^2*A*b + 8*A*b^3 - 10*a*b^2*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^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^3) + (2*A*Sec[c + d*x]^2*Tan[c + d*x])/(5*a)))/d","C",1
426,1,1234,500,6.4176459,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 \sqrt{\cos (c+d x)} \left(a^2 B \sin (c+d x)-a A 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 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(2 a b B-2 A b^2\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 a^2-2 A b a-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)}{2 (a-b) b (a+b) d}","-\frac{\left(-3 a^2 B+2 a A b+b^2 B\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 (A b-a B) \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 B+2 a A b+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 b^2 d \sqrt{a+b}}-\frac{\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}} \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{(2 A b-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}} 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*A*b*Sin[c + d*x]) + a^2*B*Sin[c + d*x]))/(b*(-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*(-2*A*b^2 + 2*a*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*(-2*a*A*b + 3*a^2*B - b^2*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]])))/(2*(a - b)*b*(a + b)*d)","C",0
427,1,1012,416,17.9853322,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 \sqrt{\cos (c+d x)} (a B \sin (c+d x)-A b \sin (c+d x))}{\left(a^2-b^2\right) d \sqrt{a+b \cos (c+d x)}}-\frac{2 (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)-4 a (a A-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)}{(b-a) (a+b) d}","\frac{2 a (A b-a B) \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 (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 b d \sqrt{a+b}}-\frac{2 (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)}{a b d \sqrt{a+b}}-\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}} \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*Sin[c + d*x]) + a*B*Sin[c + d*x]))/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (-4*a*(a*A - 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*(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]])))/((-a + b)*(a + b)*d)","C",1
428,1,1223,284,6.3637534,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)),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(a^2 B-a A 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 B-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 \sqrt{\cos (c+d x)} \left(a b B \sin (c+d x)-A b^2 \sin (c+d x)\right)}{a \left(a^2-b^2\right) d \sqrt{a+b \cos (c+d x)}}","-\frac{2 (A b-a B) \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 (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)}{a^2 d \sqrt{a+b}}+\frac{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 \sqrt{a+b}}",1,"(-2*Sqrt[Cos[c + d*x]]*(-(A*b^2*Sin[c + d*x]) + a*b*B*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)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-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)*((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
429,1,1281,305,6.505593,"\int \frac{A+B \cos (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])/(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+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\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 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))}+\frac{2 A \tan (c+d x)}{a^2}\right)}{d}","\frac{2 b (A b-a B) \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-B)+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(a^2 A+a b 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}} 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)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-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)*((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*(a^2 - b^2)*(a + b*Cos[c + d*x])) + (2*A*Tan[c + d*x])/a^2))/d","C",1
430,1,1357,393,6.7056771,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{-\frac{4 a \left(A a^4-6 b B a^3+7 A b^2 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+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^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 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 A \sec (c+d x) \tan (c+d x)}{3 a^2}\right)}{d}","\frac{2 (a+2 b) (a (A-3 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)}{3 a^3 d \sqrt{a+b}}+\frac{2 \left(a^2 A+3 a b B-4 A b^2\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 (A b-a B) \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 B+5 a^2 A b+6 a b^2 B-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}} 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)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-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)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-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)*((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^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
431,1,1396,674,6.6966537,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*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(a^3 B \sin (c+d x)-a^2 A 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 B \sin (c+d x) a^4-3 A b \sin (c+d x) a^3-10 b^2 B \sin (c+d x) a^2+7 A b^3 \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 B a^4-2 A b a^3-8 b^2 B a^2+2 A b^3 a+3 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(6 A b^4-12 a B b^3+2 a^2 A b^2+4 a^3 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(15 B a^4-6 A b a^3-26 b^2 B a^2+14 A b^3 a+3 b^4 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 (a-b)^2 b^2 (a+b)^2 d}","\frac{2 a (A b-a B) \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 B+2 a^2 A b+9 a b^2 B-6 A b^3\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 B+6 a^2 A b-5 a^2 b B+2 a A b^2+21 a b^2 B-12 A b^3+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 b^3 d (a-b) (a+b)^{3/2}}-\frac{\left(-15 a^4 B+6 a^3 A b+26 a^2 b^2 B-14 a A b^3-3 b^4 B\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 B+6 a^3 A b+26 a^2 b^2 B-14 a A b^3-3 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 b^3 d (a-b) (a+b)^{3/2}}-\frac{\sqrt{a+b} (2 A b-5 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^2*A*b*Sin[c + d*x]) + a^3*B*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) - (2*(-3*a^3*A*b*Sin[c + d*x] + 7*a*A*b^3*Sin[c + d*x] + 6*a^4*B*Sin[c + d*x] - 10*a^2*b^2*B*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)^2*(a + b*Cos[c + d*x]))))/d + ((-4*a*(-2*a^3*A*b + 2*a*A*b^3 + 5*a^4*B - 8*a^2*b^2*B + 3*b^4*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*(2*a^2*A*b^2 + 6*A*b^4 + 4*a^3*b*B - 12*a*b^3*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*(-6*a^3*A*b + 14*a*A*b^3 + 15*a^4*B - 26*a^2*b^2*B + 3*b^4*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]])))/(6*(a - b)^2*b^2*(a + b)^2*d)","C",0
432,1,1342,545,6.5337171,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*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(a^2 B \sin (c+d x)-a A 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 B \sin (c+d x) a^3-7 b^2 B \sin (c+d x) a+4 A b^3 \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(B a^3-A b a^2-b^2 B a+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(-3 B b^3+4 a A b^2-a^2 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(3 B a^3-7 b^2 B a+4 A b^3\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 B-7 a b^2 B+4 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}} 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 (A b-a B) \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 B-7 a b^2 B+4 A b^3\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 B-a^2 b B+a A b^2+6 a b^2 B-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)}{3 a b^2 d (a-b) (a+b)^{3/2}}-\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}} \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*A*b*Sin[c + d*x]) + a^2*B*Sin[c + d*x]))/(3*b*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (2*(4*A*b^3*Sin[c + d*x] + 3*a^3*B*Sin[c + d*x] - 7*a*b^2*B*Sin[c + d*x]))/(3*b*(-a^2 + b^2)^2*(a + b*Cos[c + d*x]))))/d - ((-4*a*(-(a^2*A*b) + 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*(4*a*A*b^2 - a^2*b*B - 3*b^3*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^3 + 3*a^3*B - 7*a*b^2*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 - b)^2*b*(a + b)^2*d)","C",0
433,1,1335,391,6.4549097,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*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 B \sin (c+d x)-A b \sin (c+d x))}{3 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{2 \left(A \sin (c+d x) b^3-4 a B \sin (c+d x) b^2+3 a^2 A \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(B a^3-A b a^2-b^2 B a+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(3 A a^3-4 b B a^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(A b^3-4 a B b^2+3 a^2 A 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 A-4 a b B+A b^2\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 (A b-a B) \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 A-4 a b B+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^2 d (a-b) (a+b)^{3/2}}+\frac{2 (3 a A+a B-A b-3 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 d (a-b) (a+b)^{3/2}}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*(-(A*b*Sin[c + d*x]) + a*B*Sin[c + d*x]))/(3*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) - (2*(3*a^2*A*b*Sin[c + d*x] + A*b^3*Sin[c + d*x] - 4*a*b^2*B*Sin[c + d*x]))/(3*a*(a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/d + ((-4*a*(-(a^2*A*b) + 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*(3*a^3*A + a*A*b^2 - 4*a^2*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*(3*a^2*A*b + A*b^3 - 4*a*b^2*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*(a - b)^2*(a + b)^2*d)","C",1
434,1,1384,429,6.5793258,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(A + B*Cos[c + d*x])/(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(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 \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+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-b B a^3-5 A b^2 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+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+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 b (A b-a B) \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 (A+B)+a b (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)}{3 a^2 d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{2 \left(-3 a^3 B+6 a^2 A b-a b^2 B-2 A b^3\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 B+6 a^2 A b-a b^2 B-2 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}} 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]))/(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]))/(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)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-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)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-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)*((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
435,1,1431,456,6.7229444,"\int \frac{A+B \cos (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])/(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^2 B \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{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+6 a^3 B \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 A \tan (c+d x)}{a^3}\right)}{d}-\frac{-\frac{4 a \left(-3 B a^5+9 A b a^4+5 b^2 B a^3-17 A b^3 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+6 b B a^4-15 A b^2 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+6 a^3 B b^2+3 a^4 A 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 (A b-a B) \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 B+8 a^2 A b+a b^2 B-4 A b^3\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 (A-B)-3 a^2 b (3 A+B)+2 a b^2 (3 A-B)+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)}+\frac{2 \left(3 a^4 A+6 a^3 b B-15 a^2 A b^2-2 a b^3 B+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 (a-b) (a+b)^{3/2}}",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)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-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)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-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)*((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]))/(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^3*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) + (2*A*Tan[c + d*x])/a^3))/d","C",0
436,1,1499,567,6.9498844,"\int \frac{A+B \cos (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])/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(5/2)),x]","\frac{-\frac{4 a \left(A a^6-9 b B a^5+15 A b^2 a^4+17 b^3 B a^3-32 A b^4 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+15 b^2 B a^4-28 A b^3 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+15 a^3 B b^3+8 a^4 A 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 b^3 B \sin (c+d x)-A b^4 \sin (c+d x)\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+9 a^3 B \sin (c+d x) b^3\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 b (A b-a B) \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{2 b \left(-7 a^3 B+10 a^2 A b+3 a b^2 B-6 A b^3\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{2 \left(a^4 A+8 a^3 b B-13 a^2 A b^2-4 a b^3 B+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 B)\right)-9 a^3 b (A-B)-2 a^2 b^2 (8 A+3 B)+4 a b^3 (3 A-2 B)+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)}-\frac{2 \left(-3 a^5 B+8 a^4 A b+15 a^3 b^2 B-28 a^2 A b^3-8 a b^4 B+16 A b^5\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 (a-b) (a+b)^{3/2}}",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)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-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)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-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)*((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]))/(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]))/(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
437,1,480,419,1.4365399,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^(3/2),x]","\frac{B \sqrt{\cos (c+d x)} \left(2 a \sqrt{\frac{a-b}{a+b}} \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \tan \left(\frac{1}{2} (c+d x)\right)-b \sqrt{\frac{a-b}{a+b}} \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \tan \left(\frac{1}{2} (c+d x)\right)+b \sqrt{\frac{a-b}{a+b}} \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)-4 i a \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} 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)+2 i (a-b) \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} 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)+4 i a \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} \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)}{2 b d \sqrt{\frac{a-b}{a+b}} \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{a+b \cos (c+d x)}}","\frac{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}} \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{B \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a+b \cos (c+d x)}}+\frac{a B \sin (c+d x)}{b d \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{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)}{b d}-\frac{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 b d}",1,"(B*Sqrt[Cos[c + d*x]]*((2*I)*(a - b)*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)*a*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))] + (4*I)*a*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)]*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)]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Tan[(c + d*x)/2] - b*Sqrt[(a - b)/(a + b)]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Tan[(c + d*x)/2]))/(2*b*Sqrt[(a - b)/(a + b)]*d*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[a + b*Cos[c + d*x]])","C",1
438,1,131,117,0.1434371,"\int \frac{\sqrt{\cos (c+d x)} (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(a*B + b*B*Cos[c + d*x]))/(a + b*Cos[c + d*x])^(3/2),x]","-\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} \left(F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-2 \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{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \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}} \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*B*Sqrt[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*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]))/(d*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[a + b*Cos[c + d*x]])","A",1
439,1,171,110,0.8927209,"\int \frac{a B+b B \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(a*B + b*B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)),x]","-\frac{4 B (a+b) \cos ^{\frac{3}{2}}(c+d x) \csc (c+d x) \sqrt{-\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{a-b}} \sqrt{\frac{\csc ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))}{a}} F\left(\sin ^{-1}\left(\sqrt{-\frac{a+b \cos (c+d x)}{a (\cos (c+d x)-1)}}\right)|\frac{2 a}{a-b}\right)}{a d \sqrt{a+b \cos (c+d x)} \left(-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}\right)^{3/2}}","\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}",1,"(-4*(a + b)*B*Cos[c + d*x]^(3/2)*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]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[-((a + b*Cos[c + d*x])/(a*(-1 + Cos[c + d*x])))]], (2*a)/(a - b)])/(a*d*Sqrt[a + b*Cos[c + d*x]]*(-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a))^(3/2))","A",1
440,1,212,226,2.1861687,"\int \frac{a B+b B \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(a*B + b*B*Cos[c + d*x])/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{2 B \left(\tan \left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))+a \sqrt{\cos (c+d x)} \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)-\left((a+b) \sqrt{\cos (c+d x)} \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)\right)\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 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}",1,"(2*B*(-((a + b)*Sqrt[Cos[c + d*x]]*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)]) + a*Sqrt[Cos[c + d*x]]*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)] + (a + b*Cos[c + d*x])*Tan[(c + d*x)/2]))/(a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",1
441,0,0,72,35.2372893,"\int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{2+3 \cos (c+d x)}} \, dx","Integrate[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[2 + 3*Cos[c + d*x]]),x]","\int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{2+3 \cos (c+d x)}} \, dx","-\frac{\cot (c+d x) \sqrt{-\sec (c+d x)-1} \sqrt{1-\sec (c+d x)} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{3 \cos (c+d x)+2}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right)\right|5\right)}{d}",1,"Integrate[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[2 + 3*Cos[c + d*x]]), x]","F",-1
442,0,0,70,38.259766,"\int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{-2+3 \cos (c+d x)}} \, dx","Integrate[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[-2 + 3*Cos[c + d*x]]),x]","\int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{-2+3 \cos (c+d x)}} \, dx","-\frac{\sqrt{5} \cot (c+d x) \sqrt{\sec (c+d x)-1} \sqrt{\sec (c+d x)+1} E\left(\sin ^{-1}\left(\frac{\sqrt{3 \cos (c+d x)-2}}{\sqrt{\cos (c+d x)}}\right)|\frac{1}{5}\right)}{d}",1,"Integrate[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[-2 + 3*Cos[c + d*x]]), x]","F",-1
443,0,0,93,33.1854071,"\int \frac{1+\cos (c+d x)}{\sqrt{2-3 \cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(1 + Cos[c + d*x])/(Sqrt[2 - 3*Cos[c + d*x]]*Cos[c + d*x]^(3/2)),x]","\int \frac{1+\cos (c+d x)}{\sqrt{2-3 \cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)} \, dx","\frac{\sqrt{5} \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\sec (c+d x)-1} \sqrt{\sec (c+d x)+1} E\left(\sin ^{-1}\left(\frac{\sqrt{2-3 \cos (c+d x)}}{\sqrt{-\cos (c+d x)}}\right)|\frac{1}{5}\right)}{d}",1,"Integrate[(1 + Cos[c + d*x])/(Sqrt[2 - 3*Cos[c + d*x]]*Cos[c + d*x]^(3/2)), x]","F",-1
444,0,0,95,30.1584852,"\int \frac{1+\cos (c+d x)}{\sqrt{-2-3 \cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(1 + Cos[c + d*x])/(Sqrt[-2 - 3*Cos[c + d*x]]*Cos[c + d*x]^(3/2)),x]","\int \frac{1+\cos (c+d x)}{\sqrt{-2-3 \cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)} \, dx","\frac{\sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{-\sec (c+d x)-1} \sqrt{1-\sec (c+d x)} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{-3 \cos (c+d x)-2}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right)\right|5\right)}{d}",1,"Integrate[(1 + Cos[c + d*x])/(Sqrt[-2 - 3*Cos[c + d*x]]*Cos[c + d*x]^(3/2)), x]","F",-1
445,0,0,72,37.9870669,"\int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{3+2 \cos (c+d x)}} \, dx","Integrate[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[3 + 2*Cos[c + d*x]]),x]","\int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{3+2 \cos (c+d x)}} \, dx","\frac{2 \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)+3}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right)\right|-5\right)}{3 d}",1,"Integrate[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[3 + 2*Cos[c + d*x]]), x]","F",-1
446,0,0,74,38.3789022,"\int \frac{1+\cos (c+d x)}{\sqrt{3-2 \cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(1 + Cos[c + d*x])/(Sqrt[3 - 2*Cos[c + d*x]]*Cos[c + d*x]^(3/2)),x]","\int \frac{1+\cos (c+d x)}{\sqrt{3-2 \cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)} \, dx","\frac{2 \sqrt{5} \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} E\left(\sin ^{-1}\left(\frac{\sqrt{3-2 \cos (c+d x)}}{\sqrt{\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{3 d}",1,"Integrate[(1 + Cos[c + d*x])/(Sqrt[3 - 2*Cos[c + d*x]]*Cos[c + d*x]^(3/2)), x]","F",-1
447,0,0,98,41.0632844,"\int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{-3+2 \cos (c+d x)}} \, dx","Integrate[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[-3 + 2*Cos[c + d*x]]),x]","\int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{-3+2 \cos (c+d x)}} \, dx","-\frac{2 \sqrt{5} \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} E\left(\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)-3}}{\sqrt{-\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{3 d}",1,"Integrate[(1 + Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[-3 + 2*Cos[c + d*x]]), x]","F",-1
448,0,0,96,30.5785351,"\int \frac{1+\cos (c+d x)}{\sqrt{-3-2 \cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(1 + Cos[c + d*x])/(Sqrt[-3 - 2*Cos[c + d*x]]*Cos[c + d*x]^(3/2)),x]","\int \frac{1+\cos (c+d x)}{\sqrt{-3-2 \cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)} \, dx","-\frac{2 \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} E\left(\left.\sin ^{-1}\left(\frac{\sqrt{-2 \cos (c+d x)-3}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right)\right|-5\right)}{3 d}",1,"Integrate[(1 + Cos[c + d*x])/(Sqrt[-3 - 2*Cos[c + d*x]]*Cos[c + d*x]^(3/2)), x]","F",-1
449,0,0,36,7.983384,"\int (c \cos (e+f x))^m (a+b \cos (e+f x))^n (A+B \cos (e+f x)) \, dx","Integrate[(c*Cos[e + f*x])^m*(a + b*Cos[e + f*x])^n*(A + B*Cos[e + f*x]),x]","\int (c \cos (e+f x))^m (a+b \cos (e+f x))^n (A+B \cos (e+f x)) \, dx","\text{Int}\left((A+B \cos (e+f x)) (c \cos (e+f x))^m (a+b \cos (e+f x))^n,x\right)",0,"Integrate[(c*Cos[e + f*x])^m*(a + b*Cos[e + f*x])^n*(A + B*Cos[e + f*x]), x]","A",-1
450,1,487,595,6.2022958,"\int (c \cos (e+f x))^m (a+b \cos (e+f x))^4 (A+B \cos (e+f x)) \, dx","Integrate[(c*Cos[e + f*x])^m*(a + b*Cos[e + f*x])^4*(A + B*Cos[e + f*x]),x]","-\frac{a^4 A \sin (e+f x) \cos (e+f x) (c \cos (e+f x))^m \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right)}{f (m+1) \sqrt{\sin ^2(e+f x)}}-\frac{a^3 (a B+4 A b) \sin (e+f x) \cos ^2(e+f x) (c \cos (e+f x))^m \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(e+f x)\right)}{f (m+2) \sqrt{\sin ^2(e+f x)}}-\frac{2 a^2 b (2 a B+3 A b) \sin (e+f x) \cos ^3(e+f x) (c \cos (e+f x))^m \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\cos ^2(e+f x)\right)}{f (m+3) \sqrt{\sin ^2(e+f x)}}-\frac{b^3 (4 a B+A b) \sin (e+f x) \cos ^5(e+f x) (c \cos (e+f x))^m \, _2F_1\left(\frac{1}{2},\frac{m+5}{2};\frac{m+7}{2};\cos ^2(e+f x)\right)}{f (m+5) \sqrt{\sin ^2(e+f x)}}-\frac{2 a b^2 (3 a B+2 A b) \sin (e+f x) \cos ^4(e+f x) (c \cos (e+f x))^m \, _2F_1\left(\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};\cos ^2(e+f x)\right)}{f (m+4) \sqrt{\sin ^2(e+f x)}}-\frac{b^4 B \sin (e+f x) \cos ^6(e+f x) (c \cos (e+f x))^m \, _2F_1\left(\frac{1}{2},\frac{m+6}{2};\frac{m+8}{2};\cos ^2(e+f x)\right)}{f (m+6) \sqrt{\sin ^2(e+f x)}}","\frac{b^2 \sin (e+f x) \cos (e+f x) \left(a^2 B \left(m^2+11 m+36\right)+2 a A b (m+5)^2+b^2 B (m+4)^2\right) (c \cos (e+f x))^{m+1}}{c f (m+3) (m+4) (m+5)}+\frac{b \sin (e+f x) \left(2 a^3 B \left(m^2+10 m+28\right)+a^2 A b \left(5 m^2+47 m+110\right)+4 a b^2 B \left(m^2+8 m+15\right)+A b^3 \left(m^2+8 m+15\right)\right) (c \cos (e+f x))^{m+1}}{c f (m+2) (m+4) (m+5)}-\frac{\sin (e+f x) \left(a^4 B \left(m^2+8 m+15\right)+4 a^3 A b \left(m^2+8 m+15\right)+6 a^2 b^2 B \left(m^2+7 m+10\right)+4 a A b^3 \left(m^2+7 m+10\right)+b^4 B \left(m^2+6 m+8\right)\right) (c \cos (e+f x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(e+f x)\right)}{c^2 f (m+2) (m+3) (m+5) \sqrt{\sin ^2(e+f x)}}-\frac{\sin (e+f x) \left(a^4 A \left(m^2+6 m+8\right)+4 a^3 b B \left(m^2+5 m+4\right)+6 a^2 A b^2 \left(m^2+5 m+4\right)+4 a b^3 B \left(m^2+4 m+3\right)+A b^4 \left(m^2+4 m+3\right)\right) (c \cos (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right)}{c f (m+1) (m+2) (m+4) \sqrt{\sin ^2(e+f x)}}+\frac{b \sin (e+f x) (a B (m+8)+A b (m+5)) (a+b \cos (e+f x))^2 (c \cos (e+f x))^{m+1}}{c f (m+4) (m+5)}+\frac{b B \sin (e+f x) (a+b \cos (e+f x))^3 (c \cos (e+f x))^{m+1}}{c f (m+5)}",1,"-((a^4*A*Cos[e + f*x]*(c*Cos[e + f*x])^m*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(1 + m)*Sqrt[Sin[e + f*x]^2])) - (a^3*(4*A*b + a*B)*Cos[e + f*x]^2*(c*Cos[e + f*x])^m*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(2 + m)*Sqrt[Sin[e + f*x]^2]) - (2*a^2*b*(3*A*b + 2*a*B)*Cos[e + f*x]^3*(c*Cos[e + f*x])^m*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(3 + m)*Sqrt[Sin[e + f*x]^2]) - (2*a*b^2*(2*A*b + 3*a*B)*Cos[e + f*x]^4*(c*Cos[e + f*x])^m*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(4 + m)*Sqrt[Sin[e + f*x]^2]) - (b^3*(A*b + 4*a*B)*Cos[e + f*x]^5*(c*Cos[e + f*x])^m*Hypergeometric2F1[1/2, (5 + m)/2, (7 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(5 + m)*Sqrt[Sin[e + f*x]^2]) - (b^4*B*Cos[e + f*x]^6*(c*Cos[e + f*x])^m*Hypergeometric2F1[1/2, (6 + m)/2, (8 + m)/2, Cos[e + f*x]^2]*Sin[e + f*x])/(f*(6 + m)*Sqrt[Sin[e + f*x]^2])","A",1
451,1,269,406,2.8145703,"\int (c \cos (e+f x))^m (a+b \cos (e+f x))^3 (A+B \cos (e+f x)) \, dx","Integrate[(c*Cos[e + f*x])^m*(a + b*Cos[e + f*x])^3*(A + B*Cos[e + f*x]),x]","\frac{\sin (e+f x) \cos (e+f x) (c \cos (e+f x))^m \left(\cos (e+f x) \left(b \cos (e+f x) \left(b \cos (e+f x) \left(-\frac{(3 a B+A b) \, _2F_1\left(\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};\cos ^2(e+f x)\right)}{m+4}-\frac{b B \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{m+5}{2};\frac{m+7}{2};\cos ^2(e+f x)\right)}{m+5}\right)-\frac{3 a (a B+A b) \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\cos ^2(e+f x)\right)}{m+3}\right)-\frac{a^2 (a B+3 A b) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(e+f x)\right)}{m+2}\right)-\frac{a^3 A \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right)}{m+1}\right)}{f \sqrt{\sin ^2(e+f x)}}","-\frac{\sin (e+f x) \left(b (m+1) \left(2 a^2 B (m+5)+3 a A b (m+4)+b^2 B (m+3)\right)+a^2 (m+2) (a A (m+4)+b B (m+1))\right) (c \cos (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right)}{c f (m+1) (m+2) (m+4) \sqrt{\sin ^2(e+f x)}}+\frac{b \sin (e+f x) \left(2 a^2 B (m+5)+3 a A b (m+4)+b^2 B (m+3)\right) (c \cos (e+f x))^{m+1}}{c f (m+2) (m+4)}-\frac{\sin (e+f x) \left(a^3 B (m+3)+3 a^2 A b (m+3)+3 a b^2 B (m+2)+A b^3 (m+2)\right) (c \cos (e+f x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(e+f x)\right)}{c^2 f (m+2) (m+3) \sqrt{\sin ^2(e+f x)}}+\frac{b^2 \sin (e+f x) \cos (e+f x) (a B (m+6)+A b (m+4)) (c \cos (e+f x))^{m+1}}{c f (m+3) (m+4)}+\frac{b B \sin (e+f x) (a+b \cos (e+f x))^2 (c \cos (e+f x))^{m+1}}{c f (m+4)}",1,"(Cos[e + f*x]*(c*Cos[e + f*x])^m*(-((a^3*A*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2])/(1 + m)) + Cos[e + f*x]*(-((a^2*(3*A*b + a*B)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[e + f*x]^2])/(2 + m)) + b*Cos[e + f*x]*((-3*a*(A*b + a*B)*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, Cos[e + f*x]^2])/(3 + m) + b*Cos[e + f*x]*(-(((A*b + 3*a*B)*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, Cos[e + f*x]^2])/(4 + m)) - (b*B*Cos[e + f*x]*Hypergeometric2F1[1/2, (5 + m)/2, (7 + m)/2, Cos[e + f*x]^2])/(5 + m)))))*Sin[e + f*x])/(f*Sqrt[Sin[e + f*x]^2])","A",1
452,1,217,287,1.7041327,"\int (c \cos (e+f x))^m (a+b \cos (e+f x))^2 (A+B \cos (e+f x)) \, dx","Integrate[(c*Cos[e + f*x])^m*(a + b*Cos[e + f*x])^2*(A + B*Cos[e + f*x]),x]","\frac{\sin (e+f x) \cos (e+f x) (c \cos (e+f x))^m \left(\cos (e+f x) \left(b \cos (e+f x) \left(-\frac{(2 a B+A b) \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\cos ^2(e+f x)\right)}{m+3}-\frac{b B \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};\cos ^2(e+f x)\right)}{m+4}\right)-\frac{a (a B+2 A b) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(e+f 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(e+f x)\right)}{m+1}\right)}{f \sqrt{\sin ^2(e+f x)}}","-\frac{\sin (e+f x) \left(a^2 A (m+2)+2 a b B (m+1)+A b^2 (m+1)\right) (c \cos (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right)}{c f (m+1) (m+2) \sqrt{\sin ^2(e+f x)}}-\frac{\sin (e+f x) \left(a (m+3) (a B+2 A b)+b^2 B (m+2)\right) (c \cos (e+f x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(e+f x)\right)}{c^2 f (m+2) (m+3) \sqrt{\sin ^2(e+f x)}}+\frac{b \sin (e+f x) (a B (m+4)+A b (m+3)) (c \cos (e+f x))^{m+1}}{c f (m+2) (m+3)}+\frac{b B \sin (e+f x) (a+b \cos (e+f x)) (c \cos (e+f x))^{m+1}}{c f (m+3)}",1,"(Cos[e + f*x]*(c*Cos[e + f*x])^m*(-((a^2*A*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2])/(1 + m)) + Cos[e + f*x]*(-((a*(2*A*b + a*B)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[e + f*x]^2])/(2 + m)) + b*Cos[e + f*x]*(-(((A*b + 2*a*B)*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, Cos[e + f*x]^2])/(3 + m)) - (b*B*Cos[e + f*x]*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, Cos[e + f*x]^2])/(4 + m))))*Sin[e + f*x])/(f*Sqrt[Sin[e + f*x]^2])","A",1
453,1,151,196,0.338701,"\int (c \cos (e+f x))^m (a+b \cos (e+f x)) (A+B \cos (e+f x)) \, dx","Integrate[(c*Cos[e + f*x])^m*(a + b*Cos[e + f*x])*(A + B*Cos[e + f*x]),x]","-\frac{\sin (e+f x) \cos (e+f x) (c \cos (e+f x))^m \left((a A (m+2)+b B (m+1)) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right)+(m+1) \left((a B+A b) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(e+f x)\right)-b B \sqrt{\sin ^2(e+f x)}\right)\right)}{f (m+1) (m+2) \sqrt{\sin ^2(e+f x)}}","-\frac{(a B+A b) \sin (e+f x) (c \cos (e+f x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(e+f x)\right)}{c^2 f (m+2) \sqrt{\sin ^2(e+f x)}}-\frac{\sin (e+f x) (a A (m+2)+b B (m+1)) (c \cos (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right)}{c f (m+1) (m+2) \sqrt{\sin ^2(e+f x)}}+\frac{b B \sin (e+f x) (c \cos (e+f x))^{m+1}}{c f (m+2)}",1,"-((Cos[e + f*x]*(c*Cos[e + f*x])^m*Sin[e + f*x]*((b*B*(1 + m) + a*A*(2 + m))*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2] + (1 + m)*((A*b + a*B)*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[e + f*x]^2] - b*B*Sqrt[Sin[e + f*x]^2])))/(f*(1 + m)*(2 + m)*Sqrt[Sin[e + f*x]^2]))","A",1
454,1,10482,286,26.9408816,"\int \frac{(c \cos (e+f x))^m (A+B \cos (e+f x))}{a+b \cos (e+f x)} \, dx","Integrate[((c*Cos[e + f*x])^m*(A + B*Cos[e + f*x]))/(a + b*Cos[e + f*x]),x]","\text{Result too large to show}","\frac{a c (A b-a B) \sin (e+f x) \cos ^2(e+f x)^{\frac{1-m}{2}} (c \cos (e+f x))^{m-1} F_1\left(\frac{1}{2};\frac{1-m}{2},1;\frac{3}{2};\sin ^2(e+f x),-\frac{b^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{b f \left(a^2-b^2\right)}-\frac{(A b-a B) \sin (e+f x) \cos ^2(e+f x)^{-m/2} (c \cos (e+f x))^m F_1\left(\frac{1}{2};-\frac{m}{2},1;\frac{3}{2};\sin ^2(e+f x),-\frac{b^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}-\frac{B \sin (e+f x) (c \cos (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right)}{b c f (m+1) \sqrt{\sin ^2(e+f x)}}",1,"Result too large to show","B",0
455,0,0,181,66.7526826,"\int (c \cos (e+f x))^m (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) \, dx","Integrate[(c*Cos[e + f*x])^m*(a + b*Cos[e + f*x])^(3/2)*(A + B*Cos[e + f*x]),x]","\int (c \cos (e+f x))^m (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) \, dx","\frac{2 \text{Int}\left(\frac{(c \cos (e+f x))^m \left(\frac{1}{2} c \cos (e+f x) \left(a (2 m+5) (a B+2 A b)+b^2 B (2 m+3)\right)+\frac{1}{2} b c \cos ^2(e+f x) (2 a B (m+3)+A b (2 m+5))+\frac{1}{2} a c \left(2 a A \left(m+\frac{5}{2}\right)+2 b B (m+1)\right)\right)}{\sqrt{a+b \cos (e+f x)}},x\right)}{c (2 m+5)}+\frac{2 b B \sin (e+f x) \sqrt{a+b \cos (e+f x)} (c \cos (e+f x))^{m+1}}{c f (2 m+5)}",0,"Integrate[(c*Cos[e + f*x])^m*(a + b*Cos[e + f*x])^(3/2)*(A + B*Cos[e + f*x]), x]","A",-1
456,0,0,38,9.666828,"\int (c \cos (e+f x))^m \sqrt{a+b \cos (e+f x)} (A+B \cos (e+f x)) \, dx","Integrate[(c*Cos[e + f*x])^m*Sqrt[a + b*Cos[e + f*x]]*(A + B*Cos[e + f*x]),x]","\int (c \cos (e+f x))^m \sqrt{a+b \cos (e+f x)} (A+B \cos (e+f x)) \, dx","\text{Int}\left(\sqrt{a+b \cos (e+f x)} (A+B \cos (e+f x)) (c \cos (e+f x))^m,x\right)",0,"Integrate[(c*Cos[e + f*x])^m*Sqrt[a + b*Cos[e + f*x]]*(A + B*Cos[e + f*x]), x]","A",-1
457,0,0,38,8.0973048,"\int \frac{(c \cos (e+f x))^m (A+B \cos (e+f x))}{\sqrt{a+b \cos (e+f x)}} \, dx","Integrate[((c*Cos[e + f*x])^m*(A + B*Cos[e + f*x]))/Sqrt[a + b*Cos[e + f*x]],x]","\int \frac{(c \cos (e+f x))^m (A+B \cos (e+f x))}{\sqrt{a+b \cos (e+f x)}} \, dx","\text{Int}\left(\frac{(A+B \cos (e+f x)) (c \cos (e+f x))^m}{\sqrt{a+b \cos (e+f x)}},x\right)",0,"Integrate[((c*Cos[e + f*x])^m*(A + B*Cos[e + f*x]))/Sqrt[a + b*Cos[e + f*x]], x]","A",-1
458,0,0,191,10.7196546,"\int \frac{(c \cos (e+f x))^m (A+B \cos (e+f x))}{(a+b \cos (e+f x))^{3/2}} \, dx","Integrate[((c*Cos[e + f*x])^m*(A + B*Cos[e + f*x]))/(a + b*Cos[e + f*x])^(3/2),x]","\int \frac{(c \cos (e+f x))^m (A+B \cos (e+f x))}{(a+b \cos (e+f x))^{3/2}} \, dx","\frac{2 \text{Int}\left(\frac{(c \cos (e+f x))^m \left(-\frac{1}{2} b c (2 m+3) (A b-a B) \cos ^2(e+f x)-\frac{1}{2} a c (A b-a B) \cos (e+f x)+\frac{1}{2} c \left(2 b \left(m+\frac{1}{2}\right) (A b-a B)+a (a A-b B)\right)\right)}{\sqrt{a+b \cos (e+f x)}},x\right)}{a c \left(a^2-b^2\right)}+\frac{2 b (A b-a B) \sin (e+f x) (c \cos (e+f x))^{m+1}}{a c f \left(a^2-b^2\right) \sqrt{a+b \cos (e+f x)}}",0,"Integrate[((c*Cos[e + f*x])^m*(A + B*Cos[e + f*x]))/(a + b*Cos[e + f*x])^(3/2), x]","A",-1
459,1,292,172,1.9315513,"\int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*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 B) 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+B) \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 B e^{i (c+d x)}-30 B e^{3 i (c+d x)}-5 B e^{4 i (c+d x)}-15 B e^{5 i (c+d x)}+5 B\right)}{30 d \left(1+e^{2 i (c+d x)}\right)^2}","\frac{2 a (A+B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a (3 A+5 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a (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 d}-\frac{2 a (3 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 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}",1,"(a*(-1 + E^((2*I)*c))*(1 + Cos[c + d*x])*Csc[c]*(5*A + 5*B - 3*A*E^(I*(c + d*x)) - 15*B*E^(I*(c + d*x)) - 24*A*E^((3*I)*(c + d*x)) - 30*B*E^((3*I)*(c + d*x)) - 5*A*E^((4*I)*(c + d*x)) - 5*B*E^((4*I)*(c + d*x)) - 9*A*E^((5*I)*(c + d*x)) - 15*B*E^((5*I)*(c + d*x)) - (5*I)*(A + B)*(1 + E^((2*I)*(c + d*x)))^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (3*A + 5*B)*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
460,1,225,135,1.2294495,"\int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2),x]","\frac{a (\cos (c+d x)+1) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(i \left((A+B) e^{i (c+d x)} \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)-3 A e^{i (c+d x)}-A e^{2 i (c+d x)}-3 A e^{3 i (c+d x)}+A-3 B e^{i (c+d x)}-3 B e^{3 i (c+d x)}\right)+(A+3 B) \left(1+e^{2 i (c+d x)}\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d \left(1+e^{2 i (c+d x)}\right)}","\frac{2 a (A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a (A+3 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 a (A+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 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}",1,"(a*(1 + Cos[c + d*x])*((A + 3*B)*(1 + E^((2*I)*(c + d*x)))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + I*(A - 3*A*E^(I*(c + d*x)) - 3*B*E^(I*(c + d*x)) - A*E^((2*I)*(c + d*x)) - 3*A*E^((3*I)*(c + d*x)) - 3*B*E^((3*I)*(c + d*x)) + (A + B)*E^(I*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(3/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))*Sec[(c + d*x)/2]^2*Sqrt[Sec[c + d*x]])/(3*d*(1 + E^((2*I)*(c + d*x))))","C",1
461,1,157,106,1.0587976,"\int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2),x]","\frac{2 a e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(i (A-B) 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)+3 (A+B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 A \sin (c+d x)-3 i A \cos (c+d x)+3 i B \cos (c+d x)\right)}{3 d}","\frac{2 a (A+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 a (A-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 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}",1,"(2*a*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*((-3*I)*A*Cos[c + d*x] + (3*I)*B*Cos[c + d*x] + 3*(A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + I*(A - B)*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))] + 3*A*Sin[c + d*x]))/(3*d*E^(I*d*x))","C",1
462,1,148,110,1.2740537,"\int (a+a \cos (c+d x)) (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]],x]","\frac{2 a e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-i (A+B) 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) (B \sin (c+d x)+3 i (A+B))+(3 A+B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d}","\frac{2 a (3 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 d}+\frac{2 a (A+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 a B \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(2*a*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*((3*A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - I*(A + B)*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]*((3*I)*(A + B) + B*Sin[c + d*x])))/(3*d*E^(I*d*x))","C",1
463,1,148,141,1.5968,"\int \frac{(a+a \cos (c+d x)) (A+B \cos (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{a \sqrt{\sec (c+d x)} \left(-2 i (5 A+3 B) 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 (A+B) \sin (c+d x)+6 i (5 A+3 B)+3 B \sin (2 (c+d x)))+10 (A+B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d}","\frac{2 a (A+B) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a (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 d}+\frac{2 a (5 A+3 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 B \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a*Sqrt[Sec[c + d*x]]*(10*(A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (2*I)*(5*A + 3*B)*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*B) + 10*(A + B)*Sin[c + d*x] + 3*B*Sin[2*(c + d*x)])))/(15*d)","C",1
464,1,182,172,2.1810014,"\int \frac{(a+a \cos (c+d x)) (A+B \cos (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])*(A + B*Cos[c + d*x]))/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(-84 i (A+B) 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) (42 (A+B) \sin (2 (c+d x))+5 (28 A+23 B) \sin (c+d x)+252 i (A+B)+15 B \sin (3 (c+d x)))+20 (7 A+5 B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 a (A+B) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (7 A+5 B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (7 A+5 B) \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{6 a (A+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 B \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*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (84*I)*(A + B)*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]*((252*I)*(A + B) + 5*(28*A + 23*B)*Sin[c + d*x] + 42*(A + B)*Sin[2*(c + d*x)] + 15*B*Sin[3*(c + d*x)])))/(210*d*E^(I*d*x))","C",1
465,1,299,199,2.9910757,"\int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2),x]","\frac{a^2 e^{-i c} \left(-1+e^{2 i c}\right) \csc (c) (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(2 (4 A+5 B) 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)-10 i (A+2 B) \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)-18 A e^{i (c+d x)}-54 A e^{3 i (c+d x)}-10 A e^{4 i (c+d x)}-24 A e^{5 i (c+d x)}+10 A-30 B e^{i (c+d x)}-60 B e^{3 i (c+d x)}-5 B e^{4 i (c+d x)}-30 B e^{5 i (c+d x)}+5 B\right)}{60 d \left(1+e^{2 i (c+d x)}\right)^2}","\frac{2 a^2 (7 A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{4 a^2 (4 A+5 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^2 (A+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{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{3}{2}}(c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{5 d}",1,"(a^2*(-1 + E^((2*I)*c))*(1 + Cos[c + d*x])^2*Csc[c]*(10*A + 5*B - 18*A*E^(I*(c + d*x)) - 30*B*E^(I*(c + d*x)) - 54*A*E^((3*I)*(c + d*x)) - 60*B*E^((3*I)*(c + d*x)) - 10*A*E^((4*I)*(c + d*x)) - 5*B*E^((4*I)*(c + d*x)) - 24*A*E^((5*I)*(c + d*x)) - 30*B*E^((5*I)*(c + d*x)) - (10*I)*(A + 2*B)*(1 + E^((2*I)*(c + d*x)))^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(4*A + 5*B)*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]^4*Sqrt[Sec[c + d*x]])/(60*d*E^(I*c)*(1 + E^((2*I)*(c + d*x)))^2)","C",1
466,1,279,160,2.279283,"\int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2),x]","\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(\sqrt{\sec (c+d x)} (3 \csc (c) \cos (d x) (4 A-B \cos (2 c)+B)+2 A \tan (c+d x)+6 B \cos (c) \sin (d x))-\frac{4 i \sqrt{2} 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(e^{i d x} \left(\left(-1+e^{2 i c}\right) (2 A+3 B) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+A e^{i (c+d x)} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)\right)+3 A e^{i c} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{12 d}","\frac{2 a^2 (5 A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{4 a^2 (2 A+3 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 A \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 \sec (c+d x)+a^2\right)}{3 d}-\frac{4 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)}{d}",1,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(((-4*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(3*A*E^(I*c)*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + E^(I*d*x)*((2*A + 3*B)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))] + A*E^(I*(c + d*x))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])))/(E^(I*d*x)*(-1 + E^((2*I)*c))) + Sqrt[Sec[c + d*x]]*(3*(4*A + B - B*Cos[2*c])*Cos[d*x]*Csc[c] + 6*B*Cos[c]*Sin[d*x] + 2*A*Tan[c + d*x])))/(12*d)","C",1
467,1,302,160,1.868477,"\int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2),x]","\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(\sqrt{\sec (c+d x)} (6 (A+2 B) \cos (c) \sin (d x)-3 \csc (c) \cos (d x) ((A+2 B) \cos (2 c)-A+2 B)+B \sin (2 c) \cos (2 d x)+B \cos (2 c) \sin (2 d x))+\frac{4 i \sqrt{2} 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(e^{i d x} \left(B e^{i (c+d x)} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-\left(-1+e^{2 i c}\right) (3 A+2 B) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)\right)+3 B e^{i c} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{12 d}","\frac{2 a^2 (3 A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{4 a^2 (3 A+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 B \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{3 d \sqrt{\sec (c+d x)}}+\frac{4 a^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,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(((4*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(3*B*E^(I*c)*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + E^(I*d*x)*(-((3*A + 2*B)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]) + B*E^(I*(c + d*x))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])))/(E^(I*d*x)*(-1 + E^((2*I)*c))) + Sqrt[Sec[c + d*x]]*(-3*(-A + 2*B + (A + 2*B)*Cos[2*c])*Cos[d*x]*Csc[c] + B*Cos[2*d*x]*Sin[2*c] + 6*(A + 2*B)*Cos[c]*Sin[d*x] + B*Cos[2*c]*Sin[2*d*x])))/(12*d)","C",1
468,1,153,166,1.6253093,"\int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]],x]","\frac{a^2 \sqrt{\sec (c+d x)} \left(-4 i (5 A+4 B) 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 (A+2 B) \sin (c+d x)+60 i A+3 B \sin (2 (c+d x))+48 i B)+20 (2 A+B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d}","\frac{2 a^2 (5 A+7 B) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (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 d}+\frac{4 a^2 (5 A+4 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 B \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a^2*Sqrt[Sec[c + d*x]]*(20*(2*A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (4*I)*(5*A + 4*B)*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]*((60*I)*A + (48*I)*B + 10*(A + 2*B)*Sin[c + d*x] + 3*B*Sin[2*(c + d*x)])))/(15*d)","C",1
469,1,193,201,2.272477,"\int \frac{(a+a \cos (c+d x))^2 (A+B \cos (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/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 (4 A+3 B) 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 (56 A+51 B) \sin (c+d x)+42 (A+2 B) \sin (2 (c+d x))+672 i A+15 B \sin (3 (c+d x))+504 i B)+40 (7 A+6 B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 a^2 (7 A+9 B) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (7 A+6 B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (7 A+6 B) \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 (4 A+3 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 B \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a^2*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(40*(7*A + 6*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (56*I)*(4*A + 3*B)*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]*((672*I)*A + (504*I)*B + 5*(56*A + 51*B)*Sin[c + d*x] + 42*(A + 2*B)*Sin[2*(c + d*x)] + 15*B*Sin[3*(c + d*x)])))/(210*d*E^(I*d*x))","C",1
470,1,435,244,4.1584422,"\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2),x]","\frac{a^3 \csc (c) e^{-i d x} (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(7 \sqrt{2} \left(-1+e^{2 i c}\right) (7 A+9 B) 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(10 i (13 A+21 B) \left(1+e^{2 i (c+d x)}\right)^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 A \left(84 e^{i (c+d x)}-95 e^{2 i (c+d x)}+441 e^{3 i (c+d x)}+95 e^{4 i (c+d x)}+504 e^{5 i (c+d x)}+65 e^{6 i (c+d x)}+147 e^{7 i (c+d x)}-65\right)+21 B \left(16 e^{i (c+d x)}-5 e^{2 i (c+d x)}+54 e^{3 i (c+d x)}+5 e^{4 i (c+d x)}+56 e^{5 i (c+d x)}+5 e^{6 i (c+d x)}+18 e^{7 i (c+d x)}-5\right)\right)}{2 \left(1+e^{2 i (c+d x)}\right)^3}\right)}{420 d}","\frac{4 a^3 (41 A+42 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{2 (11 A+7 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{35 d}+\frac{4 a^3 (7 A+9 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^3 (13 A+21 B) \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) \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{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}{7 d}",1,"(a^3*(1 + Cos[c + d*x])^3*Csc[c]*Sec[(c + d*x)/2]^6*(7*Sqrt[2]*(7*A + 9*B)*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))*(21*B*(-5 + 16*E^(I*(c + d*x)) - 5*E^((2*I)*(c + d*x)) + 54*E^((3*I)*(c + d*x)) + 5*E^((4*I)*(c + d*x)) + 56*E^((5*I)*(c + d*x)) + 5*E^((6*I)*(c + d*x)) + 18*E^((7*I)*(c + d*x))) + 2*A*(-65 + 84*E^(I*(c + d*x)) - 95*E^((2*I)*(c + d*x)) + 441*E^((3*I)*(c + d*x)) + 95*E^((4*I)*(c + d*x)) + 504*E^((5*I)*(c + d*x)) + 65*E^((6*I)*(c + d*x)) + 147*E^((7*I)*(c + d*x))) + (10*I)*(13*A + 21*B)*(1 + E^((2*I)*(c + d*x)))^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])*Sqrt[Sec[c + d*x]])/(2*E^(I*(c - d*x))*(1 + E^((2*I)*(c + d*x)))^3)))/(420*d*E^(I*d*x))","C",1
471,1,268,211,3.1683444,"\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*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(2 \left(-1+e^{4 i c}\right) (9 A+5 B) 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(-18 i (9 A+5 B) \cos (c+d x)+40 (3 A+5 B) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+66 A \sin (c+d x)+30 A \sin (2 (c+d x))+54 A \sin (3 (c+d x))-54 i A \cos (3 (c+d x))+45 B \sin (c+d x)+10 B \sin (2 (c+d x))+45 B \sin (3 (c+d x))-30 i B \cos (3 (c+d x))\right)\right)}{30 d}","\frac{4 a^3 (21 A+20 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 (9 A+5 B) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}{15 d}+\frac{4 a^3 (3 A+5 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{4 a^3 (9 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 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}{5 d}",1,"(a^3*Csc[c]*Sec[c]*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*((2*(9*A + 5*B)*(-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]*((-18*I)*(9*A + 5*B)*Cos[c + d*x] - (54*I)*A*Cos[3*(c + d*x)] - (30*I)*B*Cos[3*(c + d*x)] + 40*(3*A + 5*B)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 66*A*Sin[c + d*x] + 45*B*Sin[c + d*x] + 30*A*Sin[2*(c + d*x)] + 10*B*Sin[2*(c + d*x)] + 54*A*Sin[3*(c + d*x)] + 45*B*Sin[3*(c + d*x)]))/2))/(30*d*E^(I*d*x))","C",1
472,1,202,199,1.9398879,"\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*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(4 i (A-B) \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)+40 (A+B) \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)+18 A \sin (2 (c+d x))-12 i A \cos (2 (c+d x))-12 i A+B \sin (c+d x)+6 B \sin (2 (c+d x))+B \sin (3 (c+d x))+12 i B \cos (2 (c+d x))+12 i B\right)}{6 d}","\frac{4 a^3 (4 A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 (A-B) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}{3 d}+\frac{20 a^3 (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 d}-\frac{4 a^3 (A-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 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{3 d \sqrt{\sec (c+d x)}}",1,"(a^3*Sec[c + d*x]^(3/2)*(Cos[d*x] + I*Sin[d*x])*((-12*I)*A + (12*I)*B - (12*I)*A*Cos[2*(c + d*x)] + (12*I)*B*Cos[2*(c + d*x)] + 40*(A + B)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + (4*I)*(A - B)*(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] + B*Sin[c + d*x] + 18*A*Sin[2*(c + d*x)] + 6*B*Sin[2*(c + d*x)] + B*Sin[3*(c + d*x)]))/(6*d*E^(I*d*x))","C",1
473,1,207,211,1.6402523,"\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*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(-8 i (5 A+9 B) 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)+40 (5 A+3 B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+60 A \sin (c+d x)+10 A \sin (2 (c+d x))+120 i A \cos (c+d x)+3 B \sin (c+d x)+30 B \sin (2 (c+d x))+3 B \sin (3 (c+d x))+216 i B \cos (c+d x)\right)}{30 d}","\frac{4 a^3 (5 A-6 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 (5 A+9 B) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{15 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (5 A+3 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{4 a^3 (5 A+9 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 B \sin (c+d x) (a \sec (c+d x)+a)^2}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a^3*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*((120*I)*A*Cos[c + d*x] + (216*I)*B*Cos[c + d*x] + 40*(5*A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (8*I)*(5*A + 9*B)*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))] + 60*A*Sin[c + d*x] + 3*B*Sin[c + d*x] + 10*A*Sin[2*(c + d*x)] + 30*B*Sin[2*(c + d*x)] + 3*B*Sin[3*(c + d*x)]))/(30*d*E^(I*d*x))","C",1
474,1,194,211,2.4080986,"\int (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*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(-56 i (9 A+7 B) 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 (84 A+107 B) \sin (c+d x)+42 (A+3 B) \sin (2 (c+d x))+168 i (9 A+7 B)+15 B \sin (3 (c+d x)))+40 (21 A+13 B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 (7 A+11 B) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (42 A+41 B) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (21 A+13 B) \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 (9 A+7 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 B \sin (c+d x) (a \sec (c+d x)+a)^2}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a^3*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(40*(21*A + 13*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (56*I)*(9*A + 7*B)*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]*((168*I)*(9*A + 7*B) + 5*(84*A + 107*B)*Sin[c + d*x] + 42*(A + 3*B)*Sin[2*(c + d*x)] + 15*B*Sin[3*(c + d*x)])))/(210*d*E^(I*d*x))","C",1
475,1,196,244,2.7463279,"\int \frac{(a+a \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{a^3 \sqrt{\sec (c+d x)} \left(-112 i (21 A+17 B) 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 (107 A+97 B) \sin (c+d x)+14 (54 A+73 B) \sin (2 (c+d x))+90 A \sin (3 (c+d x))+7056 i A+270 B \sin (3 (c+d x))+35 B \sin (4 (c+d x))+5712 i B)+240 (13 A+11 B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{1260 d}","\frac{4 a^3 (24 A+23 B) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (9 A+13 B) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (13 A+11 B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (13 A+11 B) \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 (21 A+17 B) \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 \sin (c+d x) (a \sec (c+d x)+a)^2}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(a^3*Sqrt[Sec[c + d*x]]*(240*(13*A + 11*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (112*I)*(21*A + 17*B)*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]*((7056*I)*A + (5712*I)*B + 30*(107*A + 97*B)*Sin[c + d*x] + 14*(54*A + 73*B)*Sin[2*(c + d*x)] + 90*A*Sin[3*(c + d*x)] + 270*B*Sin[3*(c + d*x)] + 35*B*Sin[4*(c + d*x)])))/(1260*d)","C",1
476,1,650,193,7.3190767,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x])*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)-B \sin \left(\frac{d x}{2}\right)\right)}{d}-\frac{3 (A-B) \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 B \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{B \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{B \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{(A-B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{d (a \sec (c+d x)+a)}+\frac{(5 A-3 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}-\frac{3 (A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}+\frac{(5 A-3 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 d}+\frac{3 (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 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]))) + (B*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])) - (B*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 - B)*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] - B*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*B*Cos[c])*Sec[c]*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])","C",1
477,1,400,159,4.3826369,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x])*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 (B-A) \tan \left(\frac{1}{2} (c+d x)\right)+2 (3 A-B) \csc (c) \cos (d x)\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} B \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 B \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-B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d (a \sec (c+d x)+a)}+\frac{(3 A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{(A-B) \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) \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]*B*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*B*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 - B)*Cos[d*x]*Csc[c] + 2*(-A + B)*Tan[(c + d*x)/2])))/(6*a*d*(1 + Cos[c + d*x]))","C",1
478,1,200,123,1.1233797,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{a+a \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x]),x]","-\frac{\left(-1+e^{2 i c}\right) e^{-\frac{1}{2} i (4 c+d x)} \left(\csc \left(\frac{c}{2}\right)+i \sec \left(\frac{c}{2}\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left((A-B) \left(e^{i (c+d x)} \left(1+e^{i (c+d x)}\right) \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)-3 \left(1+e^{2 i (c+d x)}\right)\right)+3 i (A+B) \left(1+e^{i (c+d x)}\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{24 a d}","-\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \sec (c+d x)+a)}+\frac{(A+B) \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) \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,"-1/24*((-1 + E^((2*I)*c))*((3*I)*(A + B)*(1 + E^(I*(c + d*x)))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (A - B)*(-3*(1 + E^((2*I)*(c + d*x))) + E^(I*(c + d*x))*(1 + 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))]))*(Csc[c/2] + I*Sec[c/2])*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]])/(a*d*E^((I/2)*(4*c + d*x)))","C",1
479,1,422,125,2.5850874,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x)) \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/((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{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 B) \cos \left(\frac{1}{2} (c-d x)\right)-B \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} B \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 B \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-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \sec (c+d x)+a)}+\frac{(A-B) \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 B) \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]*B*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*B)*Cos[(c - d*x)/2] - B*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*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]))/(6*a*d*(1 + Cos[c + d*x]))","C",1
480,1,444,163,4.8843291,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x])/((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{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right) \left((12 A-13 B) \cos \left(\frac{1}{2} (c-d x)\right)+(6 A-5 B) \cos \left(\frac{1}{2} (3 c+d x)\right)-2 B \sin (c) \sin \left(\frac{3}{2} (c+d x)\right)\right)}{\sqrt{\sec (c+d x)}}-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)+6 \sqrt{2} B \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 B \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 B) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}+\frac{(A-B) \sin (c+d x)}{d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)}-\frac{(3 A-5 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 d}+\frac{3 (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 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) + (6*Sqrt[2]*B*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*B*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]*((12*A - 13*B)*Cos[(c - d*x)/2] + (6*A - 5*B)*Cos[(3*c + d*x)/2] - 2*B*Sin[c]*Sin[(3*(c + d*x))/2]))/Sqrt[Sec[c + d*x]]))/(6*a*d*(1 + Cos[c + d*x]))","C",1
481,1,518,196,3.2462165,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x])/((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(40 (A-B) \sin (2 c) \cos (2 d x)-12 (20 A-33 B) \cos (c) \sin (d x)+40 (A-B) \cos (2 c) \sin (2 d x)-120 (A-B) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)+3 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos (d x) ((20 A-33 B) \cos (2 c)+40 A-51 B)-120 (A-B) \tan \left(\frac{c}{2}\right)+12 B \sin (3 c) \cos (3 d x)+12 B \cos (3 c) \sin (3 d x)\right)+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)+200 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} B \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 B \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{(A-B) \sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)}-\frac{(5 A-7 B) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{5 (A-B) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}+\frac{5 (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 d}-\frac{3 (5 A-7 B) \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*((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]*B*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) + 200*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] - 200*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] + Sqrt[Sec[c + d*x]]*(3*(40*A - 51*B + (20*A - 33*B)*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2] + 40*(A - B)*Cos[2*d*x]*Sin[2*c] + 12*B*Cos[3*d*x]*Sin[3*c] - 120*(A - B)*Sec[c/2]*Sec[(c + d*x)/2]*Sin[(d*x)/2] - 12*(20*A - 33*B)*Cos[c]*Sin[d*x] + 40*(A - B)*Cos[2*c]*Sin[2*d*x] + 12*B*Cos[3*c]*Sin[3*d*x] - 120*(A - B)*Tan[c/2])))/(60*a*d*(1 + Cos[c + d*x]))","C",1
482,1,303,208,3.2126774,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^2,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 (4 A-B) 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)+2 i (25 A-7 B) \cos (c+d x)+8 (5 A-2 B) \cos ^3\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)-12 A \sin (c+d x)-7 A \sin (2 (c+d x))+17 i A \cos (2 (c+d x))+29 i A+B \sin (2 (c+d x))-5 i B \cos (2 (c+d x))-5 i B\right)}{6 a^2 d (\cos (c+d x)+1)^2}","-\frac{(5 A-2 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{(4 A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}-\frac{(5 A-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 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) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"-1/6*(Cos[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*((29*I)*A - (5*I)*B + (2*I)*(25*A - 7*B)*Cos[c + d*x] + (17*I)*A*Cos[2*(c + d*x)] - (5*I)*B*Cos[2*(c + d*x)] - (I*(4*A - B)*(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))])/E^(I*(c + d*x)) + 8*(5*A - 2*B)*Cos[(c + d*x)/2]^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[(c + d*x)/2] - I*Sin[(c + d*x)/2]) - 12*A*Sin[c + d*x] - 7*A*Sin[2*(c + d*x)] + B*Sin[2*(c + d*x)])*(Cos[(c + 3*d*x)/2] + I*Sin[(c + 3*d*x)/2]))/(a^2*d*E^(I*d*x)*(1 + Cos[c + d*x])^2)","C",1
483,1,256,161,2.021156,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^2,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(2 i \cos (c+d x) (i (A-B) \sin (c+d x)+(5 A+B) \cos (c+d x)+7 A-B)+8 (2 A+B) \cos ^3\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)-i 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)\right)}{6 a^2 d (\cos (c+d x)+1)^2}","\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{A \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d (\sec (c+d x)+1)}+\frac{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-B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(((-I)*A*(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))])/E^(I*(c + d*x)) + 8*(2*A + B)*Cos[(c + d*x)/2]^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[(c + d*x)/2] - I*Sin[(c + d*x)/2]) + (2*I)*Cos[c + d*x]*(7*A - B + (5*A + B)*Cos[c + d*x] + I*(A - B)*Sin[c + d*x]))*(Cos[(c + 3*d*x)/2] + I*Sin[(c + 3*d*x)/2]))/(6*a^2*d*E^(I*d*x)*(1 + Cos[c + d*x])^2)","C",1
484,1,256,168,2.3855673,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^2 \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]),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 \left(2 \cos (c+d x) (-i (A-B) \sin (c+d x)+(A-7 B) \cos (c+d x)-A-5 B)+B 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)\right)+8 (A+2 B) \sqrt{\cos (c+d x)} \cos ^3\left(\frac{1}{2} (c+d x)\right) 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)}{6 a^2 d (\cos (c+d x)+1)^2}","\frac{(A+2 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d (\sec (c+d x)+1)}+\frac{(A+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 a^2 d}-\frac{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) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(8*(A + 2*B)*Cos[(c + d*x)/2]^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[(c + d*x)/2] - I*Sin[(c + d*x)/2]) + I*((B*(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))])/E^(I*(c + d*x)) + 2*Cos[c + d*x]*(-A - 5*B + (A - 7*B)*Cos[c + d*x] - I*(A - B)*Sin[c + d*x])))*(Cos[(c + 3*d*x)/2] + I*Sin[(c + 3*d*x)/2]))/(6*a^2*d*E^(I*d*x)*(1 + Cos[c + d*x])^2)","C",1
485,1,475,176,6.4728235,"\int \frac{A+B \cos (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])/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)),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(5 (A-4 B) \cos \left(\frac{1}{2} (c-d x)\right)+4 (A-4 B) \cos \left(\frac{1}{2} (3 c+d x)\right)+3 A \cos \left(\frac{1}{2} (c+3 d x)\right)-9 B \cos \left(\frac{1}{2} (c+3 d x)\right)-3 B \cos \left(\frac{1}{2} (5 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)-8 \sqrt{2} B \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 B \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{(2 A-5 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d (\sec (c+d x)+1)}+\frac{(2 A-5 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{(A-4 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) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d (a \sec (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) - (8*Sqrt[2]*B*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) + ((5*(A - 4*B)*Cos[(c - d*x)/2] + 4*(A - 4*B)*Cos[(3*c + d*x)/2] + 3*A*Cos[(c + 3*d*x)/2] - 9*B*Cos[(c + 3*d*x)/2] - 3*B*Cos[(5*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]] - 20*B*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
486,1,777,206,6.9074577,"\int \frac{A+B \cos (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])/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)),x]","\frac{\cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\sec (c+d x)} \left(\frac{8 (A-2 B) \cos (c) \sin (d x)}{d}-\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)-B \sin \left(\frac{d x}{2}\right)\right)}{3 d}-\frac{2 (A-B) \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) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(7 A \sin \left(\frac{d x}{2}\right)-10 B \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 \cos (2 c)+3 A-2 B \cos (2 c)-5 B)}{d}+\frac{4 (7 A-10 B) \tan \left(\frac{c}{2}\right)}{3 d}+\frac{4 B \sin (2 c) \cos (2 d x)}{3 d}+\frac{4 B \cos (2 c) \sin (2 d x)}{3 d}\right)}{(a \cos (c+d x)+a)^2}-\frac{4 \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{10 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} B \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 B \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{5 (A-2 B) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}+\frac{(4 A-7 B) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)} (\sec (c+d x)+1)}-\frac{5 (A-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 a^2 d}+\frac{(4 A-7 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) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} (a \sec (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) + (7*Sqrt[2]*B*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) - (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) + (20*B*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*(3*A - 5*B + A*Cos[2*c] - 2*B*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2])/d + (4*B*Cos[2*d*x]*Sin[2*c])/(3*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(7*A*Sin[(d*x)/2] - 10*B*Sin[(d*x)/2]))/(3*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2]))/(3*d) + (8*(A - 2*B)*Cos[c]*Sin[d*x])/d + (4*B*Cos[2*c]*Sin[2*d*x])/(3*d) + (4*(7*A - 10*B)*Tan[c/2])/(3*d) - (2*(A - B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])^2","C",1
487,1,358,261,5.3942907,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[((A + B*Cos[c + d*x])*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-9 B) 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 ((1082 A-207 B) \cos (c+d x)+6 (87 A-17 B) \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-6 i B \sin (c+d x)-18 i B \sin (2 (c+d x))-6 i B \sin (3 (c+d x))-21 B \cos (3 (c+d x))-102 B)+160 (13 A-3 B) \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{(13 A-3 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(49 A-9 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{(13 A-3 B) \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) \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) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{(8 A-3 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"-1/120*(Cos[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(((-I)*(49*A - 9*B)*(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 - 3*B)*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 - 102*B + (1082*A - 207*B)*Cos[c + d*x] + 6*(87*A - 17*B)*Cos[2*(c + d*x)] + 106*A*Cos[3*(c + d*x)] - 21*B*Cos[3*(c + d*x)] + (161*I)*A*Sin[c + d*x] - (6*I)*B*Sin[c + d*x] + (148*I)*A*Sin[2*(c + d*x)] - (18*I)*B*Sin[2*(c + d*x)] + (41*I)*A*Sin[3*(c + d*x)] - (6*I)*B*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
488,1,793,222,6.9785358,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^3} \, dx","Integrate[((A + B*Cos[c + d*x])*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)-B \sin \left(\frac{d x}{2}\right)\right)}{5 d}+\frac{2 (A-B) \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)+2 B \sin \left(\frac{d x}{2}\right)\right)}{15 d}+\frac{4 (3 A+2 B) \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)+B \sin \left(\frac{d x}{2}\right)\right)}{3 d}-\frac{2 (9 A+B) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos (d x)}{5 d}+\frac{4 (3 A+B) \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} B \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 B \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+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(3 A+B) \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) \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) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{(6 A-B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 a d (a \sec (c+d x)+a)^2}",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]*B*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*B*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 + B)*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] - B*Sin[(d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(3*A*Sin[(d*x)/2] + B*Sin[(d*x)/2]))/(3*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(3*A*Sin[(d*x)/2] + 2*B*Sin[(d*x)/2]))/(15*d) + (4*(3*A + B)*Tan[c/2])/(3*d) + (4*(3*A + 2*B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (2*(A - B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",1
489,1,792,216,6.9286437,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^3 \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/((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)-B \sin \left(\frac{d x}{2}\right)\right)}{5 d}-\frac{2 (A-B) \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(2 A \sin \left(\frac{d x}{2}\right)-7 B \sin \left(\frac{d x}{2}\right)\right)}{15 d}+\frac{4 (2 A-7 B) \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)+B \sin \left(\frac{d x}{2}\right)\right)}{3 d}-\frac{2 (A-B) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos (d x)}{5 d}+\frac{4 (A+B) \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{\sqrt{2} B \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 B \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{(A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(A+B) \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) \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) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{(4 A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (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) + (Sqrt[2]*B*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) + (2*B*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 - B)*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(2*A*Sin[(d*x)/2] - 7*B*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]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + B*Sin[(d*x)/2]))/(3*d) + (4*(A + B)*Tan[c/2])/(3*d) + (4*(2*A - 7*B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (2*(A - B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",1
490,1,793,222,7.0374597,"\int \frac{A+B \cos (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])/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)),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)-B \sin \left(\frac{d x}{2}\right)\right)}{5 d}+\frac{2 (A-B) \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(7 A \sin \left(\frac{d x}{2}\right)-12 B \sin \left(\frac{d x}{2}\right)\right)}{15 d}-\frac{4 (7 A-12 B) \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 B \sin \left(\frac{d x}{2}\right)\right)}{3 d}+\frac{2 (A+9 B) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos (d x)}{5 d}+\frac{4 (A-9 B) \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} B \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 B \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+3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(A+3 B) \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) \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) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d (a \sec (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) + (3*Sqrt[2]*B*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*B*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*B)*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(7*A*Sin[(d*x)/2] - 12*B*Sin[(d*x)/2]))/(15*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - 9*B*Sin[(d*x)/2]))/(3*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2]))/(5*d) + (4*(A - 9*B)*Tan[c/2])/(3*d) - (4*(7*A - 12*B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (2*(A - B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",1
491,1,817,228,7.1476724,"\int \frac{A+B \cos (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])/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)),x]","\frac{3 \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)}{5 d (\cos (c+d x) a+a)^3}-\frac{49 \sqrt{2} B 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)}{d (\cos (c+d x) a+a)^3}-\frac{26 B \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)-B \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) \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(12 A \sin \left(\frac{d x}{2}\right)-17 B \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 (12 A-17 B) \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)-23 B \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (-9 A+39 B+10 B \cos (2 c)) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}+\frac{16 B \cos (c) \sin (d x)}{d}-\frac{4 (9 A-23 B) \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{(3 A-13 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(3 A-13 B) \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) \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{(3 A-8 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}+\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d (a \sec (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) - (49*Sqrt[2]*B*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) - (26*B*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 + 39*B + 10*B*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(9*A*Sin[(d*x)/2] - 23*B*Sin[(d*x)/2]))/(3*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(12*A*Sin[(d*x)/2] - 17*B*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]))/(5*d) + (16*B*Cos[c]*Sin[d*x])/d - (4*(9*A - 23*B)*Tan[c/2])/(3*d) + (4*(12*A - 17*B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (2*(A - B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",1
492,1,589,259,4.722412,"\int \frac{A+B \cos (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])/((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((806 A-1961 B) \cos \left(\frac{1}{2} (c-d x)\right)+(664 A-1609 B) \cos \left(\frac{1}{2} (3 c+d x)\right)+470 A \cos \left(\frac{1}{2} (c+3 d x)\right)+265 A \cos \left(\frac{1}{2} (5 c+3 d x)\right)+117 A \cos \left(\frac{1}{2} (3 c+5 d x)\right)+30 A \cos \left(\frac{1}{2} (7 c+5 d x)\right)-1165 B \cos \left(\frac{1}{2} (c+3 d x)\right)-620 B \cos \left(\frac{1}{2} (5 c+3 d x)\right)-292 B \cos \left(\frac{1}{2} (3 c+5 d x)\right)-65 B \cos \left(\frac{1}{2} (7 c+5 d x)\right)-5 B \cos \left(\frac{1}{2} (5 c+7 d x)\right)+5 B \cos \left(\frac{1}{2} (9 c+7 d x)\right)\right)}{8 \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)+238 \sqrt{2} B \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 B \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{(13 A-33 B) \sin (c+d x)}{6 a^3 d \sqrt{\sec (c+d x)}}+\frac{7 (7 A-17 B) \sin (c+d x)}{30 d \sqrt{\sec (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}-\frac{(13 A-33 B) \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) \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-2 B) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}+\frac{(A-B) \sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^3}",1,"(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) + (238*Sqrt[2]*B*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) - (((806*A - 1961*B)*Cos[(c - d*x)/2] + (664*A - 1609*B)*Cos[(3*c + d*x)/2] + 470*A*Cos[(c + 3*d*x)/2] - 1165*B*Cos[(c + 3*d*x)/2] + 265*A*Cos[(5*c + 3*d*x)/2] - 620*B*Cos[(5*c + 3*d*x)/2] + 117*A*Cos[(3*c + 5*d*x)/2] - 292*B*Cos[(3*c + 5*d*x)/2] + 30*A*Cos[(7*c + 5*d*x)/2] - 65*B*Cos[(7*c + 5*d*x)/2] - 5*B*Cos[(5*c + 7*d*x)/2] + 5*B*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]]) - 260*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] + 660*B*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
493,1,124,220,0.5766585,"\int \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(11/2),x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} (11 (8 A+9 B) \cos (c+d x)+11 (8 A+9 B) \cos (2 (c+d x))+16 A \cos (3 (c+d x))+16 A \cos (4 (c+d x))+107 A+18 B \cos (3 (c+d x))+18 B \cos (4 (c+d x))+81 B)}{315 d}","\frac{2 a (8 A+9 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a (8 A+9 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a (8 A+9 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{32 a (8 A+9 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}}",1,"(2*Sqrt[a*(1 + Cos[c + d*x])]*(107*A + 81*B + 11*(8*A + 9*B)*Cos[c + d*x] + 11*(8*A + 9*B)*Cos[2*(c + d*x)] + 16*A*Cos[3*(c + d*x)] + 18*B*Cos[3*(c + d*x)] + 16*A*Cos[4*(c + d*x)] + 18*B*Cos[4*(c + d*x)])*Sec[c + d*x]^(9/2)*Tan[(c + d*x)/2])/(315*d)","A",1
494,1,102,175,0.4506172,"\int \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2),x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} (9 (6 A+7 B) \cos (c+d x)+2 (6 A+7 B) \cos (2 (c+d x))+12 A \cos (3 (c+d x))+27 A+14 B \cos (3 (c+d x))+14 B)}{105 d}","\frac{2 a (6 A+7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a (6 A+7 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a (6 A+7 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}",1,"(2*Sqrt[a*(1 + Cos[c + d*x])]*(27*A + 14*B + 9*(6*A + 7*B)*Cos[c + d*x] + 2*(6*A + 7*B)*Cos[2*(c + d*x)] + 12*A*Cos[3*(c + d*x)] + 14*B*Cos[3*(c + d*x)])*Sec[c + d*x]^(7/2)*Tan[(c + d*x)/2])/(105*d)","A",1
495,1,78,130,0.2796332,"\int \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2),x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} ((4 A+5 B) \cos (c+d x)+(4 A+5 B) \cos (2 (c+d x))+7 A+5 B)}{15 d}","\frac{2 a (4 A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a (4 A+5 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}",1,"(2*Sqrt[a*(1 + Cos[c + d*x])]*(7*A + 5*B + (4*A + 5*B)*Cos[c + d*x] + (4*A + 5*B)*Cos[2*(c + d*x)])*Sec[c + d*x]^(5/2)*Tan[(c + d*x)/2])/(15*d)","A",1
496,1,57,85,0.177667,"\int \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2),x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} ((2 A+3 B) \cos (c+d x)+A)}{3 d}","\frac{2 a (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 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}",1,"(2*Sqrt[a*(1 + Cos[c + d*x])]*(A + (2*A + 3*B)*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Tan[(c + d*x)/2])/(3*d)","A",1
497,1,86,96,0.2092287,"\int \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*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 A \sin \left(\frac{1}{2} (c+d x)\right)+\sqrt{2} B \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}\right)}{d}","\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}+\frac{2 \sqrt{a} 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)}{d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(Sqrt[2]*B*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*A*Sin[(c + d*x)/2]))/d","A",1
498,1,103,98,0.2102348,"\int \sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x])*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} (2 A+B) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 B \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)}\right)}{2 d}","\frac{\sqrt{a} (2 A+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)}{d}+\frac{a B \sin (c+d x)}{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]*(2*A + B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*B*Sqrt[Cos[c + d*x]]*Sin[(c + d*x)/2]))/(2*d)","A",1
499,1,120,151,0.3875246,"\int \frac{\sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/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} (4 A+3 B) \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 A+2 B \cos (c+d x)+3 B)\right)}{8 d}","\frac{\sqrt{a} (4 A+3 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)}{4 d}+\frac{a (4 A+3 B) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(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]*(4*A + 3*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(4*A + 3*B + 2*B*Cos[c + d*x])*Sin[(c + d*x)/2]))/(8*d)","A",1
500,1,138,196,0.7051777,"\int \frac{\sqrt{a+a \cos (c+d x)} (A+B \cos (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/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} (6 A+5 B) \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 (6 A+5 B) \cos (c+d x)+18 A+4 B \cos (2 (c+d x))+19 B)\right)}{48 d}","\frac{a (6 A+5 B) \sin (c+d x)}{12 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} (6 A+5 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)}{8 d}+\frac{a (6 A+5 B) \sin (c+d x)}{8 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x)}{3 d \sec ^{\frac{5}{2}}(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]]*(3*Sqrt[2]*(6*A + 5*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(18*A + 19*B + 2*(6*A + 5*B)*Cos[c + d*x] + 4*B*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
501,1,146,275,0.7817434,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^{\frac{13}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*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)} ((6342 A+6193 B) \cos (c+d x)+13 (168 A+187 B) \cos (2 (c+d x))+2184 A \cos (3 (c+d x))+336 A \cos (4 (c+d x))+336 A \cos (5 (c+d x))+2478 A+2431 B \cos (3 (c+d x))+374 B \cos (4 (c+d x))+374 B \cos (5 (c+d x))+2057 B)}{3465 d}","\frac{2 a^2 (12 A+11 B) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{99 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (168 A+187 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (168 A+187 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{1155 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (168 A+187 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{32 a^2 (168 A+187 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{11 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(2478*A + 2057*B + (6342*A + 6193*B)*Cos[c + d*x] + 13*(168*A + 187*B)*Cos[2*(c + d*x)] + 2184*A*Cos[3*(c + d*x)] + 2431*B*Cos[3*(c + d*x)] + 336*A*Cos[4*(c + d*x)] + 374*B*Cos[4*(c + d*x)] + 336*A*Cos[5*(c + d*x)] + 374*B*Cos[5*(c + d*x)])*Sec[c + d*x]^(11/2)*Tan[(c + d*x)/2])/(3465*d)","A",1
502,1,124,228,0.7135318,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*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)} ((374 A+324 B) \cos (c+d x)+11 (34 A+39 B) \cos (2 (c+d x))+68 A \cos (3 (c+d x))+68 A \cos (4 (c+d x))+376 A+78 B \cos (3 (c+d x))+78 B \cos (4 (c+d x))+351 B)}{315 d}","\frac{2 a^2 (10 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^2 (34 A+39 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a^2 (34 A+39 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 (34 A+39 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{9 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(376*A + 351*B + (374*A + 324*B)*Cos[c + d*x] + 11*(34*A + 39*B)*Cos[2*(c + d*x)] + 68*A*Cos[3*(c + d*x)] + 78*B*Cos[3*(c + d*x)] + 68*A*Cos[4*(c + d*x)] + 78*B*Cos[4*(c + d*x)])*Sec[c + d*x]^(9/2)*Tan[(c + d*x)/2])/(315*d)","A",1
503,1,102,181,0.5624075,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*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)} (3 (78 A+77 B) \cos (c+d x)+(52 A+63 B) \cos (2 (c+d x))+52 A \cos (3 (c+d x))+82 A+63 B \cos (3 (c+d x))+63 B)}{105 d}","\frac{2 a^2 (8 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^2 (52 A+63 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^2 (52 A+63 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{7 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(82*A + 63*B + 3*(78*A + 77*B)*Cos[c + d*x] + (52*A + 63*B)*Cos[2*(c + d*x)] + 52*A*Cos[3*(c + d*x)] + 63*B*Cos[3*(c + d*x)])*Sec[c + d*x]^(7/2)*Tan[(c + d*x)/2])/(105*d)","A",1
504,1,80,134,0.3310437,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} (2 (9 A+5 B) \cos (c+d x)+(18 A+25 B) \cos (2 (c+d x))+24 A+25 B)}{15 d}","\frac{2 a^2 (6 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^2 (18 A+25 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{5 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*(24*A + 25*B + 2*(9*A + 5*B)*Cos[c + d*x] + (18*A + 25*B)*Cos[2*(c + d*x)])*Sec[c + d*x]^(5/2)*Tan[(c + d*x)/2])/(15*d)","A",1
505,1,106,145,0.4020078,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*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(2 \sin \left(\frac{1}{2} (c+d x)\right) ((5 A+3 B) \cos (c+d x)+A)+3 \sqrt{2} B \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^{3/2} 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)}{d}+\frac{2 a^2 (4 A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{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]*B*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(3/2) + 2*(A + (5*A + 3*B)*Cos[c + d*x])*Sin[(c + d*x)/2]))/(3*d)","A",1
506,1,107,146,0.3208951,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*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} (2 A+3 B) \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) (2 A+B \cos (c+d x))\right)}{2 d}","\frac{a^{3/2} (2 A+3 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)}{d}-\frac{a^2 (2 A-B) \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\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]*Sqrt[Sec[c + d*x]]*(Sqrt[2]*(2*A + 3*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*(2*A + B*Cos[c + d*x])*Sin[(c + d*x)/2]))/(2*d)","A",1
507,1,121,153,0.4530041,"\int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*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(\sqrt{2} (12 A+7 B) \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 A+2 B \cos (c+d x)+7 B)\right)}{8 d}","\frac{a^{3/2} (12 A+7 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)}{4 d}+\frac{a^2 (4 A+5 B) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 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]]*(Sqrt[2]*(12*A + 7*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(4*A + 7*B + 2*B*Cos[c + d*x])*Sin[(c + d*x)/2]))/(8*d)","A",1
508,1,141,200,0.5016598,"\int \frac{(a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/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} (14 A+11 B) \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 (6 A+11 B) \cos (c+d x)+42 A+4 B \cos (2 (c+d x))+37 B)\right)}{48 d}","\frac{a^{3/2} (14 A+11 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)}{8 d}+\frac{a^2 (6 A+7 B) \sin (c+d x)}{12 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (14 A+11 B) \sin (c+d x)}{8 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 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]*(14*A + 11*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + (42*A + 37*B + 2*(6*A + 11*B)*Cos[c + d*x] + 4*B*Cos[2*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(48*d)","A",1
509,1,158,247,0.7894174,"\int \frac{(a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/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(3 \sqrt{2} (88 A+75 B) \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 (88 A+93 B) \cos (c+d x)+4 (8 A+15 B) \cos (2 (c+d x))+296 A+12 B \cos (3 (c+d x))+285 B)\right)}{384 d}","\frac{a^{3/2} (88 A+75 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)}{64 d}+\frac{a^2 (88 A+75 B) \sin (c+d x)}{96 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (8 A+9 B) \sin (c+d x)}{24 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (88 A+75 B) \sin (c+d x)}{64 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a B \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 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]]*(3*Sqrt[2]*(88*A + 75*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + (296*A + 285*B + 2*(88*A + 93*B)*Cos[c + d*x] + 4*(8*A + 15*B)*Cos[2*(c + d*x)] + 12*B*Cos[3*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(384*d)","A",1
510,1,171,322,0.9065689,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{15}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*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)} (35 (5552 A+5083 B) \cos (c+d x)+14 (15167 A+15925 B) \cos (2 (c+d x))+62760 A \cos (3 (c+d x))+62760 A \cos (4 (c+d x))+8368 A \cos (5 (c+d x))+8368 A \cos (6 (c+d x))+171806 A+69225 B \cos (3 (c+d x))+69225 B \cos (4 (c+d x))+9230 B \cos (5 (c+d x))+9230 B \cos (6 (c+d x))+162955 B)}{90090 d}","\frac{2 a^3 (280 A+299 B) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{1287 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (4184 A+4615 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{9009 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^3 (4184 A+4615 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{15015 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^3 (4184 A+4615 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{45045 d \sqrt{a \cos (c+d x)+a}}+\frac{32 a^3 (4184 A+4615 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{45045 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (16 A+13 B) \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{143 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{13}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{13 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(171806*A + 162955*B + 35*(5552*A + 5083*B)*Cos[c + d*x] + 14*(15167*A + 15925*B)*Cos[2*(c + d*x)] + 62760*A*Cos[3*(c + d*x)] + 69225*B*Cos[3*(c + d*x)] + 62760*A*Cos[4*(c + d*x)] + 69225*B*Cos[4*(c + d*x)] + 8368*A*Cos[5*(c + d*x)] + 9230*B*Cos[5*(c + d*x)] + 8368*A*Cos[6*(c + d*x)] + 9230*B*Cos[6*(c + d*x)])*Sec[c + d*x]^(13/2)*Tan[(c + d*x)/2])/(90090*d)","A",1
511,1,147,275,1.2742743,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{13}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*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)} ((25070 A+24827 B) \cos (c+d x)+(9230 A+9284 B) \cos (2 (c+d x))+9230 A \cos (3 (c+d x))+1420 A \cos (4 (c+d x))+1420 A \cos (5 (c+d x))+9070 A+10439 B \cos (3 (c+d x))+1606 B \cos (4 (c+d x))+1606 B \cos (5 (c+d x))+7678 B)}{6930 d}","\frac{2 a^3 (194 A+209 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (710 A+803 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{1155 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a^3 (710 A+803 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^3 (710 A+803 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3465 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (14 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 A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{11 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(9070*A + 7678*B + (25070*A + 24827*B)*Cos[c + d*x] + (9230*A + 9284*B)*Cos[2*(c + d*x)] + 9230*A*Cos[3*(c + d*x)] + 10439*B*Cos[3*(c + d*x)] + 1420*A*Cos[4*(c + d*x)] + 1606*B*Cos[4*(c + d*x)] + 1420*A*Cos[5*(c + d*x)] + 1606*B*Cos[5*(c + d*x)])*Sec[c + d*x]^(11/2)*Tan[(c + d*x)/2])/(6930*d)","A",1
512,1,126,228,0.9470131,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*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)} ((1396 A+1215 B) \cos (c+d x)+2 (803 A+870 B) \cos (2 (c+d x))+292 A \cos (3 (c+d x))+292 A \cos (4 (c+d x))+1454 A+345 B \cos (3 (c+d x))+345 B \cos (4 (c+d x))+1395 B)}{630 d}","\frac{2 a^3 (124 A+135 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (292 A+345 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a^3 (292 A+345 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (4 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 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{9 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(1454*A + 1395*B + (1396*A + 1215*B)*Cos[c + d*x] + 2*(803*A + 870*B)*Cos[2*(c + d*x)] + 292*A*Cos[3*(c + d*x)] + 345*B*Cos[3*(c + d*x)] + 292*A*Cos[4*(c + d*x)] + 345*B*Cos[4*(c + d*x)])*Sec[c + d*x]^(9/2)*Tan[(c + d*x)/2])/(630*d)","A",1
513,1,104,181,0.6950825,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)} ((930 A+987 B) \cos (c+d x)+2 (115 A+98 B) \cos (2 (c+d x))+230 A \cos (3 (c+d x))+290 A+301 B \cos (3 (c+d x))+196 B)}{210 d}","\frac{2 a^3 (10 A+11 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^3 (230 A+301 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (10 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 A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(290*A + 196*B + (930*A + 987*B)*Cos[c + d*x] + 2*(115*A + 98*B)*Cos[2*(c + d*x)] + 230*A*Cos[3*(c + d*x)] + 301*B*Cos[3*(c + d*x)])*Sec[c + d*x]^(7/2)*Tan[(c + d*x)/2])/(210*d)","A",1
514,1,130,192,0.8439608,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*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) (2 (14 A+5 B) \cos (c+d x)+(43 A+40 B) \cos (2 (c+d x))+49 A+40 B)+30 \sqrt{2} B \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^{5/2} 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)}{d}+\frac{2 a^3 (32 A+35 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (8 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 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^(5/2)*(30*Sqrt[2]*B*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(5/2) + 2*(49*A + 40*B + 2*(14*A + 5*B)*Cos[c + d*x] + (43*A + 40*B)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(30*d)","A",1
515,1,130,193,0.7363416,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*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(3 \sqrt{2} (2 A+5 B) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{3}{2}}(c+d x)+\sin \left(\frac{1}{2} (c+d x)\right) (4 (8 A+3 B) \cos (c+d x)+4 A+3 B \cos (2 (c+d x))+3 B)\right)}{6 d}","\frac{a^{5/2} (2 A+5 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)}{d}-\frac{a^3 (14 A+3 B) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 (2 A+B) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{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}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^(3/2)*(3*Sqrt[2]*(2*A + 5*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(3/2) + (4*A + 3*B + 4*(8*A + 3*B)*Cos[c + d*x] + 3*B*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(6*d)","A",1
516,1,126,198,0.776624,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*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(\sqrt{2} (20 A+19 B) \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) ((4 A+11 B) \cos (c+d x)+8 A+B \cos (2 (c+d x))+B)\right)}{8 d}","\frac{a^{5/2} (20 A+19 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)}{4 d}-\frac{a^3 (4 A-9 B) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (4 A-B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}{d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(Sqrt[2]*(20*A + 19*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*(8*A + B + (4*A + 11*B)*Cos[c + d*x] + B*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(8*d)","A",1
517,1,141,200,0.977346,"\int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*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} (38 A+25 B) \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 (6 A+17 B) \cos (c+d x)+66 A+4 B \cos (2 (c+d x))+79 B)\right)}{48 d}","\frac{a^{5/2} (38 A+25 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)}{8 d}+\frac{a^3 (54 A+49 B) \sin (c+d x)}{24 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (2 A+3 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sqrt{\sec (c+d x)}}+\frac{a B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 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]*(38*A + 25*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(66*A + 79*B + 2*(6*A + 17*B)*Cos[c + d*x] + 4*B*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
518,1,159,247,0.9786524,"\int \frac{(a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/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} (200 A+163 B) \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) ((272 A+362 B) \cos (c+d x)+4 (8 A+23 B) \cos (2 (c+d x))+632 A+12 B \cos (3 (c+d x))+581 B)\right)}{384 d}","\frac{a^{5/2} (200 A+163 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)}{64 d}+\frac{a^3 (104 A+95 B) \sin (c+d x)}{96 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (200 A+163 B) \sin (c+d x)}{64 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (8 A+11 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{24 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{a B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{4 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]]*(3*Sqrt[2]*(200*A + 163*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + (632*A + 581*B + (272*A + 362*B)*Cos[c + d*x] + 4*(8*A + 23*B)*Cos[2*(c + d*x)] + 12*B*Cos[3*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(384*d)","A",1
519,1,181,294,1.436595,"\int \frac{(a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/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} (326 A+283 B) \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) ((3620 A+3874 B) \cos (c+d x)+4 (230 A+331 B) \cos (2 (c+d x))+120 A \cos (3 (c+d x))+5810 A+348 B \cos (3 (c+d x))+48 B \cos (4 (c+d x))+5521 B)\right)}{3840 d}","\frac{a^{5/2} (326 A+283 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)}{128 d}+\frac{a^3 (326 A+283 B) \sin (c+d x)}{192 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (170 A+157 B) \sin (c+d x)}{240 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^3 (326 A+283 B) \sin (c+d x)}{128 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a^2 (10 A+13 B) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{40 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{a B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 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]*(326*A + 283*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + (5810*A + 5521*B + (3620*A + 3874*B)*Cos[c + d*x] + 4*(230*A + 331*B)*Cos[2*(c + d*x)] + 120*A*Cos[3*(c + d*x)] + 348*B*Cos[3*(c + d*x)] + 48*B*Cos[4*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(3840*d)","A",1
520,1,272,295,9.3295247,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{11}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x])*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-B) \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) ((214 A-918 B) \cos (c+d x)-8 (157 A-69 B) \cos (2 (c+d x))+58 A \cos (3 (c+d x))-257 A \cos (4 (c+d x))-1279 A-186 B \cos (3 (c+d x))+129 B \cos (4 (c+d x))+423 B)\right)}{315 d \sqrt{a (\cos (c+d x)+1)}}","-\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 (19 A-3 B) \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) \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) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (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{2 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}}",1,"(2*Cos[(c + d*x)/2]*((-315*I)*(A - B)*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 + 423*B + (214*A - 918*B)*Cos[c + d*x] - 8*(157*A - 69*B)*Cos[2*(c + d*x)] + 58*A*Cos[3*(c + d*x)] - 186*B*Cos[3*(c + d*x)] - 257*A*Cos[4*(c + d*x)] + 129*B*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
521,1,250,250,6.838802,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2))/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(105 i (A-B) \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}{2} \sin \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{7}{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) (3 (47 A-119 B) \cos (c+d x)+(14 B-62 A) \cos (2 (c+d x))+43 A \cos (3 (c+d x))-122 A-91 B \cos (3 (c+d x))+14 B)\right)}{105 d \sqrt{a (\cos (c+d x)+1)}}","-\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 (31 A-7 B) \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) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (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{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 + d*x)/2]*((105*I)*(A - B)*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))])] - ((-122*A + 14*B + 3*(47*A - 119*B)*Cos[c + d*x] + (-62*A + 14*B)*Cos[2*(c + d*x)] + 43*A*Cos[3*(c + d*x)] - 91*B*Cos[3*(c + d*x)])*Sec[c + d*x]^(7/2)*(Cos[(c + d*x)/2] + I*Sin[(c + d*x)/2])*Sin[(c + d*x)/2])/2))/(105*d*E^((I/2)*(c + d*x))*Sqrt[a*(1 + Cos[c + d*x])])","C",1
522,1,1718,207,7.7994674,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x])*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) \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{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}}\right)}{d \sqrt{a (\cos (c+d x)+1)}}","-\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 (13 A-5 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (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{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)) + (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)*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))))/(d*Sqrt[a*(1 + Cos[c + d*x])])","C",0
523,-1,0,162,0,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/Sqrt[a + a*Cos[c + d*x]],x]","\text{\$Aborted}","-\frac{2 (A-3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (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{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
524,1,203,119,1.5804144,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \sin \left(\frac{1}{2} (c+d x)\right) \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(10 B-(A-B) \sec (c+d x) \left(\frac{1}{2} \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)-\frac{5}{4} (4 \cos (c+d x)+\cos (2 (c+d x))+1) \csc ^4\left(\frac{1}{2} (c+d x)\right) \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)\right)}{5 d \sqrt{a (\cos (c+d x)+1)}}","\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} (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}",1,"(2*Cos[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*Sin[(c + d*x)/2]*(10*B - (A - B)*Sec[c + d*x]*((-5*(1 + 4*Cos[c + d*x] + Cos[2*(c + d*x)])*Csc[(c + d*x)/2]^4*(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]]))/4 + (Hypergeometric2F1[2, 5/2, 7/2, -(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]*Sin[c + d*x]*Tan[c + d*x])/2)))/(5*d*Sqrt[a*(1 + Cos[c + d*x])])","C",0
525,1,102,140,0.2107715,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/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((A-B) \tan ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)+\sqrt{2} B \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{d \sqrt{a (\cos (c+d x)+1)}}","\frac{\sqrt{2} (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{2 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}",1,"(2*(Sqrt[2]*B*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + (A - B)*ArcTan[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]])*Cos[(c + d*x)/2]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
526,1,467,181,1.3675064,"\int \frac{A+B \cos (c+d x)}{\sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/(Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","\frac{i e^{-2 i (c+d x)} \left(1+e^{i (c+d x)}\right) \sqrt{\sec (c+d x)} \left(-(2 A-B) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)+2 \sqrt{2} A e^{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)+2 A e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)+B \left(-e^{i (c+d x)}\right)+B e^{2 i (c+d x)}-B e^{3 i (c+d x)}+\sqrt{2} B e^{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)-\sqrt{2} B e^{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)-B e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)+B\right)}{4 d \sqrt{a (\cos (c+d x)+1)}}","\frac{(2 A-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) \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 \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"((I/4)*(1 + E^(I*(c + d*x)))*(B - B*E^(I*(c + d*x)) + B*E^((2*I)*(c + d*x)) - B*E^((3*I)*(c + d*x)) - (2*A - B)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))] + Sqrt[2]*B*E^(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))])] + 2*Sqrt[2]*A*E^(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))])] - Sqrt[2]*B*E^(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))])] + 2*A*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]] - B*E^(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^((2*I)*(c + d*x))*Sqrt[a*(1 + Cos[c + d*x])])","C",1
527,1,412,230,1.4550254,"\int \frac{A+B \cos (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])/(Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)),x]","-\frac{i e^{-3 i (c+d x)} \left(1+e^{i (c+d x)}\right) \sqrt{\sec (c+d x)} \left(-(4 A-7 B) 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-B) 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)-4 A e^{i (c+d x)}+4 A e^{2 i (c+d x)}-4 A e^{3 i (c+d x)}+4 A e^{4 i (c+d x)}+4 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 B e^{i (c+d x)}-3 B e^{2 i (c+d x)}+3 B e^{3 i (c+d x)}-2 B e^{4 i (c+d x)}+B e^{5 i (c+d x)}-7 B 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)-B\right)}{16 d \sqrt{a (\cos (c+d x)+1)}}","-\frac{(4 A-7 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)}{4 \sqrt{a} d}+\frac{(4 A-B) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} (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 \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}",1,"((-1/16*I)*(1 + E^(I*(c + d*x)))*(-B - 4*A*E^(I*(c + d*x)) + 2*B*E^(I*(c + d*x)) + 4*A*E^((2*I)*(c + d*x)) - 3*B*E^((2*I)*(c + d*x)) - 4*A*E^((3*I)*(c + d*x)) + 3*B*E^((3*I)*(c + d*x)) + 4*A*E^((4*I)*(c + d*x)) - 2*B*E^((4*I)*(c + d*x)) + B*E^((5*I)*(c + d*x)) - (4*A - 7*B)*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))] - 8*Sqrt[2]*(A - B)*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))])] + 4*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*B*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
528,1,143,192,0.4664701,"\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
529,1,2966,317,10.1809491,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*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) \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) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{14 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B) \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) \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) \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) \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]*(-1/28*((A - B)*(1 - 2*Sin[c/2 + (d*x)/2]))/((1 + Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2)) + ((A - B)*(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)*(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)*(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)*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))))/(d*(a*(1 + Cos[c + d*x]))^(3/2))","C",0
530,-1,0,270,0,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2))/(a + a*Cos[c + d*x])^(3/2),x]","\text{\$Aborted}","-\frac{(15 A-11 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(9 A-5 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B) \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) \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) \sin (c+d x) \sqrt{\sec (c+d x)}}{30 a d \sqrt{a \cos (c+d x)+a}}",1,"$Aborted","F",-1
531,1,981,223,6.8323953,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*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) \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) \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) \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{(A-B) \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) \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) \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) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B) \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) \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]*(-1/12*((A - B)*(1 - 2*Sin[c/2 + (d*x)/2]))/((1 + Sin[c/2 + (d*x)/2])*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2)) + ((A - B)*(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)*(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)*(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)*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
532,1,443,176,4.4867252,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[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) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{(A+3 B) \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) \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) \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) \sqrt{\cos (c+d x)}}{\sin \left(\frac{1}{2} (c+d x)\right)-1}-\frac{20 (A-B) \sqrt{\cos (c+d x)}}{\sin \left(\frac{1}{2} (c+d x)\right)+1}+30 (A-B) \tan ^{-1}\left(\frac{1-2 \sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)-30 (A-B) \tan ^{-1}\left(\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)+1}{\sqrt{\cos (c+d x)}}\right)\right)}{10 d (a (\cos (c+d x)+1))^{3/2}}","-\frac{(7 A-3 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]^3*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(30*(A - B)*ArcTan[(1 - 2*Sin[(c + d*x)/2])/Sqrt[Cos[c + d*x]]] - 30*(A - B)*ArcTan[(1 + 2*Sin[(c + d*x)/2])/Sqrt[Cos[c + d*x]]] - (20*(A - B)*Sqrt[Cos[c + d*x]])/(-1 + Sin[(c + d*x)/2]) - (20*(A - B)*Sqrt[Cos[c + d*x]])/(1 + Sin[(c + d*x)/2]) + (5*(A - B)*(-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)*(1 + 2*Sin[(c + d*x)/2]))/(Sqrt[Cos[c + d*x]]*(-1 + Sin[(c + d*x)/2])) + ((A + 3*B)*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
533,1,196,127,1.6426508,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*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((3 A+B) 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}{2} i (A-B) \left(\sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)}\right)}{d (a (\cos (c+d x)+1))^{3/2}}","\frac{(3 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)}{2 \sqrt{2} a^{3/2} d}-\frac{(A-B) \sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"(I*Cos[(c + d*x)/2]^3*(((3*A + B)*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/2)*(A - B)*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
534,1,243,185,1.6244046,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/((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((A-B) \left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)}-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-5 B) \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+4 B \sinh ^{-1}\left(e^{i (c+d x)}\right)-4 B \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-5 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)}{2 \sqrt{2} a^{3/2} d}+\frac{2 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)}{a^{3/2} d}+\frac{(A-B) \sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}",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))]*(4*B*ArcSinh[E^(I*(c + d*x))] - Sqrt[2]*(A - 5*B)*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - 4*B*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))/E^((I/2)*(c + d*x)) + (A - B)*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
535,1,836,237,6.70863,"\int \frac{A+B \cos (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])/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{\sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{\sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{c}{2}\right)-B \sin \left(\frac{c}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{2 A \cos \left(\frac{d x}{2}\right) \sin \left(\frac{c}{2}\right)}{d}+\frac{2 B \cos \left(\frac{3 d x}{2}\right) \sin \left(\frac{3 c}{2}\right)}{d}-\frac{2 A \cos \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)}{d}+\frac{2 B \cos \left(\frac{3 c}{2}\right) \sin \left(\frac{3 d x}{2}\right)}{d}\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{(a (\cos (c+d x)+1))^{3/2}}-\frac{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 ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{3/2}}+\frac{3 i B 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 ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{3/2}}+\frac{2 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 ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{3/2}}-\frac{3 i \sqrt{2} B 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 ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{3/2}}","\frac{(2 A-3 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)}{a^{3/2} d}-\frac{(5 A-9 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(A-B) \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{(A-3 B) \sin (c+d x)}{2 a d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"((-I)*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]^3)/(d*E^((I/2)*(c + d*x))*(a*(1 + Cos[c + d*x]))^(3/2)) + ((3*I)*B*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]^3)/(d*E^((I/2)*(c + d*x))*(a*(1 + Cos[c + d*x]))^(3/2)) + ((2*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]^3)/(d*E^((I/2)*(c + d*x))*(a*(1 + Cos[c + d*x]))^(3/2)) - ((3*I)*Sqrt[2]*B*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]^3)/(d*E^((I/2)*(c + d*x))*(a*(1 + Cos[c + d*x]))^(3/2)) + (Cos[c/2 + (d*x)/2]^3*Sqrt[Sec[c + d*x]]*((-2*A*Cos[(d*x)/2]*Sin[c/2])/d + (Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[c/2] - B*Sin[c/2]))/d + (2*B*Cos[(3*d*x)/2]*Sin[(3*c)/2])/d - (2*A*Cos[c/2]*Sin[(d*x)/2])/d + (Sec[c/2]*Sec[c/2 + (d*x)/2]^2*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2]))/d + (2*B*Cos[(3*c)/2]*Sin[(3*d*x)/2])/d))/(a*(1 + Cos[c + d*x]))^(3/2)","C",1
536,1,261,317,8.4640726,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*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) (10 (2605 A-1381 B) \cos (c+d x)+108 (157 A-85 B) \cos (2 (c+d x))+9110 A \cos (3 (c+d x))+2671 A \cos (4 (c+d x))+15053 A-5030 B \cos (3 (c+d x))-1495 B \cos (4 (c+d x))-7685 B)-240 i (283 A-163 B) 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-163 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)}{16 \sqrt{2} a^{5/2} d}+\frac{(157 A-85 B) \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) \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) \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) \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) \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 - 163*B)*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 - 7685*B + 10*(2605*A - 1381*B)*Cos[c + d*x] + 108*(157*A - 85*B)*Cos[2*(c + d*x)] + 9110*A*Cos[3*(c + d*x)] - 5030*B*Cos[3*(c + d*x)] + 2671*A*Cos[4*(c + d*x)] - 1495*B*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
537,1,243,270,3.7912674,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*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-75 B) 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-825 B) \cos (c+d x)+2 (503 A-255 B) \cos (2 (c+d x))+299 A \cos (3 (c+d x))+878 A-147 B \cos (3 (c+d x))-510 B)\right)}{12 d (a (\cos (c+d x)+1))^{5/2}}","\frac{(163 A-75 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)}{16 \sqrt{2} a^{5/2} d}+\frac{(95 A-39 B) \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) \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) \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) \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 - 75*B)*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 - 510*B + (1537*A - 825*B)*Cos[c + d*x] + 2*(503*A - 255*B)*Cos[2*(c + d*x)] + 299*A*Cos[3*(c + d*x)] - 147*B*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
538,1,219,223,2.3028153,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*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)} (2 (85 A-13 B) \cos (c+d x)+(49 A-9 B) \cos (2 (c+d x))+113 A-9 B)-i (75 A-19 B) 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{(75 A-19 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)}{16 \sqrt{2} a^{5/2} d}+\frac{(49 A-9 B) \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) \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \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*(((-I)*(75*A - 19*B)*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 - 9*B + 2*(85*A - 13*B)*Cos[c + d*x] + (49*A - 9*B)*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
539,1,216,176,1.8431199,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*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+5 B) 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-B) \cos (c+d x)+13 A-5 B)\right)}{4 d (a (\cos (c+d x)+1))^{5/2}}","\frac{(19 A+5 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)}{16 \sqrt{2} a^{5/2} d}-\frac{(9 A-B) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{(A-B) \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 + 5*B)*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 - 5*B + (9*A - B)*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
540,1,213,174,1.8225594,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/((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}{4} \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) ((A+7 B) \cos (c+d x)+5 A+3 B)+i (5 A+3 B) 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 A+3 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)}{16 \sqrt{2} a^{5/2} d}+\frac{(A+7 B) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}",1,"(Cos[(c + d*x)/2]^5*((I*(5*A + 3*B)*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)) + ((5*A + 3*B + (A + 7*B)*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]))/4))/(4*d*(a*(1 + Cos[c + d*x]))^(5/2))","C",1
541,1,264,234,2.5800687,"\int \frac{A+B \cos (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])/((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) \sec ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} ((7 A-15 B) \cos (c+d x)+3 A-11 B)-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-43 B) \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+32 B \sinh ^{-1}\left(e^{i (c+d x)}\right)-32 B \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-43 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)}{16 \sqrt{2} a^{5/2} d}+\frac{2 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)}{a^{5/2} d}+\frac{(A-B) \sin (c+d x)}{4 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}+\frac{(3 A-11 B) \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*B*ArcSinh[E^(I*(c + d*x))] - Sqrt[2]*(3*A - 43*B)*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - 32*B*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))/E^((I/2)*(c + d*x)) + ((3*A - 11*B + (7*A - 15*B)*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
542,1,929,286,7.214836,"\int \frac{A+B \cos (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])/((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(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{(A-B) \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)-27 B \sin \left(\frac{d x}{2}\right)\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}+\frac{(19 A-27 B) \tan \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}+\frac{15 (B-A) \cos \left(\frac{d x}{2}\right) \sin \left(\frac{c}{2}\right)}{2 d}+\frac{4 B \cos \left(\frac{3 d x}{2}\right) \sin \left(\frac{3 c}{2}\right)}{d}-\frac{15 (A-B) \cos \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)}{2 d}+\frac{4 B \cos \left(\frac{3 c}{2}\right) \sin \left(\frac{3 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{35 i B 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{10 i \sqrt{2} B 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{(2 A-5 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)}{a^{5/2} d}-\frac{(43 A-115 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)}{16 \sqrt{2} a^{5/2} d}-\frac{(11 A-35 B) \sin (c+d x)}{16 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{(7 A-15 B) \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) \sin (c+d x)}{4 d \sec ^{\frac{5}{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)) + (((35*I)/4)*B*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)) - ((10*I)*Sqrt[2]*B*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)) + (Cos[c/2 + (d*x)/2]^5*Sqrt[Sec[c + d*x]]*((15*(-A + B)*Cos[(d*x)/2]*Sin[c/2])/(2*d) + (4*B*Cos[(3*d*x)/2]*Sin[(3*c)/2])/d - (15*(A - B)*Cos[c/2]*Sin[(d*x)/2])/(2*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]^2*(19*A*Sin[(d*x)/2] - 27*B*Sin[(d*x)/2]))/(4*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]^4*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(2*d) + (4*B*Cos[(3*c)/2]*Sin[(3*d*x)/2])/d + ((19*A - 27*B)*Sec[c/2 + (d*x)/2]*Tan[c/2])/(4*d) - ((A - B)*Sec[c/2 + (d*x)/2]^3*Tan[c/2])/(2*d)))/(a*(1 + Cos[c + d*x]))^(5/2)","C",1
543,1,267,317,5.7713407,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + a*Cos[c + d*x])^(7/2),x]","\frac{\cos ^7\left(\frac{1}{2} (c+d x)\right) \left(-\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) (4 (9415 A-3579 B) \cos (c+d x)+8 (3069 A-1145 B) \cos (2 (c+d x))+10164 A \cos (3 (c+d x))+1887 A \cos (4 (c+d x))+21641 A-3748 B \cos (3 (c+d x))-691 B \cos (4 (c+d x))-8469 B)}{96 d}+\frac{i (1015 A-363 B) 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)}{d}\right)}{8 (a (\cos (c+d x)+1))^{7/2}}","\frac{(1015 A-363 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)}{64 \sqrt{2} a^{7/2} d}+\frac{(579 A-199 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{192 a^3 d \sqrt{a \cos (c+d x)+a}}-\frac{(1887 A-691 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{192 a^3 d \sqrt{a \cos (c+d x)+a}}-\frac{(109 A-41 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{64 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(23 A-11 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{48 a d (a \cos (c+d x)+a)^{5/2}}-\frac{(A-B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}",1,"(Cos[(c + d*x)/2]^7*((I*(1015*A - 363*B)*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))])])/(d*E^((I/2)*(c + d*x))) - ((21641*A - 8469*B + 4*(9415*A - 3579*B)*Cos[c + d*x] + 8*(3069*A - 1145*B)*Cos[2*(c + d*x)] + 10164*A*Cos[3*(c + d*x)] - 3748*B*Cos[3*(c + d*x)] + 1887*A*Cos[4*(c + d*x)] - 691*B*Cos[4*(c + d*x)])*Sec[(c + d*x)/2]^5*Sec[c + d*x]^(3/2)*Tan[(c + d*x)/2])/(96*d)))/(8*(a*(1 + Cos[c + d*x]))^(7/2))","C",1
544,1,242,270,3.2772451,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + a*Cos[c + d*x])^(7/2),x]","\frac{\cos ^7\left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{16} \tan \left(\frac{1}{2} (c+d x)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} (9 (941 A-121 B) \cos (c+d x)+4 (937 A-133 B) \cos (2 (c+d x))+691 A \cos (3 (c+d x))+5284 A-103 B \cos (3 (c+d x))-532 B)-9 i (121 A-21 B) 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)}{24 d (a (\cos (c+d x)+1))^{7/2}}","-\frac{3 (121 A-21 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)}{64 \sqrt{2} a^{7/2} d}+\frac{(691 A-103 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{192 a^3 d \sqrt{a \cos (c+d x)+a}}-\frac{(199 A-43 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac{(19 A-7 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{48 a d (a \cos (c+d x)+a)^{5/2}}-\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}",1,"(Cos[(c + d*x)/2]^7*(((-9*I)*(121*A - 21*B)*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)) + ((5284*A - 532*B + 9*(941*A - 121*B)*Cos[c + d*x] + 4*(937*A - 133*B)*Cos[2*(c + d*x)] + 691*A*Cos[3*(c + d*x)] - 103*B*Cos[3*(c + d*x)])*Sec[(c + d*x)/2]^5*Sqrt[Sec[c + d*x]]*Tan[(c + d*x)/2])/16))/(24*d*(a*(1 + Cos[c + d*x]))^(7/2))","C",1
545,1,228,223,3.0697987,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^{7/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + a*Cos[c + d*x])^(7/2),x]","\frac{\cos ^7\left(\frac{1}{2} (c+d x)\right) \left(\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 ^6\left(\frac{1}{2} (c+d x)\right) ((532 A-4 B) \cos (c+d x)+(103 A+5 B) \cos (2 (c+d x))+493 A-73 B)+48 i (63 A+13 B) 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)}{384 d (a (\cos (c+d x)+1))^{7/2}}","\frac{(63 A+13 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)}{64 \sqrt{2} a^{7/2} d}-\frac{(103 A+5 B) \sin (c+d x)}{192 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{(5 A-B) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac{(A-B) \sin (c+d x)}{6 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{7/2}}",1,"(Cos[(c + d*x)/2]^7*(((48*I)*(63*A + 13*B)*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)) + (493*A - 73*B + (532*A - 4*B)*Cos[c + d*x] + (103*A + 5*B)*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^6*Sqrt[Sec[c + d*x]]*(Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2])))/(384*d*(a*(1 + Cos[c + d*x]))^(7/2))","C",1
546,1,233,221,2.9499953,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^{7/2} \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(7/2)*Sqrt[Sec[c + d*x]]),x]","\frac{\cos ^7\left(\frac{1}{2} (c+d x)\right) \left(-\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)\right) \sec ^6\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} (4 (A+35 B) \cos (c+d x)+(17 B-5 A) \cos (2 (c+d x))+73 A+59 B)}{48 d}+\frac{i (13 A+7 B) 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)}{d}\right)}{8 (a (\cos (c+d x)+1))^{7/2}}","\frac{(13 A+7 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)}{64 \sqrt{2} a^{7/2} d}-\frac{(5 A-17 B) \sin (c+d x)}{192 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{(A+3 B) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}+\frac{(A-B) \sin (c+d x)}{6 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{7/2}}",1,"(Cos[(c + d*x)/2]^7*((I*(13*A + 7*B)*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))])])/(d*E^((I/2)*(c + d*x))) - ((73*A + 59*B + 4*(A + 35*B)*Cos[c + d*x] + (-5*A + 17*B)*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^6*Sqrt[Sec[c + d*x]]*(Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2]))/(48*d)))/(8*(a*(1 + Cos[c + d*x]))^(7/2))","C",0
547,1,488,221,7.2318152,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(3/2)),x]","\frac{\cos ^7\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\sec (c+d x)} \left(\frac{(17 A+67 B) \sin \left(\frac{c}{2}\right) \cos \left(\frac{d x}{2}\right)}{12 d}+\frac{(17 A+67 B) \cos \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)}{12 d}+\frac{\sec \left(\frac{c}{2}\right) \sec ^6\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)\right)}{3 d}+\frac{(A-B) \tan \left(\frac{c}{2}\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{\sec \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \left(29 B \sin \left(\frac{d x}{2}\right)-17 A \sin \left(\frac{d x}{2}\right)\right)}{12 d}-\frac{(17 A-29 B) \tan \left(\frac{c}{2}\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{12 d}+\frac{\sec \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \left(19 A \sin \left(\frac{d x}{2}\right)-151 B \sin \left(\frac{d x}{2}\right)\right)}{24 d}+\frac{(19 A-151 B) \tan \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d}\right)}{(a (\cos (c+d x)+1))^{7/2}}+\frac{i (7 A+5 B) 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)}} \cos ^7\left(\frac{c}{2}+\frac{d x}{2}\right) \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)}{8 d (a (\cos (c+d x)+1))^{7/2}}","\frac{(7 A+5 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)}{64 \sqrt{2} a^{7/2} d}+\frac{(17 A+67 B) \sin (c+d x)}{192 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{(A-B) \sin (c+d x)}{6 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}+\frac{(A-13 B) \sin (c+d x)}{48 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}",1,"((I/8)*(7*A + 5*B)*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]^7)/(d*E^((I/2)*(c + d*x))*(a*(1 + Cos[c + d*x]))^(7/2)) + (Cos[c/2 + (d*x)/2]^7*Sqrt[Sec[c + d*x]]*(((17*A + 67*B)*Cos[(d*x)/2]*Sin[c/2])/(12*d) + ((17*A + 67*B)*Cos[c/2]*Sin[(d*x)/2])/(12*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]^2*(19*A*Sin[(d*x)/2] - 151*B*Sin[(d*x)/2]))/(24*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]^6*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2]))/(3*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]^4*(-17*A*Sin[(d*x)/2] + 29*B*Sin[(d*x)/2]))/(12*d) + ((19*A - 151*B)*Sec[c/2 + (d*x)/2]*Tan[c/2])/(24*d) - ((17*A - 29*B)*Sec[c/2 + (d*x)/2]^3*Tan[c/2])/(12*d) + ((A - B)*Sec[c/2 + (d*x)/2]^5*Tan[c/2])/(3*d)))/(a*(1 + Cos[c + d*x]))^(7/2)","C",0
548,1,281,281,3.9635184,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(5/2)),x]","\frac{\cos ^7\left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{8} \left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^6\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} (4 (25 A-181 B) \cos (c+d x)+(67 A-247 B) \cos (2 (c+d x))+97 A-541 B)-3 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-177 B) \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+128 B \sinh ^{-1}\left(e^{i (c+d x)}\right)-128 B \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)\right)}{48 d (a (\cos (c+d x)+1))^{7/2}}","\frac{(5 A-177 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)}{64 \sqrt{2} a^{7/2} d}+\frac{2 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)}{a^{7/2} d}+\frac{(5 A-49 B) \sin (c+d x)}{64 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{(5 A-17 B) \sin (c+d x)}{48 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}+\frac{(A-B) \sin (c+d x)}{6 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}",1,"(Cos[(c + d*x)/2]^7*(((-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))]*(128*B*ArcSinh[E^(I*(c + d*x))] - Sqrt[2]*(5*A - 177*B)*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - 128*B*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))/E^((I/2)*(c + d*x)) + ((97*A - 541*B + 4*(25*A - 181*B)*Cos[c + d*x] + (67*A - 247*B)*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^6*Sqrt[Sec[c + d*x]]*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2]))/8))/(48*d*(a*(1 + Cos[c + d*x]))^(7/2))","C",1
549,1,1017,333,7.6548836,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x])/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(7/2)),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{\sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{(A-B) \tan \left(\frac{c}{2}\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{\sec \left(\frac{c}{2}\right) \left(53 B \sin \left(\frac{d x}{2}\right)-41 A \sin \left(\frac{d x}{2}\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{12 d}-\frac{(41 A-53 B) \tan \left(\frac{c}{2}\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{12 d}+\frac{\sec \left(\frac{c}{2}\right) \left(379 A \sin \left(\frac{d x}{2}\right)-703 B \sin \left(\frac{d x}{2}\right)\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d}+\frac{(379 A-703 B) \tan \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d}+\frac{(427 B-247 A) \cos \left(\frac{d x}{2}\right) \sin \left(\frac{c}{2}\right)}{12 d}+\frac{8 B \cos \left(\frac{3 d x}{2}\right) \sin \left(\frac{3 c}{2}\right)}{d}-\frac{(247 A-427 B) \cos \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)}{12 d}+\frac{8 B \cos \left(\frac{3 c}{2}\right) \sin \left(\frac{3 d x}{2}\right)}{d}\right) \cos ^7\left(\frac{c}{2}+\frac{d x}{2}\right)}{(a (\cos (c+d x)+1))^{7/2}}-\frac{49 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 ^7\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d (a (\cos (c+d x)+1))^{7/2}}+\frac{189 i B 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 ^7\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d (a (\cos (c+d x)+1))^{7/2}}+\frac{8 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 ^7\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{7/2}}-\frac{28 i \sqrt{2} B 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 ^7\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{7/2}}","\frac{(2 A-7 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)}{a^{7/2} d}-\frac{(177 A-637 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)}{64 \sqrt{2} a^{7/2} d}-\frac{7 (7 A-27 B) \sin (c+d x)}{64 a^3 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{(79 A-259 B) \sin (c+d x)}{192 a^2 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}+\frac{(3 A-7 B) \sin (c+d x)}{16 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}+\frac{(A-B) \sin (c+d x)}{6 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}",1,"(((-49*I)/8)*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]^7)/(d*E^((I/2)*(c + d*x))*(a*(1 + Cos[c + d*x]))^(7/2)) + (((189*I)/8)*B*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]^7)/(d*E^((I/2)*(c + d*x))*(a*(1 + Cos[c + d*x]))^(7/2)) + ((8*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]^7)/(d*E^((I/2)*(c + d*x))*(a*(1 + Cos[c + d*x]))^(7/2)) - ((28*I)*Sqrt[2]*B*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]^7)/(d*E^((I/2)*(c + d*x))*(a*(1 + Cos[c + d*x]))^(7/2)) + (Cos[c/2 + (d*x)/2]^7*Sqrt[Sec[c + d*x]]*(((-247*A + 427*B)*Cos[(d*x)/2]*Sin[c/2])/(12*d) + (8*B*Cos[(3*d*x)/2]*Sin[(3*c)/2])/d - ((247*A - 427*B)*Cos[c/2]*Sin[(d*x)/2])/(12*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]^2*(379*A*Sin[(d*x)/2] - 703*B*Sin[(d*x)/2]))/(24*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]^6*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2]))/(3*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]^4*(-41*A*Sin[(d*x)/2] + 53*B*Sin[(d*x)/2]))/(12*d) + (8*B*Cos[(3*c)/2]*Sin[(3*d*x)/2])/d + ((379*A - 703*B)*Sec[c/2 + (d*x)/2]*Tan[c/2])/(24*d) - ((41*A - 53*B)*Sec[c/2 + (d*x)/2]^3*Tan[c/2])/(12*d) + ((A - B)*Sec[c/2 + (d*x)/2]^5*Tan[c/2])/(3*d)))/(a*(1 + Cos[c + d*x]))^(7/2)","C",0
550,1,132,180,1.8657985,"\int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2),x]","\frac{\sec ^{\frac{5}{2}}(c+d x) \left(20 (a B+A b) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-12 (3 a A+5 b B) \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) (10 (a B+A b) \cos (c+d x)+3 (3 a A+5 b B) \cos (2 (c+d x))+15 (a A+b B))\right)}{30 d}","\frac{2 (a B+A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 (3 a A+5 b B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 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 d}-\frac{2 (3 a A+5 b 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 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}",1,"(Sec[c + d*x]^(5/2)*(-12*(3*a*A + 5*b*B)*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 20*(A*b + a*B)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 2*(15*(a*A + b*B) + 10*(A*b + a*B)*Cos[c + d*x] + 3*(3*a*A + 5*b*B)*Cos[2*(c + d*x)])*Sin[c + d*x]))/(30*d)","A",1
551,1,104,143,0.8292431,"\int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left((a A+3 b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-3 (a B+A b) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\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 (a B+A b) \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 (a A+3 b 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 (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)}{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)*EllipticE[(c + d*x)/2, 2] + (a*A + 3*b*B)*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
552,1,85,111,0.2667984,"\int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2),x]","\frac{2 \sqrt{\sec (c+d x)} \left((a B+A b) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\left((a A-b B) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)+a A \sin (c+d x)\right)}{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)}{d}-\frac{2 (a A-b 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 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}",1,"(2*Sqrt[Sec[c + d*x]]*(-((a*A - b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]) + (A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + a*A*Sin[c + d*x]))/d","A",1
553,1,90,115,0.2337089,"\int (a+b \cos (c+d x)) (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(2 (3 a A+b B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 (a B+A b) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b B \sin (2 (c+d x))\right)}{3 d}","\frac{2 (3 a A+b 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 (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)}{d}+\frac{2 b B \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Sec[c + d*x]]*(6*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*(3*a*A + b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + b*B*Sin[2*(c + d*x)]))/(3*d)","A",1
554,1,108,148,0.5418797,"\int \frac{(a+b \cos (c+d x)) (A+B \cos (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) (5 a B+5 A b+3 b B \cos (c+d x))+10 (a B+A b) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 (5 a A+3 b B) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d}","\frac{2 (a B+A b) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\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 d}+\frac{2 (5 a A+3 b 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 b B \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(6*(5*a*A + 3*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 10*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (5*A*b + 5*a*B + 3*b*B*Cos[c + d*x])*Sin[2*(c + d*x)]))/(15*d)","A",1
555,1,125,180,0.9942115,"\int \frac{(a+b \cos (c+d x)) (A+B \cos (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])*(A + B*Cos[c + d*x]))/Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) (42 (a B+A b) \cos (c+d x)+70 a A+15 b B \cos (2 (c+d x))+65 b B)+20 (7 a A+5 b B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+252 (a B+A b) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 (a B+A b) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (7 a A+5 b B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 (7 a A+5 b B) \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{6 (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)}{5 d}+\frac{2 b B \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(252*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 20*(7*a*A + 5*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (70*a*A + 65*b*B + 42*(A*b + a*B)*Cos[c + d*x] + 15*b*B*Cos[2*(c + d*x)])*Sin[2*(c + d*x)]))/(210*d)","A",1
556,1,171,221,2.4152007,"\int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2),x]","\frac{\sec ^{\frac{5}{2}}(c+d x) \left(20 \left(a^2 B+2 a A b+3 b^2 B\right) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-12 \left(3 a^2 A+10 a b B+5 A b^2\right) \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) \left(3 \left(3 a^2 A+10 a b B+5 A b^2\right) \cos (2 (c+d x))+15 \left(a^2 A+2 a b B+A b^2\right)+10 a (a B+2 A b) \cos (c+d x)\right)\right)}{30 d}","\frac{2 \left(a^2 B+2 a A b+3 b^2 B\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(3 a^2 A+5 b (2 a B+A b)\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{2 \left(3 a^2 A+5 b (2 a B+A b)\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 (5 a B+7 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+b)}{5 d}",1,"(Sec[c + d*x]^(5/2)*(-12*(3*a^2*A + 5*A*b^2 + 10*a*b*B)*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 20*(2*a*A*b + a^2*B + 3*b^2*B)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 2*(15*(a^2*A + A*b^2 + 2*a*b*B) + 10*a*(2*A*b + a*B)*Cos[c + d*x] + 3*(3*a^2*A + 5*A*b^2 + 10*a*b*B)*Cos[2*(c + d*x)])*Sin[c + d*x]))/(30*d)","A",1
557,1,125,177,1.1384739,"\int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(\left(a^2 A+6 a b B+3 A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-3 \left(a^2 B+2 a A b-b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{a \sin (c+d x) (3 (a B+2 A b) \cos (c+d x)+a A)}{\cos ^{\frac{3}{2}}(c+d x)}\right)}{3 d}","\frac{2 \left(a^2 A+6 a b B+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 d}-\frac{2 \left(a^2 B+2 a A b-b^2 B\right) \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 (3 a B+5 A b) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+b)}{3 d}",1,"(2*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-3*(2*a*A*b + a^2*B - b^2*B)*EllipticE[(c + d*x)/2, 2] + (a^2*A + 3*A*b^2 + 6*a*b*B)*EllipticF[(c + d*x)/2, 2] + (a*(a*A + 3*(2*A*b + a*B)*Cos[c + d*x])*Sin[c + d*x])/Cos[c + d*x]^(3/2)))/(3*d)","A",1
558,1,124,161,0.7494952,"\int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(2 \left(3 a^2 B+6 a A b+b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\left(-6 a^2 A+12 a b B+6 A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{2 \sin (c+d x) \left(3 a^2 A+b^2 B \cos (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)}{3 d}","\frac{2 \left(3 a^2 B+6 a A b+b^2 B\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^2 A-b (2 a B+A b)\right) \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^2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 b^2 B \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*A + 6*A*b^2 + 12*a*b*B)*EllipticE[(c + d*x)/2, 2] + 2*(6*a*A*b + 3*a^2*B + b^2*B)*EllipticF[(c + d*x)/2, 2] + (2*(3*a^2*A + b^2*B*Cos[c + d*x])*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(3*d)","A",1
559,1,128,171,0.9063988,"\int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(10 \left(3 a^2 A+2 a b B+A b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 \left(5 a^2 B+10 a A b+3 b^2 B\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b \sin (2 (c+d x)) (10 a B+5 A b+3 b B \cos (c+d x))\right)}{15 d}","\frac{2 \left(3 a^2 A+2 a b B+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 d}+\frac{2 \left(5 a^2 B+10 a A b+3 b^2 B\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 b (2 a B+A b) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 b^2 B \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(6*(10*a*A*b + 5*a^2*B + 3*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 10*(3*a^2*A + A*b^2 + 2*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + b*(5*A*b + 10*a*B + 3*b*B*Cos[c + d*x])*Sin[2*(c + d*x)]))/(15*d)","A",1
560,1,161,213,1.3405938,"\int \frac{(a+b \cos (c+d x))^2 (A+B \cos (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^2*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) \left(5 \left(14 a^2 B+28 a A b+3 b^2 B \cos (2 (c+d x))+13 b^2 B\right)+42 b (2 a B+A b) \cos (c+d x)\right)+20 \left(7 a^2 B+14 a A b+5 b^2 B\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+84 \left(5 a^2 A+6 a b B+3 A b^2\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 \left(7 a^2 B+14 a A b+5 b^2 B\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left(7 a^2 B+14 a A b+5 b^2 B\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(5 a^2 A+6 a b B+3 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 d}+\frac{2 b (2 a B+A b) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b^2 B \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(84*(5*a^2*A + 3*A*b^2 + 6*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 20*(14*a*A*b + 7*a^2*B + 5*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (42*b*(A*b + 2*a*B)*Cos[c + d*x] + 5*(28*a*A*b + 14*a^2*B + 13*b^2*B + 3*b^2*B*Cos[2*(c + d*x)]))*Sin[2*(c + d*x)]))/(210*d)","A",1
561,1,225,295,3.6516438,"\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*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 \left(5 a^2 A+21 a b B+21 A b^2\right) \tan (c+d x)+21 a^2 (a B+3 A b) \tan (c+d x) \sec (c+d x)+21 \left(3 a^3 B+9 a^2 A b+15 a b^2 B+5 A b^3\right) \sin (c+d x)+5 \left(5 a^3 A+21 a^2 b B+21 a A b^2+21 b^3 B\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-21 \left(3 a^3 B+9 a^2 A b+15 a b^2 B+5 A b^3\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{105 d}","\frac{2 a \left(5 a^2 A+21 a b B+18 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 a^2 (7 a B+11 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 \left(3 a^3 B+9 a^2 A b+15 a b^2 B+5 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 \left(5 a^3 A+21 a^2 b B+21 a A b^2+21 b^3 B\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(3 a^3 B+9 a^2 A b+15 a b^2 B+5 A b^3\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 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+b)^2}{7 d}",1,"(2*Sqrt[Sec[c + d*x]]*(-21*(9*a^2*A*b + 5*A*b^3 + 3*a^3*B + 15*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 5*(5*a^3*A + 21*a*A*b^2 + 21*a^2*b*B + 21*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 21*(9*a^2*A*b + 5*A*b^3 + 3*a^3*B + 15*a*b^2*B)*Sin[c + d*x] + 5*a*(5*a^2*A + 21*A*b^2 + 21*a*b*B)*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
562,1,192,244,1.5972337,"\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{a \sin (c+d x) \left(9 \left(a^2 A+5 a b B+5 A b^2\right) \cos (2 (c+d x))+15 \left(a^2 A+3 a b B+3 A b^2\right)+10 a (a B+3 A b) \cos (c+d x)\right)}{2 \cos ^{\frac{5}{2}}(c+d x)}+5 \left(a^3 B+3 a^2 A b+9 a b^2 B+3 A b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-3 \left(3 a^3 A+15 a^2 b B+15 a A b^2-5 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d}","\frac{2 a \left(3 a^2 A+15 a b B+14 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a^2 (5 a B+9 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{2 \left(a^3 B+3 a^2 A b+9 a b^2 B+3 A b^3\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(3 a^3 A+15 a^2 b B+15 a A b^2-5 b^3 B\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 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+b)^2}{5 d}",1,"(2*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-3*(3*a^3*A + 15*a*A*b^2 + 15*a^2*b*B - 5*b^3*B)*EllipticE[(c + d*x)/2, 2] + 5*(3*a^2*A*b + 3*A*b^3 + a^3*B + 9*a*b^2*B)*EllipticF[(c + d*x)/2, 2] + (a*(15*(a^2*A + 3*A*b^2 + 3*a*b*B) + 10*a*(3*A*b + a*B)*Cos[c + d*x] + 9*(a^2*A + 5*A*b^2 + 5*a*b*B)*Cos[2*(c + d*x)])*Sin[c + d*x])/(2*Cos[c + d*x]^(5/2))))/(15*d)","A",1
563,1,166,239,1.9414565,"\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{\sin (c+d x) \left(2 a^3 A+6 a^2 (a B+3 A b) \cos (c+d x)+b^3 B \cos (2 (c+d x))+b^3 B\right)}{\cos ^{\frac{3}{2}}(c+d x)}+2 \left(a^3 A+9 a^2 b B+9 a A b^2+b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d}","\frac{2 a \left(3 a^2 B+9 a A b-2 b^2 B\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 a^2 (a A-b B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 \left(a^3 A+9 a^2 b B+9 a A b^2+b^3 B\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^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) \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 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{3 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-6*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*EllipticE[(c + d*x)/2, 2] + 2*(a^3*A + 9*a*A*b^2 + 9*a^2*b*B + b^3*B)*EllipticF[(c + d*x)/2, 2] + ((2*a^3*A + b^3*B + 6*a^2*(3*A*b + a*B)*Cos[c + d*x] + b^3*B*Cos[2*(c + d*x)])*Sin[c + d*x])/Cos[c + d*x]^(3/2)))/(3*d)","A",1
564,1,172,237,1.4357235,"\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{2 \sin (c+d x) \left(3 \left(10 a^3 A+b^3 B \cos (2 (c+d x))+b^3 B\right)+10 b^2 (3 a B+A b) \cos (c+d x)\right)}{\sqrt{\cos (c+d x)}}+20 \left(3 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+12 \left(-5 a^3 A+15 a^2 b B+15 a A b^2+3 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{30 d}","\frac{2 a^2 (5 a A-b B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 \left(3 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\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(5 a^3 A-15 a^2 b B-15 a A b^2-3 b^3 B\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 b^2 (9 a B+5 A b) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(12*(-5*a^3*A + 15*a*A*b^2 + 15*a^2*b*B + 3*b^3*B)*EllipticE[(c + d*x)/2, 2] + 20*(9*a^2*A*b + A*b^3 + 3*a^3*B + 3*a*b^2*B)*EllipticF[(c + d*x)/2, 2] + (2*(10*b^2*(A*b + 3*a*B)*Cos[c + d*x] + 3*(10*a^3*A + b^3*B + b^3*B*Cos[2*(c + d*x)]))*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(30*d)","A",1
565,1,180,245,1.3062133,"\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(b \sin (2 (c+d x)) \left(5 \left(42 a^2 B+42 a A b+3 b^2 B \cos (2 (c+d x))+13 b^2 B\right)+42 b (3 a B+A b) \cos (c+d x)\right)+20 \left(21 a^3 A+21 a^2 b B+21 a A b^2+5 b^3 B\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+84 \left(5 a^3 B+15 a^2 A b+9 a b^2 B+3 A b^3\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 b \left(18 a^2 B+21 a A b+5 b^2 B\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left(21 a^3 A+21 a^2 b B+21 a A b^2+5 b^3 B\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(5 a^3 B+15 a^2 A b+9 a b^2 B+3 A b^3\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 b^2 (11 a B+7 A b) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(84*(15*a^2*A*b + 3*A*b^3 + 5*a^3*B + 9*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 20*(21*a^3*A + 21*a*A*b^2 + 21*a^2*b*B + 5*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + b*(42*b*(A*b + 3*a*B)*Cos[c + d*x] + 5*(42*a*A*b + 42*a^2*B + 13*b^2*B + 3*b^2*B*Cos[2*(c + d*x)]))*Sin[2*(c + d*x)]))/(210*d)","A",1
566,1,219,295,1.8324721,"\int \frac{(a+b \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) \left(7 b \left(108 a^2 B+108 a A b+43 b^2 B\right) \cos (c+d x)+5 \left(84 a^3 B+252 a^2 A b+18 b^2 (3 a B+A b) \cos (2 (c+d x))+234 a b^2 B+78 A b^3+7 b^3 B \cos (3 (c+d x))\right)\right)+120 \left(7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+168 \left(15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{1260 d}","\frac{2 b \left(22 a^2 B+27 a A b+7 b^2 B\right) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left(7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\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^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\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^2 (13 a B+9 A b) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 b B \sin (c+d x) (a \sec (c+d x)+b)^2}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(168*(15*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 7*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 120*(21*a^2*A*b + 5*A*b^3 + 7*a^3*B + 15*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (7*b*(108*a*A*b + 108*a^2*B + 43*b^2*B)*Cos[c + d*x] + 5*(252*a^2*A*b + 78*A*b^3 + 84*a^3*B + 234*a*b^2*B + 18*b^2*(A*b + 3*a*B)*Cos[2*(c + d*x)] + 7*b^3*B*Cos[3*(c + d*x)]))*Sin[2*(c + d*x)]))/(1260*d)","A",1
567,1,225,210,3.4261275,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x])*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 B)+3 a b (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 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 (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 b (A b-a B) \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 \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 + a^2*(A - 3*B) + 3*a*b*(A - B))*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]))/(a^3*d)","A",1
568,1,125,126,1.2854595,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x])*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(-(a A-a B+A b) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+(A b-a B) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+a A E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a^2 d \left(\sec ^2(c+d x)-2\right)}","-\frac{2 (A b-a B) \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 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}",1,"(-2*Cos[2*(c + d*x)]*Csc[c + d*x]*(a*A*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1] - (a*A + A*b - a*B)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] + (A*b - a*B)*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*Sec[c + d*x]*Sqrt[-Tan[c + d*x]^2])/(a^2*d*(-2 + Sec[c + d*x]^2))","A",1
569,1,76,101,0.5358785,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{a+b \cos (c+d x)} \, dx","Integrate[((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x]),x]","\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) \left((a B-A b) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+A b F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a b d}","\frac{2 (A b-a B) \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 d (a+b)}+\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)}{b d}",1,"(2*Cot[c + d*x]*(A*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] + (-(A*b) + a*B)*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*Sqrt[-Tan[c + d*x]^2])/(a*b*d)","A",1
570,1,220,149,6.3585514,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]]),x]","\frac{\cot (c+d x) \left(2 A 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)-2 a 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)+b B \sec ^{\frac{7}{2}}(c+d x)-b B \sec ^{\frac{3}{2}}(c+d x)+b B \cos (2 (c+d x)) \sec ^{\frac{7}{2}}(c+d x)-b B \cos (2 (c+d x)) \sec ^{\frac{3}{2}}(c+d x)+2 b B \sqrt{-\tan ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-2 b B \sqrt{-\tan ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{b^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)}{b^2 d}-\frac{2 a (A b-a B) \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 B \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]*(-(b*B*Sec[c + d*x]^(3/2)) - b*B*Cos[2*(c + d*x)]*Sec[c + d*x]^(3/2) + b*B*Sec[c + d*x]^(7/2) + b*B*Cos[2*(c + d*x)]*Sec[c + d*x]^(7/2) - 2*b*B*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] + 2*b*B*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] + 2*A*b*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] - 2*a*B*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2]))/(b^2*d)","A",1
571,1,278,197,6.6600158,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])*Sec[c + d*x]^(3/2)),x]","\frac{2 \csc (c+d x) \left(3 a^2 B \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)+b (-3 a B+3 A b+b 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)-3 b (A b-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)-3 a A b \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)-3 a b B \sec ^2(c+d x)+3 a b B+3 A b^2 \sec ^2(c+d x)-3 A b^2+b^2 B \sin (c+d x) \tan (c+d x)\right)}{3 b^3 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 a^2 (A b-a B) \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 \left(-3 a^2 B+3 a A b-b^2 B\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 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)}{b^2 d}+\frac{2 B \sin (c+d x)}{3 b d \sqrt{\sec (c+d x)}}",1,"(2*Csc[c + d*x]*(-3*A*b^2 + 3*a*b*B + 3*A*b^2*Sec[c + d*x]^2 - 3*a*b*B*Sec[c + d*x]^2 + b^2*B*Sin[c + d*x]*Tan[c + d*x] - 3*b*(A*b - a*B)*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[-Tan[c + d*x]^2] + b*(3*A*b - 3*a*B + b*B)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[-Tan[c + d*x]^2] - 3*a*A*b*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[-Tan[c + d*x]^2] + 3*a^2*B*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[-Tan[c + d*x]^2]))/(3*b^3*d*Sec[c + d*x]^(3/2))","A",1
572,1,735,405,7.1089056,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^2,x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{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{\left(2 a^3 B-4 a^2 A b-3 a b^2 B+5 A b^3\right) \sin (c+d x)}{a^3 \left(a^2-b^2\right)}\right)}{d}+\frac{\frac{\left(6 a^3 b B-12 a^2 A b^2-9 a b^3 B+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(12 a^4 B-28 a^3 A b-24 a^2 b^2 B+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{2 \left(-4 a^4 A+30 a^3 b B-44 a^2 A b^2-27 a b^3 B+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{b (A b-a B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{a d \left(a^2-b^2\right) (a \sec (c+d x)+b)}+\frac{\left(2 a^2 A+3 a b B-5 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d \left(a^2-b^2\right)}+\frac{\left(2 a^2 A+3 a b B-5 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 a^2 d \left(a^2-b^2\right)}-\frac{\left(-2 a^3 B+4 a^2 A b+3 a b^2 B-5 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^3 d \left(a^2-b^2\right)}+\frac{\left(-2 a^3 B+4 a^2 A b+3 a b^2 B-5 A b^3\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{b \left(-5 a^3 B+7 a^2 A b+3 a b^2 B-5 A b^3\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 + 30*a^3*b*B - 27*a*b^3*B)*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)*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)*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)*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*(a^2 - b^2)*(a + b*Cos[c + d*x])) + (2*A*Tan[c + d*x])/(3*a^2)))/d","A",0
573,1,681,316,6.9047989,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x])*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 b B-3 A b^2\right) \sin (c+d x)}{a^2 \left(a^2-b^2\right)}+\frac{A b^2 \sin (c+d x)-a b B \sin (c+d x)}{a \left(a^2-b^2\right) (a+b \cos (c+d x))}\right)}{d}-\frac{\frac{\left(2 a^2 A b+a b^2 B-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))}+\frac{2 \left(4 a^3 A+4 a^2 b B-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(-4 a^3 B+10 a^2 A b+3 a b^2 B-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))}}{4 a^2 d (a-b) (a+b)}","\frac{b (A b-a B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{a d \left(a^2-b^2\right) (a \sec (c+d x)+b)}+\frac{\left(2 a^2 A+a b B-3 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d \left(a^2-b^2\right)}+\frac{(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)}{a d \left(a^2-b^2\right)}-\frac{\left(2 a^2 A+a b B-3 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^2 d \left(a^2-b^2\right)}-\frac{\left(-3 a^3 B+5 a^2 A b+a b^2 B-3 A b^3\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) (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)*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)*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)*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)*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*(a^2 - b^2)*(a + b*Cos[c + d*x]))))/d","B",0
574,1,639,260,6.8378174,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^2} \, dx","Integrate[((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^2,x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{a B \sin (c+d x)-A b \sin (c+d x)}{\left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{(a B-A b) \sin (c+d x)}{a \left(a^2-b^2\right)}\right)}{d}+\frac{\frac{2 \left(-4 a^2 A+a b B+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 b^2-a b B\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 A b-4 a^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))}}{4 a d (b-a) (a+b)}","\frac{b (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d \left(a^2-b^2\right) (a \sec (c+d x)+b)}-\frac{(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)}{b d \left(a^2-b^2\right)}-\frac{(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 d \left(a^2-b^2\right)}+\frac{\left(a^3 (-B)+3 a^2 A b-a b^2 B-A b^3\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) (a+b)^2}",1,"((2*(-4*a^2*A + 3*A*b^2 + a*b*B)*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)*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)*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) + a*B)*Sin[c + d*x])/(a*(a^2 - b^2))) + (-(A*b*Sin[c + d*x]) + a*B*Sin[c + d*x])/((a^2 - b^2)*(a + b*Cos[c + d*x]))))/d","B",0
575,1,626,258,6.8193114,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x))^2 \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{(A b-a B) \sin (c+d x)}{b \left(b^2-a^2\right)}+\frac{a^2 B \sin (c+d x)-a A b \sin (c+d x)}{b \left(b^2-a^2\right) (a+b \cos (c+d x))}\right)}{d}+\frac{\frac{(A b-a B) \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-4 b B) \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 (a B-A b) \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))}}{4 d (a-b) (a+b)}","-\frac{(A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d \left(a^2-b^2\right) (a \sec (c+d x)+b)}+\frac{\left(a^2 B+a A b-2 b^2 B\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{(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)}{b d \left(a^2-b^2\right)}-\frac{\left(a^3 B+a^2 A b-3 a b^2 B+A b^3\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) (a+b)^2}",1,"((2*(-(A*b) + a*B)*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 - 4*b*B)*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 - a*B)*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)*(a + b)*d) + (Sqrt[Sec[c + d*x]]*(((A*b - a*B)*Sin[c + d*x])/(b*(-a^2 + b^2)) + (-(a*A*b*Sin[c + d*x]) + a^2*B*Sin[c + d*x])/(b*(-a^2 + b^2)*(a + b*Cos[c + d*x]))))/d","B",0
576,1,655,284,6.888161,"\int \frac{A+B \cos (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])/((a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)),x]","\frac{\frac{2 \left(a^2 (-B)-a A b+2 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) \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 B+a A b+2 b^2 B\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 b^2-4 a b 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))}}{4 b d (b-a) (a+b)}+\frac{\sqrt{\sec (c+d x)} \left(\frac{a^2 A b \sin (c+d x)-a^3 B \sin (c+d x)}{b^2 \left(b^2-a^2\right) (a+b \cos (c+d x))}-\frac{a (a B-A b) \sin (c+d x)}{b^2 \left(a^2-b^2\right)}\right)}{d}","\frac{a (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{b d \left(a^2-b^2\right) (a \sec (c+d x)+b)}-\frac{\left(-3 a^2 B+a A b+2 b^2 B\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{\left(-3 a^3 B+a^2 A b+4 a b^2 B-2 A b^3\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{a \left(-3 a^3 B+a^2 A b+5 a b^2 B-3 A b^3\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*A*b) - a^2*B + 2*b^2*B)*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*b^2 - 4*a*b*B)*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*A*b - 3*a^2*B + 2*b^2*B)*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*b*(-a + b)*(a + b)*d) + (Sqrt[Sec[c + d*x]]*(-((a*(-(A*b) + a*B)*Sin[c + d*x])/(b^2*(a^2 - b^2))) + (a^2*A*b*Sin[c + d*x] - a^3*B*Sin[c + d*x])/(b^2*(-a^2 + b^2)*(a + b*Cos[c + d*x]))))/d","B",0
577,1,701,363,6.9921059,"\int \frac{A+B \cos (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])/((a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{a^2 (a B-A b) \sin (c+d x)}{b^3 \left(a^2-b^2\right)}-\frac{a^3 A b \sin (c+d x)-a^4 B \sin (c+d x)}{b^3 \left(b^2-a^2\right) (a+b \cos (c+d x))}+\frac{B \sin (2 (c+d x))}{3 b^2}\right)}{d}-\frac{\frac{2 \left(8 a^2 b B-12 a A b^2+4 b^3 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))}+\frac{2 \left(5 a^3 B-3 a^2 A b-8 a b^2 B+6 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(15 a^3 B-9 a^2 A b-12 a b^2 B+6 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))}}{12 b^2 d (a-b) (a+b)}","\frac{a (A b-a B) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a \sec (c+d x)+b)}-\frac{\left(-5 a^2 B+3 a A b+2 b^2 B\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}+\frac{\left(-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\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{a^2 \left(-5 a^3 B+3 a^2 A b+7 a b^2 B-5 A b^3\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}-\frac{\left(-15 a^4 B+9 a^3 A b+16 a^2 b^2 B-12 a A b^3+2 b^4 B\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)}",1,"-1/12*((2*(-3*a^2*A*b + 6*A*b^3 + 5*a^3*B - 8*a*b^2*B)*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*A*b^2 + 8*a^2*b*B + 4*b^3*B)*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 + 6*A*b^3 + 15*a^3*B - 12*a*b^2*B)*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)*b^2*(a + b)*d) + (Sqrt[Sec[c + d*x]]*((a^2*(-(A*b) + a*B)*Sin[c + d*x])/(b^3*(a^2 - b^2)) - (a^3*A*b*Sin[c + d*x] - a^4*B*Sin[c + d*x])/(b^3*(-a^2 + b^2)*(a + b*Cos[c + d*x])) + (B*Sin[2*(c + d*x)])/(3*b^2)))/d","A",0
578,1,844,480,7.2088774,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[((A + B*Cos[c + d*x])*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+9 b B a^3-29 A b^2 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{A b^2 \sin (c+d x)-a b B \sin (c+d x)}{2 a \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{-5 A \sin (c+d x) b^4+a B \sin (c+d x) b^3+11 a^2 A \sin (c+d x) b^2-7 a^3 B \sin (c+d x) b}{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+19 b^2 B a^3-95 A b^3 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+32 b B a^4-80 A b^2 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+9 a^3 B b^2+8 a^4 A 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{b (A b-a B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 a d \left(a^2-b^2\right) (a \sec (c+d x)+b)^2}+\frac{b \left(-7 a^3 B+11 a^2 A b+a b^2 B-5 A b^3\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 (a \sec (c+d x)+b)}+\frac{\left(-7 a^3 B+11 a^2 A b+a b^2 B-5 A b^3\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 d \left(a^2-b^2\right)^2}+\frac{\left(8 a^4 A+9 a^3 b B-29 a^2 A b^2-3 a b^3 B+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(8 a^4 A+9 a^3 b B-29 a^2 A b^2-3 a b^3 B+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(-15 a^5 B+35 a^4 A b+6 a^3 b^2 B-38 a^2 A b^3-3 a b^4 B+15 A b^5\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 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)*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)*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)*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)*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])/(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])/(4*a^2*(a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/d","A",0
579,1,797,405,7.1083569,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^3} \, dx","Integrate[((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{2 \left(16 A a^4-9 b B a^3-19 A b^2 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+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(3 A b^4+a B b^3-9 a^2 A b^2+5 a^3 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)}}{16 a^2 (a-b)^2 (a+b)^2 d}+\frac{\sqrt{\sec (c+d x)} \left(-\frac{\left(5 B a^3-9 A b a^2+b^2 B a+3 A b^3\right) \sin (c+d x)}{4 a^2 \left(a^2-b^2\right)^2}+\frac{a B \sin (c+d x)-A b \sin (c+d x)}{2 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{3 B \sin (c+d x) a^3-7 A b \sin (c+d x) a^2+3 b^2 B \sin (c+d x) a+A b^3 \sin (c+d x)}{4 a \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}\right)}{d}","\frac{b (A b-a B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 a d \left(a^2-b^2\right) (a \sec (c+d x)+b)^2}+\frac{b \left(-5 a^3 B+9 a^2 A b-a b^2 B-3 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^2 d \left(a^2-b^2\right)^2 (a \sec (c+d x)+b)}-\frac{\left(-3 a^3 B+7 a^2 A b-3 a b^2 B-A b^3\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 d \left(a^2-b^2\right)^2}-\frac{\left(-5 a^3 B+9 a^2 A b-a b^2 B-3 A b^3\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 d \left(a^2-b^2\right)^2}+\frac{\left(-3 a^5 B+15 a^4 A b-10 a^3 b^2 B-6 a^2 A b^3+a b^4 B+3 A b^5\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 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)*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)*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)*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]]*(-1/4*((-9*a^2*A*b + 3*A*b^3 + 5*a^3*B + a*b^2*B)*Sin[c + d*x])/(a^2*(a^2 - b^2)^2) + (-(A*b*Sin[c + d*x]) + a*B*Sin[c + d*x])/(2*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) + (-7*a^2*A*b*Sin[c + d*x] + A*b^3*Sin[c + d*x] + 3*a^3*B*Sin[c + d*x] + 3*a*b^2*B*Sin[c + d*x])/(4*a*(a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/d","A",0
580,1,784,402,6.9308131,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x))^3 \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]),x]","\frac{\sqrt{\sec (c+d x)} \left(-\frac{a A b \sin (c+d x)-a^2 B \sin (c+d x)}{2 b \left(b^2-a^2\right) (a+b \cos (c+d x))^2}+\frac{\left(a^3 B-5 a^2 A b+5 a b^2 B-A b^3\right) \sin (c+d x)}{4 a b \left(a^2-b^2\right)^2}+\frac{a^3 B \sin (c+d x)+3 a^2 A b \sin (c+d x)-7 a b^2 B \sin (c+d x)+3 A b^3 \sin (c+d x)}{4 b \left(b^2-a^2\right)^2 (a+b \cos (c+d x))}\right)}{d}+\frac{\frac{2 \left(16 a^3 A-24 a^2 b B+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(5 a^3 B-9 a^2 A b+a b^2 B+3 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(a^3 (-B)+5 a^2 A b-5 a b^2 B+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))}}{16 a d (a-b)^2 (a+b)^2}","\frac{b (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \left(a^2-b^2\right) (a \sec (c+d x)+b)^2}-\frac{\left(-3 a^3 B+7 a^2 A b-3 a b^2 B-A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a d \left(a^2-b^2\right)^2 (a \sec (c+d x)+b)}+\frac{\left(a^3 B+3 a^2 A b-7 a b^2 B+3 A b^3\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^2 d \left(a^2-b^2\right)^2}+\frac{\left(a^3 (-B)+5 a^2 A b-5 a b^2 B+A b^3\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 d \left(a^2-b^2\right)^2}-\frac{\left(a^5 B+3 a^4 A b-10 a^3 b^2 B+10 a^2 A b^3-3 a b^4 B-A b^5\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^2 d (a-b)^2 (a+b)^3}",1,"((2*(-9*a^2*A*b + 3*A*b^3 + 5*a^3*B + a*b^2*B)*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 + 8*a*A*b^2 - 24*a^2*b*B)*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 + A*b^3 - a^3*B - 5*a*b^2*B)*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*(a - b)^2*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(((-5*a^2*A*b - A*b^3 + a^3*B + 5*a*b^2*B)*Sin[c + d*x])/(4*a*b*(a^2 - b^2)^2) - (a*A*b*Sin[c + d*x] - a^2*B*Sin[c + d*x])/(2*b*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (3*a^2*A*b*Sin[c + d*x] + 3*A*b^3*Sin[c + d*x] + a^3*B*Sin[c + d*x] - 7*a*b^2*B*Sin[c + d*x])/(4*b*(-a^2 + b^2)^2*(a + b*Cos[c + d*x]))))/d","A",0
581,1,786,400,6.9252622,"\int \frac{A+B \cos (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])/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)),x]","\frac{\sqrt{\sec (c+d x)} \left(-\frac{a^3 B \sin (c+d x)-a^2 A b \sin (c+d x)}{2 b^2 \left(b^2-a^2\right) (a+b \cos (c+d x))^2}+\frac{\left(3 a^3 B+a^2 A b-9 a b^2 B+5 A b^3\right) \sin (c+d x)}{4 b^2 \left(a^2-b^2\right)^2}+\frac{-5 a^4 B \sin (c+d x)+a^3 A b \sin (c+d x)+11 a^2 b^2 B \sin (c+d x)-7 a A b^3 \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(-8 a^2 b B+24 a A b^2-16 b^3 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))}+\frac{2 \left(a^3 B-5 a^2 A b+5 a b^2 B-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(3 a^3 B+a^2 A b-9 a b^2 B+5 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))}}{16 b d (a-b)^2 (a+b)^2}","-\frac{(A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 d \left(a^2-b^2\right) (a \sec (c+d x)+b)^2}+\frac{\left(a^3 B+3 a^2 A b-7 a b^2 B+3 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 b d \left(a^2-b^2\right)^2 (a \sec (c+d x)+b)}-\frac{\left(3 a^3 B+a^2 A b-9 a b^2 B+5 A b^3\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^2 d \left(a^2-b^2\right)^2}+\frac{\left(3 a^4 B+a^3 A b-5 a^2 b^2 B-7 a A b^3+8 b^4 B\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^5 B+a^4 A b-6 a^3 b^2 B-10 a^2 A b^3+15 a b^4 B-3 A b^5\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^3 d (a-b)^2 (a+b)^3}",1,"-1/16*((2*(-5*a^2*A*b - A*b^3 + a^3*B + 5*a*b^2*B)*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^2 - 8*a^2*b*B - 16*b^3*B)*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 + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*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*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(((a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2) - (-(a^2*A*b*Sin[c + d*x]) + a^3*B*Sin[c + d*x])/(2*b^2*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (a^3*A*b*Sin[c + d*x] - 7*a*A*b^3*Sin[c + d*x] - 5*a^4*B*Sin[c + d*x] + 11*a^2*b^2*B*Sin[c + d*x])/(4*b^2*(-a^2 + b^2)^2*(a + b*Cos[c + d*x]))))/d","A",0
582,1,820,427,7.0800773,"\int \frac{A+B \cos (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])/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)),x]","\frac{\frac{2 \left(5 B a^4-A b a^3-7 b^2 B a^2-5 A b^3 a+8 b^4 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 b^4-32 a B b^3+8 a^2 A b^2+8 a^3 B 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 B a^4-3 A b a^3-29 b^2 B a^2+9 A b^3 a+8 b^4 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-b)^2 b^2 (a+b)^2 d}+\frac{\sqrt{\sec (c+d x)} \left(-\frac{a \left(7 B a^3-3 A b a^2-13 b^2 B a+9 A b^3\right) \sin (c+d x)}{4 b^3 \left(a^2-b^2\right)^2}-\frac{a^3 A b \sin (c+d x)-a^4 B \sin (c+d x)}{2 b^3 \left(b^2-a^2\right) (a+b \cos (c+d x))^2}+\frac{9 B \sin (c+d x) a^5-5 A b \sin (c+d x) a^4-15 b^2 B \sin (c+d x) a^3+11 A b^3 \sin (c+d x) a^2}{4 b^3 \left(b^2-a^2\right)^2 (a+b \cos (c+d x))}\right)}{d}","\frac{a (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 b d \left(a^2-b^2\right) (a \sec (c+d x)+b)^2}+\frac{a \left(-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 b^2 d \left(a^2-b^2\right)^2 (a \sec (c+d x)+b)}-\frac{\left(-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\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^5 B+3 a^4 A b+33 a^3 b^2 B-5 a^2 A b^3-24 a b^4 B+8 A b^5\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{a \left(-15 a^5 B+3 a^4 A b+38 a^3 b^2 B-6 a^2 A b^3-35 a b^4 B+15 A b^5\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*(-(a^3*A*b) - 5*a*A*b^3 + 5*a^4*B - 7*a^2*b^2*B + 8*b^4*B)*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*(8*a^2*A*b^2 + 16*A*b^4 + 8*a^3*b*B - 32*a*b^3*B)*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^3*A*b + 9*a*A*b^3 + 15*a^4*B - 29*a^2*b^2*B + 8*b^4*B)*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*(-3*a^2*A*b + 9*A*b^3 + 7*a^3*B - 13*a*b^2*B)*Sin[c + d*x])/(b^3*(a^2 - b^2)^2) - (a^3*A*b*Sin[c + d*x] - a^4*B*Sin[c + d*x])/(2*b^3*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (-5*a^4*A*b*Sin[c + d*x] + 11*a^2*A*b^3*Sin[c + d*x] + 9*a^5*B*Sin[c + d*x] - 15*a^3*b^2*B*Sin[c + d*x])/(4*b^3*(-a^2 + b^2)^2*(a + b*Cos[c + d*x]))))/d","A",0
583,1,865,521,7.3153364,"\int \frac{A+B \cos (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])/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{\left(11 B a^3-7 A b a^2-17 b^2 B a+13 A b^3\right) \sin (c+d x) a^2}{4 b^4 \left(a^2-b^2\right)^2}-\frac{a^5 B \sin (c+d x)-a^4 A b \sin (c+d x)}{2 b^4 \left(b^2-a^2\right) (a+b \cos (c+d x))^2}+\frac{-13 B \sin (c+d x) a^6+9 A b \sin (c+d x) a^5+19 b^2 B \sin (c+d x) a^4-15 A b^3 \sin (c+d x) a^3}{4 b^4 \left(b^2-a^2\right)^2 (a+b \cos (c+d x))}+\frac{B \sin (2 (c+d x))}{3 b^3}\right)}{d}-\frac{\frac{2 \left(35 B a^5-15 A b a^4-73 b^2 B a^3+21 A b^3 a^2+56 b^4 B a-24 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 B b^5+96 a A b^4-112 a^2 B b^3-24 a^3 A b^2+56 a^4 B 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 B a^5-45 A b a^4-195 b^2 B a^3+87 A b^3 a^2+72 b^4 B a-24 A b^5\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{a (A b-a B) \sin (c+d x)}{2 b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a \sec (c+d x)+b)^2}+\frac{a \left(-7 a^3 B+3 a^2 A b+13 a b^2 B-9 A b^3\right) \sin (c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a \sec (c+d x)+b)}-\frac{\left(-35 a^4 B+15 a^3 A b+61 a^2 b^2 B-33 a A b^3-8 b^4 B\right) \sin (c+d x)}{12 b^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}+\frac{\left(-35 a^5 B+15 a^4 A b+65 a^3 b^2 B-29 a^2 A b^3-24 a b^4 B+8 A b^5\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{a^2 \left(-35 a^5 B+15 a^4 A b+86 a^3 b^2 B-38 a^2 A b^3-63 a b^4 B+35 A b^5\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}-\frac{\left(-105 a^6 B+45 a^5 A b+223 a^4 b^2 B-99 a^3 A b^3-128 a^2 b^4 B+72 a A b^5-8 b^6 B\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}",1,"-1/48*((2*(-15*a^4*A*b + 21*a^2*A*b^3 - 24*A*b^5 + 35*a^5*B - 73*a^3*b^2*B + 56*a*b^4*B)*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^3*A*b^2 + 96*a*A*b^4 + 56*a^4*b*B - 112*a^2*b^3*B - 16*b^5*B)*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^4*A*b + 87*a^2*A*b^3 - 24*A*b^5 + 105*a^5*B - 195*a^3*b^2*B + 72*a*b^4*B)*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^2*(-7*a^2*A*b + 13*A*b^3 + 11*a^3*B - 17*a*b^2*B)*Sin[c + d*x])/(4*b^4*(a^2 - b^2)^2) - (-(a^4*A*b*Sin[c + d*x]) + a^5*B*Sin[c + d*x])/(2*b^4*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (9*a^5*A*b*Sin[c + d*x] - 15*a^3*A*b^3*Sin[c + d*x] - 13*a^6*B*Sin[c + d*x] + 19*a^4*b^2*B*Sin[c + d*x])/(4*b^4*(-a^2 + b^2)^2*(a + b*Cos[c + d*x])) + (B*Sin[2*(c + d*x)])/(3*b^3)))/d","A",0
584,1,47,64,0.0705638,"\int \frac{(a B+b B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x]),x]","\frac{2 B \sec ^{\frac{3}{2}}(c+d x) \left(\sin (c+d x)+\cos ^{\frac{3}{2}}(c+d x) F\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}",1,"(2*B*Sec[c + d*x]^(3/2)*(Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + Sin[c + d*x]))/(3*d)","A",1
585,1,46,60,0.0457094,"\int \frac{(a B+b B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x]),x]","\frac{2 B \sqrt{\sec (c+d x)} \left(\sin (c+d x)-\sqrt{\cos (c+d x)} E\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}",1,"(2*B*Sqrt[Sec[c + d*x]]*(-(Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]) + Sin[c + d*x]))/d","A",1
586,1,37,37,0.0298219,"\int \frac{(a B+b B \cos (c+d x)) \sqrt{\sec (c+d x)}}{a+b \cos (c+d x)} \, dx","Integrate[((a*B + b*B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x]),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)}{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*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d","A",1
587,1,37,37,0.0371392,"\int \frac{a B+b B \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt{\sec (c+d x)}} \, dx","Integrate[(a*B + b*B*Cos[c + d*x])/((a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]]),x]","\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}","\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,"(2*B*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",1
588,1,50,64,0.0463249,"\int \frac{a B+b B \cos (c+d x)}{(a+b \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a*B + b*B*Cos[c + d*x])/((a + b*Cos[c + d*x])*Sec[c + d*x]^(3/2)),x]","\frac{B \sqrt{\sec (c+d x)} \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)}{3 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}",1,"(B*Sqrt[Sec[c + d*x]]*(2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + Sin[2*(c + d*x)]))/(3*d)","A",1
589,1,56,64,0.0733139,"\int \frac{a B+b B \cos (c+d x)}{(a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a*B + b*B*Cos[c + d*x])/((a + b*Cos[c + d*x])*Sec[c + d*x]^(5/2)),x]","\frac{B \sqrt{\sec (c+d x)} \left(\sin (c+d x)+\sin (3 (c+d x))+12 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{10 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}",1,"(B*Sqrt[Sec[c + d*x]]*(12*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + Sin[c + d*x] + Sin[3*(c + d*x)]))/(10*d)","A",1
590,1,3321,473,23.759321,"\int \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2),x]","\text{Result too large to show}","\frac{2 \left(25 a^2 A+7 a b B-4 A b^2\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 (a-b) \sqrt{a+b} \left(a^2 (25 A-63 B)+2 a b (3 A-7 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-b) \sqrt{a+b} \left(63 a^3 B+19 a^2 A b-14 a b^2 B+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}} 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)*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]))/(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]]) + (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]]) - (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]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*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*(a + b)*(8*A*b^2 - 2*a*b*(3*A + 7*B) + a^2*(25*A + 63*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)] - (19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*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)*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*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*(a + b)*(8*A*b^2 - 2*a*b*(3*A + 7*B) + a^2*(25*A + 63*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)] - (19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*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)*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*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*(a + b)*(8*A*b^2 - 2*a*b*(3*A + 7*B) + a^2*(25*A + 63*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)] - (19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*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*((19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4) - ((a + b)*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*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*(a + b)*(8*A*b^2 - 2*a*b*(3*A + 7*B) + a^2*(25*A + 63*B))*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)*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*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*(a + b)*(8*A*b^2 - 2*a*b*(3*A + 7*B) + a^2*(25*A + 63*B))*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*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*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))*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)*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*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]))/(105*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((-2*(a + b)*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*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*(a + b)*(8*A*b^2 - 2*a*b*(3*A + 7*B) + a^2*(25*A + 63*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)] - (19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*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]))/(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
591,1,423,390,17.4725409,"\int \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*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+5 a b B-2 A b^2\right) \sin (c+d x)}{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{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(-\left(\left(9 a^2 A+5 a b B-2 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))\right)-2 (a+b) \left(9 a^2 A+5 a b B-2 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)+2 a (a+b) (9 a A+5 a B-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)\right)}{15 a^2 d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \cos (c+d x)}}","-\frac{2 (a-b) \sqrt{a+b} (9 a A-5 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)}{15 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \left(9 a^2 A+5 a b 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}} 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[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(9*a^2*A - 2*A*b^2 + 5*a*b*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*(a + b)*(9*a*A - 2*A*b + 5*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)] - (9*a^2*A - 2*A*b^2 + 5*a*b*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*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]]*((2*(9*a^2*A - 2*A*b^2 + 5*a*b*B)*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
592,1,346,324,14.4510113,"\int \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2),x]","\frac{\sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 (3 a B+A b) \sin (c+d x)}{3 a}+\frac{2}{3} A \tan (c+d x)\right)}{d}+\frac{2 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(-\left((3 a B+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))\right)+2 a (a+b) (A+3 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 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{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \cos (c+d x)}}","\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 (a-b) \sqrt{a+b} (A-3 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 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}",1,"(2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(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*(a + b)*(A + 3*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)] - (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*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*(A*b + 3*a*B)*Sin[c + d*x])/(3*a) + (2*A*Tan[c + d*x])/3))/d","A",0
593,1,635,411,17.2805308,"\int \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2),x]","\frac{2 \left(-(a (A+B)+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)+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 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)-2 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)+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)}{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)+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 \sqrt{a+b} (A b-a (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 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 d \sqrt{\sec (c+d x)}}-\frac{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}} \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*A*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*(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*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*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)] + 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)] - (b*(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)]*(-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)])","A",0
594,1,787,445,17.546695,"\int \sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sqrt[Sec[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
595,1,1121,533,18.619321,"\int \frac{\sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{B \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (2 (c+d x))}{4 d}+\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(4 A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)-4 a A b \tan ^5\left(\frac{1}{2} (c+d x)\right)-a^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+a b B \tan ^5\left(\frac{1}{2} (c+d x)\right)-8 A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 a b B \tan ^3\left(\frac{1}{2} (c+d x)\right)+8 a A 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^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)+8 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)+4 A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+4 a A b \tan \left(\frac{1}{2} (c+d x)\right)+a^2 B \tan \left(\frac{1}{2} (c+d x)\right)+a b B \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) (4 A b+a 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 A-a B+2 b 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}}+8 a A 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^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}}+8 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}}\right)}{4 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{\sqrt{a+b} \left(a^2 (-B)+4 a A b+4 b^2 B\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{(a B+4 A b) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}+\frac{\sqrt{a+b} (B (a+2 b)+4 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-b) \sqrt{a+b} (a B+4 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 a b d \sqrt{\sec (c+d x)}}+\frac{B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}",1,"(B*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[2*(c + d*x)])/(4*d) + (Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(4*a*A*b*Tan[(c + d*x)/2] + 4*A*b^2*Tan[(c + d*x)/2] + a^2*B*Tan[(c + d*x)/2] + a*b*B*Tan[(c + d*x)/2] - 8*A*b^2*Tan[(c + d*x)/2]^3 - 2*a*b*B*Tan[(c + d*x)/2]^3 - 4*a*A*b*Tan[(c + d*x)/2]^5 + 4*A*b^2*Tan[(c + d*x)/2]^5 - a^2*B*Tan[(c + d*x)/2]^5 + a*b*B*Tan[(c + d*x)/2]^5 + 8*a*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^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)] + 8*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)] + 8*a*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^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)] + 8*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)] + (a + b)*(4*A*b + 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)] - 2*b*(4*a*A - a*B + 2*b*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)]))/(4*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",0
596,1,1533,620,14.6491875,"\int \frac{\sqrt{a+b \cos (c+d x)} (A+B \cos (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Cos[c + d*x]]*(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{1}{12} B \sin (c+d x)+\frac{(6 A b+a B) \sin (2 (c+d x))}{24 b}+\frac{1}{12} B \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(6 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)-6 a^2 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 a^3 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+16 b^3 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 a b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 a^2 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)-12 a A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)-32 b^3 B \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 a^2 b B \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)-12 a^2 A 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 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 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)+6 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+6 a^2 A b \tan \left(\frac{1}{2} (c+d x)\right)-3 a^3 B \tan \left(\frac{1}{2} (c+d x)\right)+16 b^3 B \tan \left(\frac{1}{2} (c+d x)\right)+16 a b^2 B \tan \left(\frac{1}{2} (c+d x)\right)-3 a^2 b B \tan \left(\frac{1}{2} (c+d x)\right)-(a+b) \left(3 B a^2-6 A b a-16 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 b \left(B a^2+2 b (3 A-7 B) a-12 A b^2\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}}-12 a^2 A 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 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 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}}\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{\left(-3 a^2 B+6 a A b+16 b^2 B\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{24 b^2 d}-\frac{(a-b) \sqrt{a+b} \left(-3 a^2 B+6 a A b+16 b^2 B\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{\sqrt{a+b} \left(a^3 (-B)+2 a^2 A b-4 a b^2 B-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}} \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{\sqrt{a+b} (a+2 b) (-3 a B+6 A b+8 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}} 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{(2 A b-a B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b d \sqrt{\sec (c+d x)}}+\frac{B \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]]*((B*Sin[c + d*x])/12 + ((6*A*b + a*B)*Sin[2*(c + d*x)])/(24*b) + (B*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)]*(6*a^2*A*b*Tan[(c + d*x)/2] + 6*a*A*b^2*Tan[(c + d*x)/2] - 3*a^3*B*Tan[(c + d*x)/2] - 3*a^2*b*B*Tan[(c + d*x)/2] + 16*a*b^2*B*Tan[(c + d*x)/2] + 16*b^3*B*Tan[(c + d*x)/2] - 12*a*A*b^2*Tan[(c + d*x)/2]^3 + 6*a^2*b*B*Tan[(c + d*x)/2]^3 - 32*b^3*B*Tan[(c + d*x)/2]^3 - 6*a^2*A*b*Tan[(c + d*x)/2]^5 + 6*a*A*b^2*Tan[(c + d*x)/2]^5 + 3*a^3*B*Tan[(c + d*x)/2]^5 - 3*a^2*b*B*Tan[(c + d*x)/2]^5 - 16*a*b^2*B*Tan[(c + d*x)/2]^5 + 16*b^3*B*Tan[(c + d*x)/2]^5 - 12*a^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)] + 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)] + 6*a^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)] + 24*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)] - 12*a^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)] + 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)] + 6*a^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)] + 24*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)] - (a + b)*(-6*a*A*b + 3*a^2*B - 16*b^2*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*b*(-12*A*b^2 + 2*a*b*(3*A - 7*B) + a^2*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)]))/(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
597,1,3739,562,25.9868125,"\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(11/2),x]","\text{Result too large to show}","\frac{2 \left(49 a^2 A+72 a b B+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{2 \left(75 a^3 B+88 a^2 A b+9 a b^2 B-4 A b^3\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(-\left(a^3 (147 A-75 B)\right)+3 a^2 b (13 A-57 B)+6 a b^2 (A-3 B)+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-b) \sqrt{a+b} \left(147 a^4 A+246 a^3 b B+33 a^2 A b^2-18 a b^3 B+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 (9 a B+10 A b) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{63 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{9 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 + 246*a^3*b*B - 18*a*b^3*B)*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]))/(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]))/(315*a^2) + (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]]) - (82*a*b*B)/(105*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b^3*B)/(35*a*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]]) + (5*a^2*B*Sqrt[Sec[c + d*x]])/(21*Sqrt[a + b*Cos[c + d*x]]) - (31*b^2*B*Sqrt[Sec[c + d*x]])/(105*Sqrt[a + b*Cos[c + d*x]]) + (2*b^4*B*Sqrt[Sec[c + d*x]])/(35*a^2*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]]) - (82*b^2*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*Sqrt[a + b*Cos[c + d*x]]) + (2*b^4*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(35*a^2*Sqrt[a + b*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*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*(a + b)*(8*A*b^3 - 6*a*b^2*(A + 3*B) + 3*a^3*(49*A + 25*B) + 3*a^2*b*(13*A + 57*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)] - (147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B)*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)*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*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*(a + b)*(8*A*b^3 - 6*a*b^2*(A + 3*B) + 3*a^3*(49*A + 25*B) + 3*a^2*b*(13*A + 57*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)] - (147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B)*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)*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*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*(a + b)*(8*A*b^3 - 6*a*b^2*(A + 3*B) + 3*a^3*(49*A + 25*B) + 3*a^2*b*(13*A + 57*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)] - (147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B)*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*((147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4) - ((a + b)*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*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*(a + b)*(8*A*b^3 - 6*a*b^2*(A + 3*B) + 3*a^3*(49*A + 25*B) + 3*a^2*b*(13*A + 57*B))*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)*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*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*(a + b)*(8*A*b^3 - 6*a*b^2*(A + 3*B) + 3*a^3*(49*A + 25*B) + 3*a^2*b*(13*A + 57*B))*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*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(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)*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(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)*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^3 - 6*a*b^2*(A + 3*B) + 3*a^3*(49*A + 25*B) + 3*a^2*b*(13*A + 57*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 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a + b)*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*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]))/(315*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((-2*(a + b)*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*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*(a + b)*(8*A*b^3 - 6*a*b^2*(A + 3*B) + 3*a^3*(49*A + 25*B) + 3*a^2*b*(13*A + 57*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)] - (147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*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]))/(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
598,1,3318,473,23.8950911,"\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2),x]","\text{Result too large to show}","\frac{2 \left(25 a^2 A+42 a b B+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(-\left(a^2 (25 A-63 B)\right)+3 a b (19 A-7 B)+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{2 (a-b) \sqrt{a+b} \left(63 a^3 B+82 a^2 A b+21 a b^2 B-6 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}} 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+8 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 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*(-82*a^2*A*b + 6*A*b^3 - 63*a^3*B - 21*a*b^2*B)*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]))/(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]]) + (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]]) - (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]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*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*(a + b)*(-6*A*b^2 + 3*a*b*(19*A + 7*B) + a^2*(25*A + 63*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)] - (82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B)*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)*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*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*(a + b)*(-6*A*b^2 + 3*a*b*(19*A + 7*B) + a^2*(25*A + 63*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)] - (82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B)*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)*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*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*(a + b)*(-6*A*b^2 + 3*a*b*(19*A + 7*B) + a^2*(25*A + 63*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)] - (82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B)*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*((82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4) - ((a + b)*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*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*(a + b)*(-6*A*b^2 + 3*a*b*(19*A + 7*B) + a^2*(25*A + 63*B))*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)*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*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*(a + b)*(-6*A*b^2 + 3*a*b*(19*A + 7*B) + a^2*(25*A + 63*B))*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*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B)*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B)*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 + 3*a*b*(19*A + 7*B) + a^2*(25*A + 63*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 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a + b)*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*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]))/(105*a^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((-2*(a + b)*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*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*(a + b)*(-6*A*b^2 + 3*a*b*(19*A + 7*B) + a^2*(25*A + 63*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)] - (82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*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]))/(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
599,1,427,393,18.7532249,"\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*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+20 a b B+3 A b^2\right) \sin (c+d x)}{15 a}+\frac{2}{15} \sec (c+d x) (5 a B \sin (c+d x)+6 A b \sin (c+d x))+\frac{2}{5} a A \tan (c+d x) \sec (c+d x)\right)}{d}+\frac{2 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(-\left(\left(9 a^2 A+20 a b B+3 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))\right)-2 (a+b) \left(9 a^2 A+20 a b B+3 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)+2 a (a+b) (a (9 A+5 B)+3 b (A+5 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)\right)}{15 a d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \cos (c+d x)}}","\frac{2 (a-b) \sqrt{a+b} \left(9 a^2 A+20 a b B+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)}{15 a^2 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) \sqrt{a+b \cos (c+d x)}}{15 d}-\frac{2 (a-b) \sqrt{a+b} (9 a A-5 a B-3 A b+15 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}} 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 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}",1,"(2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(9*a^2*A + 3*A*b^2 + 20*a*b*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*(a + b)*(3*b*(A + 5*B) + a*(9*A + 5*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)] - (9*a^2*A + 3*A*b^2 + 20*a*b*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*a*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*(9*a^2*A + 3*A*b^2 + 20*a*b*B)*Sin[c + d*x])/(15*a) + (2*Sec[c + d*x]*(6*A*b*Sin[c + d*x] + 5*a*B*Sin[c + d*x]))/15 + (2*a*A*Sec[c + d*x]*Tan[c + d*x])/5))/d","A",0
600,1,5981,479,24.5527182,"\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2),x]","\text{Result too large to show}","\frac{2 \sqrt{a+b} \left(a^2 (A-3 B)-a (4 A b-6 b B)+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}} 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-b) \sqrt{a+b} (3 a B+4 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 d \sqrt{\sec (c+d x)}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}-\frac{2 b 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}} \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
601,1,927,509,16.7179373,"\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2),x]","\frac{2 a 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^2 A \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a A b \tan ^5\left(\frac{1}{2} (c+d x)\right)+b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-a b B \tan ^5\left(\frac{1}{2} (c+d x)\right)+4 a A b \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 b^2 B \tan ^3\left(\frac{1}{2} (c+d x)\right)+4 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 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^2 A \tan \left(\frac{1}{2} (c+d x)\right)-2 a A b \tan \left(\frac{1}{2} (c+d x)\right)+b^2 B \tan \left(\frac{1}{2} (c+d x)\right)+a b B \tan \left(\frac{1}{2} (c+d x)\right)-(a+b) (2 a A-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 \left((A+B) a^2+2 b (A-B) a-A b^2\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}}+4 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 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}}\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 a A-b B) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{\sqrt{a+b} (2 a (A-B)-b (4 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{(a-b) \sqrt{a+b} (2 a A-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)}{a d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} (3 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)}{d \sqrt{\sec (c+d x)}}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}",1,"(2*a*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^2*A*Tan[(c + d*x)/2] - 2*a*A*b*Tan[(c + d*x)/2] + a*b*B*Tan[(c + d*x)/2] + b^2*B*Tan[(c + d*x)/2] + 4*a*A*b*Tan[(c + d*x)/2]^3 - 2*b^2*B*Tan[(c + d*x)/2]^3 + 2*a^2*A*Tan[(c + d*x)/2]^5 - 2*a*A*b*Tan[(c + d*x)/2]^5 - a*b*B*Tan[(c + d*x)/2]^5 + b^2*B*Tan[(c + d*x)/2]^5 + 4*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*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*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*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)] - (a + b)*(2*a*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^2) + 2*a*b*(A - B) + a^2*(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*(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
602,1,1134,532,18.5628297,"\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]],x]","\frac{b B \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (2 (c+d x))}{4 d}+\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(4 A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)-4 a A b \tan ^5\left(\frac{1}{2} (c+d x)\right)-5 a^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+5 a b B \tan ^5\left(\frac{1}{2} (c+d x)\right)-8 A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)-10 a b B \tan ^3\left(\frac{1}{2} (c+d x)\right)+24 a A 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 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)+8 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)+4 A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+4 a A b \tan \left(\frac{1}{2} (c+d x)\right)+5 a^2 B \tan \left(\frac{1}{2} (c+d x)\right)+5 a b B \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) (4 A b+5 a 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 \left(4 (A-B) a^2+b (B-8 A) a-2 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}}+24 a A 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 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}}+8 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}}\right)}{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} \left(3 a^2 B+12 a A b+4 b^2 B\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{(5 a B+4 A b) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{\sqrt{a+b} (8 a A+5 a B+4 A b+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}} 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} (5 a B+4 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 a d \sqrt{\sec (c+d x)}}+\frac{b B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}",1,"(b*B*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[2*(c + d*x)])/(4*d) + (Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(4*a*A*b*Tan[(c + d*x)/2] + 4*A*b^2*Tan[(c + d*x)/2] + 5*a^2*B*Tan[(c + d*x)/2] + 5*a*b*B*Tan[(c + d*x)/2] - 8*A*b^2*Tan[(c + d*x)/2]^3 - 10*a*b*B*Tan[(c + d*x)/2]^3 - 4*a*A*b*Tan[(c + d*x)/2]^5 + 4*A*b^2*Tan[(c + d*x)/2]^5 - 5*a^2*B*Tan[(c + d*x)/2]^5 + 5*a*b*B*Tan[(c + d*x)/2]^5 + 24*a*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)] + 6*a^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)] + 8*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)] + 24*a*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)] + 6*a^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)] + 8*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)] + (a + b)*(4*A*b + 5*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)] + 2*(4*a^2*(A - B) - 2*b^2*B + a*b*(-8*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)]))/(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)])","B",0
603,1,1489,626,19.3465611,"\int \frac{(a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{1}{12} b B \sin (c+d x)+\frac{1}{24} (6 A b+7 a B) \sin (2 (c+d x))+\frac{1}{12} b B \sin (3 (c+d x))\right)}{d}+\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(30 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)-30 a^2 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 a^3 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+16 b^3 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 a b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 a^2 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)-60 a A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)-32 b^3 B \tan ^3\left(\frac{1}{2} (c+d x)\right)-6 a^2 b B \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)+36 a^2 A 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 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)+72 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)+30 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+30 a^2 A b \tan \left(\frac{1}{2} (c+d x)\right)+3 a^3 B \tan \left(\frac{1}{2} (c+d x)\right)+16 b^3 B \tan \left(\frac{1}{2} (c+d x)\right)+16 a b^2 B \tan \left(\frac{1}{2} (c+d x)\right)+3 a^2 b B \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(3 B a^2+30 A b a+16 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 b \left((24 A-7 B) a^2+(26 b B-6 A b) a+12 A b^2\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}}+36 a^2 A 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 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}}+72 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}}\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 B+30 a A b+16 b^2 B\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 B+30 a A b+14 a b B+12 A b^2+16 b^2 B\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 B+30 a A b+16 b^2 B\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{\sqrt{a+b} \left(a^3 (-B)+6 a^2 A b+12 a b^2 B+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}} \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{(7 a B+6 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{12 d \sqrt{\sec (c+d x)}}+\frac{b B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((b*B*Sin[c + d*x])/12 + ((6*A*b + 7*a*B)*Sin[2*(c + d*x)])/24 + (b*B*Sin[3*(c + d*x)])/12))/d + (Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(30*a^2*A*b*Tan[(c + d*x)/2] + 30*a*A*b^2*Tan[(c + d*x)/2] + 3*a^3*B*Tan[(c + d*x)/2] + 3*a^2*b*B*Tan[(c + d*x)/2] + 16*a*b^2*B*Tan[(c + d*x)/2] + 16*b^3*B*Tan[(c + d*x)/2] - 60*a*A*b^2*Tan[(c + d*x)/2]^3 - 6*a^2*b*B*Tan[(c + d*x)/2]^3 - 32*b^3*B*Tan[(c + d*x)/2]^3 - 30*a^2*A*b*Tan[(c + d*x)/2]^5 + 30*a*A*b^2*Tan[(c + d*x)/2]^5 - 3*a^3*B*Tan[(c + d*x)/2]^5 + 3*a^2*b*B*Tan[(c + d*x)/2]^5 - 16*a*b^2*B*Tan[(c + d*x)/2]^5 + 16*b^3*B*Tan[(c + d*x)/2]^5 + 36*a^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)] + 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)] - 6*a^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)] + 72*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)] + 36*a^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)] + 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)] - 6*a^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)] + 72*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)] + (a + b)*(30*a*A*b + 3*a^2*B + 16*b^2*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*b*(12*A*b^2 + a^2*(24*A - 7*B) + a*(-6*A*b + 26*b*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)]))/(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",0
604,1,1888,730,21.3999314,"\int \frac{(a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(3/2)*(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{1}{96} (8 A b+9 a B) \sin (c+d x)+\frac{\left(3 B a^2+56 A b a+48 b^2 B\right) \sin (2 (c+d x))}{192 b}+\frac{1}{96} (8 A b+9 a B) \sin (3 (c+d x))+\frac{1}{32} b B \sin (4 (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(128 A b^4 \tan ^5\left(\frac{1}{2} (c+d x)\right)-128 a A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)+24 a^2 A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)-24 a^3 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)+9 a^4 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+156 a b^3 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-156 a^2 b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-9 a^3 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)-256 A b^4 \tan ^3\left(\frac{1}{2} (c+d x)\right)-48 a^2 A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)-312 a b^3 B \tan ^3\left(\frac{1}{2} (c+d x)\right)+18 a^3 b B \tan ^3\left(\frac{1}{2} (c+d x)\right)+576 a 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)-48 a^3 A 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)+18 a^4 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)+288 b^4 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)+144 a^2 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)+128 A b^4 \tan \left(\frac{1}{2} (c+d x)\right)+128 a A b^3 \tan \left(\frac{1}{2} (c+d x)\right)+24 a^2 A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+24 a^3 A b \tan \left(\frac{1}{2} (c+d x)\right)-9 a^4 B \tan \left(\frac{1}{2} (c+d x)\right)+156 a b^3 B \tan \left(\frac{1}{2} (c+d x)\right)+156 a^2 b^2 B \tan \left(\frac{1}{2} (c+d x)\right)-9 a^3 b B \tan \left(\frac{1}{2} (c+d x)\right)-(a+b) \left(9 B a^3-24 A b a^2-156 b^2 B a-128 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}}+2 b \left(3 B a^3+2 b (28 A-57 B) a^2-4 b^2 (52 A-9 B) a-72 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}}+576 a 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}}-48 a^3 A 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}}+18 a^4 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}}+288 b^4 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}}+144 a^2 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}}\right)}{192 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 B+8 a A b+12 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{32 b d \sqrt{\sec (c+d x)}}+\frac{\left(-9 a^3 B+24 a^2 A b+156 a b^2 B+128 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{192 b^2 d}-\frac{\sqrt{a+b} \left(9 a^3 B-6 a^2 b (4 A+B)-4 a b^2 (28 A+39 B)-8 b^3 (16 A+9 B)\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^2 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(-9 a^3 B+24 a^2 A b+156 a b^2 B+128 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}} 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} \left(-3 a^4 B+8 a^3 A b-24 a^2 b^2 B-96 a A b^3-48 b^4 B\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{(8 A b-3 a B) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 b d \sqrt{\sec (c+d x)}}+\frac{B \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{4 b d \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(((8*A*b + 9*a*B)*Sin[c + d*x])/96 + ((56*a*A*b + 3*a^2*B + 48*b^2*B)*Sin[2*(c + d*x)])/(192*b) + ((8*A*b + 9*a*B)*Sin[3*(c + d*x)])/96 + (b*B*Sin[4*(c + d*x)])/32))/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^3*A*b*Tan[(c + d*x)/2] + 24*a^2*A*b^2*Tan[(c + d*x)/2] + 128*a*A*b^3*Tan[(c + d*x)/2] + 128*A*b^4*Tan[(c + d*x)/2] - 9*a^4*B*Tan[(c + d*x)/2] - 9*a^3*b*B*Tan[(c + d*x)/2] + 156*a^2*b^2*B*Tan[(c + d*x)/2] + 156*a*b^3*B*Tan[(c + d*x)/2] - 48*a^2*A*b^2*Tan[(c + d*x)/2]^3 - 256*A*b^4*Tan[(c + d*x)/2]^3 + 18*a^3*b*B*Tan[(c + d*x)/2]^3 - 312*a*b^3*B*Tan[(c + d*x)/2]^3 - 24*a^3*A*b*Tan[(c + d*x)/2]^5 + 24*a^2*A*b^2*Tan[(c + d*x)/2]^5 - 128*a*A*b^3*Tan[(c + d*x)/2]^5 + 128*A*b^4*Tan[(c + d*x)/2]^5 + 9*a^4*B*Tan[(c + d*x)/2]^5 - 9*a^3*b*B*Tan[(c + d*x)/2]^5 - 156*a^2*b^2*B*Tan[(c + d*x)/2]^5 + 156*a*b^3*B*Tan[(c + d*x)/2]^5 - 48*a^3*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)] + 576*a*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)] + 18*a^4*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)] + 144*a^2*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)] + 288*b^4*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*a^3*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)] + 576*a*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)] + 18*a^4*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)] + 144*a^2*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)] + 288*b^4*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)*(-24*a^2*A*b - 128*A*b^3 + 9*a^3*B - 156*a*b^2*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*b*(2*a^2*b*(28*A - 57*B) - 4*a*b^2*(52*A - 9*B) + 3*a^3*B - 72*b^3*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)]))/(192*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",1
605,1,4198,662,27.2433137,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{13}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(13/2),x]","\text{Result too large to show}","\frac{2 \left(81 a^2 A+209 a b B+113 A b^2\right) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{693 d}+\frac{2 \left(539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3465 a d}+\frac{2 \left(675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3465 a^2 d}+\frac{2 (a-b) \sqrt{a+b} \left(3 a^4 (225 A-539 B)-6 a^3 b (505 A-209 B)+15 a^2 b^2 (19 A-121 B)+10 a b^3 (3 A-11 B)+40 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)}{3465 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \left(1617 a^5 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 a b^4 B+40 A b^5\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)}{3465 a^4 d \sqrt{\sec (c+d x)}}+\frac{2 a (11 a B+14 A b) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{99 d}+\frac{2 a A \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{11 d}",1,"(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)*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]))/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]))/(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]))/(3465*a^2) + (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]]) - (7*a^3*B)/(15*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (31*a*b^2*B)/(35*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b^4*B)/(63*a*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]]) + (38*a^2*b*B*Sqrt[Sec[c + d*x]])/(105*Sqrt[a + b*Cos[c + d*x]]) - (124*b^3*B*Sqrt[Sec[c + d*x]])/(315*Sqrt[a + b*Cos[c + d*x]]) + (2*b^5*B*Sqrt[Sec[c + d*x]])/(63*a^2*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]]) - (7*a^2*b*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[a + b*Cos[c + d*x]]) - (31*b^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(35*Sqrt[a + b*Cos[c + d*x]]) + (2*b^5*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(63*a^2*Sqrt[a + b*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(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)*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)*(40*A*b^4 - 10*a*b^3*(3*A + 11*B) + 15*a^2*b^2*(19*A + 121*B) + 6*a^3*b*(505*A + 209*B) + 3*a^4*(225*A + 539*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)] - (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)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3465*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)*(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)*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)*(40*A*b^4 - 10*a*b^3*(3*A + 11*B) + 15*a^2*b^2*(19*A + 121*B) + 6*a^3*b*(505*A + 209*B) + 3*a^4*(225*A + 539*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)] - (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)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3465*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)*(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)*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)*(40*A*b^4 - 10*a*b^3*(3*A + 11*B) + 15*a^2*b^2*(19*A + 121*B) + 6*a^3*b*(505*A + 209*B) + 3*a^4*(225*A + 539*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)] - (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)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3465*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*((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)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4) - ((a + b)*(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)*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)*(40*A*b^4 - 10*a*b^3*(3*A + 11*B) + 15*a^2*b^2*(19*A + 121*B) + 6*a^3*b*(505*A + 209*B) + 3*a^4*(225*A + 539*B))*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)*(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)*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)*(40*A*b^4 - 10*a*b^3*(3*A + 11*B) + 15*a^2*b^2*(19*A + 121*B) + 6*a^3*b*(505*A + 209*B) + 3*a^4*(225*A + 539*B))*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*(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)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(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)*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(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)*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)*(40*A*b^4 - 10*a*b^3*(3*A + 11*B) + 15*a^2*b^2*(19*A + 121*B) + 6*a^3*b*(505*A + 209*B) + 3*a^4*(225*A + 539*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 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a + b)*(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)*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]))/(3465*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((-2*(a + b)*(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)*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)*(40*A*b^4 - 10*a*b^3*(3*A + 11*B) + 15*a^2*b^2*(19*A + 121*B) + 6*a^3*b*(505*A + 209*B) + 3*a^4*(225*A + 539*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)] - (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)*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]))/(3465*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
606,1,3755,562,26.3640484,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(11/2),x]","\text{Result too large to show}","\frac{2 \left(49 a^2 A+135 a b B+75 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 \left(75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\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(3 a^3 (49 A-25 B)-6 a^2 b (19 A-60 B)+15 a b^2 (11 A-3 B)+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(147 a^4 A+435 a^3 b B+279 a^2 A b^2+45 a b^3 B-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 (3 a B+4 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 A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{9 d}",1,"(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)*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]))/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]))/(315*a) + (2*a^2*A*Sec[c + d*x]^3*Tan[c + d*x])/9))/d + (2*((-7*a^3*A)/(15*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (31*a*A*b^2)/(35*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*A*b^4)/(63*a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (29*a^2*b*B)/(21*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (b^3*B)/(7*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (38*a^2*A*b*Sqrt[Sec[c + d*x]])/(105*Sqrt[a + b*Cos[c + d*x]]) - (124*A*b^3*Sqrt[Sec[c + d*x]])/(315*Sqrt[a + b*Cos[c + d*x]]) + (2*A*b^5*Sqrt[Sec[c + d*x]])/(63*a^2*Sqrt[a + b*Cos[c + d*x]]) + (5*a^3*B*Sqrt[Sec[c + d*x]])/(21*Sqrt[a + b*Cos[c + d*x]]) - (2*a*b^2*B*Sqrt[Sec[c + d*x]])/(21*Sqrt[a + b*Cos[c + d*x]]) - (b^4*B*Sqrt[Sec[c + d*x]])/(7*a*Sqrt[a + b*Cos[c + d*x]]) - (7*a^2*A*b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[a + b*Cos[c + d*x]]) - (31*A*b^3*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(35*Sqrt[a + b*Cos[c + d*x]]) + (2*A*b^5*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(63*a^2*Sqrt[a + b*Cos[c + d*x]]) - (29*a*b^2*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(21*Sqrt[a + b*Cos[c + d*x]]) - (b^4*B*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*(a + b)*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*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*(a + b)*(-10*A*b^3 + 15*a*b^2*(11*A + 3*B) + 3*a^3*(49*A + 25*B) + 6*a^2*b*(19*A + 60*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)] - (147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*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)*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*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*(a + b)*(-10*A*b^3 + 15*a*b^2*(11*A + 3*B) + 3*a^3*(49*A + 25*B) + 6*a^2*b*(19*A + 60*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)] - (147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*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)*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*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*(a + b)*(-10*A*b^3 + 15*a*b^2*(11*A + 3*B) + 3*a^3*(49*A + 25*B) + 6*a^2*b*(19*A + 60*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)] - (147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*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*((147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4) - ((a + b)*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*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*(a + b)*(-10*A*b^3 + 15*a*b^2*(11*A + 3*B) + 3*a^3*(49*A + 25*B) + 6*a^2*b*(19*A + 60*B))*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)*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*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*(a + b)*(-10*A*b^3 + 15*a*b^2*(11*A + 3*B) + 3*a^3*(49*A + 25*B) + 6*a^2*b*(19*A + 60*B))*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*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(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)*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(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)*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)*(-10*A*b^3 + 15*a*b^2*(11*A + 3*B) + 3*a^3*(49*A + 25*B) + 6*a^2*b*(19*A + 60*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 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a + b)*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*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]))/(315*a^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((-2*(a + b)*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*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*(a + b)*(-10*A*b^3 + 15*a*b^2*(11*A + 3*B) + 3*a^3*(49*A + 25*B) + 6*a^2*b*(19*A + 60*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)] - (147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*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]))/(315*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
607,1,3348,474,24.6715451,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(9/2),x]","\text{Result too large to show}","\frac{2 \left(25 a^2 A+77 a b B+45 A b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{105 d}+\frac{2 (a-b) \sqrt{a+b} \left(a^2 (25 A-63 B)-8 a b (15 A-7 B)+15 b^2 (A-7 B)\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 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \left(63 a^3 B+145 a^2 A b+161 a b^2 B+15 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}} 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 a (7 a B+10 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 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*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*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]))/105 + (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]]) - (3*a^3*B)/(5*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (23*a*b^2*B)/(15*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]]) + (8*a^2*b*B*Sqrt[Sec[c + d*x]])/(15*Sqrt[a + b*Cos[c + d*x]]) - (8*b^3*B*Sqrt[Sec[c + d*x]])/(15*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]]) - (3*a^2*b*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*Sqrt[a + b*Cos[c + d*x]]) - (23*b^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[a + b*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*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*(a + b)*(15*b^2*(A + 7*B) + 8*a*b*(15*A + 7*B) + a^2*(25*A + 63*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)] - (145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*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*(a + b)*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*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*(a + b)*(15*b^2*(A + 7*B) + 8*a*b*(15*A + 7*B) + a^2*(25*A + 63*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)] - (145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*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*(a + b)*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*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*(a + b)*(15*b^2*(A + 7*B) + 8*a*b*(15*A + 7*B) + a^2*(25*A + 63*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)] - (145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*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*((145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4) - ((a + b)*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*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*(a + b)*(15*b^2*(A + 7*B) + 8*a*b*(15*A + 7*B) + a^2*(25*A + 63*B))*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)*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*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*(a + b)*(15*b^2*(A + 7*B) + 8*a*b*(15*A + 7*B) + a^2*(25*A + 63*B))*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*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*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)*(15*b^2*(A + 7*B) + 8*a*b*(15*A + 7*B) + a^2*(25*A + 63*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 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a + b)*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*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]))/(105*a*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((-2*(a + b)*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*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*(a + b)*(15*b^2*(A + 7*B) + 8*a*b*(15*A + 7*B) + a^2*(25*A + 63*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)] - (145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*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]))/(105*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
608,1,7032,553,25.574007,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2),x]","\text{Result too large to show}","\frac{2 (a-b) \sqrt{a+b} \left(9 a^2 A+35 a b B+23 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 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{a+b} \left(-\left(a^3 (9 A-5 B)\right)+a^2 b (17 A-35 B)-a b^2 (23 A-45 B)+15 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 (5 a B+8 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 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^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}} \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
609,1,7700,596,26.068375,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2),x]","\text{Result too large to show}","-\frac{\left(6 a^2 B+14 a A b-3 b^2 B\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{3 d}-\frac{\sqrt{a+b} \left(-2 a^2 (A-3 B)+2 a b (7 A-9 B)-3 b^2 (6 A+B)\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 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \left(6 a^2 B+14 a A b-3 b^2 B\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 d \sqrt{\sec (c+d x)}}+\frac{2 a (a B+2 A b) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{b \sqrt{a+b} (5 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)}{d \sqrt{\sec (c+d x)}}+\frac{2 a A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}{3 d}",1,"Result too large to show","B",0
610,1,1278,607,19.5155581,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(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(2 A \sin (c+d x) a^2+\frac{1}{4} b^2 B \sin (2 (c+d x))\right)}{d}+\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(4 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-4 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+8 a^3 A \tan ^5\left(\frac{1}{2} (c+d x)\right)-8 a^2 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)+9 a b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-9 a^2 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)-8 A b^3 \tan ^3\left(\frac{1}{2} (c+d x)\right)+16 a^2 A b \tan ^3\left(\frac{1}{2} (c+d x)\right)-18 a b^2 B \tan ^3\left(\frac{1}{2} (c+d x)\right)+40 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)+8 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)+30 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)+4 A b^3 \tan \left(\frac{1}{2} (c+d x)\right)+4 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)-8 a^3 A \tan \left(\frac{1}{2} (c+d x)\right)-8 a^2 A b \tan \left(\frac{1}{2} (c+d x)\right)+9 a b^2 B \tan \left(\frac{1}{2} (c+d x)\right)+9 a^2 b B \tan \left(\frac{1}{2} (c+d x)\right)-(a+b) \left(8 A a^2-9 b B a-4 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}}+2 \left(4 (A+B) a^3+12 b (A-B) a^2+b^2 (B-12 A) a-2 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}}+40 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}}+8 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}}+30 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}}\right)}{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{\left(8 a^2 A-9 a b B-4 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{\sqrt{a+b} \left(8 a^2 (A-B)-3 a b (8 A+3 B)-2 b^2 (2 A+B)\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 d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \left(8 a^2 A-9 a b B-4 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)}{4 a d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left(15 a^2 B+20 a A b+4 b^2 B\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 (4 a A-b B) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}+\frac{2 a 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^2*A*Sin[c + d*x] + (b^2*B*Sin[2*(c + d*x)])/4))/d + (Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-8*a^3*A*Tan[(c + d*x)/2] - 8*a^2*A*b*Tan[(c + d*x)/2] + 4*a*A*b^2*Tan[(c + d*x)/2] + 4*A*b^3*Tan[(c + d*x)/2] + 9*a^2*b*B*Tan[(c + d*x)/2] + 9*a*b^2*B*Tan[(c + d*x)/2] + 16*a^2*A*b*Tan[(c + d*x)/2]^3 - 8*A*b^3*Tan[(c + d*x)/2]^3 - 18*a*b^2*B*Tan[(c + d*x)/2]^3 + 8*a^3*A*Tan[(c + d*x)/2]^5 - 8*a^2*A*b*Tan[(c + d*x)/2]^5 - 4*a*A*b^2*Tan[(c + d*x)/2]^5 + 4*A*b^3*Tan[(c + d*x)/2]^5 - 9*a^2*b*B*Tan[(c + d*x)/2]^5 + 9*a*b^2*B*Tan[(c + d*x)/2]^5 + 40*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^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)] + 8*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)] + 40*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^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)] + 8*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)] - (a + b)*(8*a^2*A - 4*A*b^2 - 9*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*(12*a^2*b*(A - B) - 2*b^3*B + a*b^2*(-12*A + B) + 4*a^3*(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)]))/(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)])","B",0
611,1,1504,624,19.5994492,"\int (a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{1}{12} B \sin (c+d x) b^2+\frac{1}{12} B \sin (3 (c+d x)) b^2+\frac{1}{24} (6 A b+13 a B) \sin (2 (c+d x)) b\right)}{d}+\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(54 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)-54 a^2 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)-33 a^3 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+16 b^3 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 a b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+33 a^2 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)-108 a A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)-32 b^3 B \tan ^3\left(\frac{1}{2} (c+d x)\right)-66 a^2 b B \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)+180 a^2 A 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 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)+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)+54 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+54 a^2 A b \tan \left(\frac{1}{2} (c+d x)\right)+33 a^3 B \tan \left(\frac{1}{2} (c+d x)\right)+16 b^3 B \tan \left(\frac{1}{2} (c+d x)\right)+16 a b^2 B \tan \left(\frac{1}{2} (c+d x)\right)+33 a^2 b B \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(33 B a^2+54 A b a+16 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(24 (A-B) a^3+(13 b B-72 A b) a^2+2 b^2 (3 A-19 B) a-12 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}}+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}}+180 a^2 A 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 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}}+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}}\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(33 a^2 B+54 a A b+16 b^2 B\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 B)+a (54 A b+26 b B)+4 b^2 (3 A+4 B)\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(33 a^2 B+54 a A b+16 b^2 B\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{\sqrt{a+b} \left(5 a^3 B+30 a^2 A b+20 a b^2 B+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}} \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 (3 a B+2 A b) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\sec (c+d x)}}+\frac{b B \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^2*B*Sin[c + d*x])/12 + (b*(6*A*b + 13*a*B)*Sin[2*(c + d*x)])/24 + (b^2*B*Sin[3*(c + d*x)])/12))/d + (Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(54*a^2*A*b*Tan[(c + d*x)/2] + 54*a*A*b^2*Tan[(c + d*x)/2] + 33*a^3*B*Tan[(c + d*x)/2] + 33*a^2*b*B*Tan[(c + d*x)/2] + 16*a*b^2*B*Tan[(c + d*x)/2] + 16*b^3*B*Tan[(c + d*x)/2] - 108*a*A*b^2*Tan[(c + d*x)/2]^3 - 66*a^2*b*B*Tan[(c + d*x)/2]^3 - 32*b^3*B*Tan[(c + d*x)/2]^3 - 54*a^2*A*b*Tan[(c + d*x)/2]^5 + 54*a*A*b^2*Tan[(c + d*x)/2]^5 - 33*a^3*B*Tan[(c + d*x)/2]^5 + 33*a^2*b*B*Tan[(c + d*x)/2]^5 - 16*a*b^2*B*Tan[(c + d*x)/2]^5 + 16*b^3*B*Tan[(c + d*x)/2]^5 + 180*a^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)] + 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)] + 30*a^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)] + 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)] + 180*a^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)] + 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)] + 30*a^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)] + 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)] + (a + b)*(54*a*A*b + 33*a^2*B + 16*b^2*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*(-12*A*b^3 + 2*a*b^2*(3*A - 19*B) + 24*a^3*(A - B) + a^2*(-72*A*b + 13*b*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)]))/(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
612,1,1857,724,20.038847,"\int \frac{(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{1}{32} B \sin (4 (c+d x)) b^2+\frac{1}{96} (8 A b+17 a B) \sin (c+d x) b+\frac{1}{96} (8 A b+17 a B) \sin (3 (c+d x)) b+\frac{1}{192} \left(59 B a^2+104 A b a+48 b^2 B\right) \sin (2 (c+d x))\right)}{d}+\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(128 A b^4 \tan ^5\left(\frac{1}{2} (c+d x)\right)-128 a A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)+264 a^2 A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)-264 a^3 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)-15 a^4 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+284 a b^3 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-284 a^2 b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+15 a^3 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)-256 A b^4 \tan ^3\left(\frac{1}{2} (c+d x)\right)-528 a^2 A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)-568 a b^3 B \tan ^3\left(\frac{1}{2} (c+d x)\right)-30 a^3 b B \tan ^3\left(\frac{1}{2} (c+d x)\right)+960 a 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)+240 a^3 A 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^4 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)+288 b^4 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)+720 a^2 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)+128 A b^4 \tan \left(\frac{1}{2} (c+d x)\right)+128 a A b^3 \tan \left(\frac{1}{2} (c+d x)\right)+264 a^2 A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+264 a^3 A b \tan \left(\frac{1}{2} (c+d x)\right)+15 a^4 B \tan \left(\frac{1}{2} (c+d x)\right)+284 a b^3 B \tan \left(\frac{1}{2} (c+d x)\right)+284 a^2 b^2 B \tan \left(\frac{1}{2} (c+d x)\right)+15 a^3 b B \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(15 B a^3+264 A b a^2+284 b^2 B a+128 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}}-2 b \left((192 A-59 B) a^3+(322 b B-104 A b) a^2+4 b^2 (76 A-9 B) a+72 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}}+960 a 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}}+240 a^3 A 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^4 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}}+288 b^4 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}}+720 a^2 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}}\right)}{192 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(5 a^2 B+24 a A b+12 b^2 B\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{32 d \sqrt{\sec (c+d x)}}+\frac{\left(15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{192 b d}+\frac{\sqrt{a+b} \left(15 a^3 B+2 a^2 b (132 A+59 B)+4 a b^2 (52 A+71 B)+8 b^3 (16 A+9 B)\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{(a-b) \sqrt{a+b} \left(15 a^3 B+264 a^2 A b+284 a b^2 B+128 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}} 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} \left(-5 a^4 B+40 a^3 A b+120 a^2 b^2 B+160 a A b^3+48 b^4 B\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{(11 a B+8 A b) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{24 d \sqrt{\sec (c+d x)}}+\frac{b B \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{4 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((b*(8*A*b + 17*a*B)*Sin[c + d*x])/96 + ((104*a*A*b + 59*a^2*B + 48*b^2*B)*Sin[2*(c + d*x)])/192 + (b*(8*A*b + 17*a*B)*Sin[3*(c + d*x)])/96 + (b^2*B*Sin[4*(c + d*x)])/32))/d + (Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(264*a^3*A*b*Tan[(c + d*x)/2] + 264*a^2*A*b^2*Tan[(c + d*x)/2] + 128*a*A*b^3*Tan[(c + d*x)/2] + 128*A*b^4*Tan[(c + d*x)/2] + 15*a^4*B*Tan[(c + d*x)/2] + 15*a^3*b*B*Tan[(c + d*x)/2] + 284*a^2*b^2*B*Tan[(c + d*x)/2] + 284*a*b^3*B*Tan[(c + d*x)/2] - 528*a^2*A*b^2*Tan[(c + d*x)/2]^3 - 256*A*b^4*Tan[(c + d*x)/2]^3 - 30*a^3*b*B*Tan[(c + d*x)/2]^3 - 568*a*b^3*B*Tan[(c + d*x)/2]^3 - 264*a^3*A*b*Tan[(c + d*x)/2]^5 + 264*a^2*A*b^2*Tan[(c + d*x)/2]^5 - 128*a*A*b^3*Tan[(c + d*x)/2]^5 + 128*A*b^4*Tan[(c + d*x)/2]^5 - 15*a^4*B*Tan[(c + d*x)/2]^5 + 15*a^3*b*B*Tan[(c + d*x)/2]^5 - 284*a^2*b^2*B*Tan[(c + d*x)/2]^5 + 284*a*b^3*B*Tan[(c + d*x)/2]^5 + 240*a^3*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)] + 960*a*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)] - 30*a^4*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)] + 720*a^2*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)] + 288*b^4*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)] + 240*a^3*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)] + 960*a*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)] - 30*a^4*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)] + 720*a^2*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)] + 288*b^4*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)*(264*a^2*A*b + 128*A*b^3 + 15*a^3*B + 284*a*b^2*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*b*(a^3*(192*A - 59*B) + 4*a*b^2*(76*A - 9*B) + 72*b^3*B + a^2*(-104*A*b + 322*b*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)]))/(192*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",0
613,1,703,839,16.0735799,"\int \frac{(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{1}{960} \left(93 a^2 B+170 a A b+88 b^2 B\right) \sin (c+d x)+\frac{1}{960} \left(93 a^2 B+170 a A b+100 b^2 B\right) \sin (3 (c+d x))+\frac{\left(15 a^3 B+590 a^2 A b+1024 a b^2 B+480 A b^3\right) \sin (2 (c+d x))}{1920 b}+\frac{1}{320} b (21 a B+10 A b) \sin (4 (c+d x))+\frac{1}{80} b^2 B \sin (5 (c+d x))\right)}{d}-\frac{-b \left(-45 a^4 B+150 a^3 A b+1692 a^2 b^2 B+2840 a A b^3+1024 b^4 B\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 (a+b) \left(45 a^4 B-30 a^3 b (5 A+3 B)+60 a^2 b^2 (5 A+11 B)+8 a b^3 (265 A+129 B)+16 b^4 (45 A+64 B)\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)-b (a+b) \left(-45 a^4 B+150 a^3 A b+1692 a^2 b^2 B+2840 a A b^3+1024 b^4 B\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)+15 \left(-3 a^5 B+10 a^4 A b-40 a^3 b^2 B-240 a^2 A b^3-240 a b^4 B-96 A b^5\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)}{1920 b^3 d \sec ^{\frac{3}{2}}(c+d x) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2} \sqrt{a+b \cos (c+d x)}}","\frac{B \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d \sqrt{\sec (c+d x)}}+\frac{(10 A b-3 a B) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{40 b d \sqrt{\sec (c+d x)}}+\frac{\left(-15 B a^2+50 A b a+64 b^2 B\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{240 b d \sqrt{\sec (c+d x)}}+\frac{\left(-45 B a^4+150 A b a^3+1692 b^2 B a^2+2840 A b^3 a+1024 b^4 B\right) \sqrt{\sec (c+d x)} \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d}+\frac{\left(-15 B a^3+50 A b a^2+172 b^2 B a+120 A b^3\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 B a^4+150 A b a^3+1692 b^2 B a^2+2840 A b^3 a+1024 b^4 B\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 B a^4-30 b (5 A+B) a^3-4 b^2 (295 A+423 B) a^2-8 b^3 (355 A+193 B) a-16 b^4 (45 A+64 B)\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 B a^5+10 A b a^4-40 b^2 B a^3-240 A b^3 a^2-240 b^4 B a-96 A b^5\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]]*(((170*a*A*b + 93*a^2*B + 88*b^2*B)*Sin[c + d*x])/960 + ((590*a^2*A*b + 480*A*b^3 + 15*a^3*B + 1024*a*b^2*B)*Sin[2*(c + d*x)])/(1920*b) + ((170*a*A*b + 93*a^2*B + 100*b^2*B)*Sin[3*(c + d*x)])/960 + (b*(10*A*b + 21*a*B)*Sin[4*(c + d*x)])/320 + (b^2*B*Sin[5*(c + d*x)])/80))/d - (-(b*(a + b)*(150*a^3*A*b + 2840*a*A*b^3 - 45*a^4*B + 1692*a^2*b^2*B + 1024*b^4*B)*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)*(45*a^4*B - 30*a^3*b*(5*A + 3*B) + 60*a^2*b^2*(5*A + 11*B) + 16*b^4*(45*A + 64*B) + 8*a*b^3*(265*A + 129*B))*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)] + 15*(10*a^4*A*b - 240*a^2*A*b^3 - 96*A*b^5 - 3*a^5*B - 40*a^3*b^2*B - 240*a*b^4*B)*((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*(150*a^3*A*b + 2840*a*A*b^3 - 45*a^4*B + 1692*a^2*b^2*B + 1024*b^4*B)*(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])/(1920*b^3*d*Sqrt[a + b*Cos[c + d*x]]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]^(3/2))","A",1
614,1,2987,403,22.264729,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x])*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} \left(9 a^2 A-10 a b 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}} 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(a^2 (9 A-5 B)-2 a b (A+5 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 - 10*a*b*B)*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]]) - (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]]) - (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]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(9*a^2*A + 8*A*b^2 - 10*a*b*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*(8*A*b^2 + 2*a*b*(A - 5*B) + a^2*(9*A + 5*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)] - (9*a^2*A + 8*A*b^2 - 10*a*b*B)*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)*(9*a^2*A + 8*A*b^2 - 10*a*b*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*(8*A*b^2 + 2*a*b*(A - 5*B) + a^2*(9*A + 5*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)] - (9*a^2*A + 8*A*b^2 - 10*a*b*B)*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)*(9*a^2*A + 8*A*b^2 - 10*a*b*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*(8*A*b^2 + 2*a*b*(A - 5*B) + a^2*(9*A + 5*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)] - (9*a^2*A + 8*A*b^2 - 10*a*b*B)*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*((9*a^2*A + 8*A*b^2 - 10*a*b*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4) - ((a + b)*(9*a^2*A + 8*A*b^2 - 10*a*b*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*(8*A*b^2 + 2*a*b*(A - 5*B) + a^2*(9*A + 5*B))*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)*(9*a^2*A + 8*A*b^2 - 10*a*b*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*(8*A*b^2 + 2*a*b*(A - 5*B) + a^2*(9*A + 5*B))*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*(9*a^2*A + 8*A*b^2 - 10*a*b*B)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (9*a^2*A + 8*A*b^2 - 10*a*b*B)*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (9*a^2*A + 8*A*b^2 - 10*a*b*B)*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))*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)*(9*a^2*A + 8*A*b^2 - 10*a*b*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]))/(15*a^3*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((-2*(a + b)*(9*a^2*A + 8*A*b^2 - 10*a*b*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*(8*A*b^2 + 2*a*b*(A - 5*B) + a^2*(9*A + 5*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)] - (9*a^2*A + 8*A*b^2 - 10*a*b*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]))/(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
615,1,355,330,15.8220378,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 (3 a B-2 A b) \sin (c+d x)}{3 a^2}+\frac{2 A \tan (c+d x)}{3 a}\right)}{d}+\frac{2 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left((2 A b-3 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 (a (A+3 B)-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)-2 (a+b) (3 a B-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)}} 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 \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \cos (c+d x)}}","-\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} (a (A-3 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)}{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*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))*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]) + (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","A",0
616,1,279,270,14.2984659,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \left(A \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))-\frac{\sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(-2 a (A+B) \sqrt{\frac{1}{\sec (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 \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) \sqrt{\frac{1}{\sec (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)}{\sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)}}\right)}{a d \sqrt{a+b \cos (c+d x)}}","\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)}}",1,"(2*(A*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*Sin[c + d*x] - (Sqrt[Cos[(c + d*x)/2]^2*Sec[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[(1 + Sec[c + d*x])^(-1)] - 2*a*(A + B)*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[(1 + Sec[c + d*x])^(-1)] + A*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*d*Sqrt[a + b*Cos[c + d*x]])","A",0
617,1,157,268,2.5838438,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \sqrt{\sec (c+d x)+1} \sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} \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)}{d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \cos (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 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}} \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*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*((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[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[1 + Sec[c + d*x]])/(d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2])","A",1
618,1,1091,487,18.4260336,"\int \frac{A+B \cos (c+d x)}{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","\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(-a \sqrt{\frac{a-b}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)+b \sqrt{\frac{a-b}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 b \sqrt{\frac{a-b}{a+b}} B \tan ^3\left(\frac{1}{2} (c+d x)\right)-4 i A 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 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)+a \sqrt{\frac{a-b}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)+b \sqrt{\frac{a-b}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)+i (a-b) B 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-a B) 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}}-4 i A 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 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}}\right)}{b \sqrt{\frac{a-b}{a+b}} 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} (2 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}} \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{a B \sin (c+d x) \sqrt{\sec (c+d x)}}{b d \sqrt{a+b \cos (c+d x)}}+\frac{B \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{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)}{b d \sqrt{\sec (c+d x)}}-\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 b d \sqrt{\sec (c+d x)}}",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)]*(a*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2] + b*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2] - 2*b*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2]^3 - a*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2]^5 + b*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - (4*I)*A*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*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)] - (4*I)*A*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*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)] + I*(a - b)*B*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 - a*B)*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)]))/(b*Sqrt[(a - b)/(a + b)]*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",1
619,1,1157,539,19.7268134,"\int \frac{A+B \cos (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])/(Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)),x]","\frac{B \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 A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+4 a A b \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 a^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 a b B \tan ^5\left(\frac{1}{2} (c+d x)\right)+8 A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)-6 a b B \tan ^3\left(\frac{1}{2} (c+d x)\right)+8 a A 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 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)-8 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)-4 A b^2 \tan \left(\frac{1}{2} (c+d x)\right)-4 a A b \tan \left(\frac{1}{2} (c+d x)\right)+3 a^2 B \tan \left(\frac{1}{2} (c+d x)\right)+3 a b B \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) (3 a B-4 A 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 (a-2 b) b 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}}+8 a A 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 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}}-8 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}}\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} \left(-3 a^2 B+4 a A b-4 b^2 B\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{(4 A b-3 a B) \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 B+4 A b+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}} 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 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)}{4 a b^2 d \sqrt{\sec (c+d x)}}+\frac{B \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 b d \sqrt{\sec (c+d x)}}",1,"(B*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*A*b*Tan[(c + d*x)/2] - 4*A*b^2*Tan[(c + d*x)/2] + 3*a^2*B*Tan[(c + d*x)/2] + 3*a*b*B*Tan[(c + d*x)/2] + 8*A*b^2*Tan[(c + d*x)/2]^3 - 6*a*b*B*Tan[(c + d*x)/2]^3 + 4*a*A*b*Tan[(c + d*x)/2]^5 - 4*A*b^2*Tan[(c + d*x)/2]^5 - 3*a^2*B*Tan[(c + d*x)/2]^5 + 3*a*b*B*Tan[(c + d*x)/2]^5 + 8*a*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)] - 6*a^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)] - 8*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)] + 8*a*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)] - 6*a^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)] - 8*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)] + (a + b)*(-4*A*b + 3*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)] - 2*(a - 2*b)*b*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)]))/(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",1
620,1,3433,433,24.4139779,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^(3/2),x]","\text{Result too large to show}","\frac{2 (a+2 b) (a (A-3 B)+4 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^3 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{2 \left(a^2 A+3 a b B-4 A b^2\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 b (A b-a B) \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 \left(-3 a^3 B+5 a^2 A b+6 a b^2 B-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}} 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)*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*(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]]) + (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]]) + (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]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(-5*a^2*A*b + 8*A*b^3 + 3*a^3*B - 6*a*b^2*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*(a^2 - a*b - 2*b^2)*(-4*A*b + a*(A + 3*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)] - (-5*a^2*A*b + 8*A*b^3 + 3*a^3*B - 6*a*b^2*B)*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)*(-5*a^2*A*b + 8*A*b^3 + 3*a^3*B - 6*a*b^2*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*(a^2 - a*b - 2*b^2)*(-4*A*b + a*(A + 3*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)] - (-5*a^2*A*b + 8*A*b^3 + 3*a^3*B - 6*a*b^2*B)*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)*(-5*a^2*A*b + 8*A*b^3 + 3*a^3*B - 6*a*b^2*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*(a^2 - a*b - 2*b^2)*(-4*A*b + a*(A + 3*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)] - (-5*a^2*A*b + 8*A*b^3 + 3*a^3*B - 6*a*b^2*B)*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*((-5*a^2*A*b + 8*A*b^3 + 3*a^3*B - 6*a*b^2*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4) - ((a + b)*(-5*a^2*A*b + 8*A*b^3 + 3*a^3*B - 6*a*b^2*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*(a^2 - a*b - 2*b^2)*(-4*A*b + a*(A + 3*B))*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)*(-5*a^2*A*b + 8*A*b^3 + 3*a^3*B - 6*a*b^2*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*(a^2 - a*b - 2*b^2)*(-4*A*b + a*(A + 3*B))*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*(-5*a^2*A*b + 8*A*b^3 + 3*a^3*B - 6*a*b^2*B)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-5*a^2*A*b + 8*A*b^3 + 3*a^3*B - 6*a*b^2*B)*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-5*a^2*A*b + 8*A*b^3 + 3*a^3*B - 6*a*b^2*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (a*(a^2 - a*b - 2*b^2)*(-4*A*b + a*(A + 3*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 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a + b)*(-5*a^2*A*b + 8*A*b^3 + 3*a^3*B - 6*a*b^2*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^3*(a^2 - b^2)*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((-2*(a + b)*(-5*a^2*A*b + 8*A*b^3 + 3*a^3*B - 6*a*b^2*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*(a^2 - a*b - 2*b^2)*(-4*A*b + a*(A + 3*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)] - (-5*a^2*A*b + 8*A*b^3 + 3*a^3*B - 6*a*b^2*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^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
621,1,433,345,18.936302,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*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 b B-2 A b^2\right) \sin (c+d x)}{a^2 \left(a^2-b^2\right)}-\frac{2 \left(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{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(-\left(\left(a^2 A+a b B-2 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))\right)-2 (a+b) \left(a^2 A+a b B-2 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)+2 a (a+b) (a (A+B)-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)\right)}{a^2 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 b (A b-a B) \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-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)}{a^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{2 \left(a^2 A+a b 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}} 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)*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*(a^2 - b^2)*(a + b*Cos[c + d*x]))))/d + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(a^2*A - 2*A*b^2 + a*b*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*(a + b)*(-2*A*b + a*(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)] - (a^2*A - 2*A*b^2 + a*b*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(a^2*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2])","A",0
622,1,305,324,13.7806565,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 \left(\frac{b (A b-a B) \sin (c+d x)}{\sqrt{\sec (c+d x)}}+\frac{\sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(-\left((A b-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))\right)+2 a (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)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+2 (a+b) (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)}}\right)}{d \left(a^3-a b^2\right) \sqrt{a+b \cos (c+d x)}}","-\frac{2 (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 (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)}{a^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{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{a+b} \sqrt{\sec (c+d x)}}",1,"(2*((b*(A*b - a*B)*Sin[c + d*x])/Sqrt[Sec[c + d*x]] + (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a + b)*(-(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]))] + 2*a*(a + b)*(A - B)*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 - 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^3 - a*b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",0
623,1,1403,476,13.9760513,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/((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 (A b-a B) \sin (c+d x)}{b \left(b^2-a^2\right)}-\frac{2 \left(a A b \sin (c+d x)-a^2 B \sin (c+d x)\right)}{b \left(b^2-a^2\right) (a+b \cos (c+d x))}\right)}{d}+\frac{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}} \left(a^2 \sqrt{\frac{a-b}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)-a b \sqrt{\frac{a-b}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)+A b^2 \sqrt{\frac{a-b}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-a A b \sqrt{\frac{a-b}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+2 a b \sqrt{\frac{a-b}{a+b}} B \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 A b^2 \sqrt{\frac{a-b}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 i a^2 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 b^2 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)-a^2 \sqrt{\frac{a-b}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)-a b \sqrt{\frac{a-b}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)+A b^2 \sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+a A b \sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-i (a-b) (a B-A b) 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}}+i (a-b) ((2 a+b) B-A b) 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}}-2 i a^2 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 b^2 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}}\right)}{b \sqrt{\frac{a-b}{a+b}} \left(a^2-b^2\right) 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{2 a (A b-a B) \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 (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 b d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{2 (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)}{a b d \sqrt{a+b} \sqrt{\sec (c+d x)}}-\frac{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}} \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 - a*B)*Sin[c + d*x])/(b*(-a^2 + b^2)) - (2*(a*A*b*Sin[c + d*x] - a^2*B*Sin[c + d*x]))/(b*(-a^2 + b^2)*(a + b*Cos[c + d*x]))))/d + (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*A*b*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2] + A*b^2*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2] - a^2*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2] - a*b*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2] - 2*A*b^2*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^3 + 2*a*b*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2]^3 - a*A*b*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^5 + A*b^2*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^5 + a^2*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - a*b*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - (2*I)*a^2*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)*b^2*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*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)*b^2*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)] - I*(a - b)*(-(A*b) + a*B)*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)] + I*(a - b)*(-(A*b) + (2*a + b)*B)*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)]))/(b*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",1
624,1,1551,560,19.5587119,"\int \frac{A+B \cos (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])/((a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)),x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{2 \left(a^2 A b \sin (c+d x)-a^3 B \sin (c+d x)\right)}{b^2 \left(b^2-a^2\right) (a+b \cos (c+d x))}-\frac{2 a (a B-A b) \sin (c+d x)}{b^2 \left(a^2-b^2\right)}\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 A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a^2 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 a^3 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+b^3 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-a b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 a^2 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)-4 a A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 b^3 B \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 a^2 b B \tan ^3\left(\frac{1}{2} (c+d x)\right)+4 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)-4 a^2 A 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 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 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)+2 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+2 a^2 A b \tan \left(\frac{1}{2} (c+d x)\right)-3 a^3 B \tan \left(\frac{1}{2} (c+d x)\right)+b^3 B \tan \left(\frac{1}{2} (c+d x)\right)+a b^2 B \tan \left(\frac{1}{2} (c+d x)\right)-3 a^2 b B \tan \left(\frac{1}{2} (c+d x)\right)-(a+b) \left(3 B a^2-2 A b a-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 b (a+b) (a B-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}}+4 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}}-4 a^2 A 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 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 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}}\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 B+2 a A b+b^2 B\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 a (A b-a B) \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 B+2 a A b+b^2 B\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{\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}} \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)}}-\frac{(2 A b-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}} 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} \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((-2*a*(-(A*b) + a*B)*Sin[c + d*x])/(b^2*(a^2 - b^2)) + (2*(a^2*A*b*Sin[c + d*x] - a^3*B*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^2*A*b*Tan[(c + d*x)/2] + 2*a*A*b^2*Tan[(c + d*x)/2] - 3*a^3*B*Tan[(c + d*x)/2] - 3*a^2*b*B*Tan[(c + d*x)/2] + a*b^2*B*Tan[(c + d*x)/2] + b^3*B*Tan[(c + d*x)/2] - 4*a*A*b^2*Tan[(c + d*x)/2]^3 + 6*a^2*b*B*Tan[(c + d*x)/2]^3 - 2*b^3*B*Tan[(c + d*x)/2]^3 - 2*a^2*A*b*Tan[(c + d*x)/2]^5 + 2*a*A*b^2*Tan[(c + d*x)/2]^5 + 3*a^3*B*Tan[(c + d*x)/2]^5 - 3*a^2*b*B*Tan[(c + d*x)/2]^5 - a*b^2*B*Tan[(c + d*x)/2]^5 + b^3*B*Tan[(c + d*x)/2]^5 - 4*a^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^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)] + 6*a^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*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)] - 4*a^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)] + 4*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)] + 6*a^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*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)] - (a + b)*(-2*a*A*b + 3*a^2*B - b^2*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*b*(a + b)*(-(A*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^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
625,1,4316,607,27.1439389,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2))/(a + b*Cos[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{2 b (A b-a B) \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{2 b \left(-7 a^3 B+10 a^2 A b+3 a b^2 B-6 A b^3\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{2 \left(a^4 A+8 a^3 b B-13 a^2 A b^2-4 a b^3 B+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 B)\right)-9 a^3 b (A-B)-2 a^2 b^2 (8 A+3 B)+4 a b^3 (3 A-2 B)+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)}}-\frac{2 \left(-3 a^5 B+8 a^4 A b+15 a^3 b^2 B-28 a^2 A b^3-8 a b^4 B+16 A b^5\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 (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*(-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)*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]))/(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]))/(3*a^3*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) + (2*A*Tan[c + d*x])/(3*a^3)))/d + (2*((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]]) - (a^2*B)/((a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (5*b^2*B)/((a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*b^4*B)/(3*a^2*(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]]) - (3*a*b*B*Sqrt[Sec[c + d*x]])/((a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) + (17*b^3*B*Sqrt[Sec[c + d*x]])/(3*a*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) - (8*b^5*B*Sqrt[Sec[c + d*x]])/(3*a^3*(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]]) - (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]]) + (5*b^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(a*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) - (8*b^5*B*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]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(-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)*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)*(-16*A*b^4 + 2*a^2*b^2*(8*A - 3*B) - 9*a^3*b*(A + B) + 4*a*b^3*(3*A + 2*B) + a^4*(A + 3*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)] - (-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)*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]*((b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(-2*(a + b)*(-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)*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)*(-16*A*b^4 + 2*a^2*b^2*(8*A - 3*B) - 9*a^3*b*(A + B) + 4*a*b^3*(3*A + 2*B) + a^4*(A + 3*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)] - (-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)*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]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(-2*(a + b)*(-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)*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)*(-16*A*b^4 + 2*a^2*b^2*(8*A - 3*B) - 9*a^3*b*(A + B) + 4*a*b^3*(3*A + 2*B) + a^4*(A + 3*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)] - (-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)*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]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1/2*((-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)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4) - ((a + b)*(-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)*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)*(-16*A*b^4 + 2*a^2*b^2*(8*A - 3*B) - 9*a^3*b*(A + B) + 4*a*b^3*(3*A + 2*B) + a^4*(A + 3*B))*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^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)*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)*(-16*A*b^4 + 2*a^2*b^2*(8*A - 3*B) - 9*a^3*b*(A + B) + 4*a*b^3*(3*A + 2*B) + a^4*(A + 3*B))*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^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)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(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)*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(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)*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)*(-16*A*b^4 + 2*a^2*b^2*(8*A - 3*B) - 9*a^3*b*(A + B) + 4*a*b^3*(3*A + 2*B) + a^4*(A + 3*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 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a + b)*(-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)*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*(a + b)*(-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)*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)*(-16*A*b^4 + 2*a^2*b^2*(8*A - 3*B) - 9*a^3*b*(A + B) + 4*a*b^3*(3*A + 2*B) + a^4*(A + 3*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)] - (-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)*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
626,1,3891,496,26.7509843,"\int \frac{(A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{2 b (A b-a B) \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{2 b \left(-5 a^3 B+8 a^2 A b+a b^2 B-4 A b^3\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^3 (A-B)-3 a^2 b (3 A+B)+2 a b^2 (3 A-B)+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)}}+\frac{2 \left(3 a^4 A+6 a^3 b B-15 a^2 A b^2-2 a b^3 B+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 (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*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B)*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]))/(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]))/(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]]) - (2*a*b*B)/((a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b^3*B)/(3*a*(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^2*B*Sqrt[Sec[c + d*x]])/((a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) - (5*b^2*B*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]) + (2*b^4*B*Sqrt[Sec[c + d*x]])/(3*a^2*(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]]) - (2*b^2*B*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*B*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*(a + b)*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*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*(a + b)*(8*A*b^3 + 3*a^2*b*(-3*A + B) + 3*a^3*(A + B) - 2*a*b^2*(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]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B)*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)*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*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*(a + b)*(8*A*b^3 + 3*a^2*b*(-3*A + B) + 3*a^3*(A + B) - 2*a*b^2*(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]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B)*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)*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*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*(a + b)*(8*A*b^3 + 3*a^2*b*(-3*A + B) + 3*a^3*(A + B) - 2*a*b^2*(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]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B)*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*((3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4) - ((a + b)*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*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*(a + b)*(8*A*b^3 + 3*a^2*b*(-3*A + B) + 3*a^3*(A + B) - 2*a*b^2*(3*A + B))*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)*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*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*(a + b)*(8*A*b^3 + 3*a^2*b*(-3*A + B) + 3*a^3*(A + B) - 2*a*b^2*(3*A + B))*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*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(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)*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(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)*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^3 + 3*a^2*b*(-3*A + B) + 3*a^3*(A + B) - 2*a*b^2*(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 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a + b)*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*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^3*(a^2 - b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((-2*(a + b)*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*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*(a + b)*(8*A*b^3 + 3*a^2*b*(-3*A + B) + 3*a^3*(A + B) - 2*a*b^2*(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]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*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^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
627,1,3493,469,24.5245016,"\int \frac{(A+B \cos (c+d x)) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{2 b (A b-a B) \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(-3 a^2 (A+B)+a b (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)}{3 a^2 d \sqrt{a+b} \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{2 \left(-3 a^3 B+6 a^2 A b-a b^2 B-2 A b^3\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(-3 a^3 B+6 a^2 A b-a b^2 B-2 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}} 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)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2) + (2*(-(A*b*Sin[c + d*x]) + a*B*Sin[c + d*x]))/(3*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) + (2*(-5*a^2*A*b*Sin[c + d*x] + A*b^3*Sin[c + d*x] + 2*a^3*B*Sin[c + d*x] + 2*a*b^2*B*Sin[c + d*x]))/(3*a*(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]]) + (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]]) - (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]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a + b)*(-6*a^2*A*b + 2*A*b^3 + 3*a^3*B + a*b^2*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*(a + b)*(-2*A*b^2 + 3*a^2*(A - B) + 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]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (-6*a^2*A*b + 2*A*b^3 + 3*a^3*B + a*b^2*B)*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)*(-6*a^2*A*b + 2*A*b^3 + 3*a^3*B + a*b^2*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*(a + b)*(-2*A*b^2 + 3*a^2*(A - B) + 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]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (-6*a^2*A*b + 2*A*b^3 + 3*a^3*B + a*b^2*B)*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)*(-6*a^2*A*b + 2*A*b^3 + 3*a^3*B + a*b^2*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*(a + b)*(-2*A*b^2 + 3*a^2*(A - B) + 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]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (-6*a^2*A*b + 2*A*b^3 + 3*a^3*B + a*b^2*B)*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]]*(((-6*a^2*A*b + 2*A*b^3 + 3*a^3*B + a*b^2*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-6*a^2*A*b + 2*A*b^3 + 3*a^3*B + a*b^2*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*(a + b)*(-2*A*b^2 + 3*a^2*(A - B) + a*b*(3*A - B))*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^2*A*b + 2*A*b^3 + 3*a^3*B + a*b^2*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*(a + b)*(-2*A*b^2 + 3*a^2*(A - B) + a*b*(3*A - B))*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^2*A*b + 2*A*b^3 + 3*a^3*B + a*b^2*B)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-6*a^2*A*b + 2*A*b^3 + 3*a^3*B + a*b^2*B)*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-6*a^2*A*b + 2*A*b^3 + 3*a^3*B + a*b^2*B)*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^2*(A - B) + 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 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-6*a^2*A*b + 2*A*b^3 + 3*a^3*B + a*b^2*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^3 - a*b^2)^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((2*(a + b)*(-6*a^2*A*b + 2*A*b^3 + 3*a^3*B + a*b^2*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*(a + b)*(-2*A*b^2 + 3*a^2*(A - B) + 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]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (-6*a^2*A*b + 2*A*b^3 + 3*a^3*B + a*b^2*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^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
628,1,528,431,18.8006518,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]),x]","\frac{\sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)} \left(-\frac{2 \left(3 a^2 A-4 a b B+A b^2\right) \sin (c+d x)}{3 a \left(a^2-b^2\right)^2}+\frac{2 \left(a^2 B \sin (c+d x)-a A b \sin (c+d x)\right)}{3 b \left(b^2-a^2\right) (a+b \cos (c+d x))^2}+\frac{2 \left(a^3 B \sin (c+d x)+2 a^2 A b \sin (c+d x)-5 a b^2 B \sin (c+d x)+2 A b^3 \sin (c+d x)\right)}{3 b \left(b^2-a^2\right)^2 (a+b \cos (c+d x))}\right)}{d}-\frac{2 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(-\left(\left(3 a^2 A-4 a b B+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))\right)-2 (a+b) \left(3 a^2 A-4 a b B+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)+2 a (a+b) (3 a A-a B+A b-3 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)\right)}{3 a d \left(a^2-b^2\right)^2 \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(3 a^2 A-4 a b B+A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 (A b-a B) \sin (c+d x)}{3 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^2 A-4 a b B+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)}{3 a^2 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}+\frac{2 (a (3 A+B)-b (A+3 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 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*(3*a^2*A + A*b^2 - 4*a*b*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)^2) + (2*(-(a*A*b*Sin[c + d*x]) + a^2*B*Sin[c + d*x]))/(3*b*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (2*(2*a^2*A*b*Sin[c + d*x] + 2*A*b^3*Sin[c + d*x] + a^3*B*Sin[c + d*x] - 5*a*b^2*B*Sin[c + d*x]))/(3*b*(-a^2 + b^2)^2*(a + b*Cos[c + d*x]))))/d - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(a + b)*(3*a^2*A + A*b^2 - 4*a*b*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*(a + b)*(3*a*A + A*b - a*B - 3*b*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)] - (3*a^2*A + A*b^2 - 4*a*b*B)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*a*(a^2 - b^2)^2*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2])","A",0
629,1,1994,602,16.2060806,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x))^{5/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x])/((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{2 \left(3 B a^3-7 b^2 B a+4 A b^3\right) \sin (c+d x)}{3 b^2 \left(b^2-a^2\right)^2}-\frac{2 \left(a^3 B \sin (c+d x)-a^2 A 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(4 B \sin (c+d x) a^4-A b \sin (c+d x) a^3-8 b^2 B \sin (c+d x) a^2+5 A b^3 \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{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}} \left(-3 a^4 \sqrt{\frac{a-b}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)-7 a b^3 \sqrt{\frac{a-b}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)+7 a^2 b^2 \sqrt{\frac{a-b}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 a^3 b \sqrt{\frac{a-b}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)+4 A b^4 \sqrt{\frac{a-b}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-4 a A b^3 \sqrt{\frac{a-b}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+14 a b^3 \sqrt{\frac{a-b}{a+b}} B \tan ^3\left(\frac{1}{2} (c+d x)\right)-6 a^3 b \sqrt{\frac{a-b}{a+b}} B \tan ^3\left(\frac{1}{2} (c+d x)\right)-8 A b^4 \sqrt{\frac{a-b}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 i a^4 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)+6 i b^4 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)-12 i a^2 b^2 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)+3 a^4 \sqrt{\frac{a-b}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)-7 a b^3 \sqrt{\frac{a-b}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)-7 a^2 b^2 \sqrt{\frac{a-b}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)+3 a^3 b \sqrt{\frac{a-b}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)+4 A b^4 \sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+4 a A b^3 \sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+i (a-b) \left(3 B a^3-7 b^2 B a+4 A b^3\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) \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}}-i (a-b) \left(6 B a^3+4 b B a^2-b^2 (A+9 B) a+3 b^3 (A-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) \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 i a^4 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}}+6 i b^4 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}}-12 i a^2 b^2 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}}\right)}{3 b^2 \sqrt{\frac{a-b}{a+b}} \left(a^2-b^2\right)^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{2 \left(3 a^3 B-7 a b^2 B+4 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}} 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} \sqrt{\sec (c+d x)}}+\frac{2 a (A b-a B) \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 a \left(3 a^3 B-7 a b^2 B+4 A b^3\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^3 B+a^2 b B-a b^2 (A+6 B)+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)}{3 a b^2 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}-\frac{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}} \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*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*Sin[c + d*x])/(3*b^2*(-a^2 + b^2)^2) - (2*(-(a^2*A*b*Sin[c + d*x]) + a^3*B*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) - (2*(-(a^3*A*b*Sin[c + d*x]) + 5*a*A*b^3*Sin[c + d*x] + 4*a^4*B*Sin[c + d*x] - 8*a^2*b^2*B*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)^2*(a + b*Cos[c + d*x]))))/d - (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)]*(4*a*A*b^3*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2] + 4*A*b^4*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2] + 3*a^4*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2] + 3*a^3*b*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2] - 7*a^2*b^2*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2] - 7*a*b^3*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2] - 8*A*b^4*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^3 - 6*a^3*b*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2]^3 + 14*a*b^3*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2]^3 - 4*a*A*b^3*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^5 + 4*A*b^4*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^5 - 3*a^4*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2]^5 + 3*a^3*b*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2]^5 + 7*a^2*b^2*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - 7*a*b^3*Sqrt[(a - b)/(a + b)]*B*Tan[(c + d*x)/2]^5 + (6*I)*a^4*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)] - (12*I)*a^2*b^2*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)] + (6*I)*b^4*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)] + (6*I)*a^4*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)] - (12*I)*a^2*b^2*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)] + (6*I)*b^4*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)] + I*(a - b)*(4*A*b^3 + 3*a^3*B - 7*a*b^2*B)*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)] - I*(a - b)*(3*b^3*(A - B) + 6*a^3*B + 4*a^2*b*B - a*b^2*(A + 9*B))*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)]))/(3*b^2*Sqrt[(a - b)/(a + b)]*(a^2 - b^2)^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
630,1,2318,733,22.3688924,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x])/((a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)),x]","\text{Result too large to show}","\frac{2 a (A b-a B) \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 a \left(-5 a^3 B+2 a^2 A b+9 a b^2 B-6 A b^3\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(15 a^3 B-a^2 (6 A b-5 b B)-a b^2 (2 A+21 B)+3 b^3 (4 A-B)\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 b^3 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}-\frac{\left(-15 a^4 B+6 a^3 A b+26 a^2 b^2 B-14 a A b^3-3 b^4 B\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(-15 a^4 B+6 a^3 A b+26 a^2 b^2 B-14 a A b^3-3 b^4 B\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{\sqrt{a+b} (2 A b-5 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]]*((-2*a*(-3*a^2*A*b + 7*A*b^3 + 6*a^3*B - 10*a*b^2*B)*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2) + (2*(-(a^3*A*b*Sin[c + d*x]) + a^4*B*Sin[c + d*x]))/(3*b^3*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (2*(-4*a^4*A*b*Sin[c + d*x] + 8*a^2*A*b^3*Sin[c + d*x] + 7*a^5*B*Sin[c + d*x] - 11*a^3*b^2*B*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)]*(6*a^4*A*b*Tan[(c + d*x)/2] + 6*a^3*A*b^2*Tan[(c + d*x)/2] - 14*a^2*A*b^3*Tan[(c + d*x)/2] - 14*a*A*b^4*Tan[(c + d*x)/2] - 15*a^5*B*Tan[(c + d*x)/2] - 15*a^4*b*B*Tan[(c + d*x)/2] + 26*a^3*b^2*B*Tan[(c + d*x)/2] + 26*a^2*b^3*B*Tan[(c + d*x)/2] - 3*a*b^4*B*Tan[(c + d*x)/2] - 3*b^5*B*Tan[(c + d*x)/2] - 12*a^3*A*b^2*Tan[(c + d*x)/2]^3 + 28*a*A*b^4*Tan[(c + d*x)/2]^3 + 30*a^4*b*B*Tan[(c + d*x)/2]^3 - 52*a^2*b^3*B*Tan[(c + d*x)/2]^3 + 6*b^5*B*Tan[(c + d*x)/2]^3 - 6*a^4*A*b*Tan[(c + d*x)/2]^5 + 6*a^3*A*b^2*Tan[(c + d*x)/2]^5 + 14*a^2*A*b^3*Tan[(c + d*x)/2]^5 - 14*a*A*b^4*Tan[(c + d*x)/2]^5 + 15*a^5*B*Tan[(c + d*x)/2]^5 - 15*a^4*b*B*Tan[(c + d*x)/2]^5 - 26*a^3*b^2*B*Tan[(c + d*x)/2]^5 + 26*a^2*b^3*B*Tan[(c + d*x)/2]^5 + 3*a*b^4*B*Tan[(c + d*x)/2]^5 - 3*b^5*B*Tan[(c + d*x)/2]^5 - 12*a^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)] + 24*a^2*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)] - 12*A*b^5*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*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)] - 60*a^3*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)] + 30*a*b^4*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)] - 12*a^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)] + 24*a^2*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)] - 12*A*b^5*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^5*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)] - 60*a^3*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)] + 30*a*b^4*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)*(-6*a^3*A*b + 14*a*A*b^3 + 15*a^4*B - 26*a^2*b^2*B + 3*b^4*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*b*(a + b)*(3*A*b^3 + 3*a*b^2*(A - 2*B) + 5*a^3*B - a^2*b*(2*A + 3*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)]))/(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
631,1,298,266,6.10507,"\int \frac{(a B+b B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((a*B + b*B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^(3/2),x]","B \left(\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{a d}-\frac{2 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(\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 \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) \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)}{a d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \cos (c+d x)}}\right)","\frac{2 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^2 d \sqrt{\sec (c+d x)}}-\frac{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)}{a d \sqrt{\sec (c+d x)}}",1,"B*((2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(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*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)] + Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(a*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]))","A",1
632,1,104,130,0.1450639,"\int \frac{(a B+b B \cos (c+d x)) \sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[((a*B + b*B*Cos[c + d*x])*Sqrt[Sec[c + d*x]])/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 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)}{d \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}","\frac{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)}{a d \sqrt{\sec (c+d x)}}",1,"(2*B*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)])/(d*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])","A",1
633,1,147,137,0.1886154,"\int \frac{a B+b B \cos (c+d x)}{(a+b \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}} \, dx","Integrate[(a*B + b*B*Cos[c + d*x])/((a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]),x]","-\frac{2 B \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\sec (c+d x)+1} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} \left(F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-2 \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{\frac{1}{\cos (c+d x)+1}} \sqrt{a+b \cos (c+d x)}}","-\frac{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}} \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*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)] - 2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])*Sqrt[1 + Sec[c + d*x]])/(d*Sqrt[(1 + Cos[c + d*x])^(-1)]*Sqrt[a + b*Cos[c + d*x]])","A",0
634,1,508,479,3.3291032,"\int \frac{a B+b B \cos (c+d x)}{(a+b \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a*B + b*B*Cos[c + d*x])/((a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)),x]","\frac{B \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)+1} \left(2 a \sqrt{\frac{a-b}{a+b}} \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \tan \left(\frac{1}{2} (c+d x)\right)-b \sqrt{\frac{a-b}{a+b}} \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \tan \left(\frac{1}{2} (c+d x)\right)+b \sqrt{\frac{a-b}{a+b}} \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)-4 i a \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} 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)+2 i (a-b) \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} 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)+4 i a \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} \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)}{4 b d \sqrt{\frac{a-b}{a+b}} \left(\frac{1}{\cos (c+d x)+1}\right)^{3/2} \sqrt{a+b \cos (c+d x)}}","\frac{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}} \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{a B \sin (c+d x) \sqrt{\sec (c+d x)}}{b d \sqrt{a+b \cos (c+d x)}}+\frac{B \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{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)}{b d \sqrt{\sec (c+d x)}}-\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 b d \sqrt{\sec (c+d x)}}",1,"(B*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]^2*Sqrt[1 + Sec[c + d*x]]*((2*I)*(a - b)*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)*a*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))] + (4*I)*a*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)]*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)]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Tan[(c + d*x)/2] - b*Sqrt[(a - b)/(a + b)]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Tan[(c + d*x)/2]))/(4*b*Sqrt[(a - b)/(a + b)]*d*((1 + Cos[c + d*x])^(-1))^(3/2)*Sqrt[a + b*Cos[c + d*x]])","C",1
635,0,0,59,9.5193121,"\int (a+b \cos (e+f x))^n (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx","Integrate[(a + b*Cos[e + f*x])^n*(A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m,x]","\int (a+b \cos (e+f x))^n (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx","(c \cos (e+f x))^m (c \sec (e+f x))^m \text{Int}\left((A+B \cos (e+f x)) (c \cos (e+f x))^{-m} (a+b \cos (e+f x))^n,x\right)",0,"Integrate[(a + b*Cos[e + f*x])^n*(A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m, x]","A",-1
636,1,317,644,4.3705891,"\int (a+b \cos (e+f x))^4 (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx","Integrate[(a + b*Cos[e + f*x])^4*(A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m,x]","\frac{\sqrt{-\tan ^2(e+f x)} \cot (e+f x) (c \sec (e+f x))^m \left(\frac{b^3 (4 a B+A b) \cos ^4(e+f x) \, _2F_1\left(\frac{1}{2},\frac{m-4}{2};\frac{m-2}{2};\sec ^2(e+f x)\right)}{m-4}+a \left(\frac{2 b^2 (3 a B+2 A b) \cos ^3(e+f x) \, _2F_1\left(\frac{1}{2},\frac{m-3}{2};\frac{m-1}{2};\sec ^2(e+f x)\right)}{m-3}+a \left(\frac{2 b (2 a B+3 A b) \cos ^2(e+f x) \, _2F_1\left(\frac{1}{2},\frac{m-2}{2};\frac{m}{2};\sec ^2(e+f x)\right)}{m-2}+a \left(\frac{(a B+4 A b) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{m-1}{2};\frac{m+1}{2};\sec ^2(e+f x)\right)}{m-1}+\frac{a A \, _2F_1\left(\frac{1}{2},\frac{m}{2};\frac{m+2}{2};\sec ^2(e+f x)\right)}{m}\right)\right)\right)+\frac{b^4 B \cos ^5(e+f x) \, _2F_1\left(\frac{1}{2},\frac{m-5}{2};\frac{m-3}{2};\sec ^2(e+f x)\right)}{m-5}\right)}{f}","-\frac{a^2 c^5 \tan (e+f x) \sec (e+f x) \left(a^2 A (2-m)^2+2 a b B (1-m)^2+A b^2 \left(m^2-m+6\right)\right) (c \sec (e+f x))^{m-5}}{f (1-m) (2-m) (3-m)}-\frac{a c^5 \tan (e+f x) \left(a^3 B \left(m^2-4 m+3\right)+4 a^2 A b \left(m^2-4 m+3\right)+a b^2 B \left(5 m^2-13 m+8\right)+2 A b^3 \left(m^2-2 m+4\right)\right) (c \sec (e+f x))^{m-5}}{f (1-m) (2-m) (4-m)}-\frac{c^6 \sin (e+f x) \left(a^4 B \left(m^2-8 m+15\right)+4 a^3 A b \left(m^2-8 m+15\right)+6 a^2 b^2 B \left(m^2-7 m+10\right)+4 a A b^3 \left(m^2-7 m+10\right)+b^4 B \left(m^2-6 m+8\right)\right) (c \sec (e+f x))^{m-6} \, _2F_1\left(\frac{1}{2},\frac{6-m}{2};\frac{8-m}{2};\cos ^2(e+f x)\right)}{f (2-m) (4-m) (6-m) \sqrt{\sin ^2(e+f x)}}-\frac{c^5 \sin (e+f x) \left(a^4 A \left(m^2-6 m+8\right)+4 a^3 b B \left(m^2-5 m+4\right)+6 a^2 A b^2 \left(m^2-5 m+4\right)+4 a b^3 B \left(m^2-4 m+3\right)+A b^4 \left(m^2-4 m+3\right)\right) (c \sec (e+f x))^{m-5} \, _2F_1\left(\frac{1}{2},\frac{5-m}{2};\frac{7-m}{2};\cos ^2(e+f x)\right)}{f (1-m) (3-m) (5-m) \sqrt{\sin ^2(e+f x)}}-\frac{a c^5 \tan (e+f x) (a B (1-m)-A b (m+2)) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-5}}{f (1-m) (2-m)}-\frac{a A c^5 \tan (e+f x) (a \sec (e+f x)+b)^3 (c \sec (e+f x))^{m-5}}{f (1-m)}",1,"(Cot[e + f*x]*((b^4*B*Cos[e + f*x]^5*Hypergeometric2F1[1/2, (-5 + m)/2, (-3 + m)/2, Sec[e + f*x]^2])/(-5 + m) + (b^3*(A*b + 4*a*B)*Cos[e + f*x]^4*Hypergeometric2F1[1/2, (-4 + m)/2, (-2 + m)/2, Sec[e + f*x]^2])/(-4 + m) + a*((2*b^2*(2*A*b + 3*a*B)*Cos[e + f*x]^3*Hypergeometric2F1[1/2, (-3 + m)/2, (-1 + m)/2, Sec[e + f*x]^2])/(-3 + m) + a*((2*b*(3*A*b + 2*a*B)*Cos[e + f*x]^2*Hypergeometric2F1[1/2, (-2 + m)/2, m/2, Sec[e + f*x]^2])/(-2 + m) + a*(((4*A*b + a*B)*Cos[e + f*x]*Hypergeometric2F1[1/2, (-1 + m)/2, (1 + m)/2, Sec[e + f*x]^2])/(-1 + m) + (a*A*Hypergeometric2F1[1/2, m/2, (2 + m)/2, Sec[e + f*x]^2])/m))))*(c*Sec[e + f*x])^m*Sqrt[-Tan[e + f*x]^2])/f","A",1
637,1,259,455,2.5207794,"\int (a+b \cos (e+f x))^3 (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx","Integrate[(a + b*Cos[e + f*x])^3*(A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m,x]","\frac{\sqrt{-\tan ^2(e+f x)} \cot (e+f x) (c \sec (e+f x))^m \left(\frac{b^2 (3 a B+A b) \cos ^3(e+f x) \, _2F_1\left(\frac{1}{2},\frac{m-3}{2};\frac{m-1}{2};\sec ^2(e+f x)\right)}{m-3}+a \left(\frac{3 b (a B+A b) \cos ^2(e+f x) \, _2F_1\left(\frac{1}{2},\frac{m-2}{2};\frac{m}{2};\sec ^2(e+f x)\right)}{m-2}+a \left(\frac{(a B+3 A b) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{m-1}{2};\frac{m+1}{2};\sec ^2(e+f x)\right)}{m-1}+\frac{a A \, _2F_1\left(\frac{1}{2},\frac{m}{2};\frac{m+2}{2};\sec ^2(e+f x)\right)}{m}\right)\right)+\frac{b^3 B \cos ^4(e+f x) \, _2F_1\left(\frac{1}{2},\frac{m-4}{2};\frac{m-2}{2};\sec ^2(e+f x)\right)}{m-4}\right)}{f}","-\frac{a c^4 \tan (e+f x) \left(a^2 A (2-m)+3 a b B (1-m)-2 A b^2 m\right) (c \sec (e+f x))^{m-4}}{f (1-m) (3-m)}-\frac{a^2 c^4 \tan (e+f x) \sec (e+f x) (a B (1-m)-A b (m+1)) (c \sec (e+f x))^{m-4}}{f (1-m) (2-m)}-\frac{c^5 \sin (e+f x) \left(a^3 A \left(m^2-6 m+8\right)+3 a^2 b B \left(m^2-5 m+4\right)+3 a A b^2 \left(m^2-5 m+4\right)+b^3 B \left(m^2-4 m+3\right)\right) (c \sec (e+f x))^{m-5} \, _2F_1\left(\frac{1}{2},\frac{5-m}{2};\frac{7-m}{2};\cos ^2(e+f x)\right)}{f (1-m) (3-m) (5-m) \sqrt{\sin ^2(e+f x)}}-\frac{c^4 \sin (e+f x) \left(a^3 B (3-m)+3 a^2 A b (3-m)+3 a b^2 B (2-m)+A b^3 (2-m)\right) (c \sec (e+f x))^{m-4} \, _2F_1\left(\frac{1}{2},\frac{4-m}{2};\frac{6-m}{2};\cos ^2(e+f x)\right)}{f (2-m) (4-m) \sqrt{\sin ^2(e+f x)}}-\frac{a A c^4 \tan (e+f x) (a \sec (e+f x)+b)^2 (c \sec (e+f x))^{m-4}}{f (1-m)}",1,"(Cot[e + f*x]*((b^3*B*Cos[e + f*x]^4*Hypergeometric2F1[1/2, (-4 + m)/2, (-2 + m)/2, Sec[e + f*x]^2])/(-4 + m) + (b^2*(A*b + 3*a*B)*Cos[e + f*x]^3*Hypergeometric2F1[1/2, (-3 + m)/2, (-1 + m)/2, Sec[e + f*x]^2])/(-3 + m) + a*((3*b*(A*b + a*B)*Cos[e + f*x]^2*Hypergeometric2F1[1/2, (-2 + m)/2, m/2, Sec[e + f*x]^2])/(-2 + m) + a*(((3*A*b + a*B)*Cos[e + f*x]*Hypergeometric2F1[1/2, (-1 + m)/2, (1 + m)/2, Sec[e + f*x]^2])/(-1 + m) + (a*A*Hypergeometric2F1[1/2, m/2, (2 + m)/2, Sec[e + f*x]^2])/m)))*(c*Sec[e + f*x])^m*Sqrt[-Tan[e + f*x]^2])/f","A",1
638,1,205,327,0.9860943,"\int (a+b \cos (e+f x))^2 (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx","Integrate[(a + b*Cos[e + f*x])^2*(A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m,x]","\frac{\sqrt{-\tan ^2(e+f x)} \cot (e+f x) (c \sec (e+f x))^m \left(\frac{b (2 a B+A b) \cos ^2(e+f x) \, _2F_1\left(\frac{1}{2},\frac{m-2}{2};\frac{m}{2};\sec ^2(e+f x)\right)}{m-2}+a \left(\frac{(a B+2 A b) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{m-1}{2};\frac{m+1}{2};\sec ^2(e+f x)\right)}{m-1}+\frac{a A \, _2F_1\left(\frac{1}{2},\frac{m}{2};\frac{m+2}{2};\sec ^2(e+f x)\right)}{m}\right)+\frac{b^2 B \cos ^3(e+f x) \, _2F_1\left(\frac{1}{2},\frac{m-3}{2};\frac{m-1}{2};\sec ^2(e+f x)\right)}{m-3}\right)}{f}","-\frac{c^4 \sin (e+f x) \left(a^2 B (3-m)+2 a A b (3-m)+b^2 B (2-m)\right) (c \sec (e+f x))^{m-4} \, _2F_1\left(\frac{1}{2},\frac{4-m}{2};\frac{6-m}{2};\cos ^2(e+f x)\right)}{f (2-m) (4-m) \sqrt{\sin ^2(e+f x)}}-\frac{c^3 \sin (e+f x) \left(a^2 A (2-m)+2 a b B (1-m)+A b^2 (1-m)\right) (c \sec (e+f x))^{m-3} \, _2F_1\left(\frac{1}{2},\frac{3-m}{2};\frac{5-m}{2};\cos ^2(e+f x)\right)}{f (1-m) (3-m) \sqrt{\sin ^2(e+f x)}}-\frac{a c^3 \tan (e+f x) (a B (1-m)-A b m) (c \sec (e+f x))^{m-3}}{f (1-m) (2-m)}-\frac{a A c^3 \tan (e+f x) (a \sec (e+f x)+b) (c \sec (e+f x))^{m-3}}{f (1-m)}",1,"(Cot[e + f*x]*((b^2*B*Cos[e + f*x]^3*Hypergeometric2F1[1/2, (-3 + m)/2, (-1 + m)/2, Sec[e + f*x]^2])/(-3 + m) + (b*(A*b + 2*a*B)*Cos[e + f*x]^2*Hypergeometric2F1[1/2, (-2 + m)/2, m/2, Sec[e + f*x]^2])/(-2 + m) + a*(((2*A*b + a*B)*Cos[e + f*x]*Hypergeometric2F1[1/2, (-1 + m)/2, (1 + m)/2, Sec[e + f*x]^2])/(-1 + m) + (a*A*Hypergeometric2F1[1/2, m/2, (2 + m)/2, Sec[e + f*x]^2])/m))*(c*Sec[e + f*x])^m*Sqrt[-Tan[e + f*x]^2])/f","A",1
639,1,163,217,0.3834543,"\int (a+b \cos (e+f x)) (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx","Integrate[(a + b*Cos[e + f*x])*(A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m,x]","\frac{\sqrt{-\tan ^2(e+f x)} \cot (e+f x) (c \sec (e+f x))^m \left((m-2) \left(m (a B+A b) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{m-1}{2};\frac{m+1}{2};\sec ^2(e+f x)\right)+a A (m-1) \, _2F_1\left(\frac{1}{2},\frac{m}{2};\frac{m+2}{2};\sec ^2(e+f x)\right)\right)+b B (m-1) m \cos ^2(e+f x) \, _2F_1\left(\frac{1}{2},\frac{m-2}{2};\frac{m}{2};\sec ^2(e+f x)\right)\right)}{f (m-2) (m-1) m}","-\frac{c^3 \sin (e+f x) (a A (2-m)+b B (1-m)) (c \sec (e+f x))^{m-3} \, _2F_1\left(\frac{1}{2},\frac{3-m}{2};\frac{5-m}{2};\cos ^2(e+f x)\right)}{f (1-m) (3-m) \sqrt{\sin ^2(e+f x)}}-\frac{c^2 (a B+A b) \sin (e+f x) (c \sec (e+f x))^{m-2} \, _2F_1\left(\frac{1}{2},\frac{2-m}{2};\frac{4-m}{2};\cos ^2(e+f x)\right)}{f (2-m) \sqrt{\sin ^2(e+f x)}}-\frac{a A c^2 \tan (e+f x) (c \sec (e+f x))^{m-2}}{f (1-m)}",1,"(Cot[e + f*x]*(b*B*(-1 + m)*m*Cos[e + f*x]^2*Hypergeometric2F1[1/2, (-2 + m)/2, m/2, Sec[e + f*x]^2] + (-2 + m)*((A*b + a*B)*m*Cos[e + f*x]*Hypergeometric2F1[1/2, (-1 + m)/2, (1 + m)/2, Sec[e + f*x]^2] + a*A*(-1 + m)*Hypergeometric2F1[1/2, m/2, (2 + m)/2, Sec[e + f*x]^2]))*(c*Sec[e + f*x])^m*Sqrt[-Tan[e + f*x]^2])/(f*(-2 + m)*(-1 + m)*m)","A",1
640,1,10630,299,26.4437836,"\int \frac{(A+B \cos (e+f x)) (c \sec (e+f x))^m}{a+b \cos (e+f x)} \, dx","Integrate[((A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m)/(a + b*Cos[e + f*x]),x]","\text{Result too large to show}","-\frac{(A b-a B) \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{m/2} (c \sec (e+f x))^{m+1} F_1\left(\frac{1}{2};\frac{m}{2},1;\frac{3}{2};\sin ^2(e+f x),-\frac{b^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{c f \left(a^2-b^2\right)}+\frac{a (A b-a B) \sin (e+f x) \cos ^2(e+f x)^{\frac{m+1}{2}} (c \sec (e+f x))^{m+1} F_1\left(\frac{1}{2};\frac{m+1}{2},1;\frac{3}{2};\sin ^2(e+f x),-\frac{b^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{b c f \left(a^2-b^2\right)}-\frac{B c \sin (e+f x) (c \sec (e+f x))^{m-1} \, _2F_1\left(\frac{1}{2},\frac{1-m}{2};\frac{3-m}{2};\cos ^2(e+f x)\right)}{b f (1-m) \sqrt{\sin ^2(e+f x)}}",1,"Result too large to show","B",0
641,0,0,210,61.5942329,"\int (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx","Integrate[(a + b*Cos[e + f*x])^(3/2)*(A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m,x]","\int (a+b \cos (e+f x))^{3/2} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx","\frac{2 (c \cos (e+f x))^m (c \sec (e+f x))^m \text{Int}\left(\frac{(c \cos (e+f x))^{-m} \left(\frac{1}{2} c \cos (e+f x) \left(a (5-2 m) (a B+2 A b)+b^2 B (3-2 m)\right)+\frac{1}{2} b c \cos ^2(e+f x) (2 a B (3-m)+A b (5-2 m))+\frac{1}{2} a c \left(2 a A \left(\frac{5}{2}-m\right)+2 b B (1-m)\right)\right)}{\sqrt{a+b \cos (e+f x)}},x\right)}{c (5-2 m)}+\frac{2 b B \sin (e+f x) \cos (e+f x) \sqrt{a+b \cos (e+f x)} (c \sec (e+f x))^m}{f (5-2 m)}",0,"Integrate[(a + b*Cos[e + f*x])^(3/2)*(A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m, x]","A",-1
642,0,0,61,13.1987618,"\int \sqrt{a+b \cos (e+f x)} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx","Integrate[Sqrt[a + b*Cos[e + f*x]]*(A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m,x]","\int \sqrt{a+b \cos (e+f x)} (A+B \cos (e+f x)) (c \sec (e+f x))^m \, dx","(c \cos (e+f x))^m (c \sec (e+f x))^m \text{Int}\left(\sqrt{a+b \cos (e+f x)} (A+B \cos (e+f x)) (c \cos (e+f x))^{-m},x\right)",0,"Integrate[Sqrt[a + b*Cos[e + f*x]]*(A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m, x]","A",-1
643,0,0,61,8.8749798,"\int \frac{(A+B \cos (e+f x)) (c \sec (e+f x))^m}{\sqrt{a+b \cos (e+f x)}} \, dx","Integrate[((A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m)/Sqrt[a + b*Cos[e + f*x]],x]","\int \frac{(A+B \cos (e+f x)) (c \sec (e+f x))^m}{\sqrt{a+b \cos (e+f x)}} \, dx","(c \cos (e+f x))^m (c \sec (e+f x))^m \text{Int}\left(\frac{(A+B \cos (e+f x)) (c \cos (e+f x))^{-m}}{\sqrt{a+b \cos (e+f x)}},x\right)",0,"Integrate[((A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m)/Sqrt[a + b*Cos[e + f*x]], x]","A",-1
644,0,0,213,11.2117073,"\int \frac{(A+B \cos (e+f x)) (c \sec (e+f x))^m}{(a+b \cos (e+f x))^{3/2}} \, dx","Integrate[((A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m)/(a + b*Cos[e + f*x])^(3/2),x]","\int \frac{(A+B \cos (e+f x)) (c \sec (e+f x))^m}{(a+b \cos (e+f x))^{3/2}} \, dx","\frac{2 (c \cos (e+f x))^m (c \sec (e+f x))^m \text{Int}\left(\frac{(c \cos (e+f x))^{-m} \left(\frac{1}{2} c \left(a^2 A-2 a b B (1-m)+A b^2 (1-2 m)\right)-\frac{1}{2} b c (3-2 m) (A b-a B) \cos ^2(e+f x)-\frac{1}{2} a c (A b-a B) \cos (e+f x)\right)}{\sqrt{a+b \cos (e+f x)}},x\right)}{a c \left(a^2-b^2\right)}+\frac{2 b (A b-a B) \sin (e+f x) \cos (e+f x) (c \sec (e+f x))^m}{a f \left(a^2-b^2\right) \sqrt{a+b \cos (e+f x)}}",0,"Integrate[((A + B*Cos[e + f*x])*(c*Sec[e + f*x])^m)/(a + b*Cos[e + f*x])^(3/2), x]","A",-1