1,1,75,114,0.1382424,"\int \cos ^5(c+d x) (a+a \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Cos[c + d*x]),x]","\frac{a \left(192 \sin ^5(c+d x)-640 \sin ^3(c+d x)+960 \sin (c+d x)+5 (45 \sin (2 (c+d x))+9 \sin (4 (c+d x))+\sin (6 (c+d x))+60 c+60 d x)\right)}{960 d}","\frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16}",1,"(a*(960*Sin[c + d*x] - 640*Sin[c + d*x]^3 + 192*Sin[c + d*x]^5 + 5*(60*c + 60*d*x + 45*Sin[2*(c + d*x)] + 9*Sin[4*(c + d*x)] + Sin[6*(c + d*x)])))/(960*d)","A",1
2,1,65,92,0.1158887,"\int \cos ^4(c+d x) (a+a \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Cos[c + d*x]),x]","\frac{a \left(96 \sin ^5(c+d x)-320 \sin ^3(c+d x)+480 \sin (c+d x)+15 (12 (c+d x)+8 \sin (2 (c+d x))+\sin (4 (c+d x)))\right)}{480 d}","\frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a x}{8}",1,"(a*(480*Sin[c + d*x] - 320*Sin[c + d*x]^3 + 96*Sin[c + d*x]^5 + 15*(12*(c + d*x) + 8*Sin[2*(c + d*x)] + Sin[4*(c + d*x)])))/(480*d)","A",1
3,1,73,76,0.0873687,"\int \cos ^3(c+d x) (a+a \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Cos[c + d*x]),x]","\frac{3 a (c+d x)}{8 d}-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{a \sin (2 (c+d x))}{4 d}+\frac{a \sin (4 (c+d x))}{32 d}","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a x}{8}",1,"(3*a*(c + d*x))/(8*d) + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d) + (a*Sin[2*(c + d*x)])/(4*d) + (a*Sin[4*(c + d*x)])/(32*d)","A",1
4,1,57,54,0.0691503,"\int \cos ^2(c+d x) (a+a \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x]),x]","\frac{a (c+d x)}{2 d}-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{a \sin (2 (c+d x))}{4 d}","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}",1,"(a*(c + d*x))/(2*d) + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d) + (a*Sin[2*(c + d*x)])/(4*d)","A",1
5,1,32,38,0.0501429,"\int \cos (c+d x) (a+a \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x]),x]","\frac{a (2 (c+d x)+4 \sin (c+d x)+\sin (2 (c+d x)))}{4 d}","\frac{a \sin (c+d x)}{d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}",1,"(a*(2*(c + d*x) + 4*Sin[c + d*x] + Sin[2*(c + d*x)]))/(4*d)","A",1
6,1,26,15,0.006277,"\int (a+a \cos (c+d x)) \, dx","Integrate[a + a*Cos[c + d*x],x]","\frac{a \sin (c) \cos (d x)}{d}+\frac{a \cos (c) \sin (d x)}{d}+a x","\frac{a \sin (c+d x)}{d}+a x",1,"a*x + (a*Cos[d*x]*Sin[c])/d + (a*Cos[c]*Sin[d*x])/d","A",1
7,1,16,16,0.0077659,"\int (a+a \cos (c+d x)) \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*Sec[c + d*x],x]","\frac{a \tanh ^{-1}(\sin (c+d x))}{d}+a x","\frac{a \tanh ^{-1}(\sin (c+d x))}{d}+a x",1,"a*x + (a*ArcTanh[Sin[c + d*x]])/d","A",1
8,1,24,24,0.0098333,"\int (a+a \cos (c+d x)) \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{a \tan (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}","\frac{a \tan (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}",1,"(a*ArcTanh[Sin[c + d*x]])/d + (a*Tan[c + d*x])/d","A",1
9,1,47,47,0.0138972,"\int (a+a \cos (c+d x)) \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{a \tan (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d}","\frac{a \tan (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d}",1,"(a*ArcTanh[Sin[c + d*x]])/(2*d) + (a*Tan[c + d*x])/d + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
10,1,60,63,0.1461853,"\int (a+a \cos (c+d x)) \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{a \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d}","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d}",1,"(a*ArcTanh[Sin[c + d*x]])/(2*d) + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d","A",1
11,1,76,85,0.1505588,"\int (a+a \cos (c+d x)) \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*Sec[c + d*x]^5,x]","\frac{a \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}+\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)}{8 d}","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a \tan (c+d x) \sec (c+d x)}{8 d}",1,"(a*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (3*a*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x]))/(8*d) + (a*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d","A",1
12,1,65,101,0.2500381,"\int (a+a \cos (c+d x)) \sec ^6(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*Sec[c + d*x]^6,x]","\frac{a \left(45 \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(24 \tan ^4(c+d x)+80 \tan ^2(c+d x)+30 \sec ^3(c+d x)+45 \sec (c+d x)+120\right)\right)}{120 d}","\frac{a \tan ^5(c+d x)}{5 d}+\frac{2 a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a \tan (c+d x) \sec (c+d x)}{8 d}",1,"(a*(45*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(120 + 45*Sec[c + d*x] + 30*Sec[c + d*x]^3 + 80*Tan[c + d*x]^2 + 24*Tan[c + d*x]^4)))/(120*d)","A",1
13,1,73,129,0.1960996,"\int \cos ^4(c+d x) (a+a \cos (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^4*(a + a*Cos[c + d*x])^2,x]","\frac{a^2 (1200 \sin (c+d x)+465 \sin (2 (c+d x))+200 \sin (3 (c+d x))+75 \sin (4 (c+d x))+24 \sin (5 (c+d x))+5 \sin (6 (c+d x))+660 d x)}{960 d}","\frac{2 a^2 \sin ^5(c+d x)}{5 d}-\frac{4 a^2 \sin ^3(c+d x)}{3 d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{11 a^2 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{11 a^2 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{11 a^2 x}{16}",1,"(a^2*(660*d*x + 1200*Sin[c + d*x] + 465*Sin[2*(c + d*x)] + 200*Sin[3*(c + d*x)] + 75*Sin[4*(c + d*x)] + 24*Sin[5*(c + d*x)] + 5*Sin[6*(c + d*x)]))/(960*d)","A",1
14,1,61,103,0.1261426,"\int \cos ^3(c+d x) (a+a \cos (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2,x]","\frac{a^2 (110 \sin (c+d x)+40 \sin (2 (c+d x))+15 \sin (3 (c+d x))+5 \sin (4 (c+d x))+\sin (5 (c+d x))+60 d x)}{80 d}","\frac{a^2 \sin ^5(c+d x)}{5 d}-\frac{a^2 \sin ^3(c+d x)}{d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{2 d}+\frac{3 a^2 \sin (c+d x) \cos (c+d x)}{4 d}+\frac{3 a^2 x}{4}",1,"(a^2*(60*d*x + 110*Sin[c + d*x] + 40*Sin[2*(c + d*x)] + 15*Sin[3*(c + d*x)] + 5*Sin[4*(c + d*x)] + Sin[5*(c + d*x)]))/(80*d)","A",1
15,1,53,87,0.1289501,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^2,x]","\frac{a^2 (144 \sin (c+d x)+48 \sin (2 (c+d x))+16 \sin (3 (c+d x))+3 \sin (4 (c+d x))+84 d x)}{96 d}","-\frac{2 a^2 \sin ^3(c+d x)}{3 d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{7 a^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{7 a^2 x}{8}",1,"(a^2*(84*d*x + 144*Sin[c + d*x] + 48*Sin[2*(c + d*x)] + 16*Sin[3*(c + d*x)] + 3*Sin[4*(c + d*x)]))/(96*d)","A",1
16,1,41,57,0.0843665,"\int \cos (c+d x) (a+a \cos (c+d x))^2 \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^2,x]","\frac{a^2 (21 \sin (c+d x)+6 \sin (2 (c+d x))+\sin (3 (c+d x))+12 d x)}{12 d}","-\frac{a^2 \sin ^3(c+d x)}{3 d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{d}+a^2 x",1,"(a^2*(12*d*x + 21*Sin[c + d*x] + 6*Sin[2*(c + d*x)] + Sin[3*(c + d*x)]))/(12*d)","A",1
17,1,34,45,0.047032,"\int (a+a \cos (c+d x))^2 \, dx","Integrate[(a + a*Cos[c + d*x])^2,x]","\frac{a^2 (6 (c+d x)+8 \sin (c+d x)+\sin (2 (c+d x)))}{4 d}","\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3 a^2 x}{2}",1,"(a^2*(6*(c + d*x) + 8*Sin[c + d*x] + Sin[2*(c + d*x)]))/(4*d)","A",1
18,1,47,34,0.0125486,"\int (a+a \cos (c+d x))^2 \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*Sec[c + d*x],x]","\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \sin (c) \cos (d x)}{d}+\frac{a^2 \cos (c) \sin (d x)}{d}+2 a^2 x","\frac{a^2 \sin (c+d x)}{d}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+2 a^2 x",1,"2*a^2*x + (a^2*ArcTanh[Sin[c + d*x]])/d + (a^2*Cos[d*x]*Sin[c])/d + (a^2*Cos[c]*Sin[d*x])/d","A",1
19,1,28,34,0.0149654,"\int (a+a \cos (c+d x))^2 \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*Sec[c + d*x]^2,x]","a^2 \left(\frac{\tan (c+d x)}{d}+\frac{2 \tanh ^{-1}(\sin (c+d x))}{d}+x\right)","\frac{a^2 \tan (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}+a^2 x",1,"a^2*(x + (2*ArcTanh[Sin[c + d*x]])/d + Tan[c + d*x]/d)","A",1
20,1,54,54,0.0120958,"\int (a+a \cos (c+d x))^2 \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*Sec[c + d*x]^3,x]","\frac{2 a^2 \tan (c+d x)}{d}+\frac{3 a^2 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{2 d}","\frac{2 a^2 \tan (c+d x)}{d}+\frac{3 a^2 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{2 d}",1,"(3*a^2*ArcTanh[Sin[c + d*x]])/(2*d) + (2*a^2*Tan[c + d*x])/d + (a^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
21,1,162,66,5.7848756,"\int (a+a \cos (c+d x))^2 \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*Sec[c + d*x]^4,x]","-\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(-2 \tan (c) \cos (c+d x)-\sec (c) (-4 \sin (2 c+d x)+3 \sin (c+2 d x)+3 \sin (3 c+2 d x)+5 \sin (2 c+3 d x)+13 \sin (d x))+12 \cos ^3(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{48 d}","\frac{a^2 \tan ^3(c+d x)}{3 d}+\frac{2 a^2 \tan (c+d x)}{d}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{d}",1,"-1/48*(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*Sec[c + d*x]^3*(12*Cos[c + d*x]^3*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(13*Sin[d*x] - 4*Sin[2*c + d*x] + 3*Sin[c + 2*d*x] + 3*Sin[3*c + 2*d*x] + 5*Sin[2*c + 3*d*x]) - 2*Cos[c + d*x]*Tan[c]))/d","B",1
22,1,797,96,6.4308682,"\int (a+a \cos (c+d x))^2 \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*Sec[c + d*x]^5,x]","-\frac{7 (\cos (c+d x) a+a)^2 \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d}+\frac{7 (\cos (c+d x) a+a)^2 \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d}+\frac{(\cos (c+d x) a+a)^2 \sin \left(\frac{d x}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{(\cos (c+d x) a+a)^2 \sin \left(\frac{d x}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{(\cos (c+d x) a+a)^2 \left(29 \cos \left(\frac{c}{2}\right)-13 \sin \left(\frac{c}{2}\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{192 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{(\cos (c+d x) a+a)^2 \left(-29 \cos \left(\frac{c}{2}\right)-13 \sin \left(\frac{c}{2}\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{192 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{(\cos (c+d x) a+a)^2 \sin \left(\frac{d x}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{12 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{(\cos (c+d x) a+a)^2 \sin \left(\frac{d x}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{12 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{(\cos (c+d x) a+a)^2 \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{64 d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^4}-\frac{(\cos (c+d x) a+a)^2 \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{64 d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^4}","\frac{2 a^2 \tan ^3(c+d x)}{3 d}+\frac{2 a^2 \tan (c+d x)}{d}+\frac{7 a^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{7 a^2 \tan (c+d x) \sec (c+d x)}{8 d}",1,"(-7*(a + a*Cos[c + d*x])^2*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^4)/(32*d) + (7*(a + a*Cos[c + d*x])^2*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^4)/(32*d) + ((a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4)/(64*d*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^4) + ((a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*Sin[(d*x)/2])/(12*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + ((a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(29*Cos[c/2] - 13*Sin[c/2]))/(192*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + ((a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*Sin[(d*x)/2])/(3*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) - ((a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4)/(64*d*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^4) + ((a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*Sin[(d*x)/2])/(12*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + ((a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(-29*Cos[c/2] - 13*Sin[c/2]))/(192*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + ((a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*Sin[(d*x)/2])/(3*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",0
23,1,73,129,0.1846536,"\int \cos ^3(c+d x) (a+a \cos (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^3*(a + a*Cos[c + d*x])^3,x]","\frac{a^3 (2520 \sin (c+d x)+945 \sin (2 (c+d x))+380 \sin (3 (c+d x))+135 \sin (4 (c+d x))+36 \sin (5 (c+d x))+5 \sin (6 (c+d x))+1380 d x)}{960 d}","\frac{3 a^3 \sin ^5(c+d x)}{5 d}-\frac{7 a^3 \sin ^3(c+d x)}{3 d}+\frac{4 a^3 \sin (c+d x)}{d}+\frac{a^3 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{23 a^3 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{23 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{23 a^3 x}{16}",1,"(a^3*(1380*d*x + 2520*Sin[c + d*x] + 945*Sin[2*(c + d*x)] + 380*Sin[3*(c + d*x)] + 135*Sin[4*(c + d*x)] + 36*Sin[5*(c + d*x)] + 5*Sin[6*(c + d*x)]))/(960*d)","A",1
24,1,63,105,0.1282861,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^3,x]","\frac{a^3 (1380 \sin (c+d x)+480 \sin (2 (c+d x))+170 \sin (3 (c+d x))+45 \sin (4 (c+d x))+6 \sin (5 (c+d x))+780 d x)}{480 d}","\frac{a^3 \sin ^5(c+d x)}{5 d}-\frac{5 a^3 \sin ^3(c+d x)}{3 d}+\frac{4 a^3 \sin (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{13 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{13 a^3 x}{8}",1,"(a^3*(780*d*x + 1380*Sin[c + d*x] + 480*Sin[2*(c + d*x)] + 170*Sin[3*(c + d*x)] + 45*Sin[4*(c + d*x)] + 6*Sin[5*(c + d*x)]))/(480*d)","A",1
25,1,51,85,0.1213078,"\int \cos (c+d x) (a+a \cos (c+d x))^3 \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^3,x]","\frac{a^3 (104 \sin (c+d x)+32 \sin (2 (c+d x))+8 \sin (3 (c+d x))+\sin (4 (c+d x))+60 d x)}{32 d}","-\frac{a^3 \sin ^3(c+d x)}{d}+\frac{4 a^3 \sin (c+d x)}{d}+\frac{a^3 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{15 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{15 a^3 x}{8}",1,"(a^3*(60*d*x + 104*Sin[c + d*x] + 32*Sin[2*(c + d*x)] + 8*Sin[3*(c + d*x)] + Sin[4*(c + d*x)]))/(32*d)","A",1
26,1,44,63,0.0693001,"\int (a+a \cos (c+d x))^3 \, dx","Integrate[(a + a*Cos[c + d*x])^3,x]","\frac{a^3 (45 \sin (c+d x)+9 \sin (2 (c+d x))+\sin (3 (c+d x))+30 c+30 d x)}{12 d}","-\frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{4 a^3 \sin (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{5 a^3 x}{2}",1,"(a^3*(30*c + 30*d*x + 45*Sin[c + d*x] + 9*Sin[2*(c + d*x)] + Sin[3*(c + d*x)]))/(12*d)","A",1
27,1,81,59,0.0748118,"\int (a+a \cos (c+d x))^3 \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*Sec[c + d*x],x]","\frac{a^3 \left(12 \sin (c+d x)+\sin (2 (c+d x))-4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+14 d x\right)}{4 d}","\frac{3 a^3 \sin (c+d x)}{d}+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{7 a^3 x}{2}",1,"(a^3*(14*d*x - 4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 12*Sin[c + d*x] + Sin[2*(c + d*x)]))/(4*d)","A",1
28,1,211,48,0.7321134,"\int (a+a \cos (c+d x))^3 \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*Sec[c + d*x]^2,x]","\frac{1}{8} a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(\frac{\sin (c) \cos (d x)}{d}+\frac{\cos (c) \sin (d x)}{d}+\frac{\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{\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{3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+3 x\right)","\frac{a^3 \sin (c+d x)}{d}+\frac{a^3 \tan (c+d x)}{d}+\frac{3 a^3 \tanh ^{-1}(\sin (c+d x))}{d}+3 a^3 x",1,"(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(3*x - (3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (Cos[d*x]*Sin[c])/d + (Cos[c]*Sin[d*x])/d + Sin[(d*x)/2]/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + Sin[(d*x)/2]/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/8","B",1
29,1,50,59,0.0281401,"\int (a+a \cos (c+d x))^3 \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*Sec[c + d*x]^3,x]","a^3 \left(\frac{3 \tan (c+d x)}{d}+\frac{7 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{\tan (c+d x) \sec (c+d x)}{2 d}+x\right)","\frac{3 a^3 \tan (c+d x)}{d}+\frac{7 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}+a^3 x",1,"a^3*(x + (7*ArcTanh[Sin[c + d*x]])/(2*d) + (3*Tan[c + d*x])/d + (Sec[c + d*x]*Tan[c + d*x])/(2*d))","A",1
30,1,154,72,5.403425,"\int (a+a \cos (c+d x))^3 \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*Sec[c + d*x]^4,x]","-\frac{a^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^3 \left(-4 \tan (c) \cos (c+d x)-\sec (c) (-20 \sin (2 c+d x)+9 \sin (c+2 d x)+9 \sin (3 c+2 d x)+22 \sin (2 c+3 d x)+50 \sin (d x))+60 \cos ^3(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{192 d}","\frac{a^3 \tan ^3(c+d x)}{3 d}+\frac{4 a^3 \tan (c+d x)}{d}+\frac{5 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{3 a^3 \tan (c+d x) \sec (c+d x)}{2 d}",1,"-1/192*(a^3*Sec[(c + d*x)/2]^6*(1 + Sec[c + d*x])^3*(60*Cos[c + d*x]^3*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(50*Sin[d*x] - 20*Sin[2*c + d*x] + 9*Sin[c + 2*d*x] + 9*Sin[3*c + 2*d*x] + 22*Sin[2*c + 3*d*x]) - 4*Cos[c + d*x]*Tan[c]))/d","B",1
31,1,797,93,6.3870121,"\int (a+a \cos (c+d x))^3 \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*Sec[c + d*x]^5,x]","-\frac{15 (\cos (c+d x) a+a)^3 \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{64 d}+\frac{15 (\cos (c+d x) a+a)^3 \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{64 d}+\frac{3 (\cos (c+d x) a+a)^3 \sin \left(\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 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{3 (\cos (c+d x) a+a)^3 \sin \left(\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{(\cos (c+d x) a+a)^3 \left(19 \cos \left(\frac{c}{2}\right)-11 \sin \left(\frac{c}{2}\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{128 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{(\cos (c+d x) a+a)^3 \left(-19 \cos \left(\frac{c}{2}\right)-11 \sin \left(\frac{c}{2}\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{128 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{(\cos (c+d x) a+a)^3 \sin \left(\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 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{(\cos (c+d x) a+a)^3 \sin \left(\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 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{(\cos (c+d x) a+a)^3 \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{128 d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^4}-\frac{(\cos (c+d x) a+a)^3 \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{128 d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^4}","\frac{a^3 \tan ^3(c+d x)}{d}+\frac{4 a^3 \tan (c+d x)}{d}+\frac{15 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{15 a^3 \tan (c+d x) \sec (c+d x)}{8 d}",1,"(-15*(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)/(64*d) + (15*(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)/(64*d) + ((a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6)/(128*d*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^4) + ((a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*Sin[(d*x)/2])/(16*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*(19*Cos[c/2] - 11*Sin[c/2]))/(128*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (3*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*Sin[(d*x)/2])/(8*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) - ((a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6)/(128*d*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^4) + ((a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*Sin[(d*x)/2])/(16*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*(-19*Cos[c/2] - 11*Sin[c/2]))/(128*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (3*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*Sin[(d*x)/2])/(8*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",0
32,1,487,114,1.4326484,"\int (a+a \cos (c+d x))^3 \sec ^6(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*Sec[c + d*x]^6,x]","-\frac{a^3 \sec (c) \sec ^5(c+d x) \left(1440 \sin (2 c+d x)-1500 \sin (c+2 d x)-1500 \sin (3 c+2 d x)-3040 \sin (2 c+3 d x)-390 \sin (3 c+4 d x)-390 \sin (5 c+4 d x)-608 \sin (4 c+5 d x)+975 \cos (2 c+3 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+975 \cos (4 c+3 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+195 \cos (4 c+5 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+195 \cos (6 c+5 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+1950 \cos (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)+1950 \cos (2 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)-975 \cos (2 c+3 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-975 \cos (4 c+3 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-195 \cos (4 c+5 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-195 \cos (6 c+5 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-4640 \sin (d x)\right)}{3840 d}","\frac{a^3 \tan ^5(c+d x)}{5 d}+\frac{5 a^3 \tan ^3(c+d x)}{3 d}+\frac{4 a^3 \tan (c+d x)}{d}+\frac{13 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{13 a^3 \tan (c+d x) \sec (c+d x)}{8 d}",1,"-1/3840*(a^3*Sec[c]*Sec[c + d*x]^5*(975*Cos[2*c + 3*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 975*Cos[4*c + 3*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 195*Cos[4*c + 5*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 195*Cos[6*c + 5*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 1950*Cos[d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 1950*Cos[2*c + d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 975*Cos[2*c + 3*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 975*Cos[4*c + 3*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 195*Cos[4*c + 5*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 195*Cos[6*c + 5*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 4640*Sin[d*x] + 1440*Sin[2*c + d*x] - 1500*Sin[c + 2*d*x] - 1500*Sin[3*c + 2*d*x] - 3040*Sin[2*c + 3*d*x] - 390*Sin[3*c + 4*d*x] - 390*Sin[5*c + 4*d*x] - 608*Sin[4*c + 5*d*x]))/d","B",1
33,1,73,127,0.1971135,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^4,x]","\frac{a^4 (5280 \sin (c+d x)+1905 \sin (2 (c+d x))+720 \sin (3 (c+d x))+225 \sin (4 (c+d x))+48 \sin (5 (c+d x))+5 \sin (6 (c+d x))+2940 d x)}{960 d}","\frac{4 a^4 \sin ^5(c+d x)}{5 d}-\frac{4 a^4 \sin ^3(c+d x)}{d}+\frac{8 a^4 \sin (c+d x)}{d}+\frac{a^4 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{41 a^4 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{49 a^4 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{49 a^4 x}{16}",1,"(a^4*(2940*d*x + 5280*Sin[c + d*x] + 1905*Sin[2*(c + d*x)] + 720*Sin[3*(c + d*x)] + 225*Sin[4*(c + d*x)] + 48*Sin[5*(c + d*x)] + 5*Sin[6*(c + d*x)]))/(960*d)","A",1
34,1,63,102,0.1469912,"\int \cos (c+d x) (a+a \cos (c+d x))^4 \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^4,x]","\frac{a^4 (1470 \sin (c+d x)+480 \sin (2 (c+d x))+145 \sin (3 (c+d x))+30 \sin (4 (c+d x))+3 \sin (5 (c+d x))+840 d x)}{240 d}","\frac{a^4 \sin ^5(c+d x)}{5 d}-\frac{8 a^4 \sin ^3(c+d x)}{3 d}+\frac{8 a^4 \sin (c+d x)}{d}+\frac{a^4 \sin (c+d x) \cos ^3(c+d x)}{d}+\frac{7 a^4 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{7 a^4 x}{2}",1,"(a^4*(840*d*x + 1470*Sin[c + d*x] + 480*Sin[2*(c + d*x)] + 145*Sin[3*(c + d*x)] + 30*Sin[4*(c + d*x)] + 3*Sin[5*(c + d*x)]))/(240*d)","A",1
35,1,56,87,0.1112184,"\int (a+a \cos (c+d x))^4 \, dx","Integrate[(a + a*Cos[c + d*x])^4,x]","\frac{a^4 (672 \sin (c+d x)+168 \sin (2 (c+d x))+32 \sin (3 (c+d x))+3 \sin (4 (c+d x))+420 c+420 d x)}{96 d}","-\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{8 a^4 \sin (c+d x)}{d}+\frac{a^4 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{27 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{35 a^4 x}{8}",1,"(a^4*(420*c + 420*d*x + 672*Sin[c + d*x] + 168*Sin[2*(c + d*x)] + 32*Sin[3*(c + d*x)] + 3*Sin[4*(c + d*x)]))/(96*d)","A",1
36,1,91,73,0.1091442,"\int (a+a \cos (c+d x))^4 \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*Sec[c + d*x],x]","\frac{a^4 \left(81 \sin (c+d x)+12 \sin (2 (c+d x))+\sin (3 (c+d x))-12 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+72 d x\right)}{12 d}","-\frac{a^4 \sin ^3(c+d x)}{3 d}+\frac{7 a^4 \sin (c+d x)}{d}+\frac{a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a^4 \sin (c+d x) \cos (c+d x)}{d}+6 a^4 x",1,"(a^4*(72*d*x - 12*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 81*Sin[c + d*x] + 12*Sin[2*(c + d*x)] + Sin[3*(c + d*x)]))/(12*d)","A",1
37,1,241,73,1.2461389,"\int (a+a \cos (c+d x))^4 \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*Sec[c + d*x]^2,x]","\frac{1}{64} a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \left(\frac{16 \sin (c) \cos (d x)}{d}+\frac{\sin (2 c) \cos (2 d x)}{d}+\frac{16 \cos (c) \sin (d x)}{d}+\frac{\cos (2 c) \sin (2 d x)}{d}+\frac{4 \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 \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{16 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{16 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+26 x\right)","\frac{4 a^4 \sin (c+d x)}{d}+\frac{a^4 \tan (c+d x)}{d}+\frac{4 a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^4 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{13 a^4 x}{2}",1,"(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*(26*x - (16*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (16*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (16*Cos[d*x]*Sin[c])/d + (Cos[2*d*x]*Sin[2*c])/d + (16*Cos[c]*Sin[d*x])/d + (Cos[2*c]*Sin[2*d*x])/d + (4*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (4*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/64","B",1
38,1,272,73,1.2374765,"\int (a+a \cos (c+d x))^4 \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*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 \sin (c) \cos (d x)}{d}+\frac{4 \cos (c) \sin (d x)}{d}+\frac{16 \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{16 \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{1}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{1}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{26 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{26 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+16 x\right)","\frac{a^4 \sin (c+d x)}{d}+\frac{4 a^4 \tan (c+d x)}{d}+\frac{13 a^4 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^4 \tan (c+d x) \sec (c+d x)}{2 d}+4 a^4 x",1,"(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*(16*x - (26*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (26*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (4*Cos[d*x]*Sin[c])/d + (4*Cos[c]*Sin[d*x])/d + 1/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (16*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - 1/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (16*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/64","B",1
39,1,61,73,0.0373334,"\int (a+a \cos (c+d x))^4 \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*Sec[c + d*x]^4,x]","a^4 \left(\frac{\tan ^3(c+d x)}{3 d}+\frac{7 \tan (c+d x)}{d}+\frac{6 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 \tan (c+d x) \sec (c+d x)}{d}+x\right)","\frac{a^4 \tan ^3(c+d x)}{3 d}+\frac{7 a^4 \tan (c+d x)}{d}+\frac{6 a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a^4 \tan (c+d x) \sec (c+d x)}{d}+a^4 x",1,"a^4*(x + (6*ArcTanh[Sin[c + d*x]])/d + (7*Tan[c + d*x])/d + (2*Sec[c + d*x]*Tan[c + d*x])/d + Tan[c + d*x]^3/(3*d))","A",1
40,1,797,96,6.3653167,"\int (a+a \cos (c+d x))^4 \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*Sec[c + d*x]^5,x]","-\frac{35 (\cos (c+d x) a+a)^4 \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{128 d}+\frac{35 (\cos (c+d x) a+a)^4 \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{128 d}+\frac{5 (\cos (c+d x) a+a)^4 \sin \left(\frac{d x}{2}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{12 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{5 (\cos (c+d x) a+a)^4 \sin \left(\frac{d x}{2}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{12 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{(\cos (c+d x) a+a)^4 \left(97 \cos \left(\frac{c}{2}\right)-65 \sin \left(\frac{c}{2}\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{768 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{(\cos (c+d x) a+a)^4 \left(-97 \cos \left(\frac{c}{2}\right)-65 \sin \left(\frac{c}{2}\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{768 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{(\cos (c+d x) a+a)^4 \sin \left(\frac{d x}{2}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{(\cos (c+d x) a+a)^4 \sin \left(\frac{d x}{2}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{(\cos (c+d x) a+a)^4 \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{256 d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^4}-\frac{(\cos (c+d x) a+a)^4 \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{256 d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^4}","\frac{4 a^4 \tan ^3(c+d x)}{3 d}+\frac{8 a^4 \tan (c+d x)}{d}+\frac{35 a^4 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{27 a^4 \tan (c+d x) \sec (c+d x)}{8 d}",1,"(-35*(a + a*Cos[c + d*x])^4*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^8)/(128*d) + (35*(a + a*Cos[c + d*x])^4*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^8)/(128*d) + ((a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8)/(256*d*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^4) + ((a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*Sin[(d*x)/2])/(24*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + ((a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(97*Cos[c/2] - 65*Sin[c/2]))/(768*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (5*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*Sin[(d*x)/2])/(12*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) - ((a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8)/(256*d*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^4) + ((a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*Sin[(d*x)/2])/(24*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + ((a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(-97*Cos[c/2] - 65*Sin[c/2]))/(768*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (5*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*Sin[(d*x)/2])/(12*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",1
41,1,498,111,1.4312113,"\int (a+a \cos (c+d x))^4 \sec ^6(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*Sec[c + d*x]^6,x]","-\frac{a^4 \sec (c) \sec ^5(c+d x) \left(960 \sin (2 c+d x)-660 \sin (c+2 d x)-660 \sin (3 c+2 d x)-1600 \sin (2 c+3 d x)+60 \sin (4 c+3 d x)-210 \sin (3 c+4 d x)-210 \sin (5 c+4 d x)-332 \sin (4 c+5 d x)+525 \cos (2 c+3 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+525 \cos (4 c+3 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+105 \cos (4 c+5 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+105 \cos (6 c+5 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+1050 \cos (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)+1050 \cos (2 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)-525 \cos (2 c+3 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-525 \cos (4 c+3 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-105 \cos (4 c+5 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-105 \cos (6 c+5 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-2360 \sin (d x)\right)}{960 d}","\frac{a^4 \tan ^5(c+d x)}{5 d}+\frac{8 a^4 \tan ^3(c+d x)}{3 d}+\frac{8 a^4 \tan (c+d x)}{d}+\frac{7 a^4 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^4 \tan (c+d x) \sec ^3(c+d x)}{d}+\frac{7 a^4 \tan (c+d x) \sec (c+d x)}{2 d}",1,"-1/960*(a^4*Sec[c]*Sec[c + d*x]^5*(525*Cos[2*c + 3*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 525*Cos[4*c + 3*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 105*Cos[4*c + 5*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 105*Cos[6*c + 5*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 1050*Cos[d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 1050*Cos[2*c + d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 525*Cos[2*c + 3*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 525*Cos[4*c + 3*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 105*Cos[4*c + 5*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 105*Cos[6*c + 5*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 2360*Sin[d*x] + 960*Sin[2*c + d*x] - 660*Sin[c + 2*d*x] - 660*Sin[3*c + 2*d*x] - 1600*Sin[2*c + 3*d*x] + 60*Sin[4*c + 3*d*x] - 210*Sin[3*c + 4*d*x] - 210*Sin[5*c + 4*d*x] - 332*Sin[4*c + 5*d*x]))/d","B",1
42,1,211,136,0.7757284,"\int (a+a \cos (c+d x))^4 \sec ^7(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*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(23520 \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) (3750 \sin (2 c+d x)+15360 \sin (c+2 d x)-1920 \sin (3 c+2 d x)+3845 \sin (2 c+3 d x)+3845 \sin (4 c+3 d x)+6912 \sin (3 c+4 d x)+735 \sin (4 c+5 d x)+735 \sin (6 c+5 d x)+1152 \sin (5 c+6 d x)-11520 \sin (c)+3750 \sin (d x))\right)}{122880 d}","\frac{4 a^4 \tan ^5(c+d x)}{5 d}+\frac{4 a^4 \tan ^3(c+d x)}{d}+\frac{8 a^4 \tan (c+d x)}{d}+\frac{49 a^4 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac{41 a^4 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{49 a^4 \tan (c+d x) \sec (c+d x)}{16 d}",1,"-1/122880*(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*Sec[c + d*x]^6*(23520*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]*(-11520*Sin[c] + 3750*Sin[d*x] + 3750*Sin[2*c + d*x] + 15360*Sin[c + 2*d*x] - 1920*Sin[3*c + 2*d*x] + 3845*Sin[2*c + 3*d*x] + 3845*Sin[4*c + 3*d*x] + 6912*Sin[3*c + 4*d*x] + 735*Sin[4*c + 5*d*x] + 735*Sin[6*c + 5*d*x] + 1152*Sin[5*c + 6*d*x])))/d","A",1
43,1,173,118,0.3207497,"\int \frac{\cos ^5(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^5/(a + a*Cos[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(-168 \sin \left(c+\frac{d x}{2}\right)-120 \sin \left(c+\frac{3 d x}{2}\right)-120 \sin \left(2 c+\frac{3 d x}{2}\right)+40 \sin \left(2 c+\frac{5 d x}{2}\right)+40 \sin \left(3 c+\frac{5 d x}{2}\right)-5 \sin \left(3 c+\frac{7 d x}{2}\right)-5 \sin \left(4 c+\frac{7 d x}{2}\right)+3 \sin \left(4 c+\frac{9 d x}{2}\right)+3 \sin \left(5 c+\frac{9 d x}{2}\right)+360 d x \cos \left(c+\frac{d x}{2}\right)-552 \sin \left(\frac{d x}{2}\right)+360 d x \cos \left(\frac{d x}{2}\right)\right)}{384 a d}","\frac{4 \sin ^3(c+d x)}{3 a d}-\frac{4 \sin (c+d x)}{a d}-\frac{\sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}+\frac{5 \sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac{15 \sin (c+d x) \cos (c+d x)}{8 a d}+\frac{15 x}{8 a}",1,"(Sec[c/2]*Sec[(c + d*x)/2]*(360*d*x*Cos[(d*x)/2] + 360*d*x*Cos[c + (d*x)/2] - 552*Sin[(d*x)/2] - 168*Sin[c + (d*x)/2] - 120*Sin[c + (3*d*x)/2] - 120*Sin[2*c + (3*d*x)/2] + 40*Sin[2*c + (5*d*x)/2] + 40*Sin[3*c + (5*d*x)/2] - 5*Sin[3*c + (7*d*x)/2] - 5*Sin[4*c + (7*d*x)/2] + 3*Sin[4*c + (9*d*x)/2] + 3*Sin[5*c + (9*d*x)/2]))/(384*a*d)","A",1
44,1,143,94,0.2759772,"\int \frac{\cos ^4(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^4/(a + a*Cos[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(21 \sin \left(c+\frac{d x}{2}\right)+18 \sin \left(c+\frac{3 d x}{2}\right)+18 \sin \left(2 c+\frac{3 d x}{2}\right)-2 \sin \left(2 c+\frac{5 d x}{2}\right)-2 \sin \left(3 c+\frac{5 d x}{2}\right)+\sin \left(3 c+\frac{7 d x}{2}\right)+\sin \left(4 c+\frac{7 d x}{2}\right)-36 d x \cos \left(c+\frac{d x}{2}\right)+69 \sin \left(\frac{d x}{2}\right)-36 d x \cos \left(\frac{d x}{2}\right)\right)}{48 a d}","-\frac{4 \sin ^3(c+d x)}{3 a d}+\frac{4 \sin (c+d x)}{a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}-\frac{3 \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{3 x}{2 a}",1,"(Sec[c/2]*Sec[(c + d*x)/2]*(-36*d*x*Cos[(d*x)/2] - 36*d*x*Cos[c + (d*x)/2] + 69*Sin[(d*x)/2] + 21*Sin[c + (d*x)/2] + 18*Sin[c + (3*d*x)/2] + 18*Sin[2*c + (3*d*x)/2] - 2*Sin[2*c + (5*d*x)/2] - 2*Sin[3*c + (5*d*x)/2] + Sin[3*c + (7*d*x)/2] + Sin[4*c + (7*d*x)/2]))/(48*a*d)","A",1
45,1,117,76,0.2307352,"\int \frac{\cos ^3(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^3/(a + a*Cos[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(-4 \sin \left(c+\frac{d x}{2}\right)-3 \sin \left(c+\frac{3 d x}{2}\right)-3 \sin \left(2 c+\frac{3 d x}{2}\right)+\sin \left(2 c+\frac{5 d x}{2}\right)+\sin \left(3 c+\frac{5 d x}{2}\right)+12 d x \cos \left(c+\frac{d x}{2}\right)-20 \sin \left(\frac{d x}{2}\right)+12 d x \cos \left(\frac{d x}{2}\right)\right)}{16 a d}","-\frac{2 \sin (c+d x)}{a d}-\frac{\sin (c+d x) \cos ^2(c+d x)}{d (a \cos (c+d x)+a)}+\frac{3 \sin (c+d x) \cos (c+d x)}{2 a d}+\frac{3 x}{2 a}",1,"(Sec[c/2]*Sec[(c + d*x)/2]*(12*d*x*Cos[(d*x)/2] + 12*d*x*Cos[c + (d*x)/2] - 20*Sin[(d*x)/2] - 4*Sin[c + (d*x)/2] - 3*Sin[c + (3*d*x)/2] - 3*Sin[2*c + (3*d*x)/2] + Sin[2*c + (5*d*x)/2] + Sin[3*c + (5*d*x)/2]))/(16*a*d)","A",1
46,1,89,43,0.1964149,"\int \frac{\cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^2/(a + a*Cos[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(c+\frac{d x}{2}\right)+\sin \left(c+\frac{3 d x}{2}\right)+\sin \left(2 c+\frac{3 d x}{2}\right)-2 d x \cos \left(c+\frac{d x}{2}\right)+5 \sin \left(\frac{d x}{2}\right)-2 d x \cos \left(\frac{d x}{2}\right)\right)}{4 a d}","\frac{\sin (c+d x)}{a d}+\frac{\sin (c+d x)}{a d (\cos (c+d x)+1)}-\frac{x}{a}",1,"(Sec[c/2]*Sec[(c + d*x)/2]*(-2*d*x*Cos[(d*x)/2] - 2*d*x*Cos[c + (d*x)/2] + 5*Sin[(d*x)/2] + Sin[c + (d*x)/2] + Sin[c + (3*d*x)/2] + Sin[2*c + (3*d*x)/2]))/(4*a*d)","B",1
47,1,57,29,0.0715643,"\int \frac{\cos (c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]/(a + a*Cos[c + d*x]),x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(d x \cos \left(\frac{1}{2} (c+d x)\right)-\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)\right)}{a d (\cos (c+d x)+1)}","\frac{x}{a}-\frac{\sin (c+d x)}{d (a \cos (c+d x)+a)}",1,"(2*Cos[(c + d*x)/2]*(d*x*Cos[(c + d*x)/2] - Sec[c/2]*Sin[(d*x)/2]))/(a*d*(1 + Cos[c + d*x]))","A",1
48,1,17,22,0.013302,"\int \frac{1}{a+a \cos (c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^(-1),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{a d}","\frac{\sin (c+d x)}{d (a \cos (c+d x)+a)}",1,"Tan[(c + d*x)/2]/(a*d)","A",1
49,1,103,38,0.1498379,"\int \frac{\sec (c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[Sec[c + d*x]/(a + a*Cos[c + d*x]),x]","-\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+\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)}{a d (\cos (c+d x)+1)}","\frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{\sin (c+d x)}{d (a \cos (c+d x)+a)}",1,"(-2*Cos[(c + d*x)/2]*(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]*Sin[(d*x)/2]))/(a*d*(1 + Cos[c + d*x]))","B",1
50,1,188,53,0.6935402,"\int \frac{\sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[Sec[c + d*x]^2/(a + a*Cos[c + d*x]),x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+\cos \left(\frac{1}{2} (c+d x)\right) \left(\frac{\sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{a d (\cos (c+d x)+1)}","\frac{2 \tan (c+d x)}{a d}-\frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{\tan (c+d x)}{d (a \cos (c+d x)+a)}",1,"(2*Cos[(c + d*x)/2]*(Sec[c/2]*Sin[(d*x)/2] + Cos[(c + d*x)/2]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Sin[d*x]/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))))/(a*d*(1 + Cos[c + d*x]))","B",1
51,1,244,83,1.2975148,"\int \frac{\sec ^3(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[Sec[c + d*x]^3/(a + a*Cos[c + d*x]),x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right) \left(-\frac{4 \sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{1}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{1}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-6 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-4 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)\right)}{2 a d (\cos (c+d x)+1)}","-\frac{2 \tan (c+d x)}{a d}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{3 \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{\tan (c+d x) \sec (c+d x)}{d (a \cos (c+d x)+a)}",1,"(Cos[(c + d*x)/2]*(-4*Sec[c/2]*Sin[(d*x)/2] + Cos[(c + d*x)/2]*(-6*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^(-2) - (Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^(-2) - (4*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
52,1,368,103,4.2713923,"\int \frac{\sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[Sec[c + d*x]^4/(a + a*Cos[c + d*x]),x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(6 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+\frac{1}{8} \sec (c) \cos \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(-12 \sin (2 c+d x)-6 \sin (c+2 d x)-6 \sin (3 c+2 d x)+20 \sin (2 c+3 d x)+9 \cos (2 c+3 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+9 \cos (4 c+3 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+27 \cos (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)+27 \cos (2 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)-9 \cos (2 c+3 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-9 \cos (4 c+3 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+48 \sin (d x)\right)\right)}{3 a d (\cos (c+d x)+1)}","\frac{4 \tan ^3(c+d x)}{3 a d}+\frac{4 \tan (c+d x)}{a d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{3 \tan (c+d x) \sec (c+d x)}{2 a d}-\frac{\tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)}",1,"(Cos[(c + d*x)/2]*(6*Sec[c/2]*Sin[(d*x)/2] + (Cos[(c + d*x)/2]*Sec[c]*Sec[c + d*x]^3*(9*Cos[2*c + 3*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 9*Cos[4*c + 3*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 27*Cos[d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 27*Cos[2*c + d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 9*Cos[2*c + 3*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 9*Cos[4*c + 3*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 48*Sin[d*x] - 12*Sin[2*c + d*x] - 6*Sin[c + 2*d*x] - 6*Sin[3*c + 2*d*x] + 20*Sin[2*c + 3*d*x]))/8))/(3*a*d*(1 + Cos[c + d*x]))","B",1
53,1,199,124,0.43348,"\int \frac{\cos ^5(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^5/(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(-156 \sin \left(c+\frac{d x}{2}\right)+342 \sin \left(c+\frac{3 d x}{2}\right)+118 \sin \left(2 c+\frac{3 d x}{2}\right)+30 \sin \left(2 c+\frac{5 d x}{2}\right)+30 \sin \left(3 c+\frac{5 d x}{2}\right)-3 \sin \left(3 c+\frac{7 d x}{2}\right)-3 \sin \left(4 c+\frac{7 d x}{2}\right)+\sin \left(4 c+\frac{9 d x}{2}\right)+\sin \left(5 c+\frac{9 d x}{2}\right)-360 d x \cos \left(c+\frac{d x}{2}\right)-120 d x \cos \left(c+\frac{3 d x}{2}\right)-120 d x \cos \left(2 c+\frac{3 d x}{2}\right)+516 \sin \left(\frac{d x}{2}\right)-360 d x \cos \left(\frac{d x}{2}\right)\right)}{192 a^2 d}","-\frac{4 \sin ^3(c+d x)}{a^2 d}+\frac{12 \sin (c+d x)}{a^2 d}-\frac{10 \sin (c+d x) \cos ^3(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{5 \sin (c+d x) \cos (c+d x)}{a^2 d}-\frac{5 x}{a^2}-\frac{\sin (c+d x) \cos ^4(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^3*(-360*d*x*Cos[(d*x)/2] - 360*d*x*Cos[c + (d*x)/2] - 120*d*x*Cos[c + (3*d*x)/2] - 120*d*x*Cos[2*c + (3*d*x)/2] + 516*Sin[(d*x)/2] - 156*Sin[c + (d*x)/2] + 342*Sin[c + (3*d*x)/2] + 118*Sin[2*c + (3*d*x)/2] + 30*Sin[2*c + (5*d*x)/2] + 30*Sin[3*c + (5*d*x)/2] - 3*Sin[3*c + (7*d*x)/2] - 3*Sin[4*c + (7*d*x)/2] + Sin[4*c + (9*d*x)/2] + Sin[5*c + (9*d*x)/2]))/(192*a^2*d)","A",1
54,1,177,114,0.2885938,"\int \frac{\cos ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^4/(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(147 \sin \left(c+\frac{d x}{2}\right)-239 \sin \left(c+\frac{3 d x}{2}\right)-63 \sin \left(2 c+\frac{3 d x}{2}\right)-15 \sin \left(2 c+\frac{5 d x}{2}\right)-15 \sin \left(3 c+\frac{5 d x}{2}\right)+3 \sin \left(3 c+\frac{7 d x}{2}\right)+3 \sin \left(4 c+\frac{7 d x}{2}\right)+252 d x \cos \left(c+\frac{d x}{2}\right)+84 d x \cos \left(c+\frac{3 d x}{2}\right)+84 d x \cos \left(2 c+\frac{3 d x}{2}\right)-381 \sin \left(\frac{d x}{2}\right)+252 d x \cos \left(\frac{d x}{2}\right)\right)}{192 a^2 d}","-\frac{16 \sin (c+d x)}{3 a^2 d}-\frac{8 \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{7 \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{7 x}{2 a^2}-\frac{\sin (c+d x) \cos ^3(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^3*(252*d*x*Cos[(d*x)/2] + 252*d*x*Cos[c + (d*x)/2] + 84*d*x*Cos[c + (3*d*x)/2] + 84*d*x*Cos[2*c + (3*d*x)/2] - 381*Sin[(d*x)/2] + 147*Sin[c + (d*x)/2] - 239*Sin[c + (3*d*x)/2] - 63*Sin[2*c + (3*d*x)/2] - 15*Sin[2*c + (5*d*x)/2] - 15*Sin[3*c + (5*d*x)/2] + 3*Sin[3*c + (7*d*x)/2] + 3*Sin[4*c + (7*d*x)/2]))/(192*a^2*d)","A",1
55,1,114,80,0.3475366,"\int \frac{\cos ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^3/(a + a*Cos[c + d*x])^2,x]","-\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(-6 (\sin (c+d x)-2 d x) \cos ^3\left(\frac{1}{2} (c+d x)\right)+\tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)-16 \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{4 \sin (c+d x)}{3 a^2 d}+\frac{2 \sin (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{2 x}{a^2}-\frac{\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]*(Sec[c/2]*Sin[(d*x)/2] - 16*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] - 6*Cos[(c + d*x)/2]^3*(-2*d*x + Sin[c + d*x]) + Cos[(c + d*x)/2]*Tan[c/2]))/(3*a^2*d*(1 + Cos[c + d*x])^2)","A",1
56,1,105,57,0.2248189,"\int \frac{\cos ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^2/(a + a*Cos[c + d*x])^2,x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(6 d x \cos ^3\left(\frac{1}{2} (c+d x)\right)+\tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)-10 \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{5 \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{x}{a^2}+\frac{\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(2*Cos[(c + d*x)/2]*(6*d*x*Cos[(c + d*x)/2]^3 + Sec[c/2]*Sin[(d*x)/2] - 10*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + Cos[(c + d*x)/2]*Tan[c/2]))/(3*a^2*d*(1 + Cos[c + d*x])^2)","A",1
57,1,60,55,0.1141296,"\int \frac{\cos (c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[Cos[c + d*x]/(a + a*Cos[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \left(-3 \sin \left(c+\frac{d x}{2}\right)+2 \sin \left(c+\frac{3 d x}{2}\right)+3 \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right)}{12 a^2 d}","\frac{2 \sin (c+d x)}{3 d \left(a^2 \cos (c+d x)+a^2\right)}-\frac{\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^3*(3*Sin[(d*x)/2] - 3*Sin[c + (d*x)/2] + 2*Sin[c + (3*d*x)/2]))/(12*a^2*d)","A",1
58,1,53,55,0.0484366,"\int \frac{1}{(a+a \cos (c+d x))^2} \, dx","Integrate[(a + a*Cos[c + d*x])^(-2),x]","\frac{\left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{3}{2} (c+d x)\right)\right) \cos \left(\frac{1}{2} (c+d x)\right)}{3 a^2 d (\cos (c+d x)+1)^2}","\frac{\sin (c+d x)}{3 d \left(a^2 \cos (c+d x)+a^2\right)}+\frac{\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*(3*Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2]))/(3*a^2*d*(1 + Cos[c + d*x])^2)","A",1
59,1,152,66,0.2890239,"\int \frac{\sec (c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[Sec[c + d*x]/(a + a*Cos[c + d*x])^2,x]","-\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(\tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)+\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(\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)+8 \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{\tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{4 \sin (c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(-2*Cos[(c + d*x)/2]*(6*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]]) + Sec[c/2]*Sin[(d*x)/2] + 8*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + Cos[(c + d*x)/2]*Tan[c/2]))/(3*a^2*d*(1 + Cos[c + d*x])^2)","B",1
60,1,239,81,1.1240749,"\int \frac{\sec ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[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(\tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)+\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(\frac{\sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+14 \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{10 \tan (c+d x)}{3 a^2 d}-\frac{2 \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{2 \tan (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{\tan (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(2*Cos[(c + d*x)/2]*(Sec[c/2]*Sin[(d*x)/2] + 14*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + 6*Cos[(c + d*x)/2]^3*(2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Sin[d*x]/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))) + Cos[(c + d*x)/2]*Tan[c/2]))/(3*a^2*d*(1 + Cos[c + d*x])^2)","B",1
61,1,292,119,1.8119054,"\int \frac{\sec ^3(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^3/(a + a*Cos[c + d*x])^2,x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(-2 \tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)-2 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+3 \cos ^3\left(\frac{1}{2} (c+d x)\right) \left(-\frac{8 \sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{1}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{1}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-14 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+14 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-40 \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{16 \tan (c+d x)}{3 a^2 d}+\frac{7 \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac{7 \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{8 \tan (c+d x) \sec (c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{\tan (c+d x) \sec (c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*(-2*Sec[c/2]*Sin[(d*x)/2] - 40*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + 3*Cos[(c + d*x)/2]^3*(-14*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 14*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^(-2) - (Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^(-2) - (8*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*Cos[(c + d*x)/2]*Tan[c/2]))/(3*a^2*d*(1 + Cos[c + d*x])^2)","B",1
62,1,343,133,3.8821359,"\int \frac{\sec ^4(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^4/(a + a*Cos[c + d*x])^2,x]","\frac{960 \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) \left(-153 \sin \left(c-\frac{d x}{2}\right)+21 \sin \left(c+\frac{d x}{2}\right)-135 \sin \left(2 c+\frac{d x}{2}\right)+25 \sin \left(c+\frac{3 d x}{2}\right)+45 \sin \left(2 c+\frac{3 d x}{2}\right)-85 \sin \left(3 c+\frac{3 d x}{2}\right)+99 \sin \left(c+\frac{5 d x}{2}\right)+21 \sin \left(2 c+\frac{5 d x}{2}\right)+33 \sin \left(3 c+\frac{5 d x}{2}\right)-45 \sin \left(4 c+\frac{5 d x}{2}\right)+57 \sin \left(2 c+\frac{7 d x}{2}\right)+18 \sin \left(3 c+\frac{7 d x}{2}\right)+24 \sin \left(4 c+\frac{7 d x}{2}\right)-15 \sin \left(5 c+\frac{7 d x}{2}\right)+24 \sin \left(3 c+\frac{9 d x}{2}\right)+11 \sin \left(4 c+\frac{9 d x}{2}\right)+13 \sin \left(5 c+\frac{9 d x}{2}\right)-3 \sin \left(\frac{d x}{2}\right)+155 \sin \left(\frac{3 d x}{2}\right)\right) \cos \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x)}{48 a^2 d (\cos (c+d x)+1)^2}","\frac{4 \tan ^3(c+d x)}{a^2 d}+\frac{12 \tan (c+d x)}{a^2 d}-\frac{5 \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{5 \tan (c+d x) \sec (c+d x)}{a^2 d}-\frac{10 \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{\tan (c+d x) \sec ^2(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(960*Cos[(c + d*x)/2]^4*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Cos[(c + d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]^3*(-3*Sin[(d*x)/2] + 155*Sin[(3*d*x)/2] - 153*Sin[c - (d*x)/2] + 21*Sin[c + (d*x)/2] - 135*Sin[2*c + (d*x)/2] + 25*Sin[c + (3*d*x)/2] + 45*Sin[2*c + (3*d*x)/2] - 85*Sin[3*c + (3*d*x)/2] + 99*Sin[c + (5*d*x)/2] + 21*Sin[2*c + (5*d*x)/2] + 33*Sin[3*c + (5*d*x)/2] - 45*Sin[4*c + (5*d*x)/2] + 57*Sin[2*c + (7*d*x)/2] + 18*Sin[3*c + (7*d*x)/2] + 24*Sin[4*c + (7*d*x)/2] - 15*Sin[5*c + (7*d*x)/2] + 24*Sin[3*c + (9*d*x)/2] + 11*Sin[4*c + (9*d*x)/2] + 13*Sin[5*c + (9*d*x)/2]))/(48*a^2*d*(1 + Cos[c + d*x])^2)","B",1
63,1,173,153,0.5724434,"\int \frac{\cos ^5(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^5/(a + a*Cos[c + d*x])^3,x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(15 (-12 \sin (c+d x)+\sin (2 (c+d x))+26 d x) \cos ^5\left(\frac{1}{2} (c+d x)\right)+46 \tan \left(\frac{c}{2}\right) \cos ^3\left(\frac{1}{2} (c+d x)\right)-3 \tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)-3 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)-508 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^4\left(\frac{1}{2} (c+d x)\right)+46 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{15 a^3 d (\cos (c+d x)+1)^3}","-\frac{152 \sin (c+d x)}{15 a^3 d}-\frac{76 \sin (c+d x) \cos ^2(c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{13 \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac{13 x}{2 a^3}-\frac{\sin (c+d x) \cos ^4(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{11 \sin (c+d x) \cos ^3(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(2*Cos[(c + d*x)/2]*(-3*Sec[c/2]*Sin[(d*x)/2] + 46*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] - 508*Cos[(c + d*x)/2]^4*Sec[c/2]*Sin[(d*x)/2] + 15*Cos[(c + d*x)/2]^5*(26*d*x - 12*Sin[c + d*x] + Sin[2*(c + d*x)]) - 3*Cos[(c + d*x)/2]*Tan[c/2] + 46*Cos[(c + d*x)/2]^3*Tan[c/2]))/(15*a^3*d*(1 + Cos[c + d*x])^3)","A",1
64,1,161,119,0.5336572,"\int \frac{\cos ^4(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^4/(a + a*Cos[c + d*x])^3,x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(20 (\sin (c+d x)-3 d x) \cos ^5\left(\frac{1}{2} (c+d x)\right)-12 \tan \left(\frac{c}{2}\right) \cos ^3\left(\frac{1}{2} (c+d x)\right)+\tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+96 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^4\left(\frac{1}{2} (c+d x)\right)-12 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{5 a^3 d (\cos (c+d x)+1)^3}","\frac{9 \sin (c+d x)}{5 a^3 d}+\frac{3 \sin (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{3 x}{a^3}-\frac{\sin (c+d x) \cos ^3(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{3 \sin (c+d x) \cos ^2(c+d x)}{5 a d (a \cos (c+d x)+a)^2}",1,"(2*Cos[(c + d*x)/2]*(Sec[c/2]*Sin[(d*x)/2] - 12*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + 96*Cos[(c + d*x)/2]^4*Sec[c/2]*Sin[(d*x)/2] + 20*Cos[(c + d*x)/2]^5*(-3*d*x + Sin[c + d*x]) + Cos[(c + d*x)/2]*Tan[c/2] - 12*Cos[(c + d*x)/2]^3*Tan[c/2]))/(5*a^3*d*(1 + Cos[c + d*x])^3)","A",1
65,1,154,96,0.2368753,"\int \frac{\cos ^3(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^3/(a + a*Cos[c + d*x])^3,x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(60 d x \cos ^5\left(\frac{1}{2} (c+d x)\right)+26 \tan \left(\frac{c}{2}\right) \cos ^3\left(\frac{1}{2} (c+d x)\right)-3 \tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)-3 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)-128 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^4\left(\frac{1}{2} (c+d x)\right)+26 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{15 a^3 d (\cos (c+d x)+1)^3}","-\frac{29 \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{x}{a^3}-\frac{\sin (c+d x) \cos ^2(c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac{7 \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(2*Cos[(c + d*x)/2]*(60*d*x*Cos[(c + d*x)/2]^5 - 3*Sec[c/2]*Sin[(d*x)/2] + 26*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] - 128*Cos[(c + d*x)/2]^4*Sec[c/2]*Sin[(d*x)/2] - 3*Cos[(c + d*x)/2]*Tan[c/2] + 26*Cos[(c + d*x)/2]^3*Tan[c/2]))/(15*a^3*d*(1 + Cos[c + d*x])^3)","A",1
66,1,86,83,0.1867621,"\int \frac{\cos ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^2/(a + a*Cos[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \left(-30 \sin \left(c+\frac{d x}{2}\right)+20 \sin \left(c+\frac{3 d x}{2}\right)-15 \sin \left(2 c+\frac{3 d x}{2}\right)+7 \sin \left(2 c+\frac{5 d x}{2}\right)+40 \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right)}{240 a^3 d}","\frac{7 \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{8 \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}+\frac{\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^5*(40*Sin[(d*x)/2] - 30*Sin[c + (d*x)/2] + 20*Sin[c + (3*d*x)/2] - 15*Sin[2*c + (3*d*x)/2] + 7*Sin[2*c + (5*d*x)/2]))/(240*a^3*d)","A",1
67,1,71,83,0.1365265,"\int \frac{\cos (c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[Cos[c + d*x]/(a + a*Cos[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \left(-5 \sin \left(c+\frac{d x}{2}\right)+5 \sin \left(c+\frac{3 d x}{2}\right)+\sin \left(2 c+\frac{5 d x}{2}\right)+5 \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right)}{80 a^3 d}","\frac{\sin (c+d x)}{5 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{\sin (c+d x)}{5 a d (a \cos (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^5*(5*Sin[(d*x)/2] - 5*Sin[c + (d*x)/2] + 5*Sin[c + (3*d*x)/2] + Sin[2*c + (5*d*x)/2]))/(80*a^3*d)","A",1
68,1,65,83,0.0807035,"\int \frac{1}{(a+a \cos (c+d x))^3} \, dx","Integrate[(a + a*Cos[c + d*x])^(-3),x]","\frac{\left(10 \sin \left(\frac{1}{2} (c+d x)\right)+5 \sin \left(\frac{3}{2} (c+d x)\right)+\sin \left(\frac{5}{2} (c+d x)\right)\right) \cos \left(\frac{1}{2} (c+d x)\right)}{15 a^3 d (\cos (c+d x)+1)^3}","\frac{2 \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{2 \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}+\frac{\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]*(10*Sin[(c + d*x)/2] + 5*Sin[(3*(c + d*x))/2] + Sin[(5*(c + d*x))/2]))/(15*a^3*d*(1 + Cos[c + d*x])^3)","A",1
69,1,201,97,0.4826409,"\int \frac{\sec (c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[Sec[c + d*x]/(a + a*Cos[c + d*x])^3,x]","-\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(14 \tan \left(\frac{c}{2}\right) \cos ^3\left(\frac{1}{2} (c+d x)\right)+3 \tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)+3 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+60 \cos ^5\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)+88 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^4\left(\frac{1}{2} (c+d x)\right)+14 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{15 a^3 d (\cos (c+d x)+1)^3}","\frac{\tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{22 \sin (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{7 \sin (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"(-2*Cos[(c + d*x)/2]*(60*Cos[(c + d*x)/2]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 3*Sec[c/2]*Sin[(d*x)/2] + 14*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + 88*Cos[(c + d*x)/2]^4*Sec[c/2]*Sin[(d*x)/2] + 3*Cos[(c + d*x)/2]*Tan[c/2] + 14*Cos[(c + d*x)/2]^3*Tan[c/2]))/(15*a^3*d*(1 + Cos[c + d*x])^3)","B",1
70,1,286,112,1.1459778,"\int \frac{\sec ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^2/(a + a*Cos[c + d*x])^3,x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(8 \tan \left(\frac{c}{2}\right) \cos ^3\left(\frac{1}{2} (c+d x)\right)+\tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+20 \cos ^5\left(\frac{1}{2} (c+d x)\right) \left(\frac{\sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+76 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^4\left(\frac{1}{2} (c+d x)\right)+8 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{5 a^3 d (\cos (c+d x)+1)^3}","\frac{24 \tan (c+d x)}{5 a^3 d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{3 \tan (c+d x)}{d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{3 \tan (c+d x)}{5 a d (a \cos (c+d x)+a)^2}-\frac{\tan (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"(2*Cos[(c + d*x)/2]*(Sec[c/2]*Sin[(d*x)/2] + 8*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + 76*Cos[(c + d*x)/2]^4*Sec[c/2]*Sin[(d*x)/2] + 20*Cos[(c + d*x)/2]^5*(3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Sin[d*x]/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))) + Cos[(c + d*x)/2]*Tan[c/2] + 8*Cos[(c + d*x)/2]^3*Tan[c/2]))/(5*a^3*d*(1 + Cos[c + d*x])^3)","B",1
71,1,343,156,3.8524164,"\int \frac{\sec ^3(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^3/(a + a*Cos[c + d*x])^3,x]","-\frac{24960 \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) \left(-4329 \sin \left(c-\frac{d x}{2}\right)+1989 \sin \left(c+\frac{d x}{2}\right)-3575 \sin \left(2 c+\frac{d x}{2}\right)-475 \sin \left(c+\frac{3 d x}{2}\right)+2005 \sin \left(2 c+\frac{3 d x}{2}\right)-2275 \sin \left(3 c+\frac{3 d x}{2}\right)+2673 \sin \left(c+\frac{5 d x}{2}\right)+105 \sin \left(2 c+\frac{5 d x}{2}\right)+1593 \sin \left(3 c+\frac{5 d x}{2}\right)-975 \sin \left(4 c+\frac{5 d x}{2}\right)+1325 \sin \left(2 c+\frac{7 d x}{2}\right)+255 \sin \left(3 c+\frac{7 d x}{2}\right)+875 \sin \left(4 c+\frac{7 d x}{2}\right)-195 \sin \left(5 c+\frac{7 d x}{2}\right)+304 \sin \left(3 c+\frac{9 d x}{2}\right)+90 \sin \left(4 c+\frac{9 d x}{2}\right)+214 \sin \left(5 c+\frac{9 d x}{2}\right)-1235 \sin \left(\frac{d x}{2}\right)+3805 \sin \left(\frac{3 d x}{2}\right)\right) \cos \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x)}{480 a^3 d (\cos (c+d x)+1)^3}","-\frac{152 \tan (c+d x)}{15 a^3 d}+\frac{13 \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{13 \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{76 \tan (c+d x) \sec (c+d x)}{15 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{11 \tan (c+d x) \sec (c+d x)}{15 a d (a \cos (c+d x)+a)^2}-\frac{\tan (c+d x) \sec (c+d x)}{5 d (a \cos (c+d x)+a)^3}",1,"-1/480*(24960*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*Sin[(d*x)/2] + 3805*Sin[(3*d*x)/2] - 4329*Sin[c - (d*x)/2] + 1989*Sin[c + (d*x)/2] - 3575*Sin[2*c + (d*x)/2] - 475*Sin[c + (3*d*x)/2] + 2005*Sin[2*c + (3*d*x)/2] - 2275*Sin[3*c + (3*d*x)/2] + 2673*Sin[c + (5*d*x)/2] + 105*Sin[2*c + (5*d*x)/2] + 1593*Sin[3*c + (5*d*x)/2] - 975*Sin[4*c + (5*d*x)/2] + 1325*Sin[2*c + (7*d*x)/2] + 255*Sin[3*c + (7*d*x)/2] + 875*Sin[4*c + (7*d*x)/2] - 195*Sin[5*c + (7*d*x)/2] + 304*Sin[3*c + (9*d*x)/2] + 90*Sin[4*c + (9*d*x)/2] + 214*Sin[5*c + (9*d*x)/2]))/(a^3*d*(1 + Cos[c + d*x])^3)","B",1
72,1,289,184,0.5843612,"\int \frac{\cos ^6(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Integrate[Cos[c + d*x]^6/(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(128730 \sin \left(c+\frac{d x}{2}\right)-140826 \sin \left(c+\frac{3 d x}{2}\right)+44310 \sin \left(2 c+\frac{3 d x}{2}\right)-60487 \sin \left(2 c+\frac{5 d x}{2}\right)+1225 \sin \left(3 c+\frac{5 d x}{2}\right)-12001 \sin \left(3 c+\frac{7 d x}{2}\right)-3185 \sin \left(4 c+\frac{7 d x}{2}\right)-315 \sin \left(4 c+\frac{9 d x}{2}\right)-315 \sin \left(5 c+\frac{9 d x}{2}\right)+35 \sin \left(5 c+\frac{11 d x}{2}\right)+35 \sin \left(6 c+\frac{11 d x}{2}\right)+102900 d x \cos \left(c+\frac{d x}{2}\right)+61740 d x \cos \left(c+\frac{3 d x}{2}\right)+61740 d x \cos \left(2 c+\frac{3 d x}{2}\right)+20580 d x \cos \left(2 c+\frac{5 d x}{2}\right)+20580 d x \cos \left(3 c+\frac{5 d x}{2}\right)+2940 d x \cos \left(3 c+\frac{7 d x}{2}\right)+2940 d x \cos \left(4 c+\frac{7 d x}{2}\right)-179830 \sin \left(\frac{d x}{2}\right)+102900 d x \cos \left(\frac{d x}{2}\right)\right)}{35840 a^4 d}","-\frac{576 \sin (c+d x)}{35 a^4 d}-\frac{43 \sin (c+d x) \cos ^3(c+d x)}{35 a^4 d (\cos (c+d x)+1)^2}-\frac{288 \sin (c+d x) \cos ^2(c+d x)}{35 a^4 d (\cos (c+d x)+1)}+\frac{21 \sin (c+d x) \cos (c+d x)}{2 a^4 d}+\frac{21 x}{2 a^4}-\frac{\sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{2 \sin (c+d x) \cos ^4(c+d x)}{5 a d (a \cos (c+d x)+a)^3}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^7*(102900*d*x*Cos[(d*x)/2] + 102900*d*x*Cos[c + (d*x)/2] + 61740*d*x*Cos[c + (3*d*x)/2] + 61740*d*x*Cos[2*c + (3*d*x)/2] + 20580*d*x*Cos[2*c + (5*d*x)/2] + 20580*d*x*Cos[3*c + (5*d*x)/2] + 2940*d*x*Cos[3*c + (7*d*x)/2] + 2940*d*x*Cos[4*c + (7*d*x)/2] - 179830*Sin[(d*x)/2] + 128730*Sin[c + (d*x)/2] - 140826*Sin[c + (3*d*x)/2] + 44310*Sin[2*c + (3*d*x)/2] - 60487*Sin[2*c + (5*d*x)/2] + 1225*Sin[3*c + (5*d*x)/2] - 12001*Sin[3*c + (7*d*x)/2] - 3185*Sin[4*c + (7*d*x)/2] - 315*Sin[4*c + (9*d*x)/2] - 315*Sin[5*c + (9*d*x)/2] + 35*Sin[5*c + (11*d*x)/2] + 35*Sin[6*c + (11*d*x)/2]))/(35840*a^4*d)","A",1
73,1,263,150,0.416102,"\int \frac{\cos ^5(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Integrate[Cos[c + d*x]^5/(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(46130 \sin \left(c+\frac{d x}{2}\right)-46116 \sin \left(c+\frac{3 d x}{2}\right)+18060 \sin \left(2 c+\frac{3 d x}{2}\right)-19292 \sin \left(2 c+\frac{5 d x}{2}\right)+2100 \sin \left(3 c+\frac{5 d x}{2}\right)-3791 \sin \left(3 c+\frac{7 d x}{2}\right)-735 \sin \left(4 c+\frac{7 d x}{2}\right)-105 \sin \left(4 c+\frac{9 d x}{2}\right)-105 \sin \left(5 c+\frac{9 d x}{2}\right)+29400 d x \cos \left(c+\frac{d x}{2}\right)+17640 d x \cos \left(c+\frac{3 d x}{2}\right)+17640 d x \cos \left(2 c+\frac{3 d x}{2}\right)+5880 d x \cos \left(2 c+\frac{5 d x}{2}\right)+5880 d x \cos \left(3 c+\frac{5 d x}{2}\right)+840 d x \cos \left(3 c+\frac{7 d x}{2}\right)+840 d x \cos \left(4 c+\frac{7 d x}{2}\right)-60830 \sin \left(\frac{d x}{2}\right)+29400 d x \cos \left(\frac{d x}{2}\right)\right)}{26880 a^4 d}","\frac{244 \sin (c+d x)}{105 a^4 d}-\frac{88 \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{4 \sin (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac{4 x}{a^4}-\frac{\sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{12 \sin (c+d x) \cos ^3(c+d x)}{35 a d (a \cos (c+d x)+a)^3}",1,"-1/26880*(Sec[c/2]*Sec[(c + d*x)/2]^7*(29400*d*x*Cos[(d*x)/2] + 29400*d*x*Cos[c + (d*x)/2] + 17640*d*x*Cos[c + (3*d*x)/2] + 17640*d*x*Cos[2*c + (3*d*x)/2] + 5880*d*x*Cos[2*c + (5*d*x)/2] + 5880*d*x*Cos[3*c + (5*d*x)/2] + 840*d*x*Cos[3*c + (7*d*x)/2] + 840*d*x*Cos[4*c + (7*d*x)/2] - 60830*Sin[(d*x)/2] + 46130*Sin[c + (d*x)/2] - 46116*Sin[c + (3*d*x)/2] + 18060*Sin[2*c + (3*d*x)/2] - 19292*Sin[2*c + (5*d*x)/2] + 2100*Sin[3*c + (5*d*x)/2] - 3791*Sin[3*c + (7*d*x)/2] - 735*Sin[4*c + (7*d*x)/2] - 105*Sin[4*c + (9*d*x)/2] - 105*Sin[5*c + (9*d*x)/2]))/(a^4*d)","A",1
74,1,224,127,0.3409435,"\int \frac{\cos ^4(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Integrate[Cos[c + d*x]^4/(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(1652 \sin \left(c+\frac{d x}{2}\right)-1428 \sin \left(c+\frac{3 d x}{2}\right)+756 \sin \left(2 c+\frac{3 d x}{2}\right)-560 \sin \left(2 c+\frac{5 d x}{2}\right)+168 \sin \left(3 c+\frac{5 d x}{2}\right)-104 \sin \left(3 c+\frac{7 d x}{2}\right)+735 d x \cos \left(c+\frac{d x}{2}\right)+441 d x \cos \left(c+\frac{3 d x}{2}\right)+441 d x \cos \left(2 c+\frac{3 d x}{2}\right)+147 d x \cos \left(2 c+\frac{5 d x}{2}\right)+147 d x \cos \left(3 c+\frac{5 d x}{2}\right)+21 d x \cos \left(3 c+\frac{7 d x}{2}\right)+21 d x \cos \left(4 c+\frac{7 d x}{2}\right)-1988 \sin \left(\frac{d x}{2}\right)+735 d x \cos \left(\frac{d x}{2}\right)\right)}{2688 a^4 d}","-\frac{43 \sin (c+d x)}{21 a^4 d (\cos (c+d x)+1)}+\frac{11 \sin (c+d x)}{21 a^4 d (\cos (c+d x)+1)^2}+\frac{x}{a^4}-\frac{\sin (c+d x) \cos ^3(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{2 \sin (c+d x) \cos ^2(c+d x)}{7 a d (a \cos (c+d x)+a)^3}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^7*(735*d*x*Cos[(d*x)/2] + 735*d*x*Cos[c + (d*x)/2] + 441*d*x*Cos[c + (3*d*x)/2] + 441*d*x*Cos[2*c + (3*d*x)/2] + 147*d*x*Cos[2*c + (5*d*x)/2] + 147*d*x*Cos[3*c + (5*d*x)/2] + 21*d*x*Cos[3*c + (7*d*x)/2] + 21*d*x*Cos[4*c + (7*d*x)/2] - 1988*Sin[(d*x)/2] + 1652*Sin[c + (d*x)/2] - 1428*Sin[c + (3*d*x)/2] + 756*Sin[2*c + (3*d*x)/2] - 560*Sin[2*c + (5*d*x)/2] + 168*Sin[3*c + (5*d*x)/2] - 104*Sin[3*c + (7*d*x)/2]))/(2688*a^4*d)","A",1
75,1,112,114,0.2707965,"\int \frac{\cos ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Integrate[Cos[c + d*x]^3/(a + a*Cos[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \left(-210 \sin \left(c+\frac{d x}{2}\right)+147 \sin \left(c+\frac{3 d x}{2}\right)-105 \sin \left(2 c+\frac{3 d x}{2}\right)+49 \sin \left(2 c+\frac{5 d x}{2}\right)-35 \sin \left(3 c+\frac{5 d x}{2}\right)+12 \sin \left(3 c+\frac{7 d x}{2}\right)+210 \sin \left(\frac{d x}{2}\right)\right) \sec ^7\left(\frac{1}{2} (c+d x)\right)}{2240 a^4 d}","\frac{12 \sin (c+d x)}{35 a^4 d (\cos (c+d x)+1)}-\frac{18 \sin (c+d x)}{35 a^4 d (\cos (c+d x)+1)^2}-\frac{\sin (c+d x) \cos ^2(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{8 \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^7*(210*Sin[(d*x)/2] - 210*Sin[c + (d*x)/2] + 147*Sin[c + (3*d*x)/2] - 105*Sin[2*c + (3*d*x)/2] + 49*Sin[2*c + (5*d*x)/2] - 35*Sin[3*c + (5*d*x)/2] + 12*Sin[3*c + (7*d*x)/2]))/(2240*a^4*d)","A",1
76,1,99,112,0.2604661,"\int \frac{\cos ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Integrate[Cos[c + d*x]^2/(a + a*Cos[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \left(-175 \sin \left(c+\frac{d x}{2}\right)+168 \sin \left(c+\frac{3 d x}{2}\right)-105 \sin \left(2 c+\frac{3 d x}{2}\right)+91 \sin \left(2 c+\frac{5 d x}{2}\right)+13 \sin \left(3 c+\frac{7 d x}{2}\right)+280 \sin \left(\frac{d x}{2}\right)\right) \sec ^7\left(\frac{1}{2} (c+d x)\right)}{6720 a^4 d}","\frac{13 \sin (c+d x)}{105 d \left(a^4 \cos (c+d x)+a^4\right)}+\frac{13 \sin (c+d x)}{105 d \left(a^2 \cos (c+d x)+a^2\right)^2}-\frac{11 \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac{\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^7*(280*Sin[(d*x)/2] - 175*Sin[c + (d*x)/2] + 168*Sin[c + (3*d*x)/2] - 105*Sin[2*c + (3*d*x)/2] + 91*Sin[2*c + (5*d*x)/2] + 13*Sin[3*c + (7*d*x)/2]))/(6720*a^4*d)","A",1
77,1,87,112,0.2351412,"\int \frac{\cos (c+d x)}{(a+a \cos (c+d x))^4} \, dx","Integrate[Cos[c + d*x]/(a + a*Cos[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \left(-35 \sin \left(c+\frac{d x}{2}\right)+2 \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 \sin \left(\frac{d x}{2}\right)\right) \sec ^7\left(\frac{1}{2} (c+d x)\right)}{1680 a^4 d}","\frac{8 \sin (c+d x)}{105 d \left(a^4 \cos (c+d x)+a^4\right)}+\frac{8 \sin (c+d x)}{105 d \left(a^2 \cos (c+d x)+a^2\right)^2}+\frac{4 \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^7*(35*Sin[(d*x)/2] - 35*Sin[c + (d*x)/2] + 2*(21*Sin[c + (3*d*x)/2] + 7*Sin[2*c + (5*d*x)/2] + Sin[3*c + (7*d*x)/2])))/(1680*a^4*d)","A",1
78,1,77,112,0.1645452,"\int \frac{1}{(a+a \cos (c+d x))^4} \, dx","Integrate[(a + a*Cos[c + d*x])^(-4),x]","\frac{\left(35 \sin \left(\frac{1}{2} (c+d x)\right)+21 \sin \left(\frac{3}{2} (c+d x)\right)+7 \sin \left(\frac{5}{2} (c+d x)\right)+\sin \left(\frac{7}{2} (c+d x)\right)\right) \cos \left(\frac{1}{2} (c+d x)\right)}{70 a^4 d (\cos (c+d x)+1)^4}","\frac{2 \sin (c+d x)}{35 d \left(a^4 \cos (c+d x)+a^4\right)}+\frac{2 \sin (c+d x)}{35 d \left(a^2 \cos (c+d x)+a^2\right)^2}+\frac{3 \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac{\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"(Cos[(c + d*x)/2]*(35*Sin[(c + d*x)/2] + 21*Sin[(3*(c + d*x))/2] + 7*Sin[(5*(c + d*x))/2] + Sin[(7*(c + d*x))/2]))/(70*a^4*d*(1 + Cos[c + d*x])^4)","A",1
79,1,185,120,0.8603794,"\int \frac{\sec (c+d x)}{(a+a \cos (c+d x))^4} \, dx","Integrate[Sec[c + d*x]/(a + a*Cos[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \left(434 \sin \left(c+\frac{d x}{2}\right)-525 \sin \left(c+\frac{3 d x}{2}\right)+147 \sin \left(2 c+\frac{3 d x}{2}\right)-203 \sin \left(2 c+\frac{5 d x}{2}\right)+21 \sin \left(3 c+\frac{5 d x}{2}\right)-32 \sin \left(3 c+\frac{7 d x}{2}\right)-686 \sin \left(\frac{d x}{2}\right)\right) \cos \left(\frac{1}{2} (c+d x)\right)-1344 \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)}{84 a^4 d (\cos (c+d x)+1)^4}","\frac{\tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{32 \sin (c+d x)}{21 a^4 d (\cos (c+d x)+1)}-\frac{11 \sin (c+d x)}{21 a^4 d (\cos (c+d x)+1)^2}-\frac{2 \sin (c+d x)}{7 a d (a \cos (c+d x)+a)^3}-\frac{\sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"(-1344*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]*(-686*Sin[(d*x)/2] + 434*Sin[c + (d*x)/2] - 525*Sin[c + (3*d*x)/2] + 147*Sin[2*c + (3*d*x)/2] - 203*Sin[2*c + (5*d*x)/2] + 21*Sin[3*c + (5*d*x)/2] - 32*Sin[3*c + (7*d*x)/2]))/(84*a^4*d*(1 + Cos[c + d*x])^4)","A",1
80,1,341,135,4.1484385,"\int \frac{\sec ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^2/(a + a*Cos[c + d*x])^4,x]","\frac{107520 \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) \left(-20524 \sin \left(c-\frac{d x}{2}\right)+14644 \sin \left(c+\frac{d x}{2}\right)-16660 \sin \left(2 c+\frac{d x}{2}\right)-4690 \sin \left(c+\frac{3 d x}{2}\right)+14378 \sin \left(2 c+\frac{3 d x}{2}\right)-9100 \sin \left(3 c+\frac{3 d x}{2}\right)+11668 \sin \left(c+\frac{5 d x}{2}\right)-630 \sin \left(2 c+\frac{5 d x}{2}\right)+9358 \sin \left(3 c+\frac{5 d x}{2}\right)-2940 \sin \left(4 c+\frac{5 d x}{2}\right)+4228 \sin \left(2 c+\frac{7 d x}{2}\right)+315 \sin \left(3 c+\frac{7 d x}{2}\right)+3493 \sin \left(4 c+\frac{7 d x}{2}\right)-420 \sin \left(5 c+\frac{7 d x}{2}\right)+664 \sin \left(3 c+\frac{9 d x}{2}\right)+105 \sin \left(4 c+\frac{9 d x}{2}\right)+559 \sin \left(5 c+\frac{9 d x}{2}\right)-10780 \sin \left(\frac{d x}{2}\right)+18788 \sin \left(\frac{3 d x}{2}\right)\right) \cos \left(\frac{1}{2} (c+d x)\right) \sec (c+d x)}{1680 a^4 d (\cos (c+d x)+1)^4}","\frac{664 \tan (c+d x)}{105 a^4 d}-\frac{4 \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{4 \tan (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac{88 \tan (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{12 \tan (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{\tan (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"(107520*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]*(-10780*Sin[(d*x)/2] + 18788*Sin[(3*d*x)/2] - 20524*Sin[c - (d*x)/2] + 14644*Sin[c + (d*x)/2] - 16660*Sin[2*c + (d*x)/2] - 4690*Sin[c + (3*d*x)/2] + 14378*Sin[2*c + (3*d*x)/2] - 9100*Sin[3*c + (3*d*x)/2] + 11668*Sin[c + (5*d*x)/2] - 630*Sin[2*c + (5*d*x)/2] + 9358*Sin[3*c + (5*d*x)/2] - 2940*Sin[4*c + (5*d*x)/2] + 4228*Sin[2*c + (7*d*x)/2] + 315*Sin[3*c + (7*d*x)/2] + 3493*Sin[4*c + (7*d*x)/2] - 420*Sin[5*c + (7*d*x)/2] + 664*Sin[3*c + (9*d*x)/2] + 105*Sin[4*c + (9*d*x)/2] + 559*Sin[5*c + (9*d*x)/2]))/(1680*a^4*d*(1 + Cos[c + d*x])^4)","B",1
81,1,455,185,6.2664809,"\int \frac{\sec ^3(c+d x)}{(a+a \cos (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^3/(a + a*Cos[c + d*x])^4,x]","-\frac{168 \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d (a \cos (c+d x)+a)^4}+\frac{168 \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right) \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d (a \cos (c+d x)+a)^4}+\frac{\sec \left(\frac{c}{2}\right) \sec (c) \left(61054 \sin \left(c-\frac{d x}{2}\right)-33614 \sin \left(c+\frac{d x}{2}\right)+51842 \sin \left(2 c+\frac{d x}{2}\right)+12460 \sin \left(c+\frac{3 d x}{2}\right)-33716 \sin \left(2 c+\frac{3 d x}{2}\right)+34300 \sin \left(3 c+\frac{3 d x}{2}\right)-39788 \sin \left(c+\frac{5 d x}{2}\right)+2940 \sin \left(2 c+\frac{5 d x}{2}\right)-26068 \sin \left(3 c+\frac{5 d x}{2}\right)+16660 \sin \left(4 c+\frac{5 d x}{2}\right)-21351 \sin \left(2 c+\frac{7 d x}{2}\right)-1295 \sin \left(3 c+\frac{7 d x}{2}\right)-14911 \sin \left(4 c+\frac{7 d x}{2}\right)+5145 \sin \left(5 c+\frac{7 d x}{2}\right)-7329 \sin \left(3 c+\frac{9 d x}{2}\right)-1225 \sin \left(4 c+\frac{9 d x}{2}\right)-5369 \sin \left(5 c+\frac{9 d x}{2}\right)+735 \sin \left(6 c+\frac{9 d x}{2}\right)-1152 \sin \left(4 c+\frac{11 d x}{2}\right)-280 \sin \left(5 c+\frac{11 d x}{2}\right)-872 \sin \left(6 c+\frac{11 d x}{2}\right)+24402 \sin \left(\frac{d x}{2}\right)-55556 \sin \left(\frac{3 d x}{2}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x)}{2240 d (a \cos (c+d x)+a)^4}","-\frac{576 \tan (c+d x)}{35 a^4 d}+\frac{21 \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{21 \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{288 \tan (c+d x) \sec (c+d x)}{35 a^4 d (\cos (c+d x)+1)}-\frac{43 \tan (c+d x) \sec (c+d x)}{35 a^4 d (\cos (c+d x)+1)^2}-\frac{2 \tan (c+d x) \sec (c+d x)}{5 a d (a \cos (c+d x)+a)^3}-\frac{\tan (c+d x) \sec (c+d x)}{7 d (a \cos (c+d x)+a)^4}",1,"(-168*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) + (168*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*(24402*Sin[(d*x)/2] - 55556*Sin[(3*d*x)/2] + 61054*Sin[c - (d*x)/2] - 33614*Sin[c + (d*x)/2] + 51842*Sin[2*c + (d*x)/2] + 12460*Sin[c + (3*d*x)/2] - 33716*Sin[2*c + (3*d*x)/2] + 34300*Sin[3*c + (3*d*x)/2] - 39788*Sin[c + (5*d*x)/2] + 2940*Sin[2*c + (5*d*x)/2] - 26068*Sin[3*c + (5*d*x)/2] + 16660*Sin[4*c + (5*d*x)/2] - 21351*Sin[2*c + (7*d*x)/2] - 1295*Sin[3*c + (7*d*x)/2] - 14911*Sin[4*c + (7*d*x)/2] + 5145*Sin[5*c + (7*d*x)/2] - 7329*Sin[3*c + (9*d*x)/2] - 1225*Sin[4*c + (9*d*x)/2] - 5369*Sin[5*c + (9*d*x)/2] + 735*Sin[6*c + (9*d*x)/2] - 1152*Sin[4*c + (11*d*x)/2] - 280*Sin[5*c + (11*d*x)/2] - 872*Sin[6*c + (11*d*x)/2]))/(2240*d*(a + a*Cos[c + d*x])^4)","B",1
82,1,345,225,0.7811818,"\int \frac{\cos ^7(c+d x)}{(a+a \cos (c+d x))^5} \, dx","Integrate[Cos[c + d*x]^7/(a + a*Cos[c + d*x])^5,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^9\left(\frac{1}{2} (c+d x)\right) \left(7194600 \sin \left(c+\frac{d x}{2}\right)-7472241 \sin \left(c+\frac{3 d x}{2}\right)+3432975 \sin \left(2 c+\frac{3 d x}{2}\right)-3871989 \sin \left(2 c+\frac{5 d x}{2}\right)+801675 \sin \left(3 c+\frac{5 d x}{2}\right)-1186056 \sin \left(3 c+\frac{7 d x}{2}\right)-17640 \sin \left(4 c+\frac{7 d x}{2}\right)-175184 \sin \left(4 c+\frac{9 d x}{2}\right)-45360 \sin \left(5 c+\frac{9 d x}{2}\right)-3465 \sin \left(5 c+\frac{11 d x}{2}\right)-3465 \sin \left(6 c+\frac{11 d x}{2}\right)+315 \sin \left(6 c+\frac{13 d x}{2}\right)+315 \sin \left(7 c+\frac{13 d x}{2}\right)+4921560 d x \cos \left(c+\frac{d x}{2}\right)+3281040 d x \cos \left(c+\frac{3 d x}{2}\right)+3281040 d x \cos \left(2 c+\frac{3 d x}{2}\right)+1406160 d x \cos \left(2 c+\frac{5 d x}{2}\right)+1406160 d x \cos \left(3 c+\frac{5 d x}{2}\right)+351540 d x \cos \left(3 c+\frac{7 d x}{2}\right)+351540 d x \cos \left(4 c+\frac{7 d x}{2}\right)+39060 d x \cos \left(4 c+\frac{9 d x}{2}\right)+39060 d x \cos \left(5 c+\frac{9 d x}{2}\right)-9163224 \sin \left(\frac{d x}{2}\right)+4921560 d x \cos \left(\frac{d x}{2}\right)\right)}{1290240 a^5 d}","-\frac{7664 \sin (c+d x)}{315 a^5 d}-\frac{3832 \sin (c+d x) \cos ^2(c+d x)}{315 d \left(a^5 \cos (c+d x)+a^5\right)}+\frac{31 \sin (c+d x) \cos (c+d x)}{2 a^5 d}+\frac{31 x}{2 a^5}-\frac{577 \sin (c+d x) \cos ^3(c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}-\frac{28 \sin (c+d x) \cos ^4(c+d x)}{45 a^2 d (a \cos (c+d x)+a)^3}-\frac{\sin (c+d x) \cos ^6(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac{17 \sin (c+d x) \cos ^5(c+d x)}{63 a d (a \cos (c+d x)+a)^4}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^9*(4921560*d*x*Cos[(d*x)/2] + 4921560*d*x*Cos[c + (d*x)/2] + 3281040*d*x*Cos[c + (3*d*x)/2] + 3281040*d*x*Cos[2*c + (3*d*x)/2] + 1406160*d*x*Cos[2*c + (5*d*x)/2] + 1406160*d*x*Cos[3*c + (5*d*x)/2] + 351540*d*x*Cos[3*c + (7*d*x)/2] + 351540*d*x*Cos[4*c + (7*d*x)/2] + 39060*d*x*Cos[4*c + (9*d*x)/2] + 39060*d*x*Cos[5*c + (9*d*x)/2] - 9163224*Sin[(d*x)/2] + 7194600*Sin[c + (d*x)/2] - 7472241*Sin[c + (3*d*x)/2] + 3432975*Sin[2*c + (3*d*x)/2] - 3871989*Sin[2*c + (5*d*x)/2] + 801675*Sin[3*c + (5*d*x)/2] - 1186056*Sin[3*c + (7*d*x)/2] - 17640*Sin[4*c + (7*d*x)/2] - 175184*Sin[4*c + (9*d*x)/2] - 45360*Sin[5*c + (9*d*x)/2] - 3465*Sin[5*c + (11*d*x)/2] - 3465*Sin[6*c + (11*d*x)/2] + 315*Sin[6*c + (13*d*x)/2] + 315*Sin[7*c + (13*d*x)/2]))/(1290240*a^5*d)","A",1
83,1,319,191,0.7292342,"\int \frac{\cos ^6(c+d x)}{(a+a \cos (c+d x))^5} \, dx","Integrate[Cos[c + d*x]^6/(a + a*Cos[c + d*x])^5,x]","-\frac{\sec \left(\frac{c}{2}\right) \sec ^9\left(\frac{1}{2} (c+d x)\right) \left(143010 \sin \left(c+\frac{d x}{2}\right)-138726 \sin \left(c+\frac{3 d x}{2}\right)+73290 \sin \left(2 c+\frac{3 d x}{2}\right)-70389 \sin \left(2 c+\frac{5 d x}{2}\right)+20475 \sin \left(3 c+\frac{5 d x}{2}\right)-21141 \sin \left(3 c+\frac{7 d x}{2}\right)+1575 \sin \left(4 c+\frac{7 d x}{2}\right)-3091 \sin \left(4 c+\frac{9 d x}{2}\right)-567 \sin \left(5 c+\frac{9 d x}{2}\right)-63 \sin \left(5 c+\frac{11 d x}{2}\right)-63 \sin \left(6 c+\frac{11 d x}{2}\right)+79380 d x \cos \left(c+\frac{d x}{2}\right)+52920 d x \cos \left(c+\frac{3 d x}{2}\right)+52920 d x \cos \left(2 c+\frac{3 d x}{2}\right)+22680 d x \cos \left(2 c+\frac{5 d x}{2}\right)+22680 d x \cos \left(3 c+\frac{5 d x}{2}\right)+5670 d x \cos \left(3 c+\frac{7 d x}{2}\right)+5670 d x \cos \left(4 c+\frac{7 d x}{2}\right)+630 d x \cos \left(4 c+\frac{9 d x}{2}\right)+630 d x \cos \left(5 c+\frac{9 d x}{2}\right)-175014 \sin \left(\frac{d x}{2}\right)+79380 d x \cos \left(\frac{d x}{2}\right)\right)}{64512 a^5 d}","\frac{181 \sin (c+d x)}{63 a^5 d}+\frac{5 \sin (c+d x)}{d \left(a^5 \cos (c+d x)+a^5\right)}-\frac{5 x}{a^5}-\frac{67 \sin (c+d x) \cos ^2(c+d x)}{63 a^3 d (a \cos (c+d x)+a)^2}-\frac{29 \sin (c+d x) \cos ^3(c+d x)}{63 a^2 d (a \cos (c+d x)+a)^3}-\frac{\sin (c+d x) \cos ^5(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac{5 \sin (c+d x) \cos ^4(c+d x)}{21 a d (a \cos (c+d x)+a)^4}",1,"-1/64512*(Sec[c/2]*Sec[(c + d*x)/2]^9*(79380*d*x*Cos[(d*x)/2] + 79380*d*x*Cos[c + (d*x)/2] + 52920*d*x*Cos[c + (3*d*x)/2] + 52920*d*x*Cos[2*c + (3*d*x)/2] + 22680*d*x*Cos[2*c + (5*d*x)/2] + 22680*d*x*Cos[3*c + (5*d*x)/2] + 5670*d*x*Cos[3*c + (7*d*x)/2] + 5670*d*x*Cos[4*c + (7*d*x)/2] + 630*d*x*Cos[4*c + (9*d*x)/2] + 630*d*x*Cos[5*c + (9*d*x)/2] - 175014*Sin[(d*x)/2] + 143010*Sin[c + (d*x)/2] - 138726*Sin[c + (3*d*x)/2] + 73290*Sin[2*c + (3*d*x)/2] - 70389*Sin[2*c + (5*d*x)/2] + 20475*Sin[3*c + (5*d*x)/2] - 21141*Sin[3*c + (7*d*x)/2] + 1575*Sin[4*c + (7*d*x)/2] - 3091*Sin[4*c + (9*d*x)/2] - 567*Sin[5*c + (9*d*x)/2] - 63*Sin[5*c + (11*d*x)/2] - 63*Sin[6*c + (11*d*x)/2]))/(a^5*d)","A",1
84,1,280,168,0.4898016,"\int \frac{\cos ^5(c+d x)}{(a+a \cos (c+d x))^5} \, dx","Integrate[Cos[c + d*x]^5/(a + a*Cos[c + d*x])^5,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^9\left(\frac{1}{2} (c+d x)\right) \left(100800 \sin \left(c+\frac{d x}{2}\right)-88284 \sin \left(c+\frac{3 d x}{2}\right)+56700 \sin \left(2 c+\frac{3 d x}{2}\right)-43236 \sin \left(2 c+\frac{5 d x}{2}\right)+18900 \sin \left(3 c+\frac{5 d x}{2}\right)-12384 \sin \left(3 c+\frac{7 d x}{2}\right)+3150 \sin \left(4 c+\frac{7 d x}{2}\right)-1726 \sin \left(4 c+\frac{9 d x}{2}\right)+39690 d x \cos \left(c+\frac{d x}{2}\right)+26460 d x \cos \left(c+\frac{3 d x}{2}\right)+26460 d x \cos \left(2 c+\frac{3 d x}{2}\right)+11340 d x \cos \left(2 c+\frac{5 d x}{2}\right)+11340 d x \cos \left(3 c+\frac{5 d x}{2}\right)+2835 d x \cos \left(3 c+\frac{7 d x}{2}\right)+2835 d x \cos \left(4 c+\frac{7 d x}{2}\right)+315 d x \cos \left(4 c+\frac{9 d x}{2}\right)+315 d x \cos \left(5 c+\frac{9 d x}{2}\right)-116676 \sin \left(\frac{d x}{2}\right)+39690 d x \cos \left(\frac{d x}{2}\right)\right)}{161280 a^5 d}","-\frac{661 \sin (c+d x)}{315 d \left(a^5 \cos (c+d x)+a^5\right)}+\frac{x}{a^5}+\frac{173 \sin (c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}-\frac{34 \sin (c+d x) \cos ^2(c+d x)}{105 a^2 d (a \cos (c+d x)+a)^3}-\frac{\sin (c+d x) \cos ^4(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac{13 \sin (c+d x) \cos ^3(c+d x)}{63 a d (a \cos (c+d x)+a)^4}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^9*(39690*d*x*Cos[(d*x)/2] + 39690*d*x*Cos[c + (d*x)/2] + 26460*d*x*Cos[c + (3*d*x)/2] + 26460*d*x*Cos[2*c + (3*d*x)/2] + 11340*d*x*Cos[2*c + (5*d*x)/2] + 11340*d*x*Cos[3*c + (5*d*x)/2] + 2835*d*x*Cos[3*c + (7*d*x)/2] + 2835*d*x*Cos[4*c + (7*d*x)/2] + 315*d*x*Cos[4*c + (9*d*x)/2] + 315*d*x*Cos[5*c + (9*d*x)/2] - 116676*Sin[(d*x)/2] + 100800*Sin[c + (d*x)/2] - 88284*Sin[c + (3*d*x)/2] + 56700*Sin[2*c + (3*d*x)/2] - 43236*Sin[2*c + (5*d*x)/2] + 18900*Sin[3*c + (5*d*x)/2] - 12384*Sin[3*c + (7*d*x)/2] + 3150*Sin[4*c + (7*d*x)/2] - 1726*Sin[4*c + (9*d*x)/2]))/(161280*a^5*d)","A",1
85,1,138,155,0.276502,"\int \frac{\cos ^4(c+d x)}{(a+a \cos (c+d x))^5} \, dx","Integrate[Cos[c + d*x]^4/(a + a*Cos[c + d*x])^5,x]","\frac{\sec \left(\frac{c}{2}\right) \left(-5040 \sin \left(c+\frac{d x}{2}\right)+3612 \sin \left(c+\frac{3 d x}{2}\right)-3360 \sin \left(2 c+\frac{3 d x}{2}\right)+1728 \sin \left(2 c+\frac{5 d x}{2}\right)-1260 \sin \left(3 c+\frac{5 d x}{2}\right)+432 \sin \left(3 c+\frac{7 d x}{2}\right)-315 \sin \left(4 c+\frac{7 d x}{2}\right)+83 \sin \left(4 c+\frac{9 d x}{2}\right)+5418 \sin \left(\frac{d x}{2}\right)\right) \sec ^9\left(\frac{1}{2} (c+d x)\right)}{80640 a^5 d}","\frac{83 \sin (c+d x)}{315 d \left(a^5 \cos (c+d x)+a^5\right)}-\frac{142 \sin (c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}+\frac{67 \sin (c+d x)}{315 a^2 d (a \cos (c+d x)+a)^3}-\frac{\sin (c+d x) \cos ^3(c+d x)}{9 d (a \cos (c+d x)+a)^5}-\frac{11 \sin (c+d x) \cos ^2(c+d x)}{63 a d (a \cos (c+d x)+a)^4}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^9*(5418*Sin[(d*x)/2] - 5040*Sin[c + (d*x)/2] + 3612*Sin[c + (3*d*x)/2] - 3360*Sin[2*c + (3*d*x)/2] + 1728*Sin[2*c + (5*d*x)/2] - 1260*Sin[3*c + (5*d*x)/2] + 432*Sin[3*c + (7*d*x)/2] - 315*Sin[4*c + (7*d*x)/2] + 83*Sin[4*c + (9*d*x)/2]))/(80640*a^5*d)","A",1
86,1,125,147,0.2398276,"\int \frac{\cos ^3(c+d x)}{(a+a \cos (c+d x))^5} \, dx","Integrate[Cos[c + d*x]^3/(a + a*Cos[c + d*x])^5,x]","\frac{\sec \left(\frac{c}{2}\right) \left(-315 \sin \left(c+\frac{d x}{2}\right)+273 \sin \left(c+\frac{3 d x}{2}\right)-147 \sin \left(2 c+\frac{3 d x}{2}\right)+117 \sin \left(2 c+\frac{5 d x}{2}\right)-63 \sin \left(3 c+\frac{5 d x}{2}\right)+45 \sin \left(3 c+\frac{7 d x}{2}\right)+5 \sin \left(4 c+\frac{9 d x}{2}\right)+315 \sin \left(\frac{d x}{2}\right)\right) \sec ^9\left(\frac{1}{2} (c+d x)\right)}{16128 a^5 d}","\frac{5 \sin (c+d x)}{63 d \left(a^5 \cos (c+d x)+a^5\right)}+\frac{5 \sin (c+d x)}{63 a^3 d (a \cos (c+d x)+a)^2}-\frac{17 \sin (c+d x)}{63 a^2 d (a \cos (c+d x)+a)^3}-\frac{\sin (c+d x) \cos ^2(c+d x)}{9 d (a \cos (c+d x)+a)^5}+\frac{\sin (c+d x)}{7 a d (a \cos (c+d x)+a)^4}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^9*(315*Sin[(d*x)/2] - 315*Sin[c + (d*x)/2] + 273*Sin[c + (3*d*x)/2] - 147*Sin[2*c + (3*d*x)/2] + 117*Sin[2*c + (5*d*x)/2] - 63*Sin[3*c + (5*d*x)/2] + 45*Sin[3*c + (7*d*x)/2] + 5*Sin[4*c + (9*d*x)/2]))/(16128*a^5*d)","A",1
87,1,110,139,0.23711,"\int \frac{\cos ^2(c+d x)}{(a+a \cos (c+d x))^5} \, dx","Integrate[Cos[c + d*x]^2/(a + a*Cos[c + d*x])^5,x]","\frac{\sec \left(\frac{c}{2}\right) \left(-45 \sin \left(c+\frac{d x}{2}\right)+54 \sin \left(c+\frac{3 d x}{2}\right)-30 \sin \left(2 c+\frac{3 d x}{2}\right)+36 \sin \left(2 c+\frac{5 d x}{2}\right)+9 \sin \left(3 c+\frac{7 d x}{2}\right)+\sin \left(4 c+\frac{9 d x}{2}\right)+81 \sin \left(\frac{d x}{2}\right)\right) \sec ^9\left(\frac{1}{2} (c+d x)\right)}{5760 a^5 d}","\frac{2 \sin (c+d x)}{45 d \left(a^5 \cos (c+d x)+a^5\right)}+\frac{2 \sin (c+d x)}{45 a^3 d (a \cos (c+d x)+a)^2}+\frac{\sin (c+d x)}{15 a^2 d (a \cos (c+d x)+a)^3}-\frac{2 \sin (c+d x)}{9 a d (a \cos (c+d x)+a)^4}+\frac{\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^9*(81*Sin[(d*x)/2] - 45*Sin[c + (d*x)/2] + 54*Sin[c + (3*d*x)/2] - 30*Sin[2*c + (3*d*x)/2] + 36*Sin[2*c + (5*d*x)/2] + 9*Sin[3*c + (7*d*x)/2] + Sin[4*c + (9*d*x)/2]))/(5760*a^5*d)","A",1
88,1,97,143,0.1913944,"\int \frac{\cos (c+d x)}{(a+a \cos (c+d x))^5} \, dx","Integrate[Cos[c + d*x]/(a + a*Cos[c + d*x])^5,x]","\frac{\sec \left(\frac{c}{2}\right) \left(-63 \sin \left(c+\frac{d x}{2}\right)+84 \sin \left(c+\frac{3 d x}{2}\right)+36 \sin \left(2 c+\frac{5 d x}{2}\right)+9 \sin \left(3 c+\frac{7 d x}{2}\right)+\sin \left(4 c+\frac{9 d x}{2}\right)+63 \sin \left(\frac{d x}{2}\right)\right) \sec ^9\left(\frac{1}{2} (c+d x)\right)}{8064 a^5 d}","\frac{2 \sin (c+d x)}{63 d \left(a^5 \cos (c+d x)+a^5\right)}+\frac{2 \sin (c+d x)}{63 a d \left(a^2 \cos (c+d x)+a^2\right)^2}+\frac{\sin (c+d x)}{21 a^2 d (a \cos (c+d x)+a)^3}+\frac{5 \sin (c+d x)}{63 a d (a \cos (c+d x)+a)^4}-\frac{\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^9*(63*Sin[(d*x)/2] - 63*Sin[c + (d*x)/2] + 84*Sin[c + (3*d*x)/2] + 36*Sin[2*c + (5*d*x)/2] + 9*Sin[3*c + (7*d*x)/2] + Sin[4*c + (9*d*x)/2]))/(8064*a^5*d)","A",1
89,1,89,143,0.1619678,"\int \frac{1}{(a+a \cos (c+d x))^5} \, dx","Integrate[(a + a*Cos[c + d*x])^(-5),x]","\frac{\left(126 \sin \left(\frac{1}{2} (c+d x)\right)+84 \sin \left(\frac{3}{2} (c+d x)\right)+36 \sin \left(\frac{5}{2} (c+d x)\right)+9 \sin \left(\frac{7}{2} (c+d x)\right)+\sin \left(\frac{9}{2} (c+d x)\right)\right) \cos \left(\frac{1}{2} (c+d x)\right)}{315 a^5 d (\cos (c+d x)+1)^5}","\frac{8 \sin (c+d x)}{315 d \left(a^5 \cos (c+d x)+a^5\right)}+\frac{8 \sin (c+d x)}{315 a d \left(a^2 \cos (c+d x)+a^2\right)^2}+\frac{4 \sin (c+d x)}{105 a^2 d (a \cos (c+d x)+a)^3}+\frac{4 \sin (c+d x)}{63 a d (a \cos (c+d x)+a)^4}+\frac{\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}",1,"(Cos[(c + d*x)/2]*(126*Sin[(c + d*x)/2] + 84*Sin[(3*(c + d*x))/2] + 36*Sin[(5*(c + d*x))/2] + 9*Sin[(7*(c + d*x))/2] + Sin[(9*(c + d*x))/2]))/(315*a^5*d*(1 + Cos[c + d*x])^5)","A",1
90,1,211,153,1.8107144,"\int \frac{\sec (c+d x)}{(a+a \cos (c+d x))^5} \, dx","Integrate[Sec[c + d*x]/(a + a*Cos[c + d*x])^5,x]","-\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(\sec \left(\frac{c}{2}\right) \left(-25515 \sin \left(c+\frac{d x}{2}\right)+29757 \sin \left(c+\frac{3 d x}{2}\right)-11235 \sin \left(2 c+\frac{3 d x}{2}\right)+14733 \sin \left(2 c+\frac{5 d x}{2}\right)-2835 \sin \left(3 c+\frac{5 d x}{2}\right)+4077 \sin \left(3 c+\frac{7 d x}{2}\right)-315 \sin \left(4 c+\frac{7 d x}{2}\right)+488 \sin \left(4 c+\frac{9 d x}{2}\right)+35973 \sin \left(\frac{d x}{2}\right)\right)+80640 \cos ^9\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)}{2520 a^5 d (\cos (c+d x)+1)^5}","\frac{\tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{488 \sin (c+d x)}{315 d \left(a^5 \cos (c+d x)+a^5\right)}-\frac{173 \sin (c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}-\frac{34 \sin (c+d x)}{105 a^2 d (a \cos (c+d x)+a)^3}-\frac{13 \sin (c+d x)}{63 a d (a \cos (c+d x)+a)^4}-\frac{\sin (c+d x)}{9 d (a \cos (c+d x)+a)^5}",1,"-1/2520*(Cos[(c + d*x)/2]*(80640*Cos[(c + d*x)/2]^9*(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]*(35973*Sin[(d*x)/2] - 25515*Sin[c + (d*x)/2] + 29757*Sin[c + (3*d*x)/2] - 11235*Sin[2*c + (3*d*x)/2] + 14733*Sin[2*c + (5*d*x)/2] - 2835*Sin[3*c + (5*d*x)/2] + 4077*Sin[3*c + (7*d*x)/2] - 315*Sin[4*c + (7*d*x)/2] + 488*Sin[4*c + (9*d*x)/2])))/(a^5*d*(1 + Cos[c + d*x])^5)","A",1
91,1,453,168,6.3671699,"\int \frac{\sec ^2(c+d x)}{(a+a \cos (c+d x))^5} \, dx","Integrate[Sec[c + d*x]^2/(a + a*Cos[c + d*x])^5,x]","\frac{160 \cos ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d (a \cos (c+d x)+a)^5}-\frac{160 \cos ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right) \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d (a \cos (c+d x)+a)^5}+\frac{\sec \left(\frac{c}{2}\right) \sec (c) \left(-56952 \sin \left(c-\frac{d x}{2}\right)+43722 \sin \left(c+\frac{d x}{2}\right)-47208 \sin \left(2 c+\frac{d x}{2}\right)-18144 \sin \left(c+\frac{3 d x}{2}\right)+41796 \sin \left(2 c+\frac{3 d x}{2}\right)-28350 \sin \left(3 c+\frac{3 d x}{2}\right)+34578 \sin \left(c+\frac{5 d x}{2}\right)-5691 \sin \left(2 c+\frac{5 d x}{2}\right)+28719 \sin \left(3 c+\frac{5 d x}{2}\right)-11550 \sin \left(4 c+\frac{5 d x}{2}\right)+15517 \sin \left(2 c+\frac{7 d x}{2}\right)-504 \sin \left(3 c+\frac{7 d x}{2}\right)+13186 \sin \left(4 c+\frac{7 d x}{2}\right)-2835 \sin \left(5 c+\frac{7 d x}{2}\right)+4149 \sin \left(3 c+\frac{9 d x}{2}\right)+252 \sin \left(4 c+\frac{9 d x}{2}\right)+3582 \sin \left(5 c+\frac{9 d x}{2}\right)-315 \sin \left(6 c+\frac{9 d x}{2}\right)+496 \sin \left(4 c+\frac{11 d x}{2}\right)+63 \sin \left(5 c+\frac{11 d x}{2}\right)+433 \sin \left(6 c+\frac{11 d x}{2}\right)-33978 \sin \left(\frac{d x}{2}\right)+52002 \sin \left(\frac{3 d x}{2}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \sec (c+d x)}{2016 d (a \cos (c+d x)+a)^5}","\frac{496 \tan (c+d x)}{63 a^5 d}-\frac{5 \tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{5 \tan (c+d x)}{d \left(a^5 \cos (c+d x)+a^5\right)}-\frac{67 \tan (c+d x)}{63 a^3 d (a \cos (c+d x)+a)^2}-\frac{29 \tan (c+d x)}{63 a^2 d (a \cos (c+d x)+a)^3}-\frac{5 \tan (c+d x)}{21 a d (a \cos (c+d x)+a)^4}-\frac{\tan (c+d x)}{9 d (a \cos (c+d x)+a)^5}",1,"(160*Cos[c/2 + (d*x)/2]^10*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]])/(d*(a + a*Cos[c + d*x])^5) - (160*Cos[c/2 + (d*x)/2]^10*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]])/(d*(a + a*Cos[c + d*x])^5) + (Cos[c/2 + (d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]*(-33978*Sin[(d*x)/2] + 52002*Sin[(3*d*x)/2] - 56952*Sin[c - (d*x)/2] + 43722*Sin[c + (d*x)/2] - 47208*Sin[2*c + (d*x)/2] - 18144*Sin[c + (3*d*x)/2] + 41796*Sin[2*c + (3*d*x)/2] - 28350*Sin[3*c + (3*d*x)/2] + 34578*Sin[c + (5*d*x)/2] - 5691*Sin[2*c + (5*d*x)/2] + 28719*Sin[3*c + (5*d*x)/2] - 11550*Sin[4*c + (5*d*x)/2] + 15517*Sin[2*c + (7*d*x)/2] - 504*Sin[3*c + (7*d*x)/2] + 13186*Sin[4*c + (7*d*x)/2] - 2835*Sin[5*c + (7*d*x)/2] + 4149*Sin[3*c + (9*d*x)/2] + 252*Sin[4*c + (9*d*x)/2] + 3582*Sin[5*c + (9*d*x)/2] - 315*Sin[6*c + (9*d*x)/2] + 496*Sin[4*c + (11*d*x)/2] + 63*Sin[5*c + (11*d*x)/2] + 433*Sin[6*c + (11*d*x)/2]))/(2016*d*(a + a*Cos[c + d*x])^5)","B",1
92,1,507,224,6.3492396,"\int \frac{\sec ^3(c+d x)}{(a+a \cos (c+d x))^5} \, dx","Integrate[Sec[c + d*x]^3/(a + a*Cos[c + d*x])^5,x]","-\frac{496 \cos ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d (a \cos (c+d x)+a)^5}+\frac{496 \cos ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right) \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d (a \cos (c+d x)+a)^5}+\frac{\sec \left(\frac{c}{2}\right) \sec (c) \left(3057654 \sin \left(c-\frac{d x}{2}\right)-1885854 \sin \left(c+\frac{d x}{2}\right)+2644362 \sin \left(2 c+\frac{d x}{2}\right)+867048 \sin \left(c+\frac{3 d x}{2}\right)-1868436 \sin \left(2 c+\frac{3 d x}{2}\right)+1821498 \sin \left(3 c+\frac{3 d x}{2}\right)-2083537 \sin \left(c+\frac{5 d x}{2}\right)+339885 \sin \left(2 c+\frac{5 d x}{2}\right)-1456687 \sin \left(3 c+\frac{5 d x}{2}\right)+966735 \sin \left(4 c+\frac{5 d x}{2}\right)-1195641 \sin \left(2 c+\frac{7 d x}{2}\right)+46515 \sin \left(3 c+\frac{7 d x}{2}\right)-874341 \sin \left(4 c+\frac{7 d x}{2}\right)+367815 \sin \left(5 c+\frac{7 d x}{2}\right)-494579 \sin \left(3 c+\frac{9 d x}{2}\right)-31815 \sin \left(4 c+\frac{9 d x}{2}\right)-374879 \sin \left(5 c+\frac{9 d x}{2}\right)+87885 \sin \left(6 c+\frac{9 d x}{2}\right)-128187 \sin \left(4 c+\frac{11 d x}{2}\right)-18585 \sin \left(5 c+\frac{11 d x}{2}\right)-99837 \sin \left(6 c+\frac{11 d x}{2}\right)+9765 \sin \left(7 c+\frac{11 d x}{2}\right)-15328 \sin \left(5 c+\frac{13 d x}{2}\right)-3150 \sin \left(6 c+\frac{13 d x}{2}\right)-12178 \sin \left(7 c+\frac{13 d x}{2}\right)+1472562 \sin \left(\frac{d x}{2}\right)-2822886 \sin \left(\frac{3 d x}{2}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x)}{40320 d (a \cos (c+d x)+a)^5}","-\frac{7664 \tan (c+d x)}{315 a^5 d}+\frac{31 \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}+\frac{31 \tan (c+d x) \sec (c+d x)}{2 a^5 d}-\frac{3832 \tan (c+d x) \sec (c+d x)}{315 d \left(a^5 \cos (c+d x)+a^5\right)}-\frac{577 \tan (c+d x) \sec (c+d x)}{315 a^3 d (a \cos (c+d x)+a)^2}-\frac{28 \tan (c+d x) \sec (c+d x)}{45 a^2 d (a \cos (c+d x)+a)^3}-\frac{17 \tan (c+d x) \sec (c+d x)}{63 a d (a \cos (c+d x)+a)^4}-\frac{\tan (c+d x) \sec (c+d x)}{9 d (a \cos (c+d x)+a)^5}",1,"(-496*Cos[c/2 + (d*x)/2]^10*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]])/(d*(a + a*Cos[c + d*x])^5) + (496*Cos[c/2 + (d*x)/2]^10*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]])/(d*(a + a*Cos[c + d*x])^5) + (Cos[c/2 + (d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]^2*(1472562*Sin[(d*x)/2] - 2822886*Sin[(3*d*x)/2] + 3057654*Sin[c - (d*x)/2] - 1885854*Sin[c + (d*x)/2] + 2644362*Sin[2*c + (d*x)/2] + 867048*Sin[c + (3*d*x)/2] - 1868436*Sin[2*c + (3*d*x)/2] + 1821498*Sin[3*c + (3*d*x)/2] - 2083537*Sin[c + (5*d*x)/2] + 339885*Sin[2*c + (5*d*x)/2] - 1456687*Sin[3*c + (5*d*x)/2] + 966735*Sin[4*c + (5*d*x)/2] - 1195641*Sin[2*c + (7*d*x)/2] + 46515*Sin[3*c + (7*d*x)/2] - 874341*Sin[4*c + (7*d*x)/2] + 367815*Sin[5*c + (7*d*x)/2] - 494579*Sin[3*c + (9*d*x)/2] - 31815*Sin[4*c + (9*d*x)/2] - 374879*Sin[5*c + (9*d*x)/2] + 87885*Sin[6*c + (9*d*x)/2] - 128187*Sin[4*c + (11*d*x)/2] - 18585*Sin[5*c + (11*d*x)/2] - 99837*Sin[6*c + (11*d*x)/2] + 9765*Sin[7*c + (11*d*x)/2] - 15328*Sin[5*c + (13*d*x)/2] - 3150*Sin[6*c + (13*d*x)/2] - 12178*Sin[7*c + (13*d*x)/2]))/(40320*d*(a + a*Cos[c + d*x])^5)","B",1
93,1,164,184,0.3632132,"\int \frac{\cos ^5(c+d x)}{(a+a \cos (c+d x))^6} \, dx","Integrate[Cos[c + d*x]^5/(a + a*Cos[c + d*x])^6,x]","\frac{\sec \left(\frac{c}{2}\right) \left(-33726 \sin \left(c+\frac{d x}{2}\right)+25080 \sin \left(c+\frac{3 d x}{2}\right)-23100 \sin \left(2 c+\frac{3 d x}{2}\right)+12540 \sin \left(2 c+\frac{5 d x}{2}\right)-11550 \sin \left(3 c+\frac{5 d x}{2}\right)+4565 \sin \left(3 c+\frac{7 d x}{2}\right)-3465 \sin \left(4 c+\frac{7 d x}{2}\right)+913 \sin \left(4 c+\frac{9 d x}{2}\right)-693 \sin \left(5 c+\frac{9 d x}{2}\right)+146 \sin \left(5 c+\frac{11 d x}{2}\right)+33726 \sin \left(\frac{d x}{2}\right)\right) \sec ^{11}\left(\frac{1}{2} (c+d x)\right)}{709632 a^6 d}","\frac{146 \sin (c+d x)}{693 a^6 d (\cos (c+d x)+1)}-\frac{268 \sin (c+d x)}{693 a^6 d (\cos (c+d x)+1)^2}+\frac{130 \sin (c+d x)}{693 a^6 d (\cos (c+d x)+1)^3}-\frac{118 \sin (c+d x) \cos ^2(c+d x)}{693 a^2 d (a \cos (c+d x)+a)^4}-\frac{\sin (c+d x) \cos ^4(c+d x)}{11 d (a \cos (c+d x)+a)^6}-\frac{14 \sin (c+d x) \cos ^3(c+d x)}{99 a d (a \cos (c+d x)+a)^5}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^11*(33726*Sin[(d*x)/2] - 33726*Sin[c + (d*x)/2] + 25080*Sin[c + (3*d*x)/2] - 23100*Sin[2*c + (3*d*x)/2] + 12540*Sin[2*c + (5*d*x)/2] - 11550*Sin[3*c + (5*d*x)/2] + 4565*Sin[3*c + (7*d*x)/2] - 3465*Sin[4*c + (7*d*x)/2] + 913*Sin[4*c + (9*d*x)/2] - 693*Sin[5*c + (9*d*x)/2] + 146*Sin[5*c + (11*d*x)/2]))/(709632*a^6*d)","A",1
94,1,151,176,0.3244722,"\int \frac{\cos ^4(c+d x)}{(a+a \cos (c+d x))^6} \, dx","Integrate[Cos[c + d*x]^4/(a + a*Cos[c + d*x])^6,x]","\frac{\sec \left(\frac{c}{2}\right) \left(-12936 \sin \left(c+\frac{d x}{2}\right)+10890 \sin \left(c+\frac{3 d x}{2}\right)-9240 \sin \left(2 c+\frac{3 d x}{2}\right)+6600 \sin \left(2 c+\frac{5 d x}{2}\right)-3465 \sin \left(3 c+\frac{5 d x}{2}\right)+2200 \sin \left(3 c+\frac{7 d x}{2}\right)-1155 \sin \left(4 c+\frac{7 d x}{2}\right)+671 \sin \left(4 c+\frac{9 d x}{2}\right)+61 \sin \left(5 c+\frac{11 d x}{2}\right)+15246 \sin \left(\frac{d x}{2}\right)\right) \sec ^{11}\left(\frac{1}{2} (c+d x)\right)}{1182720 a^6 d}","\frac{61 \sin (c+d x)}{1155 a^6 d (\cos (c+d x)+1)}+\frac{61 \sin (c+d x)}{1155 a^6 d (\cos (c+d x)+1)^2}-\frac{241 \sin (c+d x)}{1155 a^6 d (\cos (c+d x)+1)^3}+\frac{9 \sin (c+d x)}{77 a^2 d (a \cos (c+d x)+a)^4}-\frac{\sin (c+d x) \cos ^3(c+d x)}{11 d (a \cos (c+d x)+a)^6}-\frac{4 \sin (c+d x) \cos ^2(c+d x)}{33 a d (a \cos (c+d x)+a)^5}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^11*(15246*Sin[(d*x)/2] - 12936*Sin[c + (d*x)/2] + 10890*Sin[c + (3*d*x)/2] - 9240*Sin[2*c + (3*d*x)/2] + 6600*Sin[2*c + (5*d*x)/2] - 3465*Sin[3*c + (5*d*x)/2] + 2200*Sin[3*c + (7*d*x)/2] - 1155*Sin[4*c + (7*d*x)/2] + 671*Sin[4*c + (9*d*x)/2] + 61*Sin[5*c + (11*d*x)/2]))/(1182720*a^6*d)","A",1
95,1,92,158,0.2765275,"\int \cos ^4(c+d x) \sqrt{a+a \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^4*Sqrt[a + a*Cos[c + d*x]],x]","\frac{\left(1890 \sin \left(\frac{1}{2} (c+d x)\right)+420 \sin \left(\frac{3}{2} (c+d x)\right)+252 \sin \left(\frac{5}{2} (c+d x)\right)+45 \sin \left(\frac{7}{2} (c+d x)\right)+35 \sin \left(\frac{9}{2} (c+d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{2520 d}","\frac{2 a \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{32 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 a d}-\frac{64 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{32 a \sin (c+d x)}{45 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(1890*Sin[(c + d*x)/2] + 420*Sin[(3*(c + d*x))/2] + 252*Sin[(5*(c + d*x))/2] + 45*Sin[(7*(c + d*x))/2] + 35*Sin[(9*(c + d*x))/2]))/(2520*d)","A",1
96,1,80,122,0.1597888,"\int \cos ^3(c+d x) \sqrt{a+a \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^3*Sqrt[a + a*Cos[c + d*x]],x]","\frac{\left(105 \sin \left(\frac{1}{2} (c+d x)\right)+35 \sin \left(\frac{3}{2} (c+d x)\right)+7 \sin \left(\frac{5}{2} (c+d x)\right)+5 \sin \left(\frac{7}{2} (c+d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{140 d}","\frac{2 a \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}+\frac{12 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 a d}-\frac{8 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{35 d}+\frac{4 a \sin (c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(105*Sin[(c + d*x)/2] + 35*Sin[(3*(c + d*x))/2] + 7*Sin[(5*(c + d*x))/2] + 5*Sin[(7*(c + d*x))/2]))/(140*d)","A",1
97,1,68,86,0.101806,"\int \cos ^2(c+d x) \sqrt{a+a \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^2*Sqrt[a + a*Cos[c + d*x]],x]","\frac{\left(30 \sin \left(\frac{1}{2} (c+d x)\right)+5 \sin \left(\frac{3}{2} (c+d x)\right)+3 \sin \left(\frac{5}{2} (c+d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{30 d}","\frac{2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 a d}-\frac{4 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{14 a \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(30*Sin[(c + d*x)/2] + 5*Sin[(3*(c + d*x))/2] + 3*Sin[(5*(c + d*x))/2]))/(30*d)","A",1
98,1,54,56,0.0729884,"\int \cos (c+d x) \sqrt{a+a \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]*Sqrt[a + a*Cos[c + d*x]],x]","\frac{\left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{3}{2} (c+d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{3 d}","\frac{2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{2 a \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(3*Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2]))/(3*d)","A",1
99,1,29,26,0.0303427,"\int \sqrt{a+a \cos (c+d x)} \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{d}","\frac{2 a \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}",1,"(2*Sqrt[a*(1 + Cos[c + d*x])]*Tan[(c + d*x)/2])/d","A",1
100,1,50,37,0.046683,"\int \sqrt{a+a \cos (c+d x)} \sec (c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x],x]","\frac{\sqrt{2} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}","\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(Sqrt[2]*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2])/d","A",1
101,1,79,62,0.0953884,"\int \sqrt{a+a \cos (c+d x)} \sec ^2(c+d x) \, dx","Integrate[Sqrt[a + a*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(2 \sin \left(\frac{1}{2} (c+d x)\right)+\sqrt{2} \cos (c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{2 d}","\frac{a \tan (c+d x)}{d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]*(Sqrt[2]*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x] + 2*Sin[(c + d*x)/2]))/(2*d)","A",1
102,1,94,102,0.1807889,"\int \sqrt{a+a \cos (c+d x)} \sec ^3(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^3,x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \sin \left(\frac{3}{2} (c+d x)\right)+3 \sqrt{2} \cos ^2(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{8 d}","\frac{3 a \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{3 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{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])]*Sec[(c + d*x)/2]*Sec[c + d*x]^2*(3*Sqrt[2]*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^2 + Sin[(c + d*x)/2] + 3*Sin[(3*(c + d*x))/2]))/(8*d)","A",1
103,1,109,138,0.3287769,"\int \sqrt{a+a \cos (c+d x)} \sec ^4(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^4,x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(42 \sin \left(\frac{1}{2} (c+d x)\right)+5 \left(\sin \left(\frac{3}{2} (c+d x)\right)+3 \sin \left(\frac{5}{2} (c+d x)\right)\right)+30 \sqrt{2} \cos ^3(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{96 d}","\frac{5 a \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{5 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{5 a \tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^3*(30*Sqrt[2]*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^3 + 42*Sin[(c + d*x)/2] + 5*(Sin[(3*(c + d*x))/2] + 3*Sin[(5*(c + d*x))/2])))/(96*d)","A",1
104,1,93,162,0.2629022,"\int \cos ^3(c+d x) (a+a \cos (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(3/2),x]","\frac{a \left(3780 \sin \left(\frac{1}{2} (c+d x)\right)+1050 \sin \left(\frac{3}{2} (c+d x)\right)+378 \sin \left(\frac{5}{2} (c+d x)\right)+135 \sin \left(\frac{7}{2} (c+d x)\right)+35 \sin \left(\frac{9}{2} (c+d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{2520 d}","\frac{2 a^2 \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt{a \cos (c+d x)+a}}+\frac{34 a^2 \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{68 a^2 \sin (c+d x)}{45 d \sqrt{a \cos (c+d x)+a}}+\frac{68 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}-\frac{136 a \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(3780*Sin[(c + d*x)/2] + 1050*Sin[(3*(c + d*x))/2] + 378*Sin[(5*(c + d*x))/2] + 135*Sin[(7*(c + d*x))/2] + 35*Sin[(9*(c + d*x))/2]))/(2520*d)","A",1
105,1,81,116,0.1689128,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(3/2),x]","\frac{a \left(735 \sin \left(\frac{1}{2} (c+d x)\right)+175 \sin \left(\frac{3}{2} (c+d x)\right)+63 \sin \left(\frac{5}{2} (c+d x)\right)+15 \sin \left(\frac{7}{2} (c+d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{420 d}","\frac{152 a^2 \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{2 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 a d}-\frac{4 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{38 a \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(735*Sin[(c + d*x)/2] + 175*Sin[(3*(c + d*x))/2] + 63*Sin[(5*(c + d*x))/2] + 15*Sin[(7*(c + d*x))/2]))/(420*d)","A",1
106,1,67,86,0.1000979,"\int \cos (c+d x) (a+a \cos (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^(3/2),x]","\frac{a \left(20 \sin \left(\frac{1}{2} (c+d x)\right)+5 \sin \left(\frac{3}{2} (c+d x)\right)+\sin \left(\frac{5}{2} (c+d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{10 d}","\frac{8 a^2 \sin (c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{5 d}+\frac{2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(20*Sin[(c + d*x)/2] + 5*Sin[(3*(c + d*x))/2] + Sin[(5*(c + d*x))/2]))/(10*d)","A",1
107,1,55,59,0.0642438,"\int (a+a \cos (c+d x))^{3/2} \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2),x]","\frac{a \left(9 \sin \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{3}{2} (c+d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{3 d}","\frac{8 a^2 \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a \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]*(9*Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2]))/(3*d)","A",1
108,1,65,66,0.0752264,"\int (a+a \cos (c+d x))^{3/2} \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x],x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\sqrt{2} \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{d}","\frac{2 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a^2 \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(Sqrt[2]*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sin[(c + d*x)/2]))/d","A",1
109,1,81,65,0.1119095,"\int (a+a \cos (c+d x))^{3/2} \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2,x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+3 \sqrt{2} \cos (c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{2 d}","\frac{3 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{a^2 \tan (c+d x)}{d \sqrt{a \cos (c+d x)+a}}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]*(3*Sqrt[2]*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x] + 2*Sin[(c + d*x)/2]))/(2*d)","A",1
110,1,97,106,0.2320545,"\int (a+a \cos (c+d x))^{3/2} \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3,x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(-3 \sin \left(\frac{1}{2} (c+d x)\right)+7 \sin \left(\frac{3}{2} (c+d x)\right)+7 \sqrt{2} \cos ^2(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{8 d}","\frac{7 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{7 a^2 \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^2*(7*Sqrt[2]*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^2 - 3*Sin[(c + d*x)/2] + 7*Sin[(3*(c + d*x))/2]))/(8*d)","A",1
111,1,110,144,0.3478176,"\int (a+a \cos (c+d x))^{3/2} \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^4,x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \sqrt{a (\cos (c+d x)+1)} \left(54 \sin \left(\frac{1}{2} (c+d x)\right)+11 \left(\sin \left(\frac{3}{2} (c+d x)\right)+3 \sin \left(\frac{5}{2} (c+d x)\right)\right)+66 \sqrt{2} \cos ^3(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{96 d}","\frac{11 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{11 a^2 \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{11 a^2 \tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sec[c + d*x]^3*(66*Sqrt[2]*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^3 + 54*Sin[(c + d*x)/2] + 11*(Sin[(3*(c + d*x))/2] + 3*Sin[(5*(c + d*x))/2])))/(96*d)","A",1
112,1,107,203,0.4977577,"\int \cos ^3(c+d x) (a+a \cos (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^3*(a + a*Cos[c + d*x])^(5/2),x]","\frac{a^2 \left(31878 \sin \left(\frac{1}{2} (c+d x)\right)+8778 \sin \left(\frac{3}{2} (c+d x)\right)+3465 \sin \left(\frac{5}{2} (c+d x)\right)+1287 \sin \left(\frac{7}{2} (c+d x)\right)+385 \sin \left(\frac{9}{2} (c+d x)\right)+63 \sin \left(\frac{11}{2} (c+d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{11088 d}","\frac{46 a^3 \sin (c+d x) \cos ^4(c+d x)}{99 d \sqrt{a \cos (c+d x)+a}}+\frac{710 a^3 \sin (c+d x) \cos ^3(c+d x)}{693 d \sqrt{a \cos (c+d x)+a}}+\frac{284 a^3 \sin (c+d x)}{99 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \cos ^4(c+d x) \sqrt{a \cos (c+d x)+a}}{11 d}-\frac{568 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{693 d}+\frac{284 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{231 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(31878*Sin[(c + d*x)/2] + 8778*Sin[(3*(c + d*x))/2] + 3465*Sin[(5*(c + d*x))/2] + 1287*Sin[(7*(c + d*x))/2] + 385*Sin[(9*(c + d*x))/2] + 63*Sin[(11*(c + d*x))/2]))/(11088*d)","A",1
113,1,95,146,0.2875263,"\int \cos ^2(c+d x) (a+a \cos (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^2*(a + a*Cos[c + d*x])^(5/2),x]","\frac{a^2 \left(8190 \sin \left(\frac{1}{2} (c+d x)\right)+2100 \sin \left(\frac{3}{2} (c+d x)\right)+756 \sin \left(\frac{5}{2} (c+d x)\right)+225 \sin \left(\frac{7}{2} (c+d x)\right)+35 \sin \left(\frac{9}{2} (c+d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{2520 d}","\frac{832 a^3 \sin (c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{208 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{315 d}+\frac{2 \sin (c+d x) (a \cos (c+d x)+a)^{7/2}}{9 a d}-\frac{4 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{63 d}+\frac{26 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{105 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(8190*Sin[(c + d*x)/2] + 2100*Sin[(3*(c + d*x))/2] + 756*Sin[(5*(c + d*x))/2] + 225*Sin[(7*(c + d*x))/2] + 35*Sin[(9*(c + d*x))/2]))/(2520*d)","A",1
114,1,84,116,0.2288778,"\int \cos (c+d x) (a+a \cos (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^(5/2),x]","\frac{a^2 \left(315 \sin \left(\frac{1}{2} (c+d x)\right)+77 \sin \left(\frac{3}{2} (c+d x)\right)+3 \left(7 \sin \left(\frac{5}{2} (c+d x)\right)+\sin \left(\frac{7}{2} (c+d x)\right)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{84 d}","\frac{64 a^3 \sin (c+d x)}{21 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{21 d}+\frac{2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{7 d}+\frac{2 \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(315*Sin[(c + d*x)/2] + 77*Sin[(3*(c + d*x))/2] + 3*(7*Sin[(5*(c + d*x))/2] + Sin[(7*(c + d*x))/2])))/(84*d)","A",1
115,1,71,89,0.1035212,"\int (a+a \cos (c+d x))^{5/2} \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2),x]","\frac{a^2 \left(150 \sin \left(\frac{1}{2} (c+d x)\right)+25 \sin \left(\frac{3}{2} (c+d x)\right)+3 \sin \left(\frac{5}{2} (c+d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{30 d}","\frac{64 a^3 \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 a \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]*(150*Sin[(c + d*x)/2] + 25*Sin[(3*(c + d*x))/2] + 3*Sin[(5*(c + d*x))/2]))/(30*d)","A",1
116,1,89,98,0.5095177,"\int (a+a \cos (c+d x))^{5/2} \sec (c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x],x]","\frac{2 a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(\sqrt{1-\cos (c+d x)} (\cos (c+d x)+8)+3 \tanh ^{-1}\left(\sqrt{1-\cos (c+d x)}\right)\right)}{3 d \sqrt{1-\cos (c+d x)}}","\frac{2 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{14 a^3 \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}",1,"(2*a^2*Sqrt[a*(1 + Cos[c + d*x])]*(3*ArcTanh[Sqrt[1 - Cos[c + d*x]]] + Sqrt[1 - Cos[c + d*x]]*(8 + Cos[c + d*x]))*Tan[(c + d*x)/2])/(3*d*Sqrt[1 - Cos[c + d*x]])","A",1
117,1,1547,92,36.2271684,"\int (a+a \cos (c+d x))^{5/2} \sec ^2(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^2,x]","-\frac{\left(\frac{5}{32}-\frac{5 i}{32}\right) \left(1+e^{i c}\right) \left(\sqrt{2}-(1-i) e^{\frac{i c}{2}}+(16-16 i) e^{\frac{3 i c}{2}+i d x}+(20+20 i) \sqrt{2} e^{2 i c+\frac{3 i d x}{2}}-(34-34 i) e^{\frac{5 i c}{2}+2 i d x}-(20+20 i) \sqrt{2} e^{3 i c+\frac{5 i d x}{2}}+(16-16 i) e^{\frac{7 i c}{2}+3 i d x}+(4+4 i) \sqrt{2} e^{4 i c+\frac{7 i d x}{2}}-(1-i) e^{\frac{9 i c}{2}+4 i d x}+8 i e^{\frac{1}{2} i (c+d x)}-16 \sqrt{2} e^{i (c+d x)}-40 i e^{\frac{3}{2} i (c+d x)}+34 \sqrt{2} e^{2 i (c+d x)}+40 i e^{\frac{5}{2} i (c+d x)}-16 \sqrt{2} e^{3 i (c+d x)}-8 i e^{\frac{7}{2} i (c+d x)}+\sqrt{2} e^{4 i (c+d x)}-(4+4 i) \sqrt{2} e^{\frac{1}{2} i (2 c+d x)}\right) x (a (\cos (c+d x)+1))^{5/2} \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{\left((-1-i)+\sqrt{2} e^{\frac{i c}{2}}\right) \left(-1+e^{i c}\right) \left(i-2 \sqrt{2} e^{\frac{1}{2} i (c+d x)}-4 i e^{i (c+d x)}+2 \sqrt{2} e^{\frac{3}{2} i (c+d x)}+i e^{2 i (c+d x)}\right)^2}-\frac{5 i \tan ^{-1}\left(\frac{\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sqrt{2} \sin \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}{\sqrt{2} \cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}\right) (a (\cos (c+d x)+1))^{5/2} \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 \sqrt{2} d}-\frac{5 i \tan ^{-1}\left(\frac{\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sqrt{2} \sin \left(\frac{c}{4}+\frac{d x}{4}\right)+\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}{\sqrt{2} \cos \left(\frac{c}{4}+\frac{d x}{4}\right)+\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}\right) (a (\cos (c+d x)+1))^{5/2} \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 \sqrt{2} d}-\frac{5 (a (\cos (c+d x)+1))^{5/2} \log \left(-\sqrt{2} \cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sqrt{2} \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+2\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 \sqrt{2} d}-\frac{5 (a (\cos (c+d x)+1))^{5/2} \log \left(\sqrt{2} \cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sqrt{2} \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+2\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 \sqrt{2} d}+\frac{\cos \left(\frac{d x}{2}\right) (a (\cos (c+d x)+1))^{5/2} \sin \left(\frac{c}{2}\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{5 (a (\cos (c+d x)+1))^{5/2} \csc \left(\frac{c}{2}\right) \left(-d x \cos \left(\frac{c}{2}\right)+\frac{4 i \sqrt{2} \tan ^{-1}\left(\frac{2 i \cos \left(\frac{c}{2}\right)-i \left(2 \sin \left(\frac{c}{2}\right)-\sqrt{2}\right) \tan \left(\frac{d x}{4}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}\right) \cos \left(\frac{c}{2}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}+2 \log \left(2 \cos \left(\frac{d x}{2}\right) \sin \left(\frac{c}{2}\right)+2 \cos \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+\sqrt{2}\right) \sin \left(\frac{c}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 \sqrt{2} d \left(4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)\right)}+\frac{\cos \left(\frac{c}{2}\right) (a (\cos (c+d x)+1))^{5/2} \sin \left(\frac{d x}{2}\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{5 i \tan ^{-1}\left(\frac{2 i \cos \left(\frac{c}{2}\right)-i \left(2 \sin \left(\frac{c}{2}\right)-\sqrt{2}\right) \tan \left(\frac{d x}{4}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}\right) (a (\cos (c+d x)+1))^{5/2} \cot \left(\frac{c}{2}\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d \sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}+\frac{(a (\cos (c+d x)+1))^{5/2} \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}-\frac{(a (\cos (c+d x)+1))^{5/2} \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}","\frac{5 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{a^3 \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \tan (c+d x) \sqrt{a \cos (c+d x)+a}}{d}",1,"((-5/32 + (5*I)/32)*(1 + E^(I*c))*(Sqrt[2] - (1 - I)*E^((I/2)*c) + (16 - 16*I)*E^(((3*I)/2)*c + I*d*x) + (20 + 20*I)*Sqrt[2]*E^((2*I)*c + ((3*I)/2)*d*x) - (34 - 34*I)*E^(((5*I)/2)*c + (2*I)*d*x) - (20 + 20*I)*Sqrt[2]*E^((3*I)*c + ((5*I)/2)*d*x) + (16 - 16*I)*E^(((7*I)/2)*c + (3*I)*d*x) + (4 + 4*I)*Sqrt[2]*E^((4*I)*c + ((7*I)/2)*d*x) - (1 - I)*E^(((9*I)/2)*c + (4*I)*d*x) + (8*I)*E^((I/2)*(c + d*x)) - 16*Sqrt[2]*E^(I*(c + d*x)) - (40*I)*E^(((3*I)/2)*(c + d*x)) + 34*Sqrt[2]*E^((2*I)*(c + d*x)) + (40*I)*E^(((5*I)/2)*(c + d*x)) - 16*Sqrt[2]*E^((3*I)*(c + d*x)) - (8*I)*E^(((7*I)/2)*(c + d*x)) + Sqrt[2]*E^((4*I)*(c + d*x)) - (4 + 4*I)*Sqrt[2]*E^((I/2)*(2*c + d*x)))*x*(a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5)/(((-1 - I) + Sqrt[2]*E^((I/2)*c))*(-1 + E^(I*c))*(I - 2*Sqrt[2]*E^((I/2)*(c + d*x)) - (4*I)*E^(I*(c + d*x)) + 2*Sqrt[2]*E^(((3*I)/2)*(c + d*x)) + I*E^((2*I)*(c + d*x)))^2) - (((5*I)/8)*ArcTan[(Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4] - Sqrt[2]*Sin[c/4 + (d*x)/4])/(-Cos[c/4 + (d*x)/4] + Sqrt[2]*Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])]*(a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5)/(Sqrt[2]*d) - (((5*I)/8)*ArcTan[(Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4] - Sqrt[2]*Sin[c/4 + (d*x)/4])/(Cos[c/4 + (d*x)/4] + Sqrt[2]*Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])]*(a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5)/(Sqrt[2]*d) - (5*(a*(1 + Cos[c + d*x]))^(5/2)*Log[2 - Sqrt[2]*Cos[c/2 + (d*x)/2] - Sqrt[2]*Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^5)/(16*Sqrt[2]*d) - (5*(a*(1 + Cos[c + d*x]))^(5/2)*Log[2 + Sqrt[2]*Cos[c/2 + (d*x)/2] - Sqrt[2]*Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^5)/(16*Sqrt[2]*d) + (Cos[(d*x)/2]*(a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5*Sin[c/2])/(2*d) - (((5*I)/4)*ArcTan[((2*I)*Cos[c/2] - I*(-Sqrt[2] + 2*Sin[c/2])*Tan[(d*x)/4])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]]*(a*(1 + Cos[c + d*x]))^(5/2)*Cot[c/2]*Sec[c/2 + (d*x)/2]^5)/(d*Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]) + (5*(a*(1 + Cos[c + d*x]))^(5/2)*Csc[c/2]*Sec[c/2 + (d*x)/2]^5*(-(d*x*Cos[c/2]) + 2*Log[Sqrt[2] + 2*Cos[(d*x)/2]*Sin[c/2] + 2*Cos[c/2]*Sin[(d*x)/2]]*Sin[c/2] + ((4*I)*Sqrt[2]*ArcTan[((2*I)*Cos[c/2] - I*(-Sqrt[2] + 2*Sin[c/2])*Tan[(d*x)/4])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]]*Cos[c/2])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]))/(4*Sqrt[2]*d*(4*Cos[c/2]^2 + 4*Sin[c/2]^2)) + (Cos[c/2]*(a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5*Sin[(d*x)/2])/(2*d) + ((a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5)/(8*d*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) - ((a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5)/(8*d*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","C",0
118,1,1693,106,36.0079337,"\int (a+a \cos (c+d x))^{5/2} \sec ^3(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^3,x]","-\frac{\left(\frac{19}{128}-\frac{19 i}{128}\right) \left(1+e^{i c}\right) \left(\sqrt{2}-(1-i) e^{\frac{i c}{2}}+(16-16 i) e^{\frac{3 i c}{2}+i d x}+(20+20 i) \sqrt{2} e^{2 i c+\frac{3 i d x}{2}}-(34-34 i) e^{\frac{5 i c}{2}+2 i d x}-(20+20 i) \sqrt{2} e^{3 i c+\frac{5 i d x}{2}}+(16-16 i) e^{\frac{7 i c}{2}+3 i d x}+(4+4 i) \sqrt{2} e^{4 i c+\frac{7 i d x}{2}}-(1-i) e^{\frac{9 i c}{2}+4 i d x}+8 i e^{\frac{1}{2} i (c+d x)}-16 \sqrt{2} e^{i (c+d x)}-40 i e^{\frac{3}{2} i (c+d x)}+34 \sqrt{2} e^{2 i (c+d x)}+40 i e^{\frac{5}{2} i (c+d x)}-16 \sqrt{2} e^{3 i (c+d x)}-8 i e^{\frac{7}{2} i (c+d x)}+\sqrt{2} e^{4 i (c+d x)}-(4+4 i) \sqrt{2} e^{\frac{1}{2} i (2 c+d x)}\right) x (a (\cos (c+d x)+1))^{5/2} \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{\left((-1-i)+\sqrt{2} e^{\frac{i c}{2}}\right) \left(-1+e^{i c}\right) \left(i-2 \sqrt{2} e^{\frac{1}{2} i (c+d x)}-4 i e^{i (c+d x)}+2 \sqrt{2} e^{\frac{3}{2} i (c+d x)}+i e^{2 i (c+d x)}\right)^2}-\frac{19 i \tan ^{-1}\left(\frac{\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sqrt{2} \sin \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}{\sqrt{2} \cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}\right) (a (\cos (c+d x)+1))^{5/2} \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 \sqrt{2} d}-\frac{19 i \tan ^{-1}\left(\frac{\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sqrt{2} \sin \left(\frac{c}{4}+\frac{d x}{4}\right)+\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}{\sqrt{2} \cos \left(\frac{c}{4}+\frac{d x}{4}\right)+\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}\right) (a (\cos (c+d x)+1))^{5/2} \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 \sqrt{2} d}-\frac{19 (a (\cos (c+d x)+1))^{5/2} \log \left(-\sqrt{2} \cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sqrt{2} \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+2\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{64 \sqrt{2} d}-\frac{19 (a (\cos (c+d x)+1))^{5/2} \log \left(\sqrt{2} \cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sqrt{2} \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+2\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{64 \sqrt{2} d}+\frac{19 (a (\cos (c+d x)+1))^{5/2} \csc \left(\frac{c}{2}\right) \left(-d x \cos \left(\frac{c}{2}\right)+\frac{4 i \sqrt{2} \tan ^{-1}\left(\frac{2 i \cos \left(\frac{c}{2}\right)-i \left(2 \sin \left(\frac{c}{2}\right)-\sqrt{2}\right) \tan \left(\frac{d x}{4}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}\right) \cos \left(\frac{c}{2}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}+2 \log \left(2 \cos \left(\frac{d x}{2}\right) \sin \left(\frac{c}{2}\right)+2 \cos \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+\sqrt{2}\right) \sin \left(\frac{c}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 \sqrt{2} d \left(4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)\right)}-\frac{19 i \tan ^{-1}\left(\frac{2 i \cos \left(\frac{c}{2}\right)-i \left(2 \sin \left(\frac{c}{2}\right)-\sqrt{2}\right) \tan \left(\frac{d x}{4}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}\right) (a (\cos (c+d x)+1))^{5/2} \cot \left(\frac{c}{2}\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 d \sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}+\frac{(a (\cos (c+d x)+1))^{5/2} \left(11 \cos \left(\frac{c}{2}\right)-9 \sin \left(\frac{c}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{(a (\cos (c+d x)+1))^{5/2} \left(-11 \cos \left(\frac{c}{2}\right)-9 \sin \left(\frac{c}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{(a (\cos (c+d x)+1))^{5/2} \sin \left(\frac{d x}{2}\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{(a (\cos (c+d x)+1))^{5/2} \sin \left(\frac{d x}{2}\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 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{19 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{9 a^3 \tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \tan (c+d x) \sec (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d}",1,"((-19/128 + (19*I)/128)*(1 + E^(I*c))*(Sqrt[2] - (1 - I)*E^((I/2)*c) + (16 - 16*I)*E^(((3*I)/2)*c + I*d*x) + (20 + 20*I)*Sqrt[2]*E^((2*I)*c + ((3*I)/2)*d*x) - (34 - 34*I)*E^(((5*I)/2)*c + (2*I)*d*x) - (20 + 20*I)*Sqrt[2]*E^((3*I)*c + ((5*I)/2)*d*x) + (16 - 16*I)*E^(((7*I)/2)*c + (3*I)*d*x) + (4 + 4*I)*Sqrt[2]*E^((4*I)*c + ((7*I)/2)*d*x) - (1 - I)*E^(((9*I)/2)*c + (4*I)*d*x) + (8*I)*E^((I/2)*(c + d*x)) - 16*Sqrt[2]*E^(I*(c + d*x)) - (40*I)*E^(((3*I)/2)*(c + d*x)) + 34*Sqrt[2]*E^((2*I)*(c + d*x)) + (40*I)*E^(((5*I)/2)*(c + d*x)) - 16*Sqrt[2]*E^((3*I)*(c + d*x)) - (8*I)*E^(((7*I)/2)*(c + d*x)) + Sqrt[2]*E^((4*I)*(c + d*x)) - (4 + 4*I)*Sqrt[2]*E^((I/2)*(2*c + d*x)))*x*(a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5)/(((-1 - I) + Sqrt[2]*E^((I/2)*c))*(-1 + E^(I*c))*(I - 2*Sqrt[2]*E^((I/2)*(c + d*x)) - (4*I)*E^(I*(c + d*x)) + 2*Sqrt[2]*E^(((3*I)/2)*(c + d*x)) + I*E^((2*I)*(c + d*x)))^2) - (((19*I)/32)*ArcTan[(Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4] - Sqrt[2]*Sin[c/4 + (d*x)/4])/(-Cos[c/4 + (d*x)/4] + Sqrt[2]*Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])]*(a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5)/(Sqrt[2]*d) - (((19*I)/32)*ArcTan[(Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4] - Sqrt[2]*Sin[c/4 + (d*x)/4])/(Cos[c/4 + (d*x)/4] + Sqrt[2]*Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])]*(a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5)/(Sqrt[2]*d) - (19*(a*(1 + Cos[c + d*x]))^(5/2)*Log[2 - Sqrt[2]*Cos[c/2 + (d*x)/2] - Sqrt[2]*Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^5)/(64*Sqrt[2]*d) - (19*(a*(1 + Cos[c + d*x]))^(5/2)*Log[2 + Sqrt[2]*Cos[c/2 + (d*x)/2] - Sqrt[2]*Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^5)/(64*Sqrt[2]*d) - (((19*I)/16)*ArcTan[((2*I)*Cos[c/2] - I*(-Sqrt[2] + 2*Sin[c/2])*Tan[(d*x)/4])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]]*(a*(1 + Cos[c + d*x]))^(5/2)*Cot[c/2]*Sec[c/2 + (d*x)/2]^5)/(d*Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]) + (19*(a*(1 + Cos[c + d*x]))^(5/2)*Csc[c/2]*Sec[c/2 + (d*x)/2]^5*(-(d*x*Cos[c/2]) + 2*Log[Sqrt[2] + 2*Cos[(d*x)/2]*Sin[c/2] + 2*Cos[c/2]*Sin[(d*x)/2]]*Sin[c/2] + ((4*I)*Sqrt[2]*ArcTan[((2*I)*Cos[c/2] - I*(-Sqrt[2] + 2*Sin[c/2])*Tan[(d*x)/4])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]]*Cos[c/2])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]))/(16*Sqrt[2]*d*(4*Cos[c/2]^2 + 4*Sin[c/2]^2)) + ((a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5*Sin[(d*x)/2])/(16*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + ((a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5*(11*Cos[c/2] - 9*Sin[c/2]))/(32*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + ((a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5*Sin[(d*x)/2])/(16*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + ((a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5*(-11*Cos[c/2] - 9*Sin[c/2]))/(32*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","C",0
119,1,1825,144,36.1125435,"\int (a+a \cos (c+d x))^{5/2} \sec ^4(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^4,x]","-\frac{\left(\frac{25}{256}-\frac{25 i}{256}\right) \left(1+e^{i c}\right) \left(\sqrt{2}-(1-i) e^{\frac{i c}{2}}+(16-16 i) e^{\frac{3 i c}{2}+i d x}+(20+20 i) \sqrt{2} e^{2 i c+\frac{3 i d x}{2}}-(34-34 i) e^{\frac{5 i c}{2}+2 i d x}-(20+20 i) \sqrt{2} e^{3 i c+\frac{5 i d x}{2}}+(16-16 i) e^{\frac{7 i c}{2}+3 i d x}+(4+4 i) \sqrt{2} e^{4 i c+\frac{7 i d x}{2}}-(1-i) e^{\frac{9 i c}{2}+4 i d x}+8 i e^{\frac{1}{2} i (c+d x)}-16 \sqrt{2} e^{i (c+d x)}-40 i e^{\frac{3}{2} i (c+d x)}+34 \sqrt{2} e^{2 i (c+d x)}+40 i e^{\frac{5}{2} i (c+d x)}-16 \sqrt{2} e^{3 i (c+d x)}-8 i e^{\frac{7}{2} i (c+d x)}+\sqrt{2} e^{4 i (c+d x)}-(4+4 i) \sqrt{2} e^{\frac{1}{2} i (2 c+d x)}\right) x (a (\cos (c+d x)+1))^{5/2} \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{\left((-1-i)+\sqrt{2} e^{\frac{i c}{2}}\right) \left(-1+e^{i c}\right) \left(i-2 \sqrt{2} e^{\frac{1}{2} i (c+d x)}-4 i e^{i (c+d x)}+2 \sqrt{2} e^{\frac{3}{2} i (c+d x)}+i e^{2 i (c+d x)}\right)^2}-\frac{25 i \tan ^{-1}\left(\frac{\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sqrt{2} \sin \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}{\sqrt{2} \cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}\right) (a (\cos (c+d x)+1))^{5/2} \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{64 \sqrt{2} d}-\frac{25 i \tan ^{-1}\left(\frac{\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sqrt{2} \sin \left(\frac{c}{4}+\frac{d x}{4}\right)+\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}{\sqrt{2} \cos \left(\frac{c}{4}+\frac{d x}{4}\right)+\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}\right) (a (\cos (c+d x)+1))^{5/2} \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{64 \sqrt{2} d}-\frac{25 (a (\cos (c+d x)+1))^{5/2} \log \left(-\sqrt{2} \cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sqrt{2} \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+2\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{128 \sqrt{2} d}-\frac{25 (a (\cos (c+d x)+1))^{5/2} \log \left(\sqrt{2} \cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sqrt{2} \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+2\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{128 \sqrt{2} d}+\frac{25 (a (\cos (c+d x)+1))^{5/2} \csc \left(\frac{c}{2}\right) \left(-d x \cos \left(\frac{c}{2}\right)+\frac{4 i \sqrt{2} \tan ^{-1}\left(\frac{2 i \cos \left(\frac{c}{2}\right)-i \left(2 \sin \left(\frac{c}{2}\right)-\sqrt{2}\right) \tan \left(\frac{d x}{4}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}\right) \cos \left(\frac{c}{2}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}+2 \log \left(2 \cos \left(\frac{d x}{2}\right) \sin \left(\frac{c}{2}\right)+2 \cos \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+\sqrt{2}\right) \sin \left(\frac{c}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 \sqrt{2} d \left(4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)\right)}-\frac{25 i \tan ^{-1}\left(\frac{2 i \cos \left(\frac{c}{2}\right)-i \left(2 \sin \left(\frac{c}{2}\right)-\sqrt{2}\right) \tan \left(\frac{d x}{4}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}\right) (a (\cos (c+d x)+1))^{5/2} \cot \left(\frac{c}{2}\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d \sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}+\frac{5 (a (\cos (c+d x)+1))^{5/2} \left(5 \cos \left(\frac{c}{2}\right)-3 \sin \left(\frac{c}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{64 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{5 (a (\cos (c+d x)+1))^{5/2} \left(5 \cos \left(\frac{c}{2}\right)+3 \sin \left(\frac{c}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{64 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{5 (a (\cos (c+d x)+1))^{5/2} \sin \left(\frac{d x}{2}\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 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{5 (a (\cos (c+d x)+1))^{5/2} \sin \left(\frac{d x}{2}\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{(a (\cos (c+d x)+1))^{5/2} \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{48 d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}-\frac{(a (\cos (c+d x)+1))^{5/2} \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{48 d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}","\frac{25 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{25 a^3 \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{13 a^3 \tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}",1,"((-25/256 + (25*I)/256)*(1 + E^(I*c))*(Sqrt[2] - (1 - I)*E^((I/2)*c) + (16 - 16*I)*E^(((3*I)/2)*c + I*d*x) + (20 + 20*I)*Sqrt[2]*E^((2*I)*c + ((3*I)/2)*d*x) - (34 - 34*I)*E^(((5*I)/2)*c + (2*I)*d*x) - (20 + 20*I)*Sqrt[2]*E^((3*I)*c + ((5*I)/2)*d*x) + (16 - 16*I)*E^(((7*I)/2)*c + (3*I)*d*x) + (4 + 4*I)*Sqrt[2]*E^((4*I)*c + ((7*I)/2)*d*x) - (1 - I)*E^(((9*I)/2)*c + (4*I)*d*x) + (8*I)*E^((I/2)*(c + d*x)) - 16*Sqrt[2]*E^(I*(c + d*x)) - (40*I)*E^(((3*I)/2)*(c + d*x)) + 34*Sqrt[2]*E^((2*I)*(c + d*x)) + (40*I)*E^(((5*I)/2)*(c + d*x)) - 16*Sqrt[2]*E^((3*I)*(c + d*x)) - (8*I)*E^(((7*I)/2)*(c + d*x)) + Sqrt[2]*E^((4*I)*(c + d*x)) - (4 + 4*I)*Sqrt[2]*E^((I/2)*(2*c + d*x)))*x*(a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5)/(((-1 - I) + Sqrt[2]*E^((I/2)*c))*(-1 + E^(I*c))*(I - 2*Sqrt[2]*E^((I/2)*(c + d*x)) - (4*I)*E^(I*(c + d*x)) + 2*Sqrt[2]*E^(((3*I)/2)*(c + d*x)) + I*E^((2*I)*(c + d*x)))^2) - (((25*I)/64)*ArcTan[(Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4] - Sqrt[2]*Sin[c/4 + (d*x)/4])/(-Cos[c/4 + (d*x)/4] + Sqrt[2]*Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])]*(a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5)/(Sqrt[2]*d) - (((25*I)/64)*ArcTan[(Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4] - Sqrt[2]*Sin[c/4 + (d*x)/4])/(Cos[c/4 + (d*x)/4] + Sqrt[2]*Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])]*(a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5)/(Sqrt[2]*d) - (25*(a*(1 + Cos[c + d*x]))^(5/2)*Log[2 - Sqrt[2]*Cos[c/2 + (d*x)/2] - Sqrt[2]*Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^5)/(128*Sqrt[2]*d) - (25*(a*(1 + Cos[c + d*x]))^(5/2)*Log[2 + Sqrt[2]*Cos[c/2 + (d*x)/2] - Sqrt[2]*Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^5)/(128*Sqrt[2]*d) - (((25*I)/32)*ArcTan[((2*I)*Cos[c/2] - I*(-Sqrt[2] + 2*Sin[c/2])*Tan[(d*x)/4])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]]*(a*(1 + Cos[c + d*x]))^(5/2)*Cot[c/2]*Sec[c/2 + (d*x)/2]^5)/(d*Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]) + (25*(a*(1 + Cos[c + d*x]))^(5/2)*Csc[c/2]*Sec[c/2 + (d*x)/2]^5*(-(d*x*Cos[c/2]) + 2*Log[Sqrt[2] + 2*Cos[(d*x)/2]*Sin[c/2] + 2*Cos[c/2]*Sin[(d*x)/2]]*Sin[c/2] + ((4*I)*Sqrt[2]*ArcTan[((2*I)*Cos[c/2] - I*(-Sqrt[2] + 2*Sin[c/2])*Tan[(d*x)/4])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]]*Cos[c/2])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]))/(32*Sqrt[2]*d*(4*Cos[c/2]^2 + 4*Sin[c/2]^2)) + ((a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5)/(48*d*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + (5*(a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5*Sin[(d*x)/2])/(32*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (5*(a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5*(5*Cos[c/2] - 3*Sin[c/2]))/(64*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) - ((a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5)/(48*d*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (5*(a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5*Sin[(d*x)/2])/(32*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) - (5*(a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5*(5*Cos[c/2] + 3*Sin[c/2]))/(64*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","C",0
120,1,2069,182,35.8402344,"\int (a+a \cos (c+d x))^{5/2} \sec ^5(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^5,x]","\text{Result too large to show}","\frac{163 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{163 a^3 \tan (c+d x)}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{17 a^3 \tan (c+d x) \sec ^2(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{163 a^3 \tan (c+d x) \sec (c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \tan (c+d x) \sec ^3(c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}",1,"((-163/2048 + (163*I)/2048)*(1 + E^(I*c))*(Sqrt[2] - (1 - I)*E^((I/2)*c) + (16 - 16*I)*E^(((3*I)/2)*c + I*d*x) + (20 + 20*I)*Sqrt[2]*E^((2*I)*c + ((3*I)/2)*d*x) - (34 - 34*I)*E^(((5*I)/2)*c + (2*I)*d*x) - (20 + 20*I)*Sqrt[2]*E^((3*I)*c + ((5*I)/2)*d*x) + (16 - 16*I)*E^(((7*I)/2)*c + (3*I)*d*x) + (4 + 4*I)*Sqrt[2]*E^((4*I)*c + ((7*I)/2)*d*x) - (1 - I)*E^(((9*I)/2)*c + (4*I)*d*x) + (8*I)*E^((I/2)*(c + d*x)) - 16*Sqrt[2]*E^(I*(c + d*x)) - (40*I)*E^(((3*I)/2)*(c + d*x)) + 34*Sqrt[2]*E^((2*I)*(c + d*x)) + (40*I)*E^(((5*I)/2)*(c + d*x)) - 16*Sqrt[2]*E^((3*I)*(c + d*x)) - (8*I)*E^(((7*I)/2)*(c + d*x)) + Sqrt[2]*E^((4*I)*(c + d*x)) - (4 + 4*I)*Sqrt[2]*E^((I/2)*(2*c + d*x)))*x*(a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5)/(((-1 - I) + Sqrt[2]*E^((I/2)*c))*(-1 + E^(I*c))*(I - 2*Sqrt[2]*E^((I/2)*(c + d*x)) - (4*I)*E^(I*(c + d*x)) + 2*Sqrt[2]*E^(((3*I)/2)*(c + d*x)) + I*E^((2*I)*(c + d*x)))^2) - (((163*I)/512)*ArcTan[(Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4] - Sqrt[2]*Sin[c/4 + (d*x)/4])/(-Cos[c/4 + (d*x)/4] + Sqrt[2]*Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])]*(a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5)/(Sqrt[2]*d) - (((163*I)/512)*ArcTan[(Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4] - Sqrt[2]*Sin[c/4 + (d*x)/4])/(Cos[c/4 + (d*x)/4] + Sqrt[2]*Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])]*(a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5)/(Sqrt[2]*d) - (163*(a*(1 + Cos[c + d*x]))^(5/2)*Log[2 - Sqrt[2]*Cos[c/2 + (d*x)/2] - Sqrt[2]*Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^5)/(1024*Sqrt[2]*d) - (163*(a*(1 + Cos[c + d*x]))^(5/2)*Log[2 + Sqrt[2]*Cos[c/2 + (d*x)/2] - Sqrt[2]*Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^5)/(1024*Sqrt[2]*d) - (((163*I)/256)*ArcTan[((2*I)*Cos[c/2] - I*(-Sqrt[2] + 2*Sin[c/2])*Tan[(d*x)/4])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]]*(a*(1 + Cos[c + d*x]))^(5/2)*Cot[c/2]*Sec[c/2 + (d*x)/2]^5)/(d*Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]) + (163*(a*(1 + Cos[c + d*x]))^(5/2)*Csc[c/2]*Sec[c/2 + (d*x)/2]^5*(-(d*x*Cos[c/2]) + 2*Log[Sqrt[2] + 2*Cos[(d*x)/2]*Sin[c/2] + 2*Cos[c/2]*Sin[(d*x)/2]]*Sin[c/2] + ((4*I)*Sqrt[2]*ArcTan[((2*I)*Cos[c/2] - I*(-Sqrt[2] + 2*Sin[c/2])*Tan[(d*x)/4])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]]*Cos[c/2])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]))/(256*Sqrt[2]*d*(4*Cos[c/2]^2 + 4*Sin[c/2]^2)) + ((a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5*Sin[(d*x)/2])/(64*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^4) + ((a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5*(23*Cos[c/2] - 17*Sin[c/2]))/(384*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + (43*(a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5*Sin[(d*x)/2])/(256*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + ((a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5*(163*Cos[c/2] - 77*Sin[c/2]))/(512*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + ((a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5*Sin[(d*x)/2])/(64*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^4) + ((a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5*(-23*Cos[c/2] - 17*Sin[c/2]))/(384*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (43*(a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5*Sin[(d*x)/2])/(256*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + ((a*(1 + Cos[c + d*x]))^(5/2)*Sec[c/2 + (d*x)/2]^5*(-163*Cos[c/2] - 77*Sin[c/2]))/(512*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","C",0
121,1,83,119,0.2399637,"\int (a+a \cos (c+d x))^{7/2} \, dx","Integrate[(a + a*Cos[c + d*x])^(7/2),x]","\frac{a^3 \left(1225 \sin \left(\frac{1}{2} (c+d x)\right)+245 \sin \left(\frac{3}{2} (c+d x)\right)+49 \sin \left(\frac{5}{2} (c+d x)\right)+5 \sin \left(\frac{7}{2} (c+d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{140 d}","\frac{256 a^4 \sin (c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{64 a^3 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{35 d}+\frac{24 a^2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 a \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}",1,"(a^3*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(1225*Sin[(c + d*x)/2] + 245*Sin[(3*(c + d*x))/2] + 49*Sin[(5*(c + d*x))/2] + 5*Sin[(7*(c + d*x))/2]))/(140*d)","A",1
122,1,130,174,0.1994224,"\int \frac{\cos ^4(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]^4/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(-525 \sin \left(\frac{1}{2} (c+d x)\right)+175 \sin \left(\frac{3}{2} (c+d x)\right)-21 \sin \left(\frac{5}{2} (c+d x)\right)+15 \sin \left(\frac{7}{2} (c+d x)\right)-420 \log \left(\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)\right)+420 \log \left(\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)\right)\right)}{210 d \sqrt{a (\cos (c+d x)+1)}}","\frac{2 \sin (c+d x) \cos ^3(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}-\frac{2 \sin (c+d x) \cos ^2(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{62 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{105 a d}-\frac{148 \sin (c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(Cos[(c + d*x)/2]*(-420*Log[Cos[(c + d*x)/4] - Sin[(c + d*x)/4]] + 420*Log[Cos[(c + d*x)/4] + Sin[(c + d*x)/4]] - 525*Sin[(c + d*x)/2] + 175*Sin[(3*(c + d*x))/2] - 21*Sin[(5*(c + d*x))/2] + 15*Sin[(7*(c + d*x))/2]))/(210*d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
123,1,118,140,0.167181,"\int \frac{\cos ^3(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]^3/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(60 \sin \left(\frac{1}{2} (c+d x)\right)-5 \sin \left(\frac{3}{2} (c+d x)\right)+3 \sin \left(\frac{5}{2} (c+d x)\right)+30 \log \left(\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)\right)-30 \log \left(\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)\right)\right)}{15 d \sqrt{a (\cos (c+d x)+1)}}","\frac{2 \sin (c+d x) \cos ^2(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}-\frac{2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 a d}+\frac{28 \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(Cos[(c + d*x)/2]*(30*Log[Cos[(c + d*x)/4] - Sin[(c + d*x)/4]] - 30*Log[Cos[(c + d*x)/4] + Sin[(c + d*x)/4]] + 60*Sin[(c + d*x)/2] - 5*Sin[(3*(c + d*x))/2] + 3*Sin[(5*(c + d*x))/2]))/(15*d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
124,1,104,104,0.1124273,"\int \frac{\cos ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]^2/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(-3 \sin \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{3}{2} (c+d x)\right)-3 \log \left(\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)\right)+3 \log \left(\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)\right)\right)}{3 d \sqrt{a (\cos (c+d x)+1)}}","\frac{2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 a d}-\frac{4 \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} \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*Cos[(c + d*x)/2]*(-3*Log[Cos[(c + d*x)/4] - Sin[(c + d*x)/4]] + 3*Log[Cos[(c + d*x)/4] + Sin[(c + d*x)/4]] - 3*Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2]))/(3*d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
125,1,53,73,0.0449698,"\int \frac{\cos (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]/Sqrt[a + a*Cos[c + d*x]],x]","-\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(\tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{d \sqrt{a (\cos (c+d x)+1)}}","\frac{2 \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} \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*Cos[(c + d*x)/2]*(ArcTanh[Sin[(c + d*x)/2]] - 2*Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
126,1,40,46,0.012083,"\int \frac{1}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[1/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d \sqrt{a (\cos (c+d x)+1)}}","\frac{\sqrt{2} \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*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
127,1,65,85,0.052203,"\int \frac{\sec (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[Sec[c + d*x]/Sqrt[a + a*Cos[c + d*x]],x]","-\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(\tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\sqrt{2} \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 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \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*(ArcTanh[Sin[(c + d*x)/2]] - Sqrt[2]*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]])*Cos[(c + d*x)/2])/(d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
128,1,1540,108,27.5634662,"\int \frac{\sec ^2(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[Sec[c + d*x]^2/Sqrt[a + a*Cos[c + d*x]],x]","-\frac{2 \log \left(\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{a (\cos (c+d x)+1)}}+\frac{2 \log \left(\cos \left(\frac{c}{4}+\frac{d x}{4}\right)+\sin \left(\frac{c}{4}+\frac{d x}{4}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{a (\cos (c+d x)+1)}}+\frac{\log \left(-\sqrt{2} \cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sqrt{2} \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+2\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sqrt{2} d \sqrt{a (\cos (c+d x)+1)}}+\frac{\log \left(\sqrt{2} \cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sqrt{2} \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+2\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sqrt{2} d \sqrt{a (\cos (c+d x)+1)}}-\frac{\sqrt{2} \csc \left(\frac{c}{2}\right) \left(-d x \cos \left(\frac{c}{2}\right)+\frac{4 i \sqrt{2} \tan ^{-1}\left(\frac{2 i \cos \left(\frac{c}{2}\right)-i \left(2 \sin \left(\frac{c}{2}\right)-\sqrt{2}\right) \tan \left(\frac{d x}{4}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}\right) \cos \left(\frac{c}{2}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}+2 \log \left(2 \cos \left(\frac{d x}{2}\right) \sin \left(\frac{c}{2}\right)+2 \cos \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+\sqrt{2}\right) \sin \left(\frac{c}{2}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{a (\cos (c+d x)+1)} \left(4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)\right)}+\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(1+e^{i c}\right) \left(\sqrt{2}-(1-i) e^{\frac{i c}{2}}+(16-16 i) e^{\frac{3 i c}{2}+i d x}+(20+20 i) \sqrt{2} e^{2 i c+\frac{3 i d x}{2}}-(34-34 i) e^{\frac{5 i c}{2}+2 i d x}-(20+20 i) \sqrt{2} e^{3 i c+\frac{5 i d x}{2}}+(16-16 i) e^{\frac{7 i c}{2}+3 i d x}+(4+4 i) \sqrt{2} e^{4 i c+\frac{7 i d x}{2}}-(1-i) e^{\frac{9 i c}{2}+4 i d x}+8 i e^{\frac{1}{2} i (c+d x)}-16 \sqrt{2} e^{i (c+d x)}-40 i e^{\frac{3}{2} i (c+d x)}+34 \sqrt{2} e^{2 i (c+d x)}+40 i e^{\frac{5}{2} i (c+d x)}-16 \sqrt{2} e^{3 i (c+d x)}-8 i e^{\frac{7}{2} i (c+d x)}+\sqrt{2} e^{4 i (c+d x)}-(4+4 i) \sqrt{2} e^{\frac{1}{2} i (2 c+d x)}\right) x \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{\left((-1-i)+\sqrt{2} e^{\frac{i c}{2}}\right) \left(-1+e^{i c}\right) \left(i-2 \sqrt{2} e^{\frac{1}{2} i (c+d x)}-4 i e^{i (c+d x)}+2 \sqrt{2} e^{\frac{3}{2} i (c+d x)}+i e^{2 i (c+d x)}\right)^2 \sqrt{a (\cos (c+d x)+1)}}+\frac{i \tan ^{-1}\left(\frac{\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sqrt{2} \sin \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}{\sqrt{2} \cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{2} d \sqrt{a (\cos (c+d x)+1)}}+\frac{i \tan ^{-1}\left(\frac{\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sqrt{2} \sin \left(\frac{c}{4}+\frac{d x}{4}\right)+\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}{\sqrt{2} \cos \left(\frac{c}{4}+\frac{d x}{4}\right)+\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{2} d \sqrt{a (\cos (c+d x)+1)}}+\frac{2 i \tan ^{-1}\left(\frac{2 i \cos \left(\frac{c}{2}\right)-i \left(2 \sin \left(\frac{c}{2}\right)-\sqrt{2}\right) \tan \left(\frac{d x}{4}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}\right) \cot \left(\frac{c}{2}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{a (\cos (c+d x)+1)} \sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}+\frac{\cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{a (\cos (c+d x)+1)} \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}-\frac{\cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{a (\cos (c+d x)+1)} \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}","\frac{\tan (c+d x)}{d \sqrt{a \cos (c+d x)+a}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"((1/4 - I/4)*(1 + E^(I*c))*(Sqrt[2] - (1 - I)*E^((I/2)*c) + (16 - 16*I)*E^(((3*I)/2)*c + I*d*x) + (20 + 20*I)*Sqrt[2]*E^((2*I)*c + ((3*I)/2)*d*x) - (34 - 34*I)*E^(((5*I)/2)*c + (2*I)*d*x) - (20 + 20*I)*Sqrt[2]*E^((3*I)*c + ((5*I)/2)*d*x) + (16 - 16*I)*E^(((7*I)/2)*c + (3*I)*d*x) + (4 + 4*I)*Sqrt[2]*E^((4*I)*c + ((7*I)/2)*d*x) - (1 - I)*E^(((9*I)/2)*c + (4*I)*d*x) + (8*I)*E^((I/2)*(c + d*x)) - 16*Sqrt[2]*E^(I*(c + d*x)) - (40*I)*E^(((3*I)/2)*(c + d*x)) + 34*Sqrt[2]*E^((2*I)*(c + d*x)) + (40*I)*E^(((5*I)/2)*(c + d*x)) - 16*Sqrt[2]*E^((3*I)*(c + d*x)) - (8*I)*E^(((7*I)/2)*(c + d*x)) + Sqrt[2]*E^((4*I)*(c + d*x)) - (4 + 4*I)*Sqrt[2]*E^((I/2)*(2*c + d*x)))*x*Cos[c/2 + (d*x)/2])/(((-1 - I) + Sqrt[2]*E^((I/2)*c))*(-1 + E^(I*c))*(I - 2*Sqrt[2]*E^((I/2)*(c + d*x)) - (4*I)*E^(I*(c + d*x)) + 2*Sqrt[2]*E^(((3*I)/2)*(c + d*x)) + I*E^((2*I)*(c + d*x)))^2*Sqrt[a*(1 + Cos[c + d*x])]) + (I*ArcTan[(Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4] - Sqrt[2]*Sin[c/4 + (d*x)/4])/(-Cos[c/4 + (d*x)/4] + Sqrt[2]*Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])]*Cos[c/2 + (d*x)/2])/(Sqrt[2]*d*Sqrt[a*(1 + Cos[c + d*x])]) + (I*ArcTan[(Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4] - Sqrt[2]*Sin[c/4 + (d*x)/4])/(Cos[c/4 + (d*x)/4] + Sqrt[2]*Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])]*Cos[c/2 + (d*x)/2])/(Sqrt[2]*d*Sqrt[a*(1 + Cos[c + d*x])]) - (2*Cos[c/2 + (d*x)/2]*Log[Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4]])/(d*Sqrt[a*(1 + Cos[c + d*x])]) + (2*Cos[c/2 + (d*x)/2]*Log[Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4]])/(d*Sqrt[a*(1 + Cos[c + d*x])]) + (Cos[c/2 + (d*x)/2]*Log[2 - Sqrt[2]*Cos[c/2 + (d*x)/2] - Sqrt[2]*Sin[c/2 + (d*x)/2]])/(2*Sqrt[2]*d*Sqrt[a*(1 + Cos[c + d*x])]) + (Cos[c/2 + (d*x)/2]*Log[2 + Sqrt[2]*Cos[c/2 + (d*x)/2] - Sqrt[2]*Sin[c/2 + (d*x)/2]])/(2*Sqrt[2]*d*Sqrt[a*(1 + Cos[c + d*x])]) + ((2*I)*ArcTan[((2*I)*Cos[c/2] - I*(-Sqrt[2] + 2*Sin[c/2])*Tan[(d*x)/4])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]]*Cos[c/2 + (d*x)/2]*Cot[c/2])/(d*Sqrt[a*(1 + Cos[c + d*x])]*Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]) - (Sqrt[2]*Cos[c/2 + (d*x)/2]*Csc[c/2]*(-(d*x*Cos[c/2]) + 2*Log[Sqrt[2] + 2*Cos[(d*x)/2]*Sin[c/2] + 2*Cos[c/2]*Sin[(d*x)/2]]*Sin[c/2] + ((4*I)*Sqrt[2]*ArcTan[((2*I)*Cos[c/2] - I*(-Sqrt[2] + 2*Sin[c/2])*Tan[(d*x)/4])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]]*Cos[c/2])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]))/(d*Sqrt[a*(1 + Cos[c + d*x])]*(4*Cos[c/2]^2 + 4*Sin[c/2]^2)) + Cos[c/2 + (d*x)/2]/(d*Sqrt[a*(1 + Cos[c + d*x])]*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) - Cos[c/2 + (d*x)/2]/(d*Sqrt[a*(1 + Cos[c + d*x])]*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","C",0
129,1,1791,147,31.172278,"\int \frac{\sec ^3(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[Sec[c + d*x]^3/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \log \left(\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{a (\cos (c+d x)+1)}}-\frac{2 \log \left(\cos \left(\frac{c}{4}+\frac{d x}{4}\right)+\sin \left(\frac{c}{4}+\frac{d x}{4}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{a (\cos (c+d x)+1)}}-\frac{7 \log \left(-\sqrt{2} \cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sqrt{2} \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+2\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{8 \sqrt{2} d \sqrt{a (\cos (c+d x)+1)}}-\frac{7 \log \left(\sqrt{2} \cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sqrt{2} \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+2\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{8 \sqrt{2} d \sqrt{a (\cos (c+d x)+1)}}+\frac{7 \csc \left(\frac{c}{2}\right) \left(-d x \cos \left(\frac{c}{2}\right)+\frac{4 i \sqrt{2} \tan ^{-1}\left(\frac{2 i \cos \left(\frac{c}{2}\right)-i \left(2 \sin \left(\frac{c}{2}\right)-\sqrt{2}\right) \tan \left(\frac{d x}{4}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}\right) \cos \left(\frac{c}{2}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}+2 \log \left(2 \cos \left(\frac{d x}{2}\right) \sin \left(\frac{c}{2}\right)+2 \cos \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+\sqrt{2}\right) \sin \left(\frac{c}{2}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sqrt{2} d \sqrt{a (\cos (c+d x)+1)} \left(4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)\right)}-\frac{\left(\frac{7}{16}-\frac{7 i}{16}\right) \left(1+e^{i c}\right) \left(\sqrt{2}-(1-i) e^{\frac{i c}{2}}+(16-16 i) e^{\frac{3 i c}{2}+i d x}+(20+20 i) \sqrt{2} e^{2 i c+\frac{3 i d x}{2}}-(34-34 i) e^{\frac{5 i c}{2}+2 i d x}-(20+20 i) \sqrt{2} e^{3 i c+\frac{5 i d x}{2}}+(16-16 i) e^{\frac{7 i c}{2}+3 i d x}+(4+4 i) \sqrt{2} e^{4 i c+\frac{7 i d x}{2}}-(1-i) e^{\frac{9 i c}{2}+4 i d x}+8 i e^{\frac{1}{2} i (c+d x)}-16 \sqrt{2} e^{i (c+d x)}-40 i e^{\frac{3}{2} i (c+d x)}+34 \sqrt{2} e^{2 i (c+d x)}+40 i e^{\frac{5}{2} i (c+d x)}-16 \sqrt{2} e^{3 i (c+d x)}-8 i e^{\frac{7}{2} i (c+d x)}+\sqrt{2} e^{4 i (c+d x)}-(4+4 i) \sqrt{2} e^{\frac{1}{2} i (2 c+d x)}\right) x \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{\left((-1-i)+\sqrt{2} e^{\frac{i c}{2}}\right) \left(-1+e^{i c}\right) \left(i-2 \sqrt{2} e^{\frac{1}{2} i (c+d x)}-4 i e^{i (c+d x)}+2 \sqrt{2} e^{\frac{3}{2} i (c+d x)}+i e^{2 i (c+d x)}\right)^2 \sqrt{a (\cos (c+d x)+1)}}-\frac{7 i \tan ^{-1}\left(\frac{\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sqrt{2} \sin \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}{\sqrt{2} \cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{4 \sqrt{2} d \sqrt{a (\cos (c+d x)+1)}}-\frac{7 i \tan ^{-1}\left(\frac{\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sqrt{2} \sin \left(\frac{c}{4}+\frac{d x}{4}\right)+\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}{\sqrt{2} \cos \left(\frac{c}{4}+\frac{d x}{4}\right)+\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{4 \sqrt{2} d \sqrt{a (\cos (c+d x)+1)}}-\frac{7 i \tan ^{-1}\left(\frac{2 i \cos \left(\frac{c}{2}\right)-i \left(2 \sin \left(\frac{c}{2}\right)-\sqrt{2}\right) \tan \left(\frac{d x}{4}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}\right) \cot \left(\frac{c}{2}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d \sqrt{a (\cos (c+d x)+1)} \sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}+\frac{\left(3 \sin \left(\frac{c}{2}\right)-\cos \left(\frac{c}{2}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d \sqrt{a (\cos (c+d x)+1)} \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{\left(\cos \left(\frac{c}{2}\right)+3 \sin \left(\frac{c}{2}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d \sqrt{a (\cos (c+d x)+1)} \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{\sin \left(\frac{d x}{2}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d \sqrt{a (\cos (c+d x)+1)} \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{\sin \left(\frac{d x}{2}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d \sqrt{a (\cos (c+d x)+1)} \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{\tan (c+d x)}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{7 \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} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{\tan (c+d x) \sec (c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}",1,"((-7/16 + (7*I)/16)*(1 + E^(I*c))*(Sqrt[2] - (1 - I)*E^((I/2)*c) + (16 - 16*I)*E^(((3*I)/2)*c + I*d*x) + (20 + 20*I)*Sqrt[2]*E^((2*I)*c + ((3*I)/2)*d*x) - (34 - 34*I)*E^(((5*I)/2)*c + (2*I)*d*x) - (20 + 20*I)*Sqrt[2]*E^((3*I)*c + ((5*I)/2)*d*x) + (16 - 16*I)*E^(((7*I)/2)*c + (3*I)*d*x) + (4 + 4*I)*Sqrt[2]*E^((4*I)*c + ((7*I)/2)*d*x) - (1 - I)*E^(((9*I)/2)*c + (4*I)*d*x) + (8*I)*E^((I/2)*(c + d*x)) - 16*Sqrt[2]*E^(I*(c + d*x)) - (40*I)*E^(((3*I)/2)*(c + d*x)) + 34*Sqrt[2]*E^((2*I)*(c + d*x)) + (40*I)*E^(((5*I)/2)*(c + d*x)) - 16*Sqrt[2]*E^((3*I)*(c + d*x)) - (8*I)*E^(((7*I)/2)*(c + d*x)) + Sqrt[2]*E^((4*I)*(c + d*x)) - (4 + 4*I)*Sqrt[2]*E^((I/2)*(2*c + d*x)))*x*Cos[c/2 + (d*x)/2])/(((-1 - I) + Sqrt[2]*E^((I/2)*c))*(-1 + E^(I*c))*(I - 2*Sqrt[2]*E^((I/2)*(c + d*x)) - (4*I)*E^(I*(c + d*x)) + 2*Sqrt[2]*E^(((3*I)/2)*(c + d*x)) + I*E^((2*I)*(c + d*x)))^2*Sqrt[a*(1 + Cos[c + d*x])]) - (((7*I)/4)*ArcTan[(Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4] - Sqrt[2]*Sin[c/4 + (d*x)/4])/(-Cos[c/4 + (d*x)/4] + Sqrt[2]*Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])]*Cos[c/2 + (d*x)/2])/(Sqrt[2]*d*Sqrt[a*(1 + Cos[c + d*x])]) - (((7*I)/4)*ArcTan[(Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4] - Sqrt[2]*Sin[c/4 + (d*x)/4])/(Cos[c/4 + (d*x)/4] + Sqrt[2]*Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])]*Cos[c/2 + (d*x)/2])/(Sqrt[2]*d*Sqrt[a*(1 + Cos[c + d*x])]) + (2*Cos[c/2 + (d*x)/2]*Log[Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4]])/(d*Sqrt[a*(1 + Cos[c + d*x])]) - (2*Cos[c/2 + (d*x)/2]*Log[Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4]])/(d*Sqrt[a*(1 + Cos[c + d*x])]) - (7*Cos[c/2 + (d*x)/2]*Log[2 - Sqrt[2]*Cos[c/2 + (d*x)/2] - Sqrt[2]*Sin[c/2 + (d*x)/2]])/(8*Sqrt[2]*d*Sqrt[a*(1 + Cos[c + d*x])]) - (7*Cos[c/2 + (d*x)/2]*Log[2 + Sqrt[2]*Cos[c/2 + (d*x)/2] - Sqrt[2]*Sin[c/2 + (d*x)/2]])/(8*Sqrt[2]*d*Sqrt[a*(1 + Cos[c + d*x])]) - (((7*I)/2)*ArcTan[((2*I)*Cos[c/2] - I*(-Sqrt[2] + 2*Sin[c/2])*Tan[(d*x)/4])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]]*Cos[c/2 + (d*x)/2]*Cot[c/2])/(d*Sqrt[a*(1 + Cos[c + d*x])]*Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]) + (7*Cos[c/2 + (d*x)/2]*Csc[c/2]*(-(d*x*Cos[c/2]) + 2*Log[Sqrt[2] + 2*Cos[(d*x)/2]*Sin[c/2] + 2*Cos[c/2]*Sin[(d*x)/2]]*Sin[c/2] + ((4*I)*Sqrt[2]*ArcTan[((2*I)*Cos[c/2] - I*(-Sqrt[2] + 2*Sin[c/2])*Tan[(d*x)/4])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]]*Cos[c/2])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]))/(2*Sqrt[2]*d*Sqrt[a*(1 + Cos[c + d*x])]*(4*Cos[c/2]^2 + 4*Sin[c/2]^2)) + (Cos[c/2 + (d*x)/2]*Sin[(d*x)/2])/(2*d*Sqrt[a*(1 + Cos[c + d*x])]*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (Cos[c/2 + (d*x)/2]*(-Cos[c/2] + 3*Sin[c/2]))/(4*d*Sqrt[a*(1 + Cos[c + d*x])]*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (Cos[c/2 + (d*x)/2]*Sin[(d*x)/2])/(2*d*Sqrt[a*(1 + Cos[c + d*x])]*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (Cos[c/2 + (d*x)/2]*(Cos[c/2] + 3*Sin[c/2]))/(4*d*Sqrt[a*(1 + Cos[c + d*x])]*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","C",0
130,1,1921,181,30.13118,"\int \frac{\sec ^4(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[Sec[c + d*x]^4/Sqrt[a + a*Cos[c + d*x]],x]","-\frac{2 \log \left(\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{a (\cos (c+d x)+1)}}+\frac{2 \log \left(\cos \left(\frac{c}{4}+\frac{d x}{4}\right)+\sin \left(\frac{c}{4}+\frac{d x}{4}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{a (\cos (c+d x)+1)}}+\frac{9 \log \left(-\sqrt{2} \cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sqrt{2} \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+2\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{16 \sqrt{2} d \sqrt{a (\cos (c+d x)+1)}}+\frac{9 \log \left(\sqrt{2} \cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sqrt{2} \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+2\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{16 \sqrt{2} d \sqrt{a (\cos (c+d x)+1)}}-\frac{9 \csc \left(\frac{c}{2}\right) \left(-d x \cos \left(\frac{c}{2}\right)+\frac{4 i \sqrt{2} \tan ^{-1}\left(\frac{2 i \cos \left(\frac{c}{2}\right)-i \left(2 \sin \left(\frac{c}{2}\right)-\sqrt{2}\right) \tan \left(\frac{d x}{4}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}\right) \cos \left(\frac{c}{2}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}+2 \log \left(2 \cos \left(\frac{d x}{2}\right) \sin \left(\frac{c}{2}\right)+2 \cos \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+\sqrt{2}\right) \sin \left(\frac{c}{2}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{4 \sqrt{2} d \sqrt{a (\cos (c+d x)+1)} \left(4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)\right)}+\frac{\left(\frac{9}{32}-\frac{9 i}{32}\right) \left(1+e^{i c}\right) \left(\sqrt{2}-(1-i) e^{\frac{i c}{2}}+(16-16 i) e^{\frac{3 i c}{2}+i d x}+(20+20 i) \sqrt{2} e^{2 i c+\frac{3 i d x}{2}}-(34-34 i) e^{\frac{5 i c}{2}+2 i d x}-(20+20 i) \sqrt{2} e^{3 i c+\frac{5 i d x}{2}}+(16-16 i) e^{\frac{7 i c}{2}+3 i d x}+(4+4 i) \sqrt{2} e^{4 i c+\frac{7 i d x}{2}}-(1-i) e^{\frac{9 i c}{2}+4 i d x}+8 i e^{\frac{1}{2} i (c+d x)}-16 \sqrt{2} e^{i (c+d x)}-40 i e^{\frac{3}{2} i (c+d x)}+34 \sqrt{2} e^{2 i (c+d x)}+40 i e^{\frac{5}{2} i (c+d x)}-16 \sqrt{2} e^{3 i (c+d x)}-8 i e^{\frac{7}{2} i (c+d x)}+\sqrt{2} e^{4 i (c+d x)}-(4+4 i) \sqrt{2} e^{\frac{1}{2} i (2 c+d x)}\right) x \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{\left((-1-i)+\sqrt{2} e^{\frac{i c}{2}}\right) \left(-1+e^{i c}\right) \left(i-2 \sqrt{2} e^{\frac{1}{2} i (c+d x)}-4 i e^{i (c+d x)}+2 \sqrt{2} e^{\frac{3}{2} i (c+d x)}+i e^{2 i (c+d x)}\right)^2 \sqrt{a (\cos (c+d x)+1)}}+\frac{9 i \tan ^{-1}\left(\frac{\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sqrt{2} \sin \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}{\sqrt{2} \cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{8 \sqrt{2} d \sqrt{a (\cos (c+d x)+1)}}+\frac{9 i \tan ^{-1}\left(\frac{\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sqrt{2} \sin \left(\frac{c}{4}+\frac{d x}{4}\right)+\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}{\sqrt{2} \cos \left(\frac{c}{4}+\frac{d x}{4}\right)+\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{8 \sqrt{2} d \sqrt{a (\cos (c+d x)+1)}}+\frac{9 i \tan ^{-1}\left(\frac{2 i \cos \left(\frac{c}{2}\right)-i \left(2 \sin \left(\frac{c}{2}\right)-\sqrt{2}\right) \tan \left(\frac{d x}{4}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}\right) \cot \left(\frac{c}{2}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d \sqrt{a (\cos (c+d x)+1)} \sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}+\frac{\left(7 \cos \left(\frac{c}{2}\right)-9 \sin \left(\frac{c}{2}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d \sqrt{a (\cos (c+d x)+1)} \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{\left(-7 \cos \left(\frac{c}{2}\right)-9 \sin \left(\frac{c}{2}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d \sqrt{a (\cos (c+d x)+1)} \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{\sin \left(\frac{d x}{2}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d \sqrt{a (\cos (c+d x)+1)} \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{\sin \left(\frac{d x}{2}\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d \sqrt{a (\cos (c+d x)+1)} \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{6 d \sqrt{a (\cos (c+d x)+1)} \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}-\frac{\cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{6 d \sqrt{a (\cos (c+d x)+1)} \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}","\frac{7 \tan (c+d x)}{8 d \sqrt{a \cos (c+d x)+a}}-\frac{9 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{\tan (c+d x) \sec ^2(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{\tan (c+d x) \sec (c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}",1,"((9/32 - (9*I)/32)*(1 + E^(I*c))*(Sqrt[2] - (1 - I)*E^((I/2)*c) + (16 - 16*I)*E^(((3*I)/2)*c + I*d*x) + (20 + 20*I)*Sqrt[2]*E^((2*I)*c + ((3*I)/2)*d*x) - (34 - 34*I)*E^(((5*I)/2)*c + (2*I)*d*x) - (20 + 20*I)*Sqrt[2]*E^((3*I)*c + ((5*I)/2)*d*x) + (16 - 16*I)*E^(((7*I)/2)*c + (3*I)*d*x) + (4 + 4*I)*Sqrt[2]*E^((4*I)*c + ((7*I)/2)*d*x) - (1 - I)*E^(((9*I)/2)*c + (4*I)*d*x) + (8*I)*E^((I/2)*(c + d*x)) - 16*Sqrt[2]*E^(I*(c + d*x)) - (40*I)*E^(((3*I)/2)*(c + d*x)) + 34*Sqrt[2]*E^((2*I)*(c + d*x)) + (40*I)*E^(((5*I)/2)*(c + d*x)) - 16*Sqrt[2]*E^((3*I)*(c + d*x)) - (8*I)*E^(((7*I)/2)*(c + d*x)) + Sqrt[2]*E^((4*I)*(c + d*x)) - (4 + 4*I)*Sqrt[2]*E^((I/2)*(2*c + d*x)))*x*Cos[c/2 + (d*x)/2])/(((-1 - I) + Sqrt[2]*E^((I/2)*c))*(-1 + E^(I*c))*(I - 2*Sqrt[2]*E^((I/2)*(c + d*x)) - (4*I)*E^(I*(c + d*x)) + 2*Sqrt[2]*E^(((3*I)/2)*(c + d*x)) + I*E^((2*I)*(c + d*x)))^2*Sqrt[a*(1 + Cos[c + d*x])]) + (((9*I)/8)*ArcTan[(Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4] - Sqrt[2]*Sin[c/4 + (d*x)/4])/(-Cos[c/4 + (d*x)/4] + Sqrt[2]*Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])]*Cos[c/2 + (d*x)/2])/(Sqrt[2]*d*Sqrt[a*(1 + Cos[c + d*x])]) + (((9*I)/8)*ArcTan[(Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4] - Sqrt[2]*Sin[c/4 + (d*x)/4])/(Cos[c/4 + (d*x)/4] + Sqrt[2]*Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])]*Cos[c/2 + (d*x)/2])/(Sqrt[2]*d*Sqrt[a*(1 + Cos[c + d*x])]) - (2*Cos[c/2 + (d*x)/2]*Log[Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4]])/(d*Sqrt[a*(1 + Cos[c + d*x])]) + (2*Cos[c/2 + (d*x)/2]*Log[Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4]])/(d*Sqrt[a*(1 + Cos[c + d*x])]) + (9*Cos[c/2 + (d*x)/2]*Log[2 - Sqrt[2]*Cos[c/2 + (d*x)/2] - Sqrt[2]*Sin[c/2 + (d*x)/2]])/(16*Sqrt[2]*d*Sqrt[a*(1 + Cos[c + d*x])]) + (9*Cos[c/2 + (d*x)/2]*Log[2 + Sqrt[2]*Cos[c/2 + (d*x)/2] - Sqrt[2]*Sin[c/2 + (d*x)/2]])/(16*Sqrt[2]*d*Sqrt[a*(1 + Cos[c + d*x])]) + (((9*I)/4)*ArcTan[((2*I)*Cos[c/2] - I*(-Sqrt[2] + 2*Sin[c/2])*Tan[(d*x)/4])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]]*Cos[c/2 + (d*x)/2]*Cot[c/2])/(d*Sqrt[a*(1 + Cos[c + d*x])]*Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]) - (9*Cos[c/2 + (d*x)/2]*Csc[c/2]*(-(d*x*Cos[c/2]) + 2*Log[Sqrt[2] + 2*Cos[(d*x)/2]*Sin[c/2] + 2*Cos[c/2]*Sin[(d*x)/2]]*Sin[c/2] + ((4*I)*Sqrt[2]*ArcTan[((2*I)*Cos[c/2] - I*(-Sqrt[2] + 2*Sin[c/2])*Tan[(d*x)/4])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]]*Cos[c/2])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]))/(4*Sqrt[2]*d*Sqrt[a*(1 + Cos[c + d*x])]*(4*Cos[c/2]^2 + 4*Sin[c/2]^2)) + Cos[c/2 + (d*x)/2]/(6*d*Sqrt[a*(1 + Cos[c + d*x])]*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) - (Cos[c/2 + (d*x)/2]*Sin[(d*x)/2])/(4*d*Sqrt[a*(1 + Cos[c + d*x])]*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (Cos[c/2 + (d*x)/2]*(7*Cos[c/2] - 9*Sin[c/2]))/(8*d*Sqrt[a*(1 + Cos[c + d*x])]*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) - Cos[c/2 + (d*x)/2]/(6*d*Sqrt[a*(1 + Cos[c + d*x])]*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) - (Cos[c/2 + (d*x)/2]*Sin[(d*x)/2])/(4*d*Sqrt[a*(1 + Cos[c + d*x])]*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (Cos[c/2 + (d*x)/2]*(-7*Cos[c/2] - 9*Sin[c/2]))/(8*d*Sqrt[a*(1 + Cos[c + d*x])]*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","C",0
131,1,226,183,1.3553762,"\int \frac{\cos ^4(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^4/(a + a*Cos[c + d*x])^(3/2),x]","\frac{\cos ^3\left(\frac{1}{2} (c+d x)\right) \left(200 \sin \left(\frac{c}{2}\right) \cos \left(\frac{d x}{2}\right)-20 \sin \left(\frac{3 c}{2}\right) \cos \left(\frac{3 d x}{2}\right)+4 \sin \left(\frac{5 c}{2}\right) \cos \left(\frac{5 d x}{2}\right)+200 \cos \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)-20 \cos \left(\frac{3 c}{2}\right) \sin \left(\frac{3 d x}{2}\right)+4 \cos \left(\frac{5 c}{2}\right) \sin \left(\frac{5 d x}{2}\right)+\frac{5}{\left(\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)\right)^2}-\frac{5}{\left(\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)\right)^2}+150 \log \left(\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)\right)-150 \log \left(\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)\right)\right)}{10 d (a (\cos (c+d x)+1))^{3/2}}","-\frac{15 \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{13 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{10 a^2 d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{9 \sin (c+d x) \cos ^2(c+d x)}{10 a d \sqrt{a \cos (c+d x)+a}}+\frac{31 \sin (c+d x)}{5 a d \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]^3*(150*Log[Cos[(c + d*x)/4] - Sin[(c + d*x)/4]] - 150*Log[Cos[(c + d*x)/4] + Sin[(c + d*x)/4]] + 200*Cos[(d*x)/2]*Sin[c/2] - 20*Cos[(3*d*x)/2]*Sin[(3*c)/2] + 4*Cos[(5*d*x)/2]*Sin[(5*c)/2] + 200*Cos[c/2]*Sin[(d*x)/2] - 20*Cos[(3*c)/2]*Sin[(3*d*x)/2] + 4*Cos[(5*c)/2]*Sin[(5*d*x)/2] + 5/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^2 - 5/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^2))/(10*d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
132,1,196,145,0.9630303,"\int \frac{\cos ^3(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^3/(a + a*Cos[c + d*x])^(3/2),x]","\frac{\cos ^3\left(\frac{1}{2} (c+d x)\right) \left(-72 \sin \left(\frac{c}{2}\right) \cos \left(\frac{d x}{2}\right)+8 \sin \left(\frac{3 c}{2}\right) \cos \left(\frac{3 d x}{2}\right)-72 \cos \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+8 \cos \left(\frac{3 c}{2}\right) \sin \left(\frac{3 d x}{2}\right)-\frac{3}{\left(\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)\right)^2}+\frac{3}{\left(\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)\right)^2}-66 \log \left(\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)\right)+66 \log \left(\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)\right)\right)}{6 d (a (\cos (c+d x)+1))^{3/2}}","\frac{11 \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 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{6 a^2 d}-\frac{\sin (c+d x) \cos ^2(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{13 \sin (c+d x)}{3 a d \sqrt{a \cos (c+d x)+a}}",1,"(Cos[(c + d*x)/2]^3*(-66*Log[Cos[(c + d*x)/4] - Sin[(c + d*x)/4]] + 66*Log[Cos[(c + d*x)/4] + Sin[(c + d*x)/4]] - 72*Cos[(d*x)/2]*Sin[c/2] + 8*Cos[(3*d*x)/2]*Sin[(3*c)/2] - 72*Cos[c/2]*Sin[(d*x)/2] + 8*Cos[(3*c)/2]*Sin[(3*d*x)/2] - 3/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^2 + 3/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^2))/(6*d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
133,1,164,105,0.4635679,"\int \frac{\cos ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^2/(a + a*Cos[c + d*x])^(3/2),x]","\frac{\cos ^3\left(\frac{1}{2} (c+d x)\right) \left(16 \sin \left(\frac{c}{2}\right) \cos \left(\frac{d x}{2}\right)+16 \cos \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+\frac{1}{\left(\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)\right)^2}-\frac{1}{\left(\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)\right)^2}+14 \log \left(\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)\right)-14 \log \left(\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)\right)\right)}{2 d (a (\cos (c+d x)+1))^{3/2}}","-\frac{7 \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 \sin (c+d x)}{a d \sqrt{a \cos (c+d x)+a}}+\frac{\sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]^3*(14*Log[Cos[(c + d*x)/4] - Sin[(c + d*x)/4]] - 14*Log[Cos[(c + d*x)/4] + Sin[(c + d*x)/4]] + 16*Cos[(d*x)/2]*Sin[c/2] + 16*Cos[c/2]*Sin[(d*x)/2] + (Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^(-2) - (Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^(-2)))/(2*d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
134,1,54,77,0.0975752,"\int \frac{\cos (c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]/(a + a*Cos[c + d*x])^(3/2),x]","\frac{3 \cos ^3\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 (c+d x)}{d (a (\cos (c+d x)+1))^{3/2}}","\frac{3 \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{\sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(3*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 - Sin[c + d*x]/2)/(d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
135,1,63,77,0.0675529,"\int \frac{1}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(a + a*Cos[c + d*x])^(-3/2),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(\tan \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{d (a (\cos (c+d x)+1))^{3/2}}","\frac{\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{\sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]^2*(ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2] + Tan[(c + d*x)/2]))/(d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
136,1,1787,114,23.7489559,"\int \frac{\sec (c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]/(a + a*Cos[c + d*x])^(3/2),x]","\frac{5 \log \left(\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{3/2}}-\frac{5 \log \left(\cos \left(\frac{c}{4}+\frac{d x}{4}\right)+\sin \left(\frac{c}{4}+\frac{d x}{4}\right)\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{3/2}}-\frac{\sqrt{2} \log \left(-\sqrt{2} \cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sqrt{2} \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+2\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{3/2}}+\frac{4 \sqrt{2} \csc \left(\frac{c}{2}\right) \left(-d x \cos \left(\frac{c}{2}\right)+\frac{4 i \sqrt{2} \tan ^{-1}\left(\frac{2 i \cos \left(\frac{c}{2}\right)-i \left(2 \sin \left(\frac{c}{2}\right)-\sqrt{2}\right) \tan \left(\frac{d x}{4}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}\right) \cos \left(\frac{c}{2}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}+2 \log \left(2 \cos \left(\frac{d x}{2}\right) \sin \left(\frac{c}{2}\right)+2 \cos \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+\sqrt{2}\right) \sin \left(\frac{c}{2}\right)\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{3/2} \left(4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)\right)}-\frac{8 i \tan ^{-1}\left(\frac{2 i \cos \left(\frac{c}{2}\right)-i \left(2 \sin \left(\frac{c}{2}\right)-\sqrt{2}\right) \tan \left(\frac{d x}{4}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}\right) \cot \left(\frac{c}{2}\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{3/2} \sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}-\frac{\left(\frac{1}{2}+\frac{i}{2}\right) \log \left(\sqrt{2} \cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sqrt{2} \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+2\right) \left(\sqrt{2} \cos \left(\frac{c}{4}\right)+(1+i) \cos \left(\frac{c}{4}\right)-i \sqrt{2} \sin \left(\frac{c}{4}\right)-(1-i) \sin \left(\frac{c}{4}\right)\right) \left(\sqrt{2} \cos \left(\frac{c}{4}\right)-(1+i) \cos \left(\frac{c}{4}\right)-i \sqrt{2} \sin \left(\frac{c}{4}\right)+(1-i) \sin \left(\frac{c}{4}\right)\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{2} d (a (\cos (c+d x)+1))^{3/2} \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right)}+\frac{(1-i) \tan ^{-1}\left(\frac{\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sqrt{2} \sin \left(\frac{c}{4}+\frac{d x}{4}\right)+\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}{\sqrt{2} \cos \left(\frac{c}{4}+\frac{d x}{4}\right)+\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}\right) \left(\sqrt{2} \cos \left(\frac{c}{4}\right)+(1+i) \cos \left(\frac{c}{4}\right)-i \sqrt{2} \sin \left(\frac{c}{4}\right)-(1-i) \sin \left(\frac{c}{4}\right)\right) \left(\sqrt{2} \cos \left(\frac{c}{4}\right)-(1+i) \cos \left(\frac{c}{4}\right)-i \sqrt{2} \sin \left(\frac{c}{4}\right)+(1-i) \sin \left(\frac{c}{4}\right)\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{2} d (a (\cos (c+d x)+1))^{3/2} \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right)}-\frac{(1-i) \left(1+e^{i c}\right) \left(\sqrt{2}-(1-i) e^{\frac{i c}{2}}+(16-16 i) e^{\frac{3 i c}{2}+i d x}+(20+20 i) \sqrt{2} e^{2 i c+\frac{3 i d x}{2}}-(34-34 i) e^{\frac{5 i c}{2}+2 i d x}-(20+20 i) \sqrt{2} e^{3 i c+\frac{5 i d x}{2}}+(16-16 i) e^{\frac{7 i c}{2}+3 i d x}+(4+4 i) \sqrt{2} e^{4 i c+\frac{7 i d x}{2}}-(1-i) e^{\frac{9 i c}{2}+4 i d x}+8 i e^{\frac{1}{2} i (c+d x)}-16 \sqrt{2} e^{i (c+d x)}-40 i e^{\frac{3}{2} i (c+d x)}+34 \sqrt{2} e^{2 i (c+d x)}+40 i e^{\frac{5}{2} i (c+d x)}-16 \sqrt{2} e^{3 i (c+d x)}-8 i e^{\frac{7}{2} i (c+d x)}+\sqrt{2} e^{4 i (c+d x)}-(4+4 i) \sqrt{2} e^{\frac{1}{2} i (2 c+d x)}\right) x \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{\left((-1-i)+\sqrt{2} e^{\frac{i c}{2}}\right) \left(-1+e^{i c}\right) \left(i-2 \sqrt{2} e^{\frac{1}{2} i (c+d x)}-4 i e^{i (c+d x)}+2 \sqrt{2} e^{\frac{3}{2} i (c+d x)}+i e^{2 i (c+d x)}\right)^2 (a (\cos (c+d x)+1))^{3/2}}-\frac{2 i \sqrt{2} \tan ^{-1}\left(\frac{\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sqrt{2} \sin \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}{\sqrt{2} \cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{3/2}}-\frac{\cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d (a (\cos (c+d x)+1))^{3/2} \left(\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)\right)^2}+\frac{\cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d (a (\cos (c+d x)+1))^{3/2} \left(\cos \left(\frac{c}{4}+\frac{d x}{4}\right)+\sin \left(\frac{c}{4}+\frac{d x}{4}\right)\right)^2}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{5 \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{\sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"((-1 + I)*(1 + E^(I*c))*(Sqrt[2] - (1 - I)*E^((I/2)*c) + (16 - 16*I)*E^(((3*I)/2)*c + I*d*x) + (20 + 20*I)*Sqrt[2]*E^((2*I)*c + ((3*I)/2)*d*x) - (34 - 34*I)*E^(((5*I)/2)*c + (2*I)*d*x) - (20 + 20*I)*Sqrt[2]*E^((3*I)*c + ((5*I)/2)*d*x) + (16 - 16*I)*E^(((7*I)/2)*c + (3*I)*d*x) + (4 + 4*I)*Sqrt[2]*E^((4*I)*c + ((7*I)/2)*d*x) - (1 - I)*E^(((9*I)/2)*c + (4*I)*d*x) + (8*I)*E^((I/2)*(c + d*x)) - 16*Sqrt[2]*E^(I*(c + d*x)) - (40*I)*E^(((3*I)/2)*(c + d*x)) + 34*Sqrt[2]*E^((2*I)*(c + d*x)) + (40*I)*E^(((5*I)/2)*(c + d*x)) - 16*Sqrt[2]*E^((3*I)*(c + d*x)) - (8*I)*E^(((7*I)/2)*(c + d*x)) + Sqrt[2]*E^((4*I)*(c + d*x)) - (4 + 4*I)*Sqrt[2]*E^((I/2)*(2*c + d*x)))*x*Cos[c/2 + (d*x)/2]^3)/(((-1 - I) + Sqrt[2]*E^((I/2)*c))*(-1 + E^(I*c))*(I - 2*Sqrt[2]*E^((I/2)*(c + d*x)) - (4*I)*E^(I*(c + d*x)) + 2*Sqrt[2]*E^(((3*I)/2)*(c + d*x)) + I*E^((2*I)*(c + d*x)))^2*(a*(1 + Cos[c + d*x]))^(3/2)) - ((2*I)*Sqrt[2]*ArcTan[(Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4] - Sqrt[2]*Sin[c/4 + (d*x)/4])/(-Cos[c/4 + (d*x)/4] + Sqrt[2]*Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])]*Cos[c/2 + (d*x)/2]^3)/(d*(a*(1 + Cos[c + d*x]))^(3/2)) + (5*Cos[c/2 + (d*x)/2]^3*Log[Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4]])/(d*(a*(1 + Cos[c + d*x]))^(3/2)) - (5*Cos[c/2 + (d*x)/2]^3*Log[Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4]])/(d*(a*(1 + Cos[c + d*x]))^(3/2)) - (Sqrt[2]*Cos[c/2 + (d*x)/2]^3*Log[2 - Sqrt[2]*Cos[c/2 + (d*x)/2] - Sqrt[2]*Sin[c/2 + (d*x)/2]])/(d*(a*(1 + Cos[c + d*x]))^(3/2)) + ((1 - I)*ArcTan[(Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4] - Sqrt[2]*Sin[c/4 + (d*x)/4])/(Cos[c/4 + (d*x)/4] + Sqrt[2]*Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])]*Cos[c/2 + (d*x)/2]^3*((1 + I)*Cos[c/4] + Sqrt[2]*Cos[c/4] - (1 - I)*Sin[c/4] - I*Sqrt[2]*Sin[c/4])*((-1 - I)*Cos[c/4] + Sqrt[2]*Cos[c/4] + (1 - I)*Sin[c/4] - I*Sqrt[2]*Sin[c/4]))/(Sqrt[2]*d*(a*(1 + Cos[c + d*x]))^(3/2)*(Cos[c/2] + Sin[c/2])) - ((1/2 + I/2)*Cos[c/2 + (d*x)/2]^3*Log[2 + Sqrt[2]*Cos[c/2 + (d*x)/2] - Sqrt[2]*Sin[c/2 + (d*x)/2]]*((1 + I)*Cos[c/4] + Sqrt[2]*Cos[c/4] - (1 - I)*Sin[c/4] - I*Sqrt[2]*Sin[c/4])*((-1 - I)*Cos[c/4] + Sqrt[2]*Cos[c/4] + (1 - I)*Sin[c/4] - I*Sqrt[2]*Sin[c/4]))/(Sqrt[2]*d*(a*(1 + Cos[c + d*x]))^(3/2)*(Cos[c/2] + Sin[c/2])) - ((8*I)*ArcTan[((2*I)*Cos[c/2] - I*(-Sqrt[2] + 2*Sin[c/2])*Tan[(d*x)/4])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]]*Cos[c/2 + (d*x)/2]^3*Cot[c/2])/(d*(a*(1 + Cos[c + d*x]))^(3/2)*Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]) + (4*Sqrt[2]*Cos[c/2 + (d*x)/2]^3*Csc[c/2]*(-(d*x*Cos[c/2]) + 2*Log[Sqrt[2] + 2*Cos[(d*x)/2]*Sin[c/2] + 2*Cos[c/2]*Sin[(d*x)/2]]*Sin[c/2] + ((4*I)*Sqrt[2]*ArcTan[((2*I)*Cos[c/2] - I*(-Sqrt[2] + 2*Sin[c/2])*Tan[(d*x)/4])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]]*Cos[c/2])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]))/(d*(a*(1 + Cos[c + d*x]))^(3/2)*(4*Cos[c/2]^2 + 4*Sin[c/2]^2)) - Cos[c/2 + (d*x)/2]^3/(2*d*(a*(1 + Cos[c + d*x]))^(3/2)*(Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])^2) + Cos[c/2 + (d*x)/2]^3/(2*d*(a*(1 + Cos[c + d*x]))^(3/2)*(Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4])^2)","C",0
137,1,103,144,0.4841705,"\int \frac{\sec ^2(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^2/(a + a*Cos[c + d*x])^(3/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) (2 \sec (c+d x)+3)+9 \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-6 \sqrt{2} \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a d \sqrt{a (\cos (c+d x)+1)}}","-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}+\frac{9 \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 \tan (c+d x)}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{\tan (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(9*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2] - 6*Sqrt[2]*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2] + (3 + 2*Sec[c + d*x])*Tan[(c + d*x)/2])/(2*a*d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
138,1,1941,185,28.4291629,"\int \frac{\sec ^3(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^3/(a + a*Cos[c + d*x])^(3/2),x]","\frac{13 \log \left(\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{3/2}}-\frac{13 \log \left(\cos \left(\frac{c}{4}+\frac{d x}{4}\right)+\sin \left(\frac{c}{4}+\frac{d x}{4}\right)\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{3/2}}-\frac{19 \log \left(-\sqrt{2} \cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sqrt{2} \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+2\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 \sqrt{2} d (a (\cos (c+d x)+1))^{3/2}}-\frac{19 \log \left(\sqrt{2} \cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sqrt{2} \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+2\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 \sqrt{2} d (a (\cos (c+d x)+1))^{3/2}}+\frac{19 \csc \left(\frac{c}{2}\right) \left(-d x \cos \left(\frac{c}{2}\right)+\frac{4 i \sqrt{2} \tan ^{-1}\left(\frac{2 i \cos \left(\frac{c}{2}\right)-i \left(2 \sin \left(\frac{c}{2}\right)-\sqrt{2}\right) \tan \left(\frac{d x}{4}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}\right) \cos \left(\frac{c}{2}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}+2 \log \left(2 \cos \left(\frac{d x}{2}\right) \sin \left(\frac{c}{2}\right)+2 \cos \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+\sqrt{2}\right) \sin \left(\frac{c}{2}\right)\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{2} d (a (\cos (c+d x)+1))^{3/2} \left(4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)\right)}-\frac{19 i \tan ^{-1}\left(\frac{2 i \cos \left(\frac{c}{2}\right)-i \left(2 \sin \left(\frac{c}{2}\right)-\sqrt{2}\right) \tan \left(\frac{d x}{4}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}\right) \cot \left(\frac{c}{2}\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{3/2} \sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}+\frac{\left(7 \sin \left(\frac{c}{2}\right)-5 \cos \left(\frac{c}{2}\right)\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d (a (\cos (c+d x)+1))^{3/2} \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{\left(5 \cos \left(\frac{c}{2}\right)+7 \sin \left(\frac{c}{2}\right)\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d (a (\cos (c+d x)+1))^{3/2} \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{\left(\frac{19}{8}-\frac{19 i}{8}\right) \left(1+e^{i c}\right) \left(\sqrt{2}-(1-i) e^{\frac{i c}{2}}+(16-16 i) e^{\frac{3 i c}{2}+i d x}+(20+20 i) \sqrt{2} e^{2 i c+\frac{3 i d x}{2}}-(34-34 i) e^{\frac{5 i c}{2}+2 i d x}-(20+20 i) \sqrt{2} e^{3 i c+\frac{5 i d x}{2}}+(16-16 i) e^{\frac{7 i c}{2}+3 i d x}+(4+4 i) \sqrt{2} e^{4 i c+\frac{7 i d x}{2}}-(1-i) e^{\frac{9 i c}{2}+4 i d x}+8 i e^{\frac{1}{2} i (c+d x)}-16 \sqrt{2} e^{i (c+d x)}-40 i e^{\frac{3}{2} i (c+d x)}+34 \sqrt{2} e^{2 i (c+d x)}+40 i e^{\frac{5}{2} i (c+d x)}-16 \sqrt{2} e^{3 i (c+d x)}-8 i e^{\frac{7}{2} i (c+d x)}+\sqrt{2} e^{4 i (c+d x)}-(4+4 i) \sqrt{2} e^{\frac{1}{2} i (2 c+d x)}\right) x \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{\left((-1-i)+\sqrt{2} e^{\frac{i c}{2}}\right) \left(-1+e^{i c}\right) \left(i-2 \sqrt{2} e^{\frac{1}{2} i (c+d x)}-4 i e^{i (c+d x)}+2 \sqrt{2} e^{\frac{3}{2} i (c+d x)}+i e^{2 i (c+d x)}\right)^2 (a (\cos (c+d x)+1))^{3/2}}-\frac{19 i \tan ^{-1}\left(\frac{\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sqrt{2} \sin \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}{\sqrt{2} \cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sqrt{2} d (a (\cos (c+d x)+1))^{3/2}}-\frac{19 i \tan ^{-1}\left(\frac{\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sqrt{2} \sin \left(\frac{c}{4}+\frac{d x}{4}\right)+\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}{\sqrt{2} \cos \left(\frac{c}{4}+\frac{d x}{4}\right)+\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sqrt{2} d (a (\cos (c+d x)+1))^{3/2}}-\frac{\cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d (a (\cos (c+d x)+1))^{3/2} \left(\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)\right)^2}+\frac{\cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d (a (\cos (c+d x)+1))^{3/2} \left(\cos \left(\frac{c}{4}+\frac{d x}{4}\right)+\sin \left(\frac{c}{4}+\frac{d x}{4}\right)\right)^2}+\frac{\sin \left(\frac{d x}{2}\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{3/2} \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{\sin \left(\frac{d x}{2}\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{3/2} \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{19 \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 \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 \tan (c+d x)}{4 a d \sqrt{a \cos (c+d x)+a}}+\frac{\tan (c+d x) \sec (c+d x)}{a d \sqrt{a \cos (c+d x)+a}}-\frac{\tan (c+d x) \sec (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"((-19/8 + (19*I)/8)*(1 + E^(I*c))*(Sqrt[2] - (1 - I)*E^((I/2)*c) + (16 - 16*I)*E^(((3*I)/2)*c + I*d*x) + (20 + 20*I)*Sqrt[2]*E^((2*I)*c + ((3*I)/2)*d*x) - (34 - 34*I)*E^(((5*I)/2)*c + (2*I)*d*x) - (20 + 20*I)*Sqrt[2]*E^((3*I)*c + ((5*I)/2)*d*x) + (16 - 16*I)*E^(((7*I)/2)*c + (3*I)*d*x) + (4 + 4*I)*Sqrt[2]*E^((4*I)*c + ((7*I)/2)*d*x) - (1 - I)*E^(((9*I)/2)*c + (4*I)*d*x) + (8*I)*E^((I/2)*(c + d*x)) - 16*Sqrt[2]*E^(I*(c + d*x)) - (40*I)*E^(((3*I)/2)*(c + d*x)) + 34*Sqrt[2]*E^((2*I)*(c + d*x)) + (40*I)*E^(((5*I)/2)*(c + d*x)) - 16*Sqrt[2]*E^((3*I)*(c + d*x)) - (8*I)*E^(((7*I)/2)*(c + d*x)) + Sqrt[2]*E^((4*I)*(c + d*x)) - (4 + 4*I)*Sqrt[2]*E^((I/2)*(2*c + d*x)))*x*Cos[c/2 + (d*x)/2]^3)/(((-1 - I) + Sqrt[2]*E^((I/2)*c))*(-1 + E^(I*c))*(I - 2*Sqrt[2]*E^((I/2)*(c + d*x)) - (4*I)*E^(I*(c + d*x)) + 2*Sqrt[2]*E^(((3*I)/2)*(c + d*x)) + I*E^((2*I)*(c + d*x)))^2*(a*(1 + Cos[c + d*x]))^(3/2)) - (((19*I)/2)*ArcTan[(Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4] - Sqrt[2]*Sin[c/4 + (d*x)/4])/(-Cos[c/4 + (d*x)/4] + Sqrt[2]*Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])]*Cos[c/2 + (d*x)/2]^3)/(Sqrt[2]*d*(a*(1 + Cos[c + d*x]))^(3/2)) - (((19*I)/2)*ArcTan[(Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4] - Sqrt[2]*Sin[c/4 + (d*x)/4])/(Cos[c/4 + (d*x)/4] + Sqrt[2]*Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])]*Cos[c/2 + (d*x)/2]^3)/(Sqrt[2]*d*(a*(1 + Cos[c + d*x]))^(3/2)) + (13*Cos[c/2 + (d*x)/2]^3*Log[Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4]])/(d*(a*(1 + Cos[c + d*x]))^(3/2)) - (13*Cos[c/2 + (d*x)/2]^3*Log[Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4]])/(d*(a*(1 + Cos[c + d*x]))^(3/2)) - (19*Cos[c/2 + (d*x)/2]^3*Log[2 - Sqrt[2]*Cos[c/2 + (d*x)/2] - Sqrt[2]*Sin[c/2 + (d*x)/2]])/(4*Sqrt[2]*d*(a*(1 + Cos[c + d*x]))^(3/2)) - (19*Cos[c/2 + (d*x)/2]^3*Log[2 + Sqrt[2]*Cos[c/2 + (d*x)/2] - Sqrt[2]*Sin[c/2 + (d*x)/2]])/(4*Sqrt[2]*d*(a*(1 + Cos[c + d*x]))^(3/2)) - ((19*I)*ArcTan[((2*I)*Cos[c/2] - I*(-Sqrt[2] + 2*Sin[c/2])*Tan[(d*x)/4])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]]*Cos[c/2 + (d*x)/2]^3*Cot[c/2])/(d*(a*(1 + Cos[c + d*x]))^(3/2)*Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]) + (19*Cos[c/2 + (d*x)/2]^3*Csc[c/2]*(-(d*x*Cos[c/2]) + 2*Log[Sqrt[2] + 2*Cos[(d*x)/2]*Sin[c/2] + 2*Cos[c/2]*Sin[(d*x)/2]]*Sin[c/2] + ((4*I)*Sqrt[2]*ArcTan[((2*I)*Cos[c/2] - I*(-Sqrt[2] + 2*Sin[c/2])*Tan[(d*x)/4])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]]*Cos[c/2])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]))/(Sqrt[2]*d*(a*(1 + Cos[c + d*x]))^(3/2)*(4*Cos[c/2]^2 + 4*Sin[c/2]^2)) - Cos[c/2 + (d*x)/2]^3/(2*d*(a*(1 + Cos[c + d*x]))^(3/2)*(Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])^2) + Cos[c/2 + (d*x)/2]^3/(2*d*(a*(1 + Cos[c + d*x]))^(3/2)*(Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4])^2) + (Cos[c/2 + (d*x)/2]^3*Sin[(d*x)/2])/(d*(a*(1 + Cos[c + d*x]))^(3/2)*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (Cos[c/2 + (d*x)/2]^3*(-5*Cos[c/2] + 7*Sin[c/2]))/(2*d*(a*(1 + Cos[c + d*x]))^(3/2)*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (Cos[c/2 + (d*x)/2]^3*Sin[(d*x)/2])/(d*(a*(1 + Cos[c + d*x]))^(3/2)*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (Cos[c/2 + (d*x)/2]^3*(5*Cos[c/2] + 7*Sin[c/2]))/(2*d*(a*(1 + Cos[c + d*x]))^(3/2)*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","C",0
139,1,587,183,6.3504508,"\int \frac{\cos ^4(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^4/(a + a*Cos[c + d*x])^(5/2),x]","-\frac{40 \sin \left(\frac{c}{2}\right) \cos \left(\frac{d x}{2}\right) \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{5/2}}+\frac{8 \sin \left(\frac{3 c}{2}\right) \cos \left(\frac{3 d x}{2}\right) \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (a (\cos (c+d x)+1))^{5/2}}-\frac{40 \cos \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{5/2}}+\frac{8 \cos \left(\frac{3 c}{2}\right) \sin \left(\frac{3 d x}{2}\right) \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (a (\cos (c+d x)+1))^{5/2}}-\frac{29 \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d (a (\cos (c+d x)+1))^{5/2} \left(\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)\right)^2}+\frac{29 \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d (a (\cos (c+d x)+1))^{5/2} \left(\sin \left(\frac{c}{4}+\frac{d x}{4}\right)+\cos \left(\frac{c}{4}+\frac{d x}{4}\right)\right)^2}+\frac{\cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d (a (\cos (c+d x)+1))^{5/2} \left(\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)\right)^4}-\frac{\cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d (a (\cos (c+d x)+1))^{5/2} \left(\sin \left(\frac{c}{4}+\frac{d x}{4}\right)+\cos \left(\frac{c}{4}+\frac{d x}{4}\right)\right)^4}-\frac{163 \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right) \log \left(\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)\right)}{4 d (a (\cos (c+d x)+1))^{5/2}}+\frac{163 \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right) \log \left(\sin \left(\frac{c}{4}+\frac{d x}{4}\right)+\cos \left(\frac{c}{4}+\frac{d x}{4}\right)\right)}{4 d (a (\cos (c+d x)+1))^{5/2}}","\frac{163 \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{95 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{48 a^3 d}-\frac{197 \sin (c+d x)}{24 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{\sin (c+d x) \cos ^3(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{17 \sin (c+d x) \cos ^2(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"(-163*Cos[c/2 + (d*x)/2]^5*Log[Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4]])/(4*d*(a*(1 + Cos[c + d*x]))^(5/2)) + (163*Cos[c/2 + (d*x)/2]^5*Log[Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4]])/(4*d*(a*(1 + Cos[c + d*x]))^(5/2)) - (40*Cos[(d*x)/2]*Cos[c/2 + (d*x)/2]^5*Sin[c/2])/(d*(a*(1 + Cos[c + d*x]))^(5/2)) + (8*Cos[(3*d*x)/2]*Cos[c/2 + (d*x)/2]^5*Sin[(3*c)/2])/(3*d*(a*(1 + Cos[c + d*x]))^(5/2)) - (40*Cos[c/2]*Cos[c/2 + (d*x)/2]^5*Sin[(d*x)/2])/(d*(a*(1 + Cos[c + d*x]))^(5/2)) + (8*Cos[(3*c)/2]*Cos[c/2 + (d*x)/2]^5*Sin[(3*d*x)/2])/(3*d*(a*(1 + Cos[c + d*x]))^(5/2)) + Cos[c/2 + (d*x)/2]^5/(8*d*(a*(1 + Cos[c + d*x]))^(5/2)*(Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])^4) - (29*Cos[c/2 + (d*x)/2]^5)/(8*d*(a*(1 + Cos[c + d*x]))^(5/2)*(Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])^2) - Cos[c/2 + (d*x)/2]^5/(8*d*(a*(1 + Cos[c + d*x]))^(5/2)*(Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4])^4) + (29*Cos[c/2 + (d*x)/2]^5)/(8*d*(a*(1 + Cos[c + d*x]))^(5/2)*(Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4])^2)","B",1
140,1,216,145,4.04613,"\int \frac{\cos ^3(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^3/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\cos ^5\left(\frac{1}{2} (c+d x)\right) \left(128 \sin \left(\frac{c}{2}\right) \cos \left(\frac{d x}{2}\right)+128 \cos \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+\frac{21}{\left(\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)\right)^2}-\frac{21}{\left(\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)\right)^2}-\frac{1}{\left(\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)\right)^4}+\frac{1}{\left(\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)\right)^4}+150 \log \left(\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)\right)-150 \log \left(\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)\right)\right)}{8 d (a (\cos (c+d x)+1))^{5/2}}","-\frac{75 \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{9 \sin (c+d x)}{4 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{\sin (c+d x) \cos ^2(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}+\frac{13 \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]^5*(150*Log[Cos[(c + d*x)/4] - Sin[(c + d*x)/4]] - 150*Log[Cos[(c + d*x)/4] + Sin[(c + d*x)/4]] + 128*Cos[(d*x)/2]*Sin[c/2] + 128*Cos[c/2]*Sin[(d*x)/2] - (Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^(-4) + 21/(Cos[(c + d*x)/4] - Sin[(c + d*x)/4])^2 + (Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^(-4) - 21/(Cos[(c + d*x)/4] + Sin[(c + d*x)/4])^2))/(8*d*(a*(1 + Cos[c + d*x]))^(5/2))","A",1
141,1,103,107,1.1318183,"\int \frac{\cos ^2(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^2/(a + a*Cos[c + d*x])^(5/2),x]","\frac{-18 \sin (c+d x)-13 \sin (2 (c+d x))-152 \cos ^5\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{4} (c+d x)\right)+\cos \left(\frac{1}{4} (c+d x)\right)\right)\right)}{32 d (a (\cos (c+d x)+1))^{5/2}}","\frac{19 \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{13 \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}+\frac{\sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(-152*Cos[(c + d*x)/2]^5*(Log[Cos[(c + d*x)/4] - Sin[(c + d*x)/4]] - Log[Cos[(c + d*x)/4] + Sin[(c + d*x)/4]]) - 18*Sin[c + d*x] - 13*Sin[2*(c + d*x)])/(32*d*(a*(1 + Cos[c + d*x]))^(5/2))","A",1
142,1,65,107,0.248616,"\int \frac{\cos (c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]/(a + a*Cos[c + d*x])^(5/2),x]","\frac{2 \sin (c+d x)+5 \sin (2 (c+d x))+40 \cos ^5\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{32 d (a (\cos (c+d x)+1))^{5/2}}","\frac{5 \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 \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(40*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^5 + 2*Sin[c + d*x] + 5*Sin[2*(c + d*x)])/(32*d*(a*(1 + Cos[c + d*x]))^(5/2))","A",1
143,1,65,107,0.1445389,"\int \frac{1}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(a + a*Cos[c + d*x])^(-5/2),x]","\frac{14 \sin (c+d x)+3 \sin (2 (c+d x))+24 \cos ^5\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{32 d (a (\cos (c+d x)+1))^{5/2}}","\frac{3 \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 \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}+\frac{\sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(24*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^5 + 14*Sin[c + d*x] + 3*Sin[2*(c + d*x)])/(32*d*(a*(1 + Cos[c + d*x]))^(5/2))","A",1
144,1,1919,144,24.1376019,"\int \frac{\sec (c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]/(a + a*Cos[c + d*x])^(5/2),x]","\frac{43 \log \left(\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)\right) \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d (a (\cos (c+d x)+1))^{5/2}}-\frac{43 \log \left(\cos \left(\frac{c}{4}+\frac{d x}{4}\right)+\sin \left(\frac{c}{4}+\frac{d x}{4}\right)\right) \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d (a (\cos (c+d x)+1))^{5/2}}-\frac{2 \sqrt{2} \log \left(-\sqrt{2} \cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sqrt{2} \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+2\right) \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{5/2}}+\frac{8 \sqrt{2} \csc \left(\frac{c}{2}\right) \left(-d x \cos \left(\frac{c}{2}\right)+\frac{4 i \sqrt{2} \tan ^{-1}\left(\frac{2 i \cos \left(\frac{c}{2}\right)-i \left(2 \sin \left(\frac{c}{2}\right)-\sqrt{2}\right) \tan \left(\frac{d x}{4}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}\right) \cos \left(\frac{c}{2}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}+2 \log \left(2 \cos \left(\frac{d x}{2}\right) \sin \left(\frac{c}{2}\right)+2 \cos \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+\sqrt{2}\right) \sin \left(\frac{c}{2}\right)\right) \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{5/2} \left(4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)\right)}-\frac{16 i \tan ^{-1}\left(\frac{2 i \cos \left(\frac{c}{2}\right)-i \left(2 \sin \left(\frac{c}{2}\right)-\sqrt{2}\right) \tan \left(\frac{d x}{4}\right)}{\sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}\right) \cot \left(\frac{c}{2}\right) \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{5/2} \sqrt{4 \cos ^2\left(\frac{c}{2}\right)+4 \sin ^2\left(\frac{c}{2}\right)-2}}-\frac{(1+i) \log \left(\sqrt{2} \cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sqrt{2} \sin \left(\frac{c}{2}+\frac{d x}{2}\right)+2\right) \left(\sqrt{2} \cos \left(\frac{c}{4}\right)+(1+i) \cos \left(\frac{c}{4}\right)-i \sqrt{2} \sin \left(\frac{c}{4}\right)-(1-i) \sin \left(\frac{c}{4}\right)\right) \left(\sqrt{2} \cos \left(\frac{c}{4}\right)-(1+i) \cos \left(\frac{c}{4}\right)-i \sqrt{2} \sin \left(\frac{c}{4}\right)+(1-i) \sin \left(\frac{c}{4}\right)\right) \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{2} d (a (\cos (c+d x)+1))^{5/2} \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right)}+\frac{(1-i) \sqrt{2} \tan ^{-1}\left(\frac{\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sqrt{2} \sin \left(\frac{c}{4}+\frac{d x}{4}\right)+\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}{\sqrt{2} \cos \left(\frac{c}{4}+\frac{d x}{4}\right)+\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}\right) \left(\sqrt{2} \cos \left(\frac{c}{4}\right)+(1+i) \cos \left(\frac{c}{4}\right)-i \sqrt{2} \sin \left(\frac{c}{4}\right)-(1-i) \sin \left(\frac{c}{4}\right)\right) \left(\sqrt{2} \cos \left(\frac{c}{4}\right)-(1+i) \cos \left(\frac{c}{4}\right)-i \sqrt{2} \sin \left(\frac{c}{4}\right)+(1-i) \sin \left(\frac{c}{4}\right)\right) \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{5/2} \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right)}-\frac{11 \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d (a (\cos (c+d x)+1))^{5/2} \left(\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)\right)^2}+\frac{11 \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d (a (\cos (c+d x)+1))^{5/2} \left(\cos \left(\frac{c}{4}+\frac{d x}{4}\right)+\sin \left(\frac{c}{4}+\frac{d x}{4}\right)\right)^2}-\frac{(2-2 i) \left(1+e^{i c}\right) \left(\sqrt{2}-(1-i) e^{\frac{i c}{2}}+(16-16 i) e^{\frac{3 i c}{2}+i d x}+(20+20 i) \sqrt{2} e^{2 i c+\frac{3 i d x}{2}}-(34-34 i) e^{\frac{5 i c}{2}+2 i d x}-(20+20 i) \sqrt{2} e^{3 i c+\frac{5 i d x}{2}}+(16-16 i) e^{\frac{7 i c}{2}+3 i d x}+(4+4 i) \sqrt{2} e^{4 i c+\frac{7 i d x}{2}}-(1-i) e^{\frac{9 i c}{2}+4 i d x}+8 i e^{\frac{1}{2} i (c+d x)}-16 \sqrt{2} e^{i (c+d x)}-40 i e^{\frac{3}{2} i (c+d x)}+34 \sqrt{2} e^{2 i (c+d x)}+40 i e^{\frac{5}{2} i (c+d x)}-16 \sqrt{2} e^{3 i (c+d x)}-8 i e^{\frac{7}{2} i (c+d x)}+\sqrt{2} e^{4 i (c+d x)}-(4+4 i) \sqrt{2} e^{\frac{1}{2} i (2 c+d x)}\right) x \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{\left((-1-i)+\sqrt{2} e^{\frac{i c}{2}}\right) \left(-1+e^{i c}\right) \left(i-2 \sqrt{2} e^{\frac{1}{2} i (c+d x)}-4 i e^{i (c+d x)}+2 \sqrt{2} e^{\frac{3}{2} i (c+d x)}+i e^{2 i (c+d x)}\right)^2 (a (\cos (c+d x)+1))^{5/2}}-\frac{4 i \sqrt{2} \tan ^{-1}\left(\frac{\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sqrt{2} \sin \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}{\sqrt{2} \cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)}\right) \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (a (\cos (c+d x)+1))^{5/2}}-\frac{\cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d (a (\cos (c+d x)+1))^{5/2} \left(\cos \left(\frac{c}{4}+\frac{d x}{4}\right)-\sin \left(\frac{c}{4}+\frac{d x}{4}\right)\right)^4}+\frac{\cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d (a (\cos (c+d x)+1))^{5/2} \left(\cos \left(\frac{c}{4}+\frac{d x}{4}\right)+\sin \left(\frac{c}{4}+\frac{d x}{4}\right)\right)^4}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{43 \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{11 \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"((-2 + 2*I)*(1 + E^(I*c))*(Sqrt[2] - (1 - I)*E^((I/2)*c) + (16 - 16*I)*E^(((3*I)/2)*c + I*d*x) + (20 + 20*I)*Sqrt[2]*E^((2*I)*c + ((3*I)/2)*d*x) - (34 - 34*I)*E^(((5*I)/2)*c + (2*I)*d*x) - (20 + 20*I)*Sqrt[2]*E^((3*I)*c + ((5*I)/2)*d*x) + (16 - 16*I)*E^(((7*I)/2)*c + (3*I)*d*x) + (4 + 4*I)*Sqrt[2]*E^((4*I)*c + ((7*I)/2)*d*x) - (1 - I)*E^(((9*I)/2)*c + (4*I)*d*x) + (8*I)*E^((I/2)*(c + d*x)) - 16*Sqrt[2]*E^(I*(c + d*x)) - (40*I)*E^(((3*I)/2)*(c + d*x)) + 34*Sqrt[2]*E^((2*I)*(c + d*x)) + (40*I)*E^(((5*I)/2)*(c + d*x)) - 16*Sqrt[2]*E^((3*I)*(c + d*x)) - (8*I)*E^(((7*I)/2)*(c + d*x)) + Sqrt[2]*E^((4*I)*(c + d*x)) - (4 + 4*I)*Sqrt[2]*E^((I/2)*(2*c + d*x)))*x*Cos[c/2 + (d*x)/2]^5)/(((-1 - I) + Sqrt[2]*E^((I/2)*c))*(-1 + E^(I*c))*(I - 2*Sqrt[2]*E^((I/2)*(c + d*x)) - (4*I)*E^(I*(c + d*x)) + 2*Sqrt[2]*E^(((3*I)/2)*(c + d*x)) + I*E^((2*I)*(c + d*x)))^2*(a*(1 + Cos[c + d*x]))^(5/2)) - ((4*I)*Sqrt[2]*ArcTan[(Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4] - Sqrt[2]*Sin[c/4 + (d*x)/4])/(-Cos[c/4 + (d*x)/4] + Sqrt[2]*Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])]*Cos[c/2 + (d*x)/2]^5)/(d*(a*(1 + Cos[c + d*x]))^(5/2)) + (43*Cos[c/2 + (d*x)/2]^5*Log[Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4]])/(4*d*(a*(1 + Cos[c + d*x]))^(5/2)) - (43*Cos[c/2 + (d*x)/2]^5*Log[Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4]])/(4*d*(a*(1 + Cos[c + d*x]))^(5/2)) - (2*Sqrt[2]*Cos[c/2 + (d*x)/2]^5*Log[2 - Sqrt[2]*Cos[c/2 + (d*x)/2] - Sqrt[2]*Sin[c/2 + (d*x)/2]])/(d*(a*(1 + Cos[c + d*x]))^(5/2)) + ((1 - I)*Sqrt[2]*ArcTan[(Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4] - Sqrt[2]*Sin[c/4 + (d*x)/4])/(Cos[c/4 + (d*x)/4] + Sqrt[2]*Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])]*Cos[c/2 + (d*x)/2]^5*((1 + I)*Cos[c/4] + Sqrt[2]*Cos[c/4] - (1 - I)*Sin[c/4] - I*Sqrt[2]*Sin[c/4])*((-1 - I)*Cos[c/4] + Sqrt[2]*Cos[c/4] + (1 - I)*Sin[c/4] - I*Sqrt[2]*Sin[c/4]))/(d*(a*(1 + Cos[c + d*x]))^(5/2)*(Cos[c/2] + Sin[c/2])) - ((1 + I)*Cos[c/2 + (d*x)/2]^5*Log[2 + Sqrt[2]*Cos[c/2 + (d*x)/2] - Sqrt[2]*Sin[c/2 + (d*x)/2]]*((1 + I)*Cos[c/4] + Sqrt[2]*Cos[c/4] - (1 - I)*Sin[c/4] - I*Sqrt[2]*Sin[c/4])*((-1 - I)*Cos[c/4] + Sqrt[2]*Cos[c/4] + (1 - I)*Sin[c/4] - I*Sqrt[2]*Sin[c/4]))/(Sqrt[2]*d*(a*(1 + Cos[c + d*x]))^(5/2)*(Cos[c/2] + Sin[c/2])) - ((16*I)*ArcTan[((2*I)*Cos[c/2] - I*(-Sqrt[2] + 2*Sin[c/2])*Tan[(d*x)/4])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]]*Cos[c/2 + (d*x)/2]^5*Cot[c/2])/(d*(a*(1 + Cos[c + d*x]))^(5/2)*Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]) + (8*Sqrt[2]*Cos[c/2 + (d*x)/2]^5*Csc[c/2]*(-(d*x*Cos[c/2]) + 2*Log[Sqrt[2] + 2*Cos[(d*x)/2]*Sin[c/2] + 2*Cos[c/2]*Sin[(d*x)/2]]*Sin[c/2] + ((4*I)*Sqrt[2]*ArcTan[((2*I)*Cos[c/2] - I*(-Sqrt[2] + 2*Sin[c/2])*Tan[(d*x)/4])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]]*Cos[c/2])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]))/(d*(a*(1 + Cos[c + d*x]))^(5/2)*(4*Cos[c/2]^2 + 4*Sin[c/2]^2)) - Cos[c/2 + (d*x)/2]^5/(8*d*(a*(1 + Cos[c + d*x]))^(5/2)*(Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])^4) - (11*Cos[c/2 + (d*x)/2]^5)/(8*d*(a*(1 + Cos[c + d*x]))^(5/2)*(Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])^2) + Cos[c/2 + (d*x)/2]^5/(8*d*(a*(1 + Cos[c + d*x]))^(5/2)*(Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4])^4) + (11*Cos[c/2 + (d*x)/2]^5)/(8*d*(a*(1 + Cos[c + d*x]))^(5/2)*(Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4])^2)","C",0
145,1,2051,174,24.0976091,"\int \frac{\sec ^2(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^2/(a + a*Cos[c + d*x])^(5/2),x]","\text{Result too large to show}","-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}+\frac{115 \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 \tan (c+d x)}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{15 \tan (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{\tan (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"((5 - 5*I)*(1 + E^(I*c))*(Sqrt[2] - (1 - I)*E^((I/2)*c) + (16 - 16*I)*E^(((3*I)/2)*c + I*d*x) + (20 + 20*I)*Sqrt[2]*E^((2*I)*c + ((3*I)/2)*d*x) - (34 - 34*I)*E^(((5*I)/2)*c + (2*I)*d*x) - (20 + 20*I)*Sqrt[2]*E^((3*I)*c + ((5*I)/2)*d*x) + (16 - 16*I)*E^(((7*I)/2)*c + (3*I)*d*x) + (4 + 4*I)*Sqrt[2]*E^((4*I)*c + ((7*I)/2)*d*x) - (1 - I)*E^(((9*I)/2)*c + (4*I)*d*x) + (8*I)*E^((I/2)*(c + d*x)) - 16*Sqrt[2]*E^(I*(c + d*x)) - (40*I)*E^(((3*I)/2)*(c + d*x)) + 34*Sqrt[2]*E^((2*I)*(c + d*x)) + (40*I)*E^(((5*I)/2)*(c + d*x)) - 16*Sqrt[2]*E^((3*I)*(c + d*x)) - (8*I)*E^(((7*I)/2)*(c + d*x)) + Sqrt[2]*E^((4*I)*(c + d*x)) - (4 + 4*I)*Sqrt[2]*E^((I/2)*(2*c + d*x)))*x*Cos[c/2 + (d*x)/2]^5)/(((-1 - I) + Sqrt[2]*E^((I/2)*c))*(-1 + E^(I*c))*(I - 2*Sqrt[2]*E^((I/2)*(c + d*x)) - (4*I)*E^(I*(c + d*x)) + 2*Sqrt[2]*E^(((3*I)/2)*(c + d*x)) + I*E^((2*I)*(c + d*x)))^2*(a*(1 + Cos[c + d*x]))^(5/2)) + ((10*I)*Sqrt[2]*ArcTan[(Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4] - Sqrt[2]*Sin[c/4 + (d*x)/4])/(-Cos[c/4 + (d*x)/4] + Sqrt[2]*Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])]*Cos[c/2 + (d*x)/2]^5)/(d*(a*(1 + Cos[c + d*x]))^(5/2)) - (115*Cos[c/2 + (d*x)/2]^5*Log[Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4]])/(4*d*(a*(1 + Cos[c + d*x]))^(5/2)) + (115*Cos[c/2 + (d*x)/2]^5*Log[Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4]])/(4*d*(a*(1 + Cos[c + d*x]))^(5/2)) + (5*Sqrt[2]*Cos[c/2 + (d*x)/2]^5*Log[2 - Sqrt[2]*Cos[c/2 + (d*x)/2] - Sqrt[2]*Sin[c/2 + (d*x)/2]])/(d*(a*(1 + Cos[c + d*x]))^(5/2)) - ((5 - 5*I)*ArcTan[(Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4] - Sqrt[2]*Sin[c/4 + (d*x)/4])/(Cos[c/4 + (d*x)/4] + Sqrt[2]*Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])]*Cos[c/2 + (d*x)/2]^5*((1 + I)*Cos[c/4] + Sqrt[2]*Cos[c/4] - (1 - I)*Sin[c/4] - I*Sqrt[2]*Sin[c/4])*((-1 - I)*Cos[c/4] + Sqrt[2]*Cos[c/4] + (1 - I)*Sin[c/4] - I*Sqrt[2]*Sin[c/4]))/(Sqrt[2]*d*(a*(1 + Cos[c + d*x]))^(5/2)*(Cos[c/2] + Sin[c/2])) + ((5/2 + (5*I)/2)*Cos[c/2 + (d*x)/2]^5*Log[2 + Sqrt[2]*Cos[c/2 + (d*x)/2] - Sqrt[2]*Sin[c/2 + (d*x)/2]]*((1 + I)*Cos[c/4] + Sqrt[2]*Cos[c/4] - (1 - I)*Sin[c/4] - I*Sqrt[2]*Sin[c/4])*((-1 - I)*Cos[c/4] + Sqrt[2]*Cos[c/4] + (1 - I)*Sin[c/4] - I*Sqrt[2]*Sin[c/4]))/(Sqrt[2]*d*(a*(1 + Cos[c + d*x]))^(5/2)*(Cos[c/2] + Sin[c/2])) + ((40*I)*ArcTan[((2*I)*Cos[c/2] - I*(-Sqrt[2] + 2*Sin[c/2])*Tan[(d*x)/4])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]]*Cos[c/2 + (d*x)/2]^5*Cot[c/2])/(d*(a*(1 + Cos[c + d*x]))^(5/2)*Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]) - (20*Sqrt[2]*Cos[c/2 + (d*x)/2]^5*Csc[c/2]*(-(d*x*Cos[c/2]) + 2*Log[Sqrt[2] + 2*Cos[(d*x)/2]*Sin[c/2] + 2*Cos[c/2]*Sin[(d*x)/2]]*Sin[c/2] + ((4*I)*Sqrt[2]*ArcTan[((2*I)*Cos[c/2] - I*(-Sqrt[2] + 2*Sin[c/2])*Tan[(d*x)/4])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]]*Cos[c/2])/Sqrt[-2 + 4*Cos[c/2]^2 + 4*Sin[c/2]^2]))/(d*(a*(1 + Cos[c + d*x]))^(5/2)*(4*Cos[c/2]^2 + 4*Sin[c/2]^2)) + Cos[c/2 + (d*x)/2]^5/(8*d*(a*(1 + Cos[c + d*x]))^(5/2)*(Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])^4) + (19*Cos[c/2 + (d*x)/2]^5)/(8*d*(a*(1 + Cos[c + d*x]))^(5/2)*(Cos[c/4 + (d*x)/4] - Sin[c/4 + (d*x)/4])^2) - Cos[c/2 + (d*x)/2]^5/(8*d*(a*(1 + Cos[c + d*x]))^(5/2)*(Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4])^4) - (19*Cos[c/2 + (d*x)/2]^5)/(8*d*(a*(1 + Cos[c + d*x]))^(5/2)*(Cos[c/4 + (d*x)/4] + Sin[c/4 + (d*x)/4])^2) + (4*Cos[c/2 + (d*x)/2]^5)/(d*(a*(1 + Cos[c + d*x]))^(5/2)*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) - (4*Cos[c/2 + (d*x)/2]^5)/(d*(a*(1 + Cos[c + d*x]))^(5/2)*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","C",0
146,1,490,111,6.1687618,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x]),x]","a \left(-\frac{3 \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)}{10 d}-\frac{5 \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)}{21 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{23 \sin (c) \cos (d x)}{84 d}+\frac{\sin (2 c) \cos (2 d x)}{10 d}+\frac{\sin (3 c) \cos (3 d x)}{28 d}+\frac{23 \cos (c) \sin (d x)}{84 d}+\frac{\cos (2 c) \sin (2 d x)}{10 d}+\frac{\cos (3 c) \sin (3 d x)}{28 d}-\frac{3 \cot (c)}{5 d}\right)\right)","\frac{10 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{6 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{10 a \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*((-3*Cot[c])/(5*d) + (23*Cos[d*x]*Sin[c])/(84*d) + (Cos[2*d*x]*Sin[2*c])/(10*d) + (Cos[3*d*x]*Sin[3*c])/(28*d) + (23*Cos[c]*Sin[d*x])/(84*d) + (Cos[2*c]*Sin[2*d*x])/(10*d) + (Cos[3*c]*Sin[3*d*x])/(28*d)) - (5*(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*(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
147,1,232,87,5.6173492,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]),x]","\frac{a (\cos (c+d x)+1) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(-18 \cos (c) \sqrt{\sec ^2(c)} \sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)} \csc \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)-20 \sin (c) \sqrt{\csc ^2(c)} \cos (c+d x) \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \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)+2 \cos (c+d x) (10 \sin (c+d x)+3 \sin (2 (c+d x))-18 \cot (c))+\frac{9 \csc (c) \sec (c) \left(3 \cos \left(c-\tan ^{-1}(\tan (c))-d x\right)+\cos \left(c+\tan ^{-1}(\tan (c))+d x\right)\right)}{\sqrt{\sec ^2(c)}}\right)}{60 d \sqrt{\cos (c+d x)}}","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{6 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(a*(1 + Cos[c + d*x])*Sec[(c + d*x)/2]^2*((9*(3*Cos[c - d*x - ArcTan[Tan[c]]] + Cos[c + d*x + ArcTan[Tan[c]]])*Csc[c]*Sec[c])/Sqrt[Sec[c]^2] - 20*Cos[c + d*x]*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Csc[c]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]]*Sin[c] + 2*Cos[c + d*x]*(-18*Cot[c] + 10*Sin[c + d*x] + 3*Sin[2*(c + d*x)]) - 18*Cos[c]*Csc[d*x + ArcTan[Tan[c]]]*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sqrt[Sec[c]^2]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2]))/(60*d*Sqrt[Cos[c + d*x]])","C",0
148,1,222,61,5.0061138,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]),x]","\frac{a (\cos (c+d x)+1) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(-6 \cos (c) \sqrt{\sec ^2(c)} \sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)} \csc \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)-4 \sin (c) \sqrt{\csc ^2(c)} \cos (c+d x) \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \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)-4 \cos (c+d x) (3 \cot (c)-\sin (c+d x))+\frac{3 \csc (c) \sec (c) \left(3 \cos \left(c-\tan ^{-1}(\tan (c))-d x\right)+\cos \left(c+\tan ^{-1}(\tan (c))+d x\right)\right)}{\sqrt{\sec ^2(c)}}\right)}{12 d \sqrt{\cos (c+d x)}}","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(a*(1 + Cos[c + d*x])*Sec[(c + d*x)/2]^2*((3*(3*Cos[c - d*x - ArcTan[Tan[c]]] + Cos[c + d*x + ArcTan[Tan[c]]])*Csc[c]*Sec[c])/Sqrt[Sec[c]^2] - 4*Cos[c + d*x]*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Csc[c]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]]*Sin[c] - 4*Cos[c + d*x]*(3*Cot[c] - Sin[c + d*x]) - 6*Cos[c]*Csc[d*x + ArcTan[Tan[c]]]*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sqrt[Sec[c]^2]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2]))/(12*d*Sqrt[Cos[c + d*x]])","C",0
149,1,155,35,24.6445238,"\int \frac{a+a \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(a + a*Cos[c + d*x])/Sqrt[Cos[c + d*x]],x]","\frac{a \sqrt{\cos (c+d x)} (\cos (c+d x)+1) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(-\frac{\tan \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{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)}}-2 \sin (c) \sqrt{\csc ^2(c)} \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \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)+\tan \left(\tan ^{-1}(\tan (c))+d x\right)\right)}{2 d}","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(a*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[(c + d*x)/2]^2*(-2*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Csc[c]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]]*Sin[c] + Tan[d*x + ArcTan[Tan[c]]] - (HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Tan[d*x + ArcTan[Tan[c]]])/Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2]))/(2*d)","C",0
150,1,209,57,9.6735366,"\int \frac{a+a \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])/Cos[c + d*x]^(3/2),x]","\frac{a (\cos (c+d x)+1) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(2 \cos (c) \sqrt{\sec ^2(c)} \sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)} \csc \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)-4 \sin (c) \sqrt{\csc ^2(c)} \cos (c+d x) \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \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)+4 \csc (c) \cos (d x)-\frac{\csc (c) \sec (c) \left(3 \cos \left(c-\tan ^{-1}(\tan (c))-d x\right)+\cos \left(c+\tan ^{-1}(\tan (c))+d x\right)\right)}{\sqrt{\sec ^2(c)}}\right)}{4 d \sqrt{\cos (c+d x)}}","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(a*(1 + Cos[c + d*x])*Sec[(c + d*x)/2]^2*(4*Cos[d*x]*Csc[c] - ((3*Cos[c - d*x - ArcTan[Tan[c]]] + Cos[c + d*x + ArcTan[Tan[c]]])*Csc[c]*Sec[c])/Sqrt[Sec[c]^2] - 4*Cos[c + d*x]*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Csc[c]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]]*Sin[c] + 2*Cos[c]*Csc[d*x + ArcTan[Tan[c]]]*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sqrt[Sec[c]^2]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2]))/(4*d*Sqrt[Cos[c + d*x]])","C",0
151,1,444,83,6.1543179,"\int \frac{a+a \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])/Cos[c + d*x]^(5/2),x]","a \left(\frac{\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{\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{\sec (c) \sin (d x) \sec ^2(c+d x)}{3 d}+\frac{\sec (c) (\sin (c)+3 \sin (d x)) \sec (c+d x)}{3 d}+\frac{\csc (c) \sec (c)}{d}\right)\right)","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 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*((Csc[c]*Sec[c])/d + (Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (Sec[c]*Sec[c + d*x]*(Sin[c] + 3*Sin[d*x]))/(3*d)) - ((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]) + ((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
152,1,477,111,6.1733664,"\int \frac{a+a \cos (c+d x)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])/Cos[c + d*x]^(7/2),x]","a \left(\frac{3 \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)}{10 d}-\frac{\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{\sec (c) \sin (d x) \sec ^3(c+d x)}{5 d}+\frac{\sec (c) (3 \sin (c)+5 \sin (d x)) \sec ^2(c+d x)}{15 d}+\frac{\sec (c) (5 \sin (c)+9 \sin (d x)) \sec (c+d x)}{15 d}+\frac{3 \csc (c) \sec (c)}{5 d}\right)\right)","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{6 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{6 a \sin (c+d x)}{5 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*((3*Csc[c]*Sec[c])/(5*d) + (Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(5*d) + (Sec[c]*Sec[c + d*x]^2*(3*Sin[c] + 5*Sin[d*x]))/(15*d) + (Sec[c]*Sec[c + d*x]*(5*Sin[c] + 9*Sin[d*x]))/(15*d)) - ((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*(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
153,1,532,147,6.1361471,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2,x]","-\frac{4 \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)}{15 d}-\frac{5 \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)}{21 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{23 \sin (c) \cos (d x)}{84 d}+\frac{37 \sin (2 c) \cos (2 d x)}{360 d}+\frac{\sin (3 c) \cos (3 d x)}{28 d}+\frac{\sin (4 c) \cos (4 d x)}{144 d}+\frac{23 \cos (c) \sin (d x)}{84 d}+\frac{37 \cos (2 c) \sin (2 d x)}{360 d}+\frac{\cos (3 c) \sin (3 d x)}{28 d}+\frac{\cos (4 c) \sin (4 d x)}{144 d}-\frac{8 \cot (c)}{15 d}\right)","\frac{20 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{32 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{4 a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{32 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{20 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*((-8*Cot[c])/(15*d) + (23*Cos[d*x]*Sin[c])/(84*d) + (37*Cos[2*d*x]*Sin[2*c])/(360*d) + (Cos[3*d*x]*Sin[3*c])/(28*d) + (Cos[4*d*x]*Sin[4*c])/(144*d) + (23*Cos[c]*Sin[d*x])/(84*d) + (37*Cos[2*c]*Sin[2*d*x])/(360*d) + (Cos[3*c]*Sin[3*d*x])/(28*d) + (Cos[4*c]*Sin[4*d*x])/(144*d)) - (5*(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]) - (4*(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
154,1,500,121,6.1284977,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2,x]","-\frac{3 \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)}{10 d}-\frac{2 \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)}{7 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{17 \sin (c) \cos (d x)}{56 d}+\frac{\sin (2 c) \cos (2 d x)}{10 d}+\frac{\sin (3 c) \cos (3 d x)}{56 d}+\frac{17 \cos (c) \sin (d x)}{56 d}+\frac{\cos (2 c) \sin (2 d x)}{10 d}+\frac{\cos (3 c) \sin (3 d x)}{56 d}-\frac{3 \cot (c)}{5 d}\right)","\frac{8 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d}+\frac{12 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{4 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{8 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{7 d}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*((-3*Cot[c])/(5*d) + (17*Cos[d*x]*Sin[c])/(56*d) + (Cos[2*d*x]*Sin[2*c])/(10*d) + (Cos[3*d*x]*Sin[3*c])/(56*d) + (17*Cos[c]*Sin[d*x])/(56*d) + (Cos[2*c]*Sin[2*d*x])/(10*d) + (Cos[3*c]*Sin[3*d*x])/(56*d)) - (2*(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]) - (3*(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
155,1,235,95,5.6219743,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2 \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2,x]","\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(-24 \cos (c) \sqrt{\sec ^2(c)} \sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)} \csc \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)-20 \sin (c) \sqrt{\csc ^2(c)} \cos (c+d x) \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \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)+\cos (c+d x) (20 \sin (c+d x)+3 \sin (2 (c+d x))-48 \cot (c))+\frac{12 \csc (c) \sec (c) \left(3 \cos \left(c-\tan ^{-1}(\tan (c))-d x\right)+\cos \left(c+\tan ^{-1}(\tan (c))+d x\right)\right)}{\sqrt{\sec ^2(c)}}\right)}{60 d \sqrt{\cos (c+d x)}}","\frac{4 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{16 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{4 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*((12*(3*Cos[c - d*x - ArcTan[Tan[c]]] + Cos[c + d*x + ArcTan[Tan[c]]])*Csc[c]*Sec[c])/Sqrt[Sec[c]^2] - 20*Cos[c + d*x]*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Csc[c]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]]*Sin[c] + Cos[c + d*x]*(-48*Cot[c] + 20*Sin[c + d*x] + 3*Sin[2*(c + d*x)]) - 24*Cos[c]*Csc[d*x + ArcTan[Tan[c]]]*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sqrt[Sec[c]^2]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2]))/(60*d*Sqrt[Cos[c + d*x]])","C",0
156,1,224,67,5.1146954,"\int \frac{(a+a \cos (c+d x))^2}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(a + a*Cos[c + d*x])^2/Sqrt[Cos[c + d*x]],x]","\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(-6 \cos (c) \sqrt{\sec ^2(c)} \sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)} \csc \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)-8 \sin (c) \sqrt{\csc ^2(c)} \cos (c+d x) \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \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)+2 \cos (c+d x) (\sin (c+d x)-6 \cot (c))+\frac{3 \csc (c) \sec (c) \left(3 \cos \left(c-\tan ^{-1}(\tan (c))-d x\right)+\cos \left(c+\tan ^{-1}(\tan (c))+d x\right)\right)}{\sqrt{\sec ^2(c)}}\right)}{12 d \sqrt{\cos (c+d x)}}","\frac{8 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*((3*(3*Cos[c - d*x - ArcTan[Tan[c]]] + Cos[c + d*x + ArcTan[Tan[c]]])*Csc[c]*Sec[c])/Sqrt[Sec[c]^2] - 8*Cos[c + d*x]*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Csc[c]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]]*Sin[c] + 2*Cos[c + d*x]*(-6*Cot[c] + Sin[c + d*x]) - 6*Cos[c]*Csc[d*x + ArcTan[Tan[c]]]*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sqrt[Sec[c]^2]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2]))/(12*d*Sqrt[Cos[c + d*x]])","C",0
157,1,39,44,0.1665325,"\int \frac{(a+a \cos (c+d x))^2}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^2/Cos[c + d*x]^(3/2),x]","\frac{2 a^2 \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{\sin (c+d x)}{\sqrt{\cos (c+d x)}}\right)}{d}","\frac{4 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(2*a^2*(2*EllipticF[(c + d*x)/2, 2] + Sin[c + d*x]/Sqrt[Cos[c + d*x]]))/d","A",1
158,1,454,91,6.1560382,"\int \frac{(a+a \cos (c+d x))^2}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^2/Cos[c + d*x]^(5/2),x]","\frac{\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 \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{\sec (c) \sin (d x) \sec ^2(c+d x)}{6 d}+\frac{\sec (c) (\sin (c)+6 \sin (d x)) \sec (c+d x)}{6 d}+\frac{\csc (c) \sec (c)}{d}\right)","\frac{8 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*((Csc[c]*Sec[c])/d + (Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(6*d) + (Sec[c]*Sec[c + d*x]*(Sin[c] + 6*Sin[d*x]))/(6*d)) - (2*(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*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
159,1,487,121,6.1992928,"\int \frac{(a+a \cos (c+d x))^2}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^2/Cos[c + d*x]^(7/2),x]","\frac{2 \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^2 \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{5 d}-\frac{\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{\sec (c) \sin (d x) \sec ^3(c+d x)}{10 d}+\frac{\sec (c) (3 \sin (c)+10 \sin (d x)) \sec ^2(c+d x)}{30 d}+\frac{\sec (c) (5 \sin (c)+12 \sin (d x)) \sec (c+d x)}{15 d}+\frac{4 \csc (c) \sec (c)}{5 d}\right)","\frac{4 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{16 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{16 a^2 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*((4*Csc[c]*Sec[c])/(5*d) + (Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(10*d) + (Sec[c]*Sec[c + d*x]^2*(3*Sin[c] + 10*Sin[d*x]))/(30*d) + (Sec[c]*Sec[c + d*x]*(5*Sin[c] + 12*Sin[d*x]))/(15*d)) - ((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*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
160,1,532,147,6.1365251,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3,x]","-\frac{17 \csc (c) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \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)}{60 d}-\frac{11 \csc (c) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \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)}{42 d \sqrt{\cot ^2(c)+1}}+\sqrt{\cos (c+d x)} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \left(\frac{97 \sin (c) \cos (d x)}{336 d}+\frac{73 \sin (2 c) \cos (2 d x)}{720 d}+\frac{3 \sin (3 c) \cos (3 d x)}{112 d}+\frac{\sin (4 c) \cos (4 d x)}{288 d}+\frac{97 \cos (c) \sin (d x)}{336 d}+\frac{73 \cos (2 c) \sin (2 d x)}{720 d}+\frac{3 \cos (3 c) \sin (3 d x)}{112 d}+\frac{\cos (4 c) \sin (4 d x)}{288 d}-\frac{17 \cot (c)}{30 d}\right)","\frac{44 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{68 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^3 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{6 a^3 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{68 a^3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{44 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*((-17*Cot[c])/(30*d) + (97*Cos[d*x]*Sin[c])/(336*d) + (73*Cos[2*d*x]*Sin[2*c])/(720*d) + (3*Cos[3*d*x]*Sin[3*c])/(112*d) + (Cos[4*d*x]*Sin[4*c])/(288*d) + (97*Cos[c]*Sin[d*x])/(336*d) + (73*Cos[2*c]*Sin[2*d*x])/(720*d) + (3*Cos[3*c]*Sin[3*d*x])/(112*d) + (Cos[4*c]*Sin[4*d*x])/(288*d)) - (11*(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*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
161,1,500,121,6.1237005,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3 \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3,x]","-\frac{7 \csc (c) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \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)}{20 d}-\frac{13 \csc (c) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \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)}{42 d \sqrt{\cot ^2(c)+1}}+\sqrt{\cos (c+d x)} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \left(\frac{107 \sin (c) \cos (d x)}{336 d}+\frac{3 \sin (2 c) \cos (2 d x)}{40 d}+\frac{\sin (3 c) \cos (3 d x)}{112 d}+\frac{107 \cos (c) \sin (d x)}{336 d}+\frac{3 \cos (2 c) \sin (2 d x)}{40 d}+\frac{\cos (3 c) \sin (3 d x)}{112 d}-\frac{7 \cot (c)}{10 d}\right)","\frac{52 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{28 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^3 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{6 a^3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{52 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*((-7*Cot[c])/(10*d) + (107*Cos[d*x]*Sin[c])/(336*d) + (3*Cos[2*d*x]*Sin[2*c])/(40*d) + (Cos[3*d*x]*Sin[3*c])/(112*d) + (107*Cos[c]*Sin[d*x])/(336*d) + (3*Cos[2*c]*Sin[2*d*x])/(40*d) + (Cos[3*c]*Sin[3*d*x])/(112*d)) - (13*(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*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
162,1,233,91,5.7105372,"\int \frac{(a+a \cos (c+d x))^3}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(a + a*Cos[c + d*x])^3/Sqrt[Cos[c + d*x]],x]","\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(-18 \cos (c) \sqrt{\sec ^2(c)} \sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)} \csc \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)-20 \sin (c) \sqrt{\csc ^2(c)} \cos (c+d x) \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \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)+\cos (c+d x) (10 \sin (c+d x)+\sin (2 (c+d x))-36 \cot (c))+\frac{9 \csc (c) \sec (c) \left(3 \cos \left(c-\tan ^{-1}(\tan (c))-d x\right)+\cos \left(c+\tan ^{-1}(\tan (c))+d x\right)\right)}{\sqrt{\sec ^2(c)}}\right)}{40 d \sqrt{\cos (c+d x)}}","\frac{4 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{36 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{d}",1,"(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*((9*(3*Cos[c - d*x - ArcTan[Tan[c]]] + Cos[c + d*x + ArcTan[Tan[c]]])*Csc[c]*Sec[c])/Sqrt[Sec[c]^2] - 20*Cos[c + d*x]*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Csc[c]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]]*Sin[c] + Cos[c + d*x]*(-36*Cot[c] + 10*Sin[c + d*x] + Sin[2*(c + d*x)]) - 18*Cos[c]*Csc[d*x + ArcTan[Tan[c]]]*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sqrt[Sec[c]^2]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2]))/(40*d*Sqrt[Cos[c + d*x]])","C",0
163,1,240,91,4.6701939,"\int \frac{(a+a \cos (c+d x))^3}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^3/Cos[c + d*x]^(3/2),x]","\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(-6 \cos (c) \sqrt{\sec ^2(c)} \sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)} \csc \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)-20 \sin (c) \sqrt{\csc ^2(c)} \cos (c+d x) \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \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)+\sin (2 (c+d x))-3 \csc (c) \cos (d x)-9 \csc (c) \cos (2 c+d x)+9 \cot (c) \sqrt{\sec ^2(c)} \cos \left(c-\tan ^{-1}(\tan (c))-d x\right)+3 \cot (c) \sqrt{\sec ^2(c)} \cos \left(c+\tan ^{-1}(\tan (c))+d x\right)\right)}{24 d \sqrt{\cos (c+d x)}}","\frac{20 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(-3*Cos[d*x]*Csc[c] - 9*Cos[2*c + d*x]*Csc[c] + 9*Cos[c - d*x - ArcTan[Tan[c]]]*Cot[c]*Sqrt[Sec[c]^2] + 3*Cos[c + d*x + ArcTan[Tan[c]]]*Cot[c]*Sqrt[Sec[c]^2] - 20*Cos[c + d*x]*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Csc[c]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]]*Sin[c] + Sin[2*(c + d*x)] - 6*Cos[c]*Csc[d*x + ArcTan[Tan[c]]]*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sqrt[Sec[c]^2]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2]))/(24*d*Sqrt[Cos[c + d*x]])","C",0
164,1,463,91,6.1964711,"\int \frac{(a+a \cos (c+d x))^3}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^3/Cos[c + d*x]^(5/2),x]","\frac{\csc (c) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \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)}{4 d}-\frac{5 \csc (c) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \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)}{6 d \sqrt{\cot ^2(c)+1}}+\sqrt{\cos (c+d x)} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \left(\frac{\sec (c) \sin (d x) \sec ^2(c+d x)}{12 d}+\frac{\sec (c) (\sin (c)+9 \sin (d x)) \sec (c+d x)}{12 d}-\frac{(\cos (2 c)-5) \csc (c) \sec (c)}{8 d}\right)","\frac{20 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^3 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{6 a^3 \sin (c+d x)}{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*(-1/8*((-5 + Cos[2*c])*Csc[c]*Sec[c])/d + (Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(12*d) + (Sec[c]*Sec[c + d*x]*(Sin[c] + 9*Sin[d*x]))/(12*d)) - (5*(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*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
165,1,485,117,6.2147733,"\int \frac{(a+a \cos (c+d x))^3}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^3/Cos[c + d*x]^(7/2),x]","\frac{9 \csc (c) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \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)}{20 d}-\frac{\csc (c) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \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)}{2 d \sqrt{\cot ^2(c)+1}}+\sqrt{\cos (c+d x)} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \left(\frac{\sec (c) \sin (d x) \sec ^3(c+d x)}{20 d}+\frac{\sec (c) (\sin (c)+5 \sin (d x)) \sec ^2(c+d x)}{20 d}+\frac{\sec (c) (5 \sin (c)+18 \sin (d x)) \sec (c+d x)}{20 d}+\frac{9 \csc (c) \sec (c)}{10 d}\right)","\frac{4 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{36 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^3 \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^3 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{36 a^3 \sin (c+d x)}{5 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*((9*Csc[c]*Sec[c])/(10*d) + (Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(20*d) + (Sec[c]*Sec[c + d*x]^2*(Sin[c] + 5*Sin[d*x]))/(20*d) + (Sec[c]*Sec[c + d*x]*(5*Sin[c] + 18*Sin[d*x]))/(20*d)) - ((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]) + (9*(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
166,1,515,147,6.2455188,"\int \frac{(a+a \cos (c+d x))^3}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^3/Cos[c + d*x]^(9/2),x]","\frac{7 \csc (c) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \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)}{20 d}-\frac{13 \csc (c) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \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)}{42 d \sqrt{\cot ^2(c)+1}}+\sqrt{\cos (c+d x)} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^3 \left(\frac{\sec (c) \sin (d x) \sec ^4(c+d x)}{28 d}+\frac{\sec (c) (5 \sin (c)+21 \sin (d x)) \sec ^3(c+d x)}{140 d}+\frac{\sec (c) (63 \sin (c)+130 \sin (d x)) \sec ^2(c+d x)}{420 d}+\frac{\sec (c) (65 \sin (c)+147 \sin (d x)) \sec (c+d x)}{210 d}+\frac{7 \csc (c) \sec (c)}{10 d}\right)","\frac{52 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{28 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{52 a^3 \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{6 a^3 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a^3 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{28 a^3 \sin (c+d x)}{5 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*((7*Csc[c]*Sec[c])/(10*d) + (Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(28*d) + (Sec[c]*Sec[c + d*x]^3*(5*Sin[c] + 21*Sin[d*x]))/(140*d) + (Sec[c]*Sec[c + d*x]^2*(63*Sin[c] + 130*Sin[d*x]))/(420*d) + (Sec[c]*Sec[c + d*x]*(65*Sin[c] + 147*Sin[d*x]))/(210*d)) - (13*(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*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
167,1,271,173,3.6207727,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^4,x]","\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \left(\frac{59136 \sec (c) \left(\csc (c) \sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)} \left(3 \cos \left(c-\tan ^{-1}(\tan (c))-d x\right)+\cos \left(c+\tan ^{-1}(\tan (c))+d x\right)\right)-2 \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)\right)}{\sqrt{\sec ^2(c)} \sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)}}-108480 \sin (c) \sqrt{\csc ^2(c)} \cos (c+d x) \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \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)+\cos (c+d x) (122610 \sin (c+d x)+45584 \sin (2 (c+d x))+14445 \sin (3 (c+d x))+3080 \sin (4 (c+d x))+315 \sin (5 (c+d x))-236544 \cot (c))\right)}{443520 d \sqrt{\cos (c+d x)}}","\frac{904 a^4 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{128 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^4 \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{11 d}+\frac{8 a^4 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{150 a^4 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{77 d}+\frac{128 a^4 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{904 a^4 \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}",1,"(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*(-108480*Cos[c + d*x]*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Csc[c]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]]*Sin[c] + Cos[c + d*x]*(-236544*Cot[c] + 122610*Sin[c + d*x] + 45584*Sin[2*(c + d*x)] + 14445*Sin[3*(c + d*x)] + 3080*Sin[4*(c + d*x)] + 315*Sin[5*(c + d*x)]) + (59136*Sec[c]*(-2*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]] + (3*Cos[c - d*x - ArcTan[Tan[c]]] + Cos[c + d*x + ArcTan[Tan[c]]])*Csc[c]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2]))/(Sqrt[Sec[c]^2]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2])))/(443520*d*Sqrt[Cos[c + d*x]])","C",0
168,1,532,147,6.150679,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^4 \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^4,x]","-\frac{19 \csc (c) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^4 \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)}{60 d}-\frac{2 \csc (c) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^4 \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)}{7 d \sqrt{\cot ^2(c)+1}}+\sqrt{\cos (c+d x)} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^4 \left(\frac{17 \sin (c) \cos (d x)}{56 d}+\frac{127 \sin (2 c) \cos (2 d x)}{1440 d}+\frac{\sin (3 c) \cos (3 d x)}{56 d}+\frac{\sin (4 c) \cos (4 d x)}{576 d}+\frac{17 \cos (c) \sin (d x)}{56 d}+\frac{127 \cos (2 c) \sin (2 d x)}{1440 d}+\frac{\cos (3 c) \sin (3 d x)}{56 d}+\frac{\cos (4 c) \sin (4 d x)}{576 d}-\frac{19 \cot (c)}{30 d}\right)","\frac{32 a^4 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d}+\frac{152 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^4 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{8 a^4 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{122 a^4 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{32 a^4 \sin (c+d x) \sqrt{\cos (c+d x)}}{7 d}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*((-19*Cot[c])/(30*d) + (17*Cos[d*x]*Sin[c])/(56*d) + (127*Cos[2*d*x]*Sin[2*c])/(1440*d) + (Cos[3*d*x]*Sin[3*c])/(56*d) + (Cos[4*d*x]*Sin[4*c])/(576*d) + (17*Cos[c]*Sin[d*x])/(56*d) + (127*Cos[2*c]*Sin[2*d*x])/(1440*d) + (Cos[3*c]*Sin[3*d*x])/(56*d) + (Cos[4*c]*Sin[4*d*x])/(576*d)) - (2*(a + a*Cos[c + d*x])^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[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]) - (19*(a + a*Cos[c + d*x])^4*Csc[c]*Sec[c/2 + (d*x)/2]^8*((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
169,1,500,121,6.1675177,"\int \frac{(a+a \cos (c+d x))^4}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(a + a*Cos[c + d*x])^4/Sqrt[Cos[c + d*x]],x]","-\frac{2 \csc (c) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^4 \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{5 d}-\frac{17 \csc (c) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^4 \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)}{42 d \sqrt{\cot ^2(c)+1}}+\sqrt{\cos (c+d x)} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^4 \left(\frac{191 \sin (c) \cos (d x)}{672 d}+\frac{\sin (2 c) \cos (2 d x)}{20 d}+\frac{\sin (3 c) \cos (3 d x)}{224 d}+\frac{191 \cos (c) \sin (d x)}{672 d}+\frac{\cos (2 c) \sin (2 d x)}{20 d}+\frac{\cos (3 c) \sin (3 d x)}{224 d}-\frac{4 \cot (c)}{5 d}\right)","\frac{136 a^4 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{64 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^4 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{8 a^4 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{94 a^4 \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*((-4*Cot[c])/(5*d) + (191*Cos[d*x]*Sin[c])/(672*d) + (Cos[2*d*x]*Sin[2*c])/(20*d) + (Cos[3*d*x]*Sin[3*c])/(224*d) + (191*Cos[c]*Sin[d*x])/(672*d) + (Cos[2*c]*Sin[2*d*x])/(20*d) + (Cos[3*c]*Sin[3*d*x])/(224*d)) - (17*(a + a*Cos[c + d*x])^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[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]) - (2*(a + a*Cos[c + d*x])^4*Csc[c]*Sec[c/2 + (d*x)/2]^8*((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
170,1,497,119,6.2011211,"\int \frac{(a+a \cos (c+d x))^4}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^4/Cos[c + d*x]^(3/2),x]","-\frac{7 \csc (c) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^4 \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)}{20 d}-\frac{2 \csc (c) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^4 \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 ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \cos (c+d x)+a)^4 \left(\frac{\sin (c) \cos (d x)}{6 d}+\frac{\sin (2 c) \cos (2 d x)}{80 d}+\frac{\cos (c) \sin (d x)}{6 d}+\frac{\cos (2 c) \sin (2 d x)}{80 d}+\frac{\sec (c) \sin (d x) \sec (c+d x)}{8 d}-\frac{(33 \cos (2 c)+23) \csc (c) \sec (c)}{80 d}\right)","\frac{32 a^4 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{56 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^4 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{8 a^4 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a^4 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(-1/80*((23 + 33*Cos[2*c])*Csc[c]*Sec[c])/d + (Cos[d*x]*Sin[c])/(6*d) + (Cos[2*d*x]*Sin[2*c])/(80*d) + (Cos[c]*Sin[d*x])/(6*d) + (Sec[c]*Sec[c + d*x]*Sin[d*x])/(8*d) + (Cos[2*c]*Sin[2*d*x])/(80*d)) - (2*(a + a*Cos[c + d*x])^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[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]) - (7*(a + a*Cos[c + d*x])^4*Csc[c]*Sec[c/2 + (d*x)/2]^8*((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
171,1,70,98,0.3162154,"\int \frac{(a+a \cos (c+d x))^4}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^4/Cos[c + d*x]^(5/2),x]","\frac{a^4 \left(5 \sin (c+d x)+24 \sin (2 (c+d x))+\sin (3 (c+d x))+80 \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{6 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{40 a^4 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^4 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{8 a^4 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(a^4*(80*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 5*Sin[c + d*x] + 24*Sin[2*(c + d*x)] + Sin[3*(c + d*x)]))/(6*d*Cos[c + d*x]^(3/2))","A",1
172,1,283,121,4.3404519,"\int \frac{(a+a \cos (c+d x))^4}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^4/Cos[c + d*x]^(7/2),x]","\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \left(-\frac{168 \sec (c) \cos ^2(c+d x) \left(\csc (c) \sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)} \left(3 \cos \left(c-\tan ^{-1}(\tan (c))-d x\right)+\cos \left(c+\tan ^{-1}(\tan (c))+d x\right)\right)-2 \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)\right)}{\sqrt{\sec ^2(c)} \sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)}}-640 \sin (c) \sqrt{\csc ^2(c)} \cos ^3(c+d x) \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \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)+\csc (c) (141 \cos (2 c+d x)+40 \cos (c+2 d x)-40 \cos (3 c+2 d x)+183 \cos (2 c+3 d x)-15 \cos (4 c+3 d x)+363 \cos (d x))\right)}{960 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{32 a^4 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{56 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 a^4 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{66 a^4 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}",1,"(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*((363*Cos[d*x] + 141*Cos[2*c + d*x] + 40*Cos[c + 2*d*x] - 40*Cos[3*c + 2*d*x] + 183*Cos[2*c + 3*d*x] - 15*Cos[4*c + 3*d*x])*Csc[c] - 640*Cos[c + d*x]^3*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Csc[c]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]]*Sin[c] - (168*Cos[c + d*x]^2*Sec[c]*(-2*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]] + (3*Cos[c - d*x - ArcTan[Tan[c]]] + Cos[c + d*x + ArcTan[Tan[c]]])*Csc[c]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2]))/(Sqrt[Sec[c]^2]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2])))/(960*d*Cos[c + d*x]^(5/2))","C",0
173,1,298,147,5.1416935,"\int \frac{(a+a \cos (c+d x))^4}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^4/Cos[c + d*x]^(9/2),x]","\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \left(-\frac{1344 \sec (c) \cos ^3(c+d x) \left(\csc (c) \sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)} \left(3 \cos \left(c-\tan ^{-1}(\tan (c))-d x\right)+\cos \left(c+\tan ^{-1}(\tan (c))+d x\right)\right)-2 \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)\right)}{\sqrt{\sec ^2(c)} \sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)}}-2720 \sin (c) \sqrt{\csc ^2(c)} \cos ^4(c+d x) \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \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)+\csc (c) (-295 \cos (2 c+d x)+2184 \cos (c+2 d x)+504 \cos (3 c+2 d x)+235 \cos (2 c+3 d x)-235 \cos (4 c+3 d x)+672 \cos (3 c+4 d x)+2016 \cos (c)+295 \cos (d x))\right)}{6720 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{136 a^4 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{64 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{94 a^4 \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{8 a^4 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{64 a^4 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}",1,"(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*((2016*Cos[c] + 295*Cos[d*x] - 295*Cos[2*c + d*x] + 2184*Cos[c + 2*d*x] + 504*Cos[3*c + 2*d*x] + 235*Cos[2*c + 3*d*x] - 235*Cos[4*c + 3*d*x] + 672*Cos[3*c + 4*d*x])*Csc[c] - 2720*Cos[c + d*x]^4*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Csc[c]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]]*Sin[c] - (1344*Cos[c + d*x]^3*Sec[c]*(-2*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]] + (3*Cos[c - d*x - ArcTan[Tan[c]]] + Cos[c + d*x + ArcTan[Tan[c]]])*Csc[c]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2]))/(Sqrt[Sec[c]^2]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2])))/(6720*d*Cos[c + d*x]^(7/2))","C",0
174,1,315,128,1.8085147,"\int \frac{\cos ^{\frac{7}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x]),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(-\frac{2 \csc (c) \sqrt{\cos (c+d x)} \left(5 \sin (2 c) \sin (d x)+10 \sin ^2(c) \cos (d x)-6 \cos (c) \left(\sin ^2(c) \cos (2 d x)-8\right)-3 \sin (c) \cos (2 c) \sin (2 d x)+30 \sin \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)+15\right)}{d}+\frac{2 i \sqrt{2} e^{-i (c+d x)} \left(63 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+25 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+63 \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{15 a (\cos (c+d x)+1)}","-\frac{5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{21 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{7 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}-\frac{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"(Cos[(c + d*x)/2]^2*(((2*I)*Sqrt[2]*(63*(1 + E^((2*I)*(c + d*x))) + 63*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 25*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) - (2*Sqrt[Cos[c + d*x]]*Csc[c]*(15 + 10*Cos[d*x]*Sin[c]^2 - 6*Cos[c]*(-8 + Cos[2*d*x]*Sin[c]^2) + 30*Sec[(c + d*x)/2]*Sin[c/2]*Sin[(d*x)/2] + 5*Sin[2*c]*Sin[d*x] - 3*Cos[2*c]*Sin[c]*Sin[2*d*x]))/d))/(15*a*(1 + Cos[c + d*x]))","C",1
175,1,289,100,1.2845352,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x]),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(\frac{2 \csc (c) \sqrt{\cos (c+d x)} \left(\sin (2 c) \sin (d x)+2 \sin ^2(c) \cos (d x)+6 \sin \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)+6 \cos (c)+3\right)}{d}-\frac{2 i \sqrt{2} e^{-i (c+d x)} \left(9 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+9 \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{3 a (\cos (c+d x)+1)}","\frac{5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"(Cos[(c + d*x)/2]^2*(((-2*I)*Sqrt[2]*(9*(1 + E^((2*I)*(c + d*x))) + 9*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) + (2*Sqrt[Cos[c + d*x]]*Csc[c]*(3 + 6*Cos[c] + 2*Cos[d*x]*Sin[c]^2 + 6*Sec[(c + d*x)/2]*Sin[c/2]*Sin[(d*x)/2] + Sin[2*c]*Sin[d*x]))/d))/(3*a*(1 + Cos[c + d*x]))","C",1
176,1,264,72,2.6421585,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^(3/2)/(a + a*Cos[c + d*x]),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(-\frac{2 \sqrt{\cos (c+d x)} \left(\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)+2 \cot (c)+\csc (c)\right)}{d}+\frac{2 i \sqrt{2} e^{-i (c+d x)} \left(3 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+\left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+3 \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{a (\cos (c+d x)+1)}","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \cos (c+d x)+a)}",1,"(Cos[(c + d*x)/2]^2*(((2*I)*Sqrt[2]*(3*(1 + E^((2*I)*(c + d*x))) + 3*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) - (2*Sqrt[Cos[c + d*x]]*(2*Cot[c] + Csc[c] + Sec[c/2]*Sec[(c + d*x)/2]*Sin[(d*x)/2]))/d))/(a*(1 + Cos[c + d*x]))","C",1
177,1,256,70,1.0542097,"\int \frac{\sqrt{\cos (c+d x)}}{a+a \cos (c+d x)} \, dx","Integrate[Sqrt[Cos[c + d*x]]/(a + a*Cos[c + d*x]),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(\frac{2 \sqrt{\cos (c+d x)} \left(\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)+\csc (c)\right)}{d}-\frac{2 i \sqrt{2} e^{-i (c+d x)} \left(\left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+\left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+e^{2 i (c+d x)}+1\right)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{a (\cos (c+d x)+1)}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \cos (c+d x)+a)}",1,"(Cos[(c + d*x)/2]^2*(((-2*I)*Sqrt[2]*(1 + E^((2*I)*(c + d*x)) + (-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) + (2*Sqrt[Cos[c + d*x]]*(Csc[c] + Sec[c/2]*Sec[(c + d*x)/2]*Sin[(d*x)/2]))/d))/(a*(1 + Cos[c + d*x]))","C",1
178,1,257,70,1.0387247,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(-\frac{2 \sqrt{\cos (c+d x)} \left(\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)+\csc (c)\right)}{d}+\frac{2 i \sqrt{2} e^{-i (c+d x)} \left(\left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-\left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+e^{2 i (c+d x)}+1\right)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{a (\cos (c+d x)+1)}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \cos (c+d x)+a)}",1,"(Cos[(c + d*x)/2]^2*(((2*I)*Sqrt[2]*(1 + E^((2*I)*(c + d*x)) + (-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) - (2*Sqrt[Cos[c + d*x]]*(Csc[c] + Sec[c/2]*Sec[(c + d*x)/2]*Sin[(d*x)/2]))/d))/(a*(1 + Cos[c + d*x]))","C",1
179,1,297,96,2.0906285,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))} \, dx","Integrate[1/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(2 \cos \left(\frac{1}{2} (c-d x)\right)+\cos \left(\frac{1}{2} (3 c+d x)\right)+3 \cos \left(\frac{1}{2} (c+3 d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right)}{2 d \sqrt{\cos (c+d x)}}-\frac{2 i \sqrt{2} e^{-i (c+d x)} \left(3 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-\left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+3 \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{a (\cos (c+d x)+1)}","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{3 \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x)}{d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)}",1,"(Cos[(c + d*x)/2]^2*(((-2*I)*Sqrt[2]*(3*(1 + E^((2*I)*(c + d*x))) + 3*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) + ((2*Cos[(c - d*x)/2] + Cos[(3*c + d*x)/2] + 3*Cos[(c + 3*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2])/(2*d*Sqrt[Cos[c + d*x]])))/(a*(1 + Cos[c + d*x]))","C",1
180,1,332,124,3.9385615,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(-\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(10 \cos \left(\frac{1}{2} (c-d x)\right)+8 \cos \left(\frac{1}{2} (3 c+d x)\right)+4 \cos \left(\frac{1}{2} (c+3 d x)\right)+5 \cos \left(\frac{1}{2} (5 c+3 d x)\right)+9 \cos \left(\frac{1}{2} (3 c+5 d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right)}{4 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 i \sqrt{2} e^{-i (c+d x)} \left(9 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+9 \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{3 a (\cos (c+d x)+1)}","\frac{5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{\sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{5 \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{3 \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}",1,"(Cos[(c + d*x)/2]^2*(((2*I)*Sqrt[2]*(9*(1 + E^((2*I)*(c + d*x))) + 9*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) - ((10*Cos[(c - d*x)/2] + 8*Cos[(3*c + d*x)/2] + 4*Cos[(c + 3*d*x)/2] + 5*Cos[(5*c + 3*d*x)/2] + 9*Cos[(3*c + 5*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2])/(4*d*Cos[c + d*x]^(3/2))))/(3*a*(1 + Cos[c + d*x]))","C",1
181,1,367,160,2.6250498,"\int \frac{\cos ^{\frac{9}{2}}(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^(9/2)/(a + a*Cos[c + d*x])^2,x]","\frac{\cos ^4\left(\frac{1}{2} (c+d x)\right) \left(-\frac{2 \csc (c) \sqrt{\cos (c+d x)} \left(40 \sin ^2(c) \cos (d x)-6 \sin (c) \sin (2 c) \cos (2 d x)+8 \cos (c) (5 \sin (c) \sin (d x)+27)-6 \sin (c) \cos (2 c) \sin (2 d x)-10 \sin \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right)+240 \sin \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)-5 \sin (c) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)+120\right)}{3 d}+\frac{4 i \sqrt{2} e^{-i (c+d x)} \left(56 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+25 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+56 \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{5 a^2 (\cos (c+d x)+1)^2}","-\frac{5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{56 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{3 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{56 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a^2 d}-\frac{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d}-\frac{\sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]^4*(((4*I)*Sqrt[2]*(56*(1 + E^((2*I)*(c + d*x))) + 56*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 25*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) - (2*Sqrt[Cos[c + d*x]]*Csc[c]*(120 + 40*Cos[d*x]*Sin[c]^2 - 6*Cos[2*d*x]*Sin[c]*Sin[2*c] + 240*Sec[(c + d*x)/2]*Sin[c/2]*Sin[(d*x)/2] - 10*Sec[(c + d*x)/2]^3*Sin[c/2]*Sin[(d*x)/2] + 8*Cos[c]*(27 + 5*Sin[c]*Sin[d*x]) - 6*Cos[2*c]*Sin[c]*Sin[2*d*x] - 5*Sec[(c + d*x)/2]^2*Sin[c]*Tan[c/2]))/(3*d)))/(5*a^2*(1 + Cos[c + d*x])^2)","C",1
182,1,337,138,2.0278814,"\int \frac{\cos ^{\frac{7}{2}}(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^2,x]","\frac{\cos ^4\left(\frac{1}{2} (c+d x)\right) \left(\frac{\csc (c) \sqrt{\cos (c+d x)} \left(72 \cos \left(\frac{1}{2} (c-d x)\right)+54 \cos \left(\frac{1}{2} (3 c+d x)\right)+33 \cos \left(\frac{1}{2} (c+3 d x)\right)+9 \cos \left(\frac{1}{2} (5 c+3 d x)\right)+\cos \left(\frac{1}{2} (3 c+5 d x)\right)-\cos \left(\frac{1}{2} (7 c+5 d x)\right)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right)}{2 d}-\frac{4 i \sqrt{2} e^{-i (c+d x)} \left(21 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+10 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+21 \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{3 a^2 (\cos (c+d x)+1)^2}","\frac{10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{7 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{7 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{10 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}-\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]^4*(((-4*I)*Sqrt[2]*(21*(1 + E^((2*I)*(c + d*x))) + 21*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 10*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) + (Sqrt[Cos[c + d*x]]*(72*Cos[(c - d*x)/2] + 54*Cos[(3*c + d*x)/2] + 33*Cos[(c + 3*d*x)/2] + 9*Cos[(5*c + 3*d*x)/2] + Cos[(3*c + 5*d*x)/2] - Cos[(7*c + 5*d*x)/2])*Csc[c]*Sec[(c + d*x)/2]^3)/(2*d)))/(3*a^2*(1 + Cos[c + d*x])^2)","C",1
183,1,319,112,2.8825781,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^2,x]","\frac{\cos ^4\left(\frac{1}{2} (c+d x)\right) \left(-\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sqrt{\cos (c+d x)} \left(20 \cos \left(\frac{1}{2} (c-d x)\right)+16 \cos \left(\frac{1}{2} (3 c+d x)\right)+9 \cos \left(\frac{1}{2} (c+3 d x)\right)+3 \cos \left(\frac{1}{2} (5 c+3 d x)\right)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right)}{2 d}+\frac{4 i \sqrt{2} e^{-i (c+d x)} \left(12 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+12 \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{3 a^2 (\cos (c+d x)+1)^2}","-\frac{5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d (\cos (c+d x)+1)}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]^4*(((4*I)*Sqrt[2]*(12*(1 + E^((2*I)*(c + d*x))) + 12*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) - (Sqrt[Cos[c + d*x]]*(20*Cos[(c - d*x)/2] + 16*Cos[(3*c + d*x)/2] + 9*Cos[(c + 3*d*x)/2] + 3*Cos[(5*c + 3*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^3)/(2*d)))/(3*a^2*(1 + Cos[c + d*x])^2)","C",1
184,1,640,109,6.318632,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^2,x]","-\frac{4 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos ^4\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} (a \cos (c+d x)+a)^2}-\frac{i \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{2 e^{2 i d x} \sqrt{e^{-i d x} \left(2 i \sin (c) \left(-1+e^{2 i d x}\right)+2 \cos (c) \left(1+e^{2 i d x}\right)\right)} \sqrt{i \sin (2 c) e^{2 i d x}+\cos (2 c) e^{2 i d x}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)}{3 i d \cos (c) \left(1+e^{2 i d x}\right)-3 d \sin (c) \left(-1+e^{2 i d x}\right)}-\frac{2 \sqrt{e^{-i d x} \left(2 i \sin (c) \left(-1+e^{2 i d x}\right)+2 \cos (c) \left(1+e^{2 i d x}\right)\right)} \sqrt{i \sin (2 c) e^{2 i d x}+\cos (2 c) e^{2 i d x}+1} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)}{d \sin (c) \left(-1+e^{2 i d x}\right)-i d \cos (c) \left(1+e^{2 i d x}\right)}\right)}{2 (a \cos (c+d x)+a)^2}+\frac{\sqrt{\cos (c+d x)} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \left(-\frac{2 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 \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) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{4 \csc (c)}{d}\right)}{(a \cos (c+d x)+a)^2}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d (\cos (c+d x)+1)}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \cos (c+d x)+a)^2}",1,"((-1/2*I)*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*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*Csc[c])/d + (4*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*Sin[(d*x)/2])/(3*d) - (2*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(a + a*Cos[c + d*x])^2","C",1
185,1,63,57,0.2149448,"\int \frac{\sqrt{\cos (c+d x)}}{(a+a \cos (c+d x))^2} \, dx","Integrate[Sqrt[Cos[c + d*x]]/(a + a*Cos[c + d*x])^2,x]","\frac{4 \cos ^4\left(\frac{1}{2} (c+d x)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d (\cos (c+d x)+1)^2}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \cos (c+d x)+a)^2}",1,"(4*Cos[(c + d*x)/2]^4*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])^2)","A",1
186,1,304,109,2.0432135,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2),x]","\frac{\cos ^4\left(\frac{1}{2} (c+d x)\right) \left(-\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sqrt{\cos (c+d x)} \left(7 \cos \left(\frac{1}{2} (c-d x)\right)+2 \cos \left(\frac{1}{2} (3 c+d x)\right)+3 \cos \left(\frac{1}{2} (c+3 d x)\right)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right)}{2 d}+\frac{4 i \sqrt{2} e^{-i (c+d x)} \left(3 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-2 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+3 \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{3 a^2 (\cos (c+d x)+1)^2}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d (\cos (c+d x)+1)}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]^4*(((4*I)*Sqrt[2]*(3*(1 + E^((2*I)*(c + d*x))) + 3*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 2*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) - (Sqrt[Cos[c + d*x]]*(7*Cos[(c - d*x)/2] + 2*Cos[(3*c + d*x)/2] + 3*Cos[(c + 3*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^3)/(2*d)))/(3*a^2*(1 + Cos[c + d*x])^2)","C",1
187,1,334,136,1.9602502,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2} \, dx","Integrate[1/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2),x]","\frac{\cos ^4\left(\frac{1}{2} (c+d x)\right) \left(\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(29 \cos \left(\frac{1}{2} (c-d x)\right)+19 \cos \left(\frac{1}{2} (3 c+d x)\right)+31 \cos \left(\frac{1}{2} (c+3 d x)\right)+5 \cos \left(\frac{1}{2} (5 c+3 d x)\right)+12 \cos \left(\frac{1}{2} (3 c+5 d x)\right)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right)}{4 d \sqrt{\cos (c+d x)}}-\frac{4 i \sqrt{2} e^{-i (c+d x)} \left(12 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+12 \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{3 a^2 (\cos (c+d x)+1)^2}","-\frac{5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{4 \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{5 \sin (c+d x)}{3 a^2 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1)}-\frac{\sin (c+d x)}{3 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]^4*(((-4*I)*Sqrt[2]*(12*(1 + E^((2*I)*(c + d*x))) + 12*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) + ((29*Cos[(c - d*x)/2] + 19*Cos[(3*c + d*x)/2] + 31*Cos[(c + 3*d*x)/2] + 5*Cos[(5*c + 3*d*x)/2] + 12*Cos[(3*c + 5*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^3)/(4*d*Sqrt[Cos[c + d*x]])))/(3*a^2*(1 + Cos[c + d*x])^2)","C",1
188,1,364,162,5.8464296,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^2} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2),x]","\frac{\cos ^4\left(\frac{1}{2} (c+d x)\right) \left(-\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(82 \cos \left(\frac{1}{2} (c-d x)\right)+65 \cos \left(\frac{1}{2} (3 c+d x)\right)+68 \cos \left(\frac{1}{2} (c+3 d x)\right)+37 \cos \left(\frac{1}{2} (5 c+3 d x)\right)+53 \cos \left(\frac{1}{2} (3 c+5 d x)\right)+10 \cos \left(\frac{1}{2} (7 c+5 d x)\right)+21 \cos \left(\frac{1}{2} (5 c+7 d x)\right)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right)}{8 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 i \sqrt{2} e^{-i (c+d x)} \left(21 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-10 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+21 \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{3 a^2 (\cos (c+d x)+1)^2}","\frac{10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{7 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{7 \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x) (\cos (c+d x)+1)}+\frac{10 \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{7 \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]^4*(((4*I)*Sqrt[2]*(21*(1 + E^((2*I)*(c + d*x))) + 21*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 10*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) - ((82*Cos[(c - d*x)/2] + 65*Cos[(3*c + d*x)/2] + 68*Cos[(c + 3*d*x)/2] + 37*Cos[(5*c + 3*d*x)/2] + 53*Cos[(3*c + 5*d*x)/2] + 10*Cos[(7*c + 5*d*x)/2] + 21*Cos[(5*c + 7*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^3)/(8*d*Cos[c + d*x]^(3/2))))/(3*a^2*(1 + Cos[c + d*x])^2)","C",1
189,1,388,207,2.7470799,"\int \frac{\cos ^{\frac{11}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^(11/2)/(a + a*Cos[c + d*x])^3,x]","\frac{2 \cos ^6\left(\frac{1}{2} (c+d x)\right) \left(-\sqrt{\cos (c+d x)} \left(\frac{1}{16} \sec \left(\frac{c}{2}\right) \left(-770 \sin \left(c+\frac{d x}{2}\right)+840 \sin \left(c+\frac{3 d x}{2}\right)-150 \sin \left(2 c+\frac{3 d x}{2}\right)+238 \sin \left(2 c+\frac{5 d x}{2}\right)+40 \sin \left(3 c+\frac{5 d x}{2}\right)+5 \sin \left(3 c+\frac{7 d x}{2}\right)+5 \sin \left(4 c+\frac{7 d x}{2}\right)-\sin \left(4 c+\frac{9 d x}{2}\right)-\sin \left(5 c+\frac{9 d x}{2}\right)+1210 \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right)+264 \cot (c)+198 \csc (c)\right)+\frac{42 i \sqrt{2} e^{-i (c+d x)} \left(11 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+11 \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{5 a^3 d (\cos (c+d x)+1)^3}","-\frac{21 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{231 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{63 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{77 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{10 a^3 d}-\frac{21 \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a^3 d}-\frac{\sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{4 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{5 a d (a \cos (c+d x)+a)^2}",1,"(2*Cos[(c + d*x)/2]^6*(((42*I)*Sqrt[2]*(11*(1 + E^((2*I)*(c + d*x))) + 11*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) - Sqrt[Cos[c + d*x]]*(264*Cot[c] + 198*Csc[c] + (Sec[c/2]*Sec[(c + d*x)/2]^5*(1210*Sin[(d*x)/2] - 770*Sin[c + (d*x)/2] + 840*Sin[c + (3*d*x)/2] - 150*Sin[2*c + (3*d*x)/2] + 238*Sin[2*c + (5*d*x)/2] + 40*Sin[3*c + (5*d*x)/2] + 5*Sin[3*c + (7*d*x)/2] + 5*Sin[4*c + (7*d*x)/2] - Sin[4*c + (9*d*x)/2] - Sin[5*c + (9*d*x)/2]))/16)))/(5*a^3*d*(1 + Cos[c + d*x])^3)","C",1
190,1,369,181,1.8959948,"\int \frac{\cos ^{\frac{9}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^(9/2)/(a + a*Cos[c + d*x])^3,x]","\frac{\cos ^6\left(\frac{1}{2} (c+d x)\right) \left(\frac{\csc (c) \sqrt{\cos (c+d x)} \left(1961 \cos \left(\frac{1}{2} (c-d x)\right)+1609 \cos \left(\frac{1}{2} (3 c+d x)\right)+1165 \cos \left(\frac{1}{2} (c+3 d x)\right)+620 \cos \left(\frac{1}{2} (5 c+3 d x)\right)+292 \cos \left(\frac{1}{2} (3 c+5 d x)\right)+65 \cos \left(\frac{1}{2} (7 c+5 d x)\right)+5 \cos \left(\frac{1}{2} (5 c+7 d x)\right)-5 \cos \left(\frac{1}{2} (9 c+7 d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right)}{12 d}-\frac{4 i \sqrt{2} e^{-i (c+d x)} \left(119 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+55 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+119 \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{5 a^3 (\cos (c+d x)+1)^3}","\frac{11 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{119 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{119 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{11 \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a^3 d}-\frac{\sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 a d (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]^6*(((-4*I)*Sqrt[2]*(119*(1 + E^((2*I)*(c + d*x))) + 119*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 55*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) + (Sqrt[Cos[c + d*x]]*(1961*Cos[(c - d*x)/2] + 1609*Cos[(3*c + d*x)/2] + 1165*Cos[(c + 3*d*x)/2] + 620*Cos[(5*c + 3*d*x)/2] + 292*Cos[(3*c + 5*d*x)/2] + 65*Cos[(7*c + 5*d*x)/2] + 5*Cos[(5*c + 7*d*x)/2] - 5*Cos[(9*c + 7*d*x)/2])*Csc[c]*Sec[(c + d*x)/2]^5)/(12*d)))/(5*a^3*(1 + Cos[c + d*x])^3)","C",1
191,1,349,155,4.1957234,"\int \frac{\cos ^{\frac{7}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^3,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) \sqrt{\cos (c+d x)} \left(806 \cos \left(\frac{1}{2} (c-d x)\right)+664 \cos \left(\frac{1}{2} (3 c+d x)\right)+470 \cos \left(\frac{1}{2} (c+3 d x)\right)+265 \cos \left(\frac{1}{2} (5 c+3 d x)\right)+117 \cos \left(\frac{1}{2} (3 c+5 d x)\right)+30 \cos \left(\frac{1}{2} (7 c+5 d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{4 i \sqrt{2} e^{-i (c+d x)} \left(147 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+65 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+147 \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{15 a^3 (\cos (c+d x)+1)^3}","-\frac{13 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{49 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{13 \sin (c+d x) \sqrt{\cos (c+d x)}}{6 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{8 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]^6*(((4*I)*Sqrt[2]*(147*(1 + E^((2*I)*(c + d*x))) + 147*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 65*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) - (Sqrt[Cos[c + d*x]]*(806*Cos[(c - d*x)/2] + 664*Cos[(3*c + d*x)/2] + 470*Cos[(c + 3*d*x)/2] + 265*Cos[(5*c + 3*d*x)/2] + 117*Cos[(3*c + 5*d*x)/2] + 30*Cos[(7*c + 5*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^5)/(8*d)))/(15*a^3*(1 + Cos[c + d*x])^3)","C",1
192,1,705,155,6.4133008,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^3,x]","-\frac{2 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos ^6\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} (a \cos (c+d x)+a)^3}-\frac{9 i \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{2 e^{2 i d x} \sqrt{e^{-i d x} \left(2 i \sin (c) \left(-1+e^{2 i d x}\right)+2 \cos (c) \left(1+e^{2 i d x}\right)\right)} \sqrt{i \sin (2 c) e^{2 i d x}+\cos (2 c) e^{2 i d x}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)}{3 i d \cos (c) \left(1+e^{2 i d x}\right)-3 d \sin (c) \left(-1+e^{2 i d x}\right)}-\frac{2 \sqrt{e^{-i d x} \left(2 i \sin (c) \left(-1+e^{2 i d x}\right)+2 \cos (c) \left(1+e^{2 i d x}\right)\right)} \sqrt{i \sin (2 c) e^{2 i d x}+\cos (2 c) e^{2 i d x}+1} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)}{d \sin (c) \left(-1+e^{2 i d x}\right)-i d \cos (c) \left(1+e^{2 i d x}\right)}\right)}{10 (a \cos (c+d x)+a)^3}+\frac{\sqrt{\cos (c+d x)} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{2 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{2 \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{12 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{12 \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{36 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{36 \csc (c)}{5 d}\right)}{(a \cos (c+d x)+a)^3}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{9 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{9 \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{5 a d (a \cos (c+d x)+a)^2}",1,"(((-9*I)/10)*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*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]]*((36*Csc[c])/(5*d) + (36*Sec[c/2]*Sec[c/2 + (d*x)/2]*Sin[(d*x)/2])/(5*d) - (12*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*Sin[(d*x)/2])/(5*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*Sin[(d*x)/2])/(5*d) - (12*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(5*d) + (2*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",1
193,1,334,155,3.9494573,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^3,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) \sqrt{\cos (c+d x)} \left(14 \cos \left(\frac{1}{2} (c-d x)\right)+16 \cos \left(\frac{1}{2} (3 c+d x)\right)+20 \cos \left(\frac{1}{2} (c+3 d x)\right)-5 \cos \left(\frac{1}{2} (5 c+3 d x)\right)+3 \cos \left(\frac{1}{2} (3 c+5 d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right)}{8 d}-\frac{4 i \sqrt{2} e^{-i (c+d x)} \left(3 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+3 \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{15 a^3 (\cos (c+d x)+1)^3}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]^6*(((-4*I)*Sqrt[2]*(3*(1 + E^((2*I)*(c + d*x))) + 3*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) + (Sqrt[Cos[c + d*x]]*(14*Cos[(c - d*x)/2] + 16*Cos[(3*c + d*x)/2] + 20*Cos[(c + 3*d*x)/2] - 5*Cos[(5*c + 3*d*x)/2] + 3*Cos[(3*c + 5*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^5)/(8*d)))/(15*a^3*(1 + Cos[c + d*x])^3)","C",1
194,1,334,155,3.3929924,"\int \frac{\sqrt{\cos (c+d x)}}{(a+a \cos (c+d x))^3} \, dx","Integrate[Sqrt[Cos[c + d*x]]/(a + a*Cos[c + d*x])^3,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) \sqrt{\cos (c+d x)} \left(4 \cos \left(\frac{1}{2} (c-d x)\right)+26 \cos \left(\frac{1}{2} (3 c+d x)\right)+10 \cos \left(\frac{1}{2} (c+3 d x)\right)+5 \cos \left(\frac{1}{2} (5 c+3 d x)\right)+3 \cos \left(\frac{1}{2} (3 c+5 d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{4 i \sqrt{2} e^{-i (c+d x)} \left(3 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+3 \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{15 a^3 (\cos (c+d x)+1)^3}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]^6*(((4*I)*Sqrt[2]*(3*(1 + E^((2*I)*(c + d*x))) + 3*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) - (Sqrt[Cos[c + d*x]]*(4*Cos[(c - d*x)/2] + 26*Cos[(3*c + d*x)/2] + 10*Cos[(c + 3*d*x)/2] + 5*Cos[(5*c + 3*d*x)/2] + 3*Cos[(3*c + 5*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^5)/(8*d)))/(15*a^3*(1 + Cos[c + d*x])^3)","C",1
195,1,705,155,6.3733465,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3),x]","-\frac{2 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos ^6\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} (a \cos (c+d x)+a)^3}+\frac{9 i \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{2 e^{2 i d x} \sqrt{e^{-i d x} \left(2 i \sin (c) \left(-1+e^{2 i d x}\right)+2 \cos (c) \left(1+e^{2 i d x}\right)\right)} \sqrt{i \sin (2 c) e^{2 i d x}+\cos (2 c) e^{2 i d x}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)}{3 i d \cos (c) \left(1+e^{2 i d x}\right)-3 d \sin (c) \left(-1+e^{2 i d x}\right)}-\frac{2 \sqrt{e^{-i d x} \left(2 i \sin (c) \left(-1+e^{2 i d x}\right)+2 \cos (c) \left(1+e^{2 i d x}\right)\right)} \sqrt{i \sin (2 c) e^{2 i d x}+\cos (2 c) e^{2 i d x}+1} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)}{d \sin (c) \left(-1+e^{2 i d x}\right)-i d \cos (c) \left(1+e^{2 i d x}\right)}\right)}{10 (a \cos (c+d x)+a)^3}+\frac{\sqrt{\cos (c+d x)} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \left(-\frac{2 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{2 \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{8 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{8 \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{36 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{36 \csc (c)}{5 d}\right)}{(a \cos (c+d x)+a)^3}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{9 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{9 \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{5 a d (a \cos (c+d x)+a)^2}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}",1,"(((9*I)/10)*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*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]]*((-36*Csc[c])/(5*d) - (36*Sec[c/2]*Sec[c/2 + (d*x)/2]*Sin[(d*x)/2])/(5*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*Sin[(d*x)/2])/(5*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*Sin[(d*x)/2])/(5*d) - (8*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(5*d) - (2*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(a + a*Cos[c + d*x])^3","C",1
196,1,364,181,2.0095256,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^3} \, dx","Integrate[1/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3),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) \left(1284 \cos \left(\frac{1}{2} (c-d x)\right)+921 \cos \left(\frac{1}{2} (3 c+d x)\right)+1243 \cos \left(\frac{1}{2} (c+3 d x)\right)+374 \cos \left(\frac{1}{2} (5 c+3 d x)\right)+670 \cos \left(\frac{1}{2} (3 c+5 d x)\right)+65 \cos \left(\frac{1}{2} (7 c+5 d x)\right)+147 \cos \left(\frac{1}{2} (5 c+7 d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right)}{16 d \sqrt{\cos (c+d x)}}-\frac{4 i \sqrt{2} e^{-i (c+d x)} \left(147 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-65 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+147 \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{15 a^3 (\cos (c+d x)+1)^3}","-\frac{13 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{49 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{49 \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{13 \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}-\frac{8 \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]^6*(((-4*I)*Sqrt[2]*(147*(1 + E^((2*I)*(c + d*x))) + 147*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 65*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) + ((1284*Cos[(c - d*x)/2] + 921*Cos[(3*c + d*x)/2] + 1243*Cos[(c + 3*d*x)/2] + 374*Cos[(5*c + 3*d*x)/2] + 670*Cos[(3*c + 5*d*x)/2] + 65*Cos[(7*c + 5*d*x)/2] + 147*Cos[(5*c + 7*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^5)/(16*d*Sqrt[Cos[c + d*x]])))/(15*a^3*(1 + Cos[c + d*x])^3)","C",1
197,1,394,207,2.6434029,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^3} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^3),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) \left(5134 \cos \left(\frac{1}{2} (c-d x)\right)+4148 \cos \left(\frac{1}{2} (3 c+d x)\right)+4664 \cos \left(\frac{1}{2} (c+3 d x)\right)+2476 \cos \left(\frac{1}{2} (5 c+3 d x)\right)+3340 \cos \left(\frac{1}{2} (3 c+5 d x)\right)+944 \cos \left(\frac{1}{2} (7 c+5 d x)\right)+1620 \cos \left(\frac{1}{2} (5 c+7 d x)\right)+165 \cos \left(\frac{1}{2} (9 c+7 d x)\right)+357 \cos \left(\frac{1}{2} (7 c+9 d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right)}{96 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 i \sqrt{2} e^{-i (c+d x)} \left(119 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-55 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+119 \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{5 a^3 (\cos (c+d x)+1)^3}","\frac{11 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{119 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{119 \sin (c+d x)}{30 d \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}+\frac{11 \sin (c+d x)}{2 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{119 \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{2 \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]^6*(((4*I)*Sqrt[2]*(119*(1 + E^((2*I)*(c + d*x))) + 119*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 55*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) - ((5134*Cos[(c - d*x)/2] + 4148*Cos[(3*c + d*x)/2] + 4664*Cos[(c + 3*d*x)/2] + 2476*Cos[(5*c + 3*d*x)/2] + 3340*Cos[(3*c + 5*d*x)/2] + 944*Cos[(7*c + 5*d*x)/2] + 1620*Cos[(5*c + 7*d*x)/2] + 165*Cos[(9*c + 7*d*x)/2] + 357*Cos[(7*c + 9*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^5)/(96*d*Cos[c + d*x]^(3/2))))/(5*a^3*(1 + Cos[c + d*x])^3)","C",1
198,1,105,154,0.347047,"\int \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^(5/2)*Sqrt[a + a*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} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \left(14 \sin \left(\frac{1}{2} (c+d x)\right)+3 \sin \left(\frac{3}{2} (c+d x)\right)+2 \sin \left(\frac{5}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}\right)}{48 d}","\frac{a \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{5 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{5 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{5 a \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(15*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(14*Sin[(c + d*x)/2] + 3*Sin[(3*(c + d*x))/2] + 2*Sin[(5*(c + d*x))/2])))/(48*d)","A",1
199,1,91,116,0.2074093,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^(3/2)*Sqrt[a + a*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} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{3}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}\right)}{8 d}","\frac{a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}+\frac{3 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{3 a \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(3*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(2*Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(8*d)","A",1
200,1,77,72,0.0926359,"\int \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)} \, dx","Integrate[Sqrt[Cos[c + d*x]]*Sqrt[a + a*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} \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)}\right)}{2 d}","\frac{\sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{a \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]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*Sin[(c + d*x)/2]))/(2*d)","A",1
201,1,50,37,0.0482493,"\int \frac{\sqrt{a+a \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]/Sqrt[Cos[c + d*x]],x]","\frac{\sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{d}","\frac{2 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}",1,"(Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2])/d","A",1
202,1,39,36,0.0496673,"\int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]/Cos[c + d*x]^(3/2),x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{d \sqrt{\cos (c+d x)}}","\frac{2 a \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(2*Sqrt[a*(1 + Cos[c + d*x])]*Tan[(c + d*x)/2])/(d*Sqrt[Cos[c + d*x]])","A",1
203,1,51,77,0.0885163,"\int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]/Cos[c + d*x]^(5/2),x]","\frac{2 (2 \cos (c+d x)+1) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 a \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{4 a \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(2*Sqrt[a*(1 + Cos[c + d*x])]*(1 + 2*Cos[c + d*x])*Tan[(c + d*x)/2])/(3*d*Cos[c + d*x]^(3/2))","A",1
204,1,66,115,0.0925826,"\int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]/Cos[c + d*x]^(7/2),x]","\frac{2 \left(5 \sin \left(\frac{1}{2} (c+d x)\right)+2 \sin \left(\frac{5}{2} (c+d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{15 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{8 a \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{16 a \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(5*Sin[(c + d*x)/2] + 2*Sin[(5*(c + d*x))/2]))/(15*d*Cos[c + d*x]^(5/2))","A",1
205,1,66,153,0.1312456,"\int \frac{\sqrt{a+a \cos (c+d x)}}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]/Cos[c + d*x]^(9/2),x]","\frac{2 \left(7 \sin \left(\frac{3}{2} (c+d x)\right)+2 \sin \left(\frac{7}{2} (c+d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{35 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{16 a \sin (c+d x)}{35 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{12 a \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{32 a \sin (c+d x)}{35 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(7*Sin[(3*(c + d*x))/2] + 2*Sin[(7*(c + d*x))/2]))/(35*d*Cos[c + d*x]^(7/2))","A",1
206,1,106,160,0.3725709,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*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(33 \sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \left(26 \sin \left(\frac{1}{2} (c+d x)\right)+9 \sin \left(\frac{3}{2} (c+d x)\right)+2 \sin \left(\frac{5}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}\right)}{48 d}","\frac{11 a^{3/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{11 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{11 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(33*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(26*Sin[(c + d*x)/2] + 9*Sin[(3*(c + d*x))/2] + 2*Sin[(5*(c + d*x))/2])))/(48*d)","A",1
207,1,92,120,0.2287706,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2} \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*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(7 \sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \left(6 \sin \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{3}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}\right)}{8 d}","\frac{7 a^{3/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}+\frac{7 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(7*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(6*Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(8*d)","A",1
208,1,79,75,0.110575,"\int \frac{(a+a \cos (c+d x))^{3/2}}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)/Sqrt[Cos[c + d*x]],x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} \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)}\right)}{2 d}","\frac{3 a^{3/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(3*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*Sin[(c + d*x)/2]))/(2*d)","A",1
209,1,85,76,0.148506,"\int \frac{(a+a \cos (c+d x))^{3/2}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(3/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\sqrt{2} \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^{3/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{2 a^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*Sin[(c + d*x)/2]))/(d*Sqrt[Cos[c + d*x]])","A",1
210,1,52,81,0.1174747,"\int \frac{(a+a \cos (c+d x))^{3/2}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(5/2),x]","\frac{2 a (5 \cos (c+d x)+1) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 a^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{10 a^2 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(2*a*Sqrt[a*(1 + Cos[c + d*x])]*(1 + 5*Cos[c + d*x])*Tan[(c + d*x)/2])/(3*d*Cos[c + d*x]^(3/2))","A",1
211,1,62,121,0.1493194,"\int \frac{(a+a \cos (c+d x))^{3/2}}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(7/2),x]","\frac{2 a (3 \cos (c+d x)+3 \cos (2 (c+d x))+4) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{5 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{6 a^2 \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{12 a^2 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(2*a*Sqrt[a*(1 + Cos[c + d*x])]*(4 + 3*Cos[c + d*x] + 3*Cos[2*(c + d*x)])*Tan[(c + d*x)/2])/(5*d*Cos[c + d*x]^(5/2))","A",1
212,1,72,161,0.2225322,"\int \frac{(a+a \cos (c+d x))^{3/2}}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(9/2),x]","\frac{2 a (117 \cos (c+d x)+26 \cos (2 (c+d x))+26 \cos (3 (c+d x))+41) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{105 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{104 a^2 \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{26 a^2 \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{208 a^2 \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(2*a*Sqrt[a*(1 + Cos[c + d*x])]*(41 + 117*Cos[c + d*x] + 26*Cos[2*(c + d*x)] + 26*Cos[3*(c + d*x)])*Tan[(c + d*x)/2])/(105*d*Cos[c + d*x]^(7/2))","A",1
213,1,182,200,4.4034563,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a (\cos (c+d x)+1))^{5/2} \left(-6 \sin ^4(c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(-\frac{1}{2},\frac{3}{2},2;1,\frac{9}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)-24 \sin ^2(c+d x) (\cos (c+d x)+3) \, _2F_1\left(-\frac{1}{2},\frac{3}{2};\frac{9}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)+7 (28 \cos (c+d x)+3 \cos (2 (c+d x))+89) \, _2F_1\left(-\frac{3}{2},\frac{1}{2};\frac{7}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}{420 d}","\frac{163 a^{5/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{64 d}+\frac{17 a^3 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{24 d \sqrt{a \cos (c+d x)+a}}+\frac{163 a^3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 d \sqrt{a \cos (c+d x)+a}}+\frac{163 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{64 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{4 d}",1,"((a*(1 + Cos[c + d*x]))^(5/2)*Sec[(c + d*x)/2]^4*(7*(89 + 28*Cos[c + d*x] + 3*Cos[2*(c + d*x)])*Hypergeometric2F1[-3/2, 1/2, 7/2, 2*Sin[(c + d*x)/2]^2] - 24*(3 + Cos[c + d*x])*Hypergeometric2F1[-1/2, 3/2, 9/2, 2*Sin[(c + d*x)/2]^2]*Sin[c + d*x]^2 - 6*Csc[(c + d*x)/2]^2*HypergeometricPFQ[{-1/2, 3/2, 2}, {1, 9/2}, 2*Sin[(c + d*x)/2]^2]*Sin[c + d*x]^4)*Tan[(c + d*x)/2])/(420*d)","C",0
214,1,182,160,4.300072,"\int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2} \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a (\cos (c+d x)+1))^{5/2} \left(-2 \sin ^4(c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(\frac{1}{2},\frac{3}{2},2;1,\frac{9}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)-8 \sin ^2(c+d x) (\cos (c+d x)+3) \, _2F_1\left(\frac{1}{2},\frac{3}{2};\frac{9}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)+7 (28 \cos (c+d x)+3 \cos (2 (c+d x))+89) \, _2F_1\left(-\frac{1}{2},\frac{1}{2};\frac{7}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}{420 d}","\frac{25 a^{5/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{8 d}+\frac{13 a^3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{12 d \sqrt{a \cos (c+d x)+a}}+\frac{25 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{8 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}",1,"((a*(1 + Cos[c + d*x]))^(5/2)*Sec[(c + d*x)/2]^4*(7*(89 + 28*Cos[c + d*x] + 3*Cos[2*(c + d*x)])*Hypergeometric2F1[-1/2, 1/2, 7/2, 2*Sin[(c + d*x)/2]^2] - 8*(3 + Cos[c + d*x])*Hypergeometric2F1[1/2, 3/2, 9/2, 2*Sin[(c + d*x)/2]^2]*Sin[c + d*x]^2 - 2*Csc[(c + d*x)/2]^2*HypergeometricPFQ[{1/2, 3/2, 2}, {1, 9/2}, 2*Sin[(c + d*x)/2]^2]*Sin[c + d*x]^4)*Tan[(c + d*x)/2])/(420*d)","C",0
215,1,182,120,4.1726152,"\int \frac{(a+a \cos (c+d x))^{5/2}}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)/Sqrt[Cos[c + d*x]],x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a (\cos (c+d x)+1))^{5/2} \left(2 \sin ^4(c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(\frac{3}{2},\frac{3}{2},2;1,\frac{9}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)+8 \sin ^2(c+d x) (\cos (c+d x)+3) \, _2F_1\left(\frac{3}{2},\frac{3}{2};\frac{9}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)+7 (28 \cos (c+d x)+3 \cos (2 (c+d x))+89) \, _2F_1\left(\frac{1}{2},\frac{1}{2};\frac{7}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}{420 d}","\frac{19 a^{5/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 d}+\frac{9 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}{2 d}",1,"((a*(1 + Cos[c + d*x]))^(5/2)*Sec[(c + d*x)/2]^4*(7*(89 + 28*Cos[c + d*x] + 3*Cos[2*(c + d*x)])*Hypergeometric2F1[1/2, 1/2, 7/2, 2*Sin[(c + d*x)/2]^2] + 8*(3 + Cos[c + d*x])*Hypergeometric2F1[3/2, 3/2, 9/2, 2*Sin[(c + d*x)/2]^2]*Sin[c + d*x]^2 + 2*Csc[(c + d*x)/2]^2*HypergeometricPFQ[{3/2, 3/2, 2}, {1, 9/2}, 2*Sin[(c + d*x)/2]^2]*Sin[c + d*x]^4)*Tan[(c + d*x)/2])/(420*d)","C",0
216,1,182,114,4.2440923,"\int \frac{(a+a \cos (c+d x))^{5/2}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(3/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a (\cos (c+d x)+1))^{5/2} \left(6 \sin ^4(c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(\frac{3}{2},2,\frac{5}{2};1,\frac{9}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)+24 \sin ^2(c+d x) (\cos (c+d x)+3) \, _2F_1\left(\frac{3}{2},\frac{5}{2};\frac{9}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)+7 (28 \cos (c+d x)+3 \cos (2 (c+d x))+89) \, _2F_1\left(\frac{1}{2},\frac{3}{2};\frac{7}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}{420 d}","\frac{5 a^{5/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}-\frac{a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{d \sqrt{\cos (c+d x)}}",1,"((a*(1 + Cos[c + d*x]))^(5/2)*Sec[(c + d*x)/2]^4*(7*(89 + 28*Cos[c + d*x] + 3*Cos[2*(c + d*x)])*Hypergeometric2F1[1/2, 3/2, 7/2, 2*Sin[(c + d*x)/2]^2] + 24*(3 + Cos[c + d*x])*Hypergeometric2F1[3/2, 5/2, 9/2, 2*Sin[(c + d*x)/2]^2]*Sin[c + d*x]^2 + 6*Csc[(c + d*x)/2]^2*HypergeometricPFQ[{3/2, 2, 5/2}, {1, 9/2}, 2*Sin[(c + d*x)/2]^2]*Sin[c + d*x]^4)*Tan[(c + d*x)/2])/(420*d)","C",0
217,1,356,118,9.9262979,"\int \frac{(a+a \cos (c+d x))^{5/2}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(5/2),x]","\frac{\csc ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right) (a (\cos (c+d x)+1))^{5/2} \left(256 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \cos ^4\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(\frac{3}{2},2,\frac{7}{2};1,\frac{9}{2};2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)+512 \left(\sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-3 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+2\right) \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \, _2F_1\left(\frac{3}{2},\frac{7}{2};\frac{9}{2};2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)+\frac{21 \sqrt{2} \sin ^{-1}\left(\sqrt{2} \sqrt{\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)-10 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+15\right)}{\sqrt{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}-14 \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)} \left(12 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-31 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+30 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+45\right)\right)}{672 d}","\frac{2 a^{5/2} \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{d}+\frac{14 a^3 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"((a*(1 + Cos[c + d*x]))^(5/2)*Csc[c/2 + (d*x)/2]^3*Sec[c/2 + (d*x)/2]^5*(256*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{3/2, 2, 7/2}, {1, 9/2}, 2*Sin[c/2 + (d*x)/2]^2]*Sin[c/2 + (d*x)/2]^6 + 512*Hypergeometric2F1[3/2, 7/2, 9/2, 2*Sin[c/2 + (d*x)/2]^2]*Sin[c/2 + (d*x)/2]^6*(2 - 3*Sin[c/2 + (d*x)/2]^2 + Sin[c/2 + (d*x)/2]^4) + (21*Sqrt[2]*ArcSin[Sqrt[2]*Sqrt[Sin[c/2 + (d*x)/2]^2]]*(15 - 10*Sin[c/2 + (d*x)/2]^2 + 3*Sin[c/2 + (d*x)/2]^4))/Sqrt[Sin[c/2 + (d*x)/2]^2] - 14*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]*(45 + 30*Sin[c/2 + (d*x)/2]^2 - 31*Sin[c/2 + (d*x)/2]^4 + 12*Sin[c/2 + (d*x)/2]^6)))/(672*d)","C",0
218,1,64,121,0.1804576,"\int \frac{(a+a \cos (c+d x))^{5/2}}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(7/2),x]","\frac{a^2 (28 \cos (c+d x)+43 \cos (2 (c+d x))+49) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{15 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{22 a^3 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{86 a^3 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(49 + 28*Cos[c + d*x] + 43*Cos[2*(c + d*x)])*Tan[(c + d*x)/2])/(15*d*Cos[c + d*x]^(5/2))","A",1
219,1,74,161,5.2564969,"\int \frac{(a+a \cos (c+d x))^{5/2}}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(9/2),x]","\frac{a^2 (93 \cos (c+d x)+23 \cos (2 (c+d x))+23 \cos (3 (c+d x))+29) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{21 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{46 a^3 \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{6 a^3 \sin (c+d x)}{7 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{92 a^3 \sin (c+d x)}{21 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(29 + 93*Cos[c + d*x] + 23*Cos[2*(c + d*x)] + 23*Cos[3*(c + d*x)])*Tan[(c + d*x)/2])/(21*d*Cos[c + d*x]^(7/2))","A",1
220,1,84,201,5.3524286,"\int \frac{(a+a \cos (c+d x))^{5/2}}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(11/2),x]","\frac{a^2 (698 \cos (c+d x)+803 \cos (2 (c+d x))+146 \cos (3 (c+d x))+146 \cos (4 (c+d x))+727) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{315 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{584 a^3 \sin (c+d x)}{315 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{146 a^3 \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{38 a^3 \sin (c+d x)}{63 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{1168 a^3 \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(727 + 698*Cos[c + d*x] + 803*Cos[2*(c + d*x)] + 146*Cos[3*(c + d*x)] + 146*Cos[4*(c + d*x)])*Tan[(c + d*x)/2])/(315*d*Cos[c + d*x]^(9/2))","A",1
221,1,51,38,0.0942973,"\int \frac{(a+a \cos (c+d x))^{3/2}}{\cos ^{\frac{5}{4}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(5/4),x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a (\cos (c+d x)+1))^{3/2}}{d \sqrt[4]{\cos (c+d x)}}","\frac{4 a^2 \sin (c+d x)}{d \sqrt[4]{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(2*(a*(1 + Cos[c + d*x]))^(3/2)*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(d*Cos[c + d*x]^(1/4))","A",1
222,1,50,37,0.0651958,"\int \frac{\sqrt{a+a \cos (e+f x)}}{\sqrt{\cos (e+f x)}} \, dx","Integrate[Sqrt[a + a*Cos[e + f*x]]/Sqrt[Cos[e + f*x]],x]","\frac{\sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (e+f x)\right)\right) \sec \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\cos (e+f x)+1)}}{f}","\frac{2 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a \cos (e+f x)+a}}\right)}{f}",1,"(Sqrt[2]*ArcSin[Sqrt[2]*Sin[(e + f*x)/2]]*Sqrt[a*(1 + Cos[e + f*x])]*Sec[(e + f*x)/2])/f","A",1
223,1,188,38,3.6137192,"\int \frac{\sqrt{a-a \cos (e+f x)}}{\sqrt{-\cos (e+f x)}} \, dx","Integrate[Sqrt[a - a*Cos[e + f*x]]/Sqrt[-Cos[e + f*x]],x]","\frac{\sqrt{\cos (e)-i \sin (e)} \sqrt{-\cos (e+f x)} \left(\cot \left(\frac{1}{2} (e+f x)\right)+i\right) \sqrt{a-a \cos (e+f x)} \left(\tanh ^{-1}\left(\frac{e^{i f x}}{\sqrt{\cos (e)-i \sin (e)} \sqrt{e^{2 i f x} (\cos (e)+i \sin (e))-i \sin (e)+\cos (e)}}\right)+\tanh ^{-1}\left(\frac{\sqrt{e^{2 i f x} (\cos (e)+i \sin (e))-i \sin (e)+\cos (e)}}{\sqrt{\cos (e)-i \sin (e)}}\right)\right)}{\sqrt{2} f \sqrt{\cos (e+f x) (\cos (f x)+i \sin (f x))}}","-\frac{2 \sqrt{a} \sin ^{-1}\left(\frac{\sqrt{a} \sin (e+f x)}{\sqrt{a-a \cos (e+f x)}}\right)}{f}",1,"((ArcTanh[E^(I*f*x)/(Sqrt[Cos[e] - I*Sin[e]]*Sqrt[Cos[e] + E^((2*I)*f*x)*(Cos[e] + I*Sin[e]) - I*Sin[e]])] + ArcTanh[Sqrt[Cos[e] + E^((2*I)*f*x)*(Cos[e] + I*Sin[e]) - I*Sin[e]]/Sqrt[Cos[e] - I*Sin[e]]])*Sqrt[-Cos[e + f*x]]*Sqrt[a - a*Cos[e + f*x]]*(I + Cot[(e + f*x)/2])*Sqrt[Cos[e] - I*Sin[e]])/(Sqrt[2]*f*Sqrt[Cos[e + f*x]*(Cos[f*x] + I*Sin[f*x])])","C",1
224,1,289,171,1.2483219,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]^(5/2)/Sqrt[a + a*Cos[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} \left(-4 \sqrt{1+e^{2 i (c+d x)}} \sin \left(\frac{1}{2} (c+d x)\right)+2 \sqrt{1+e^{2 i (c+d x)}} \sin \left(\frac{3}{2} (c+d x)\right)-7 \sin \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)+7 i \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)+7 \sinh ^{-1}\left(e^{i (c+d x)}\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)-i \cos \left(\frac{1}{2} (c+d x)\right)\right)+8 \sqrt{2} \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)-i \cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{4 d \sqrt{1+e^{2 i (c+d x)}} \sqrt{a (\cos (c+d x)+1)}}","\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a \cos (c+d x)+a}}+\frac{7 \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} \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,"(Cos[(c + d*x)/2]*Sqrt[Cos[c + d*x]]*((7*I)*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]*Cos[(c + d*x)/2] - 4*Sqrt[1 + E^((2*I)*(c + d*x))]*Sin[(c + d*x)/2] - 7*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]*Sin[(c + d*x)/2] + 7*ArcSinh[E^(I*(c + d*x))]*((-I)*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 8*Sqrt[2]*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])]*((-I)*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 2*Sqrt[1 + E^((2*I)*(c + d*x))]*Sin[(3*(c + d*x))/2]))/(4*d*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[a*(1 + Cos[c + d*x])])","C",1
225,1,227,128,1.2858955,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]^(3/2)/Sqrt[a + a*Cos[c + d*x]],x]","-\frac{i e^{-\frac{1}{2} i (c+d x)} \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} \left(-\sqrt{2} e^{i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)-4 e^{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} \left(\sqrt{1+e^{2 i (c+d x)}} \left(-1+e^{i (c+d x)}\right)+e^{i (c+d x)} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)\right)}{\sqrt{2} d \sqrt{1+e^{2 i (c+d x)}} \sqrt{a (\cos (c+d x)+1)}}","-\frac{\sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} \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,"((-I)*(-(Sqrt[2]*E^(I*(c + d*x))*ArcSinh[E^(I*(c + d*x))]) - 4*E^(I*(c + d*x))*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] + Sqrt[2]*((-1 + E^(I*(c + d*x)))*Sqrt[1 + E^((2*I)*(c + d*x))] + E^(I*(c + d*x))*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))*Cos[(c + d*x)/2]*Sqrt[Cos[c + d*x]])/(Sqrt[2]*d*E^((I/2)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[a*(1 + Cos[c + d*x])])","C",1
226,1,161,95,0.4366419,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[Sqrt[Cos[c + d*x]]/Sqrt[a + a*Cos[c + d*x]],x]","\frac{i \left(1+e^{i (c+d x)}\right) \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \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)}{\sqrt{2} d \sqrt{1+e^{2 i (c+d x)}} \sqrt{a (\cos (c+d x)+1)}}","\frac{2 \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \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,"(I*(1 + E^(I*(c + d*x)))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(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))]]))/(Sqrt[2]*d*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[a*(1 + Cos[c + d*x])])","C",1
227,1,51,56,0.0524409,"\int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]]),x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \tan ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)}{d \sqrt{a (\cos (c+d x)+1)}}","\frac{\sqrt{2} \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*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
228,1,180,93,2.4725984,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx","Integrate[1/(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(\frac{1}{2} \cos (c+d x) (\cos (c+d x)+2) \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)-\frac{1}{10} \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)}{d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)}}","\frac{2 \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} \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]*((Cos[c + d*x]*(2 + Cos[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]]))/2 - (Hypergeometric2F1[2, 5/2, 7/2, -(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]*Sin[c + d*x]*Tan[c + d*x])/10))/(d*Cos[c + d*x]^(3/2)*Sqrt[a*(1 + Cos[c + d*x])])","C",0
229,1,473,131,7.6709343,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]),x]","-\frac{2 \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)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\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)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right)+7 \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3 \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}} \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(\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}} \left(7 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-3\right)+\left(3-6 \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)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\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{2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{2 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} \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*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*(1 - 2*Sin[c/2 + (d*x)/2]^2)^3*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*(15 - 20*Sin[c/2 + (d*x)/2]^2 + 8*Sin[c/2 + (d*x)/2]^4)*(ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*(3 - 6*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 + 7*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
230,1,1540,169,10.0175346,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx","Integrate[1/(Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]),x]","-\frac{2 \cot \left(\frac{c}{2}+\frac{d x}{2}\right) \csc ^6\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)}{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{2 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{26 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} \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*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
231,1,286,126,0.9010351,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{\sqrt{1+\cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]^(5/2)/Sqrt[1 + Cos[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \left(-4 \sqrt{1+e^{2 i (c+d x)}} \sin \left(\frac{1}{2} (c+d x)\right)+2 \sqrt{1+e^{2 i (c+d x)}} \sin \left(\frac{3}{2} (c+d x)\right)-7 \sin \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)+7 i \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)+7 \sinh ^{-1}\left(e^{i (c+d x)}\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)-i \cos \left(\frac{1}{2} (c+d x)\right)\right)+8 \sqrt{2} \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)-i \cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{4 d \sqrt{1+e^{2 i (c+d x)}}}","\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{\cos (c+d x)+1}}-\frac{\sqrt{2} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}+\frac{7 \sin ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{\cos (c+d x)+1}}\right)}{4 d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{\cos (c+d x)+1}}",1,"(Cos[(c + d*x)/2]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*((7*I)*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]*Cos[(c + d*x)/2] - 4*Sqrt[1 + E^((2*I)*(c + d*x))]*Sin[(c + d*x)/2] - 7*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]*Sin[(c + d*x)/2] + 7*ArcSinh[E^(I*(c + d*x))]*((-I)*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 8*Sqrt[2]*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])]*((-I)*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 2*Sqrt[1 + E^((2*I)*(c + d*x))]*Sin[(3*(c + d*x))/2]))/(4*d*Sqrt[1 + E^((2*I)*(c + d*x))])","C",0
232,1,224,85,0.8626528,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{\sqrt{1+\cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]^(3/2)/Sqrt[1 + Cos[c + d*x]],x]","-\frac{i e^{-\frac{1}{2} i (c+d x)} \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \left(-\sqrt{2} e^{i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)-4 e^{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} \left(\sqrt{1+e^{2 i (c+d x)}} \left(-1+e^{i (c+d x)}\right)+e^{i (c+d x)} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)\right)}{\sqrt{2} d \sqrt{1+e^{2 i (c+d x)}}}","\frac{\sqrt{2} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}-\frac{\sin ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{\cos (c+d x)+1}}\right)}{d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{\cos (c+d x)+1}}",1,"((-I)*(-(Sqrt[2]*E^(I*(c + d*x))*ArcSinh[E^(I*(c + d*x))]) - 4*E^(I*(c + d*x))*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] + Sqrt[2]*((-1 + E^(I*(c + d*x)))*Sqrt[1 + E^((2*I)*(c + d*x))] + E^(I*(c + d*x))*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))*Cos[(c + d*x)/2]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])])/(Sqrt[2]*d*E^((I/2)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))])","C",0
233,1,135,54,0.262083,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{1+\cos (c+d x)}} \, dx","Integrate[Sqrt[Cos[c + d*x]]/Sqrt[1 + Cos[c + d*x]],x]","-\frac{i \left(1+e^{i (c+d x)}\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \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)}{d \sqrt{1+e^{2 i (c+d x)}}}","\frac{2 \sin ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{\cos (c+d x)+1}}\right)}{d}-\frac{\sqrt{2} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}",1,"((-I)*(1 + E^(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))]])*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])])/(d*Sqrt[1 + E^((2*I)*(c + d*x))])","C",1
234,1,49,27,0.040213,"\int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{1+\cos (c+d x)}} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]]),x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \tan ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)}{d \sqrt{\cos (c+d x)+1}}","\frac{\sqrt{2} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}",1,"(2*ArcTan[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]]*Cos[(c + d*x)/2])/(d*Sqrt[1 + Cos[c + d*x]])","A",1
235,1,178,62,1.7946867,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{1+\cos (c+d x)}} \, dx","Integrate[1/(Cos[c + d*x]^(3/2)*Sqrt[1 + 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(\frac{1}{2} \cos (c+d x) (\cos (c+d x)+2) \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)-\frac{1}{10} \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)}{d \cos ^{\frac{3}{2}}(c+d x) \sqrt{\cos (c+d x)+1}}","\frac{2 \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{\cos (c+d x)+1}}-\frac{\sqrt{2} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}",1,"(2*Cos[(c + d*x)/2]*Sin[(c + d*x)/2]*((Cos[c + d*x]*(2 + Cos[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]]))/2 - (Hypergeometric2F1[2, 5/2, 7/2, -(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]*Sin[c + d*x]*Tan[c + d*x])/10))/(d*Cos[c + d*x]^(3/2)*Sqrt[1 + Cos[c + d*x]])","C",0
236,1,471,98,6.6343008,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{1+\cos (c+d x)}} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*Sqrt[1 + Cos[c + d*x]]),x]","-\frac{2 \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)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\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)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right)+7 \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3 \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}} \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(\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}} \left(7 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-3\right)+\left(3-6 \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)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right)\right)\right)}{63 d \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{7/2} \sqrt{\cos (c+d x)+1}}","\frac{2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{\cos (c+d x)+1}}+\frac{\sqrt{2} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}-\frac{2 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{\cos (c+d x)+1}}",1,"(-2*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*(1 - 2*Sin[c/2 + (d*x)/2]^2)^3*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*(15 - 20*Sin[c/2 + (d*x)/2]^2 + 8*Sin[c/2 + (d*x)/2]^4)*(ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*(3 - 6*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 + 7*Sin[c/2 + (d*x)/2]^2))))/(63*d*Sqrt[1 + Cos[c + d*x]]*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2))","C",0
237,1,1538,134,7.918409,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{1+\cos (c+d x)}} \, dx","Integrate[1/(Cos[c + d*x]^(7/2)*Sqrt[1 + Cos[c + d*x]]),x]","-\frac{2 \cot \left(\frac{c}{2}+\frac{d x}{2}\right) \csc ^6\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)}{675 d \sqrt{\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{2 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{\cos (c+d x)+1}}+\frac{2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{\cos (c+d x)+1}}-\frac{\sqrt{2} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}+\frac{26 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{\cos (c+d x)+1}}",1,"(-2*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[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
238,1,229,174,5.3581452,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(3/2),x]","\frac{\cos ^3\left(\frac{1}{2} (c+d x)\right) \left(\frac{\left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{3}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)} \sec ^2\left(\frac{1}{2} (c+d x)\right)}{d}+\frac{3 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(2 \sinh ^{-1}\left(e^{i (c+d x)}\right)+3 \sqrt{2} \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)-2 \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{\sqrt{2} d \sqrt{1+e^{2 i (c+d x)}}}\right)}{(a (\cos (c+d x)+1))^{3/2}}","-\frac{3 \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}+\frac{9 \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{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac{3 \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*(((3*I)*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(2*ArcSinh[E^(I*(c + d*x))] + 3*Sqrt[2]*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - 2*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))/(Sqrt[2]*d*Sqrt[1 + E^((2*I)*(c + d*x))]) + (Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]^2*(2*Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2]))/d))/(a*(1 + Cos[c + d*x]))^(3/2)","C",1
239,1,215,134,3.7874024,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[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{\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(4 \sinh ^{-1}\left(e^{i (c+d x)}\right)+5 \sqrt{2} \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)-4 \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{2 \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{3/2} d}-\frac{5 \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{\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*ArcSinh[E^(I*(c + d*x))] + 5*Sqrt[2]*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - 4*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))/(Sqrt[2]*d*Sqrt[1 + E^((2*I)*(c + d*x))]) - (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
240,1,118,97,0.3362668,"\int \frac{\sqrt{\cos (c+d x)}}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[Sqrt[Cos[c + d*x]]/(a + a*Cos[c + d*x])^(3/2),x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)+1} \left(\sqrt{\cos (c+d x)+1} \sin ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right)}}\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}}\right)}{2 d (a (\cos (c+d x)+1))^{3/2}}","\frac{\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{\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]*Sqrt[1 + Cos[c + d*x]]*(ArcSin[Sin[(c + d*x)/2]/Sqrt[Cos[(c + d*x)/2]^2]]*Sqrt[1 + Cos[c + d*x]] + 2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sin[(c + d*x)/2]))/(2*d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
241,1,106,97,0.5811918,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{3/2}} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(3/2)),x]","-\frac{\sin \left(\frac{1}{2} (c+d x)\right) \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} \left(3 \cot ^2\left(\frac{1}{2} (c+d x)\right) \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)+2\right)}{2 d (a (\cos (c+d x)+1))^{3/2}}","\frac{3 \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{\sin (c+d x) \sqrt{\cos (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"-1/2*(Cos[(c + d*x)/2]*Sqrt[Cos[c + d*x]]*(2 + 3*ArcTanh[Sqrt[-(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]]*Cot[(c + d*x)/2]^2*Sqrt[2 - 2*Sec[c + d*x]])*Sin[(c + d*x)/2])/(d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
242,1,456,137,7.3098072,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx","Integrate[1/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(3/2)),x]","\frac{2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(\frac{4 \sin ^2\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{5}{2};1,\frac{9}{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)}{70 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-35}-\frac{1}{6} \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2 \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}} \csc ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \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}} \left(124 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-350 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+298 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-75\right)-3 \left(34 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-100 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+91 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-25\right) \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)\right)\right)}{d \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{3/2} (a (\cos (c+d x)+1))^{3/2}}","-\frac{7 \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 \sin (c+d x)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{\sin (c+d x)}{2 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"(2*Cos[c/2 + (d*x)/2]^3*Sec[(c + d*x)/2]^2*Sin[c/2 + (d*x)/2]*((4*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 5/2}, {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]^2)/(-35 + 70*Sin[c/2 + (d*x)/2]^2) - (Csc[c/2 + (d*x)/2]^6*(1 - 2*Sin[c/2 + (d*x)/2]^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)]]*(-25 + 91*Sin[c/2 + (d*x)/2]^2 - 100*Sin[c/2 + (d*x)/2]^4 + 34*Sin[c/2 + (d*x)/2]^6) + Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*(-75 + 298*Sin[c/2 + (d*x)/2]^2 - 350*Sin[c/2 + (d*x)/2]^4 + 124*Sin[c/2 + (d*x)/2]^6)))/6))/(d*(a*(1 + Cos[c + d*x]))^(3/2)*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2))","C",0
243,1,589,177,9.235351,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(3/2)),x]","\frac{\cot ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \csc ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(-80 \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right) \cos ^6\left(\frac{1}{2} (c+d x)\right) \, _4F_3\left(2,2,2,\frac{7}{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)+120 \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) \cos ^4\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(2,2,\frac{7}{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)+21 \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3 \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}} \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}} \left(12960 \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)-58336 \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)+103992 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-89856 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+37165 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-5880\right)-15 \left(696 \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)-3232 \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)+5972 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-5391 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+2347 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-392\right) \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)\right)\right)}{945 d \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{7/2} (a (\cos (c+d x)+1))^{3/2}}","\frac{11 \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 \sin (c+d x)}{6 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{\sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{19 \sin (c+d x)}{6 a d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(Cot[c/2 + (d*x)/2]^3*Csc[c/2 + (d*x)/2]^4*Sec[(c + d*x)/2]^2*(-80*Cos[(c + d*x)/2]^6*HypergeometricPFQ[{2, 2, 2, 7/2}, {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 + 120*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 7/2}, {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) + 21*(1 - 2*Sin[c/2 + (d*x)/2]^2)^3*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*(-15*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*(-392 + 2347*Sin[c/2 + (d*x)/2]^2 - 5391*Sin[c/2 + (d*x)/2]^4 + 5972*Sin[c/2 + (d*x)/2]^6 - 3232*Sin[c/2 + (d*x)/2]^8 + 696*Sin[c/2 + (d*x)/2]^10) + Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*(-5880 + 37165*Sin[c/2 + (d*x)/2]^2 - 89856*Sin[c/2 + (d*x)/2]^4 + 103992*Sin[c/2 + (d*x)/2]^6 - 58336*Sin[c/2 + (d*x)/2]^8 + 12960*Sin[c/2 + (d*x)/2]^10))))/(945*d*(a*(1 + Cos[c + d*x]))^(3/2)*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2))","C",0
244,1,385,214,6.7134671,"\int \frac{\cos ^{\frac{7}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{\cos (c+d x)} \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{8 \sin \left(\frac{c}{2}\right) \cos \left(\frac{d x}{2}\right)}{d}+\frac{8 \cos \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)}{d}-\frac{\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{\tan \left(\frac{c}{2}\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{23 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}+\frac{23 \tan \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}\right)}{(a (\cos (c+d x)+1))^{5/2}}+\frac{5 i e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right) \left(8 \sinh ^{-1}\left(e^{i (c+d x)}\right)+\frac{23 \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)}{\sqrt{2}}-8 \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{2 \sqrt{2} d \sqrt{1+e^{2 i (c+d x)}} (a (\cos (c+d x)+1))^{5/2}}","-\frac{5 \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}+\frac{115 \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{35 \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{15 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"(((5*I)/2)*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(8*ArcSinh[E^(I*(c + d*x))] + (23*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/Sqrt[2] - 8*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Cos[c/2 + (d*x)/2]^5)/(Sqrt[2]*d*Sqrt[1 + E^((2*I)*(c + d*x))]*(a*(1 + Cos[c + d*x]))^(5/2)) + (Cos[c/2 + (d*x)/2]^5*Sqrt[Cos[c + d*x]]*((8*Cos[(d*x)/2]*Sin[c/2])/d + (8*Cos[c/2]*Sin[(d*x)/2])/d + (23*Sec[c/2]*Sec[c/2 + (d*x)/2]^2*Sin[(d*x)/2])/(4*d) - (Sec[c/2]*Sec[c/2 + (d*x)/2]^4*Sin[(d*x)/2])/(2*d) + (23*Sec[c/2 + (d*x)/2]*Tan[c/2])/(4*d) - (Sec[c/2 + (d*x)/2]^3*Tan[c/2])/(2*d)))/(a*(1 + Cos[c + d*x]))^(5/2)","C",0
245,1,349,174,6.6430479,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\cos (c+d x)} \left(\frac{\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{\tan \left(\frac{c}{2}\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{15 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}-\frac{15 \tan \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}\right)}{(a (\cos (c+d x)+1))^{5/2}}-\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)} \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right) \left(32 \sinh ^{-1}\left(e^{i (c+d x)}\right)+43 \sqrt{2} \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)-32 \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{4 \sqrt{2} d \sqrt{1+e^{2 i (c+d x)}} (a (\cos (c+d x)+1))^{5/2}}","\frac{2 \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{5/2} d}-\frac{43 \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{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}-\frac{11 \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}",1,"((-1/4*I)*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(32*ArcSinh[E^(I*(c + d*x))] + 43*Sqrt[2]*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - 32*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Cos[c/2 + (d*x)/2]^5)/(Sqrt[2]*d*Sqrt[1 + E^((2*I)*(c + d*x))]*(a*(1 + Cos[c + d*x]))^(5/2)) + (Cos[c/2 + (d*x)/2]^5*Sqrt[Cos[c + d*x]]*((-15*Sec[c/2]*Sec[c/2 + (d*x)/2]^2*Sin[(d*x)/2])/(4*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]^4*Sin[(d*x)/2])/(2*d) - (15*Sec[c/2 + (d*x)/2]*Tan[c/2])/(4*d) + (Sec[c/2 + (d*x)/2]^3*Tan[c/2])/(2*d)))/(a*(1 + Cos[c + d*x]))^(5/2)","C",1
246,1,149,137,0.8155986,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[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(3 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right)} \sin ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right)}}\right)+\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \left(5-2 \tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}{4 d \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right)} (a (\cos (c+d x)+1))^{5/2}}","\frac{3 \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{7 \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{\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*(3*ArcSin[Sin[(c + d*x)/2]/Sqrt[Cos[(c + d*x)/2]^2]]*Sqrt[Cos[(c + d*x)/2]^2] + Sqrt[2]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sin[(c + d*x)/2]*(5 - 2*Tan[(c + d*x)/2]^2)))/(4*d*Sqrt[Cos[(c + d*x)/2]^2]*(a*(1 + Cos[c + d*x]))^(5/2))","A",1
247,1,122,137,1.0439419,"\int \frac{\sqrt{\cos (c+d x)}}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[Sqrt[Cos[c + d*x]]/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\sin ^3\left(\frac{1}{2} (c+d x)\right) \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} \left(6 \csc ^2\left(\frac{1}{2} (c+d x)\right)-5 \cot ^4\left(\frac{1}{2} (c+d x)\right) \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)-2\right)}{8 d (a (\cos (c+d x)+1))^{5/2}}","\frac{5 \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{\sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}+\frac{\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]*Sqrt[Cos[c + d*x]]*(-2 + 6*Csc[(c + d*x)/2]^2 - 5*ArcTanh[Sqrt[-(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]]*Cot[(c + d*x)/2]^4*Sqrt[2 - 2*Sec[c + d*x]])*Sin[(c + d*x)/2]^3)/(8*d*(a*(1 + Cos[c + d*x]))^(5/2))","A",1
248,1,134,137,1.2844132,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{5/2}} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(5/2)),x]","-\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(\cos (c+d x) (9 \cos (c+d x)+13) \sqrt{2-2 \sec (c+d x)}-76 \cos ^4\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right) (-\sec (c+d x))}\right)\right)}{32 \sqrt{2} a^2 d \sqrt{\cos (c+d x)-1} \sqrt{a (\cos (c+d x)+1)}}","\frac{19 \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 \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"-1/32*(Sec[(c + d*x)/2]^2*(-76*ArcTanh[Sqrt[-(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]]*Cos[(c + d*x)/2]^4 + Cos[c + d*x]*(13 + 9*Cos[c + d*x])*Sqrt[2 - 2*Sec[c + d*x]])*Tan[(c + d*x)/2])/(Sqrt[2]*a^2*d*Sqrt[-1 + Cos[c + d*x]]*Sqrt[a*(1 + Cos[c + d*x])])","A",1
249,1,506,177,7.9031525,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}} \, dx","Integrate[1/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(5/2)),x]","\frac{2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right) \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(\frac{8 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \cos ^6\left(\frac{1}{2} (c+d x)\right) \, _4F_3\left(2,2,2,\frac{5}{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)}{315 \left(2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1\right)}+\frac{1}{120} \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2 \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}} \csc ^8\left(\frac{c}{2}+\frac{d x}{2}\right) \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}} \left(15344 \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)-66122 \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)+109737 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-87764 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+33980 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-5145\right)-15 \left(824 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-2021 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+1465 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-343\right) \cos ^4\left(\frac{1}{2} (c+d x)\right) \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)\right)\right)}{d \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{3/2} (a (\cos (c+d x)+1))^{5/2}}","-\frac{75 \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 \sin (c+d x)}{16 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{13 \sin (c+d x)}{16 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}",1,"(2*Cos[c/2 + (d*x)/2]^5*Sec[(c + d*x)/2]^4*Sin[c/2 + (d*x)/2]*((8*Cos[(c + d*x)/2]^6*HypergeometricPFQ[{2, 2, 2, 5/2}, {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]^2)/(315*(-1 + 2*Sin[c/2 + (d*x)/2]^2)) + (Csc[c/2 + (d*x)/2]^8*(1 - 2*Sin[c/2 + (d*x)/2]^2)^2*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*(-15*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Cos[(c + d*x)/2]^4*(-343 + 1465*Sin[c/2 + (d*x)/2]^2 - 2021*Sin[c/2 + (d*x)/2]^4 + 824*Sin[c/2 + (d*x)/2]^6) + Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*(-5145 + 33980*Sin[c/2 + (d*x)/2]^2 - 87764*Sin[c/2 + (d*x)/2]^4 + 109737*Sin[c/2 + (d*x)/2]^6 - 66122*Sin[c/2 + (d*x)/2]^8 + 15344*Sin[c/2 + (d*x)/2]^10)))/120))/(d*(a*(1 + Cos[c + d*x]))^(5/2)*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2))","C",0
250,1,639,217,10.7886774,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(5/2)),x]","-\frac{\cot ^5\left(\frac{c}{2}+\frac{d x}{2}\right) \csc ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(640 \sin ^{12}\left(\frac{c}{2}+\frac{d x}{2}\right) \cos ^8\left(\frac{1}{2} (c+d x)\right) \, _5F_4\left(2,2,2,2,\frac{7}{2};1,1,1,\frac{13}{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)-1280 \left(5 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-6\right) \sin ^{12}\left(\frac{c}{2}+\frac{d x}{2}\right) \cos ^6\left(\frac{1}{2} (c+d x)\right) \, _4F_3\left(2,2,2,\frac{7}{2};1,1,\frac{13}{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)+33 \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3 \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}} \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}} \left(4344400 \sin ^{14}\left(\frac{c}{2}+\frac{d x}{2}\right)-26448512 \sin ^{12}\left(\frac{c}{2}+\frac{d x}{2}\right)+68243596 \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)-96281836 \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)+79946462 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-38990350 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+10333785 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1148175\right)-105 \left(33208 \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)-140732 \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)+234156 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-188110 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+72902 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-10935\right) \cos ^4\left(\frac{1}{2} (c+d x)\right) \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)\right)\right)}{41580 d \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{7/2} (a (\cos (c+d x)+1))^{5/2}}","\frac{163 \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 \sin (c+d x)}{48 a^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{299 \sin (c+d x)}{48 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{17 \sin (c+d x)}{16 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}",1,"-1/41580*(Cot[c/2 + (d*x)/2]^5*Csc[c/2 + (d*x)/2]^4*Sec[(c + d*x)/2]^4*(640*Cos[(c + d*x)/2]^8*HypergeometricPFQ[{2, 2, 2, 2, 7/2}, {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 - 1280*Cos[(c + d*x)/2]^6*HypergeometricPFQ[{2, 2, 2, 7/2}, {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) + 33*(1 - 2*Sin[c/2 + (d*x)/2]^2)^3*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*(-105*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Cos[(c + d*x)/2]^4*(-10935 + 72902*Sin[c/2 + (d*x)/2]^2 - 188110*Sin[c/2 + (d*x)/2]^4 + 234156*Sin[c/2 + (d*x)/2]^6 - 140732*Sin[c/2 + (d*x)/2]^8 + 33208*Sin[c/2 + (d*x)/2]^10) + Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*(-1148175 + 10333785*Sin[c/2 + (d*x)/2]^2 - 38990350*Sin[c/2 + (d*x)/2]^4 + 79946462*Sin[c/2 + (d*x)/2]^6 - 96281836*Sin[c/2 + (d*x)/2]^8 + 68243596*Sin[c/2 + (d*x)/2]^10 - 26448512*Sin[c/2 + (d*x)/2]^12 + 4344400*Sin[c/2 + (d*x)/2]^14))))/(d*(a*(1 + Cos[c + d*x]))^(5/2)*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(7/2))","C",0
251,1,448,254,6.7514082,"\int \frac{\cos ^{\frac{9}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx","Integrate[Cos[c + d*x]^(9/2)/(a + a*Cos[c + d*x])^(7/2),x]","\frac{\sqrt{\cos (c+d x)} \cos ^7\left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{16 \sin \left(\frac{c}{2}\right) \cos \left(\frac{d x}{2}\right)}{d}+\frac{16 \cos \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)}{d}+\frac{\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{\tan \left(\frac{c}{2}\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{15 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}-\frac{15 \tan \left(\frac{c}{2}\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}+\frac{523 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d}+\frac{523 \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{7 i e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \cos ^7\left(\frac{c}{2}+\frac{d x}{2}\right) \left(64 \sinh ^{-1}\left(e^{i (c+d x)}\right)+91 \sqrt{2} \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)-64 \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{8 \sqrt{2} d \sqrt{1+e^{2 i (c+d x)}} (a (\cos (c+d x)+1))^{7/2}}","-\frac{7 \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{7/2} d}+\frac{637 \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{189 \sin (c+d x) \sqrt{\cos (c+d x)}}{64 a^3 d \sqrt{a \cos (c+d x)+a}}-\frac{259 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}-\frac{7 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{5/2}}",1,"(((7*I)/8)*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(64*ArcSinh[E^(I*(c + d*x))] + 91*Sqrt[2]*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - 64*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Cos[c/2 + (d*x)/2]^7)/(Sqrt[2]*d*Sqrt[1 + E^((2*I)*(c + d*x))]*(a*(1 + Cos[c + d*x]))^(7/2)) + (Cos[c/2 + (d*x)/2]^7*Sqrt[Cos[c + d*x]]*((16*Cos[(d*x)/2]*Sin[c/2])/d + (16*Cos[c/2]*Sin[(d*x)/2])/d + (523*Sec[c/2]*Sec[c/2 + (d*x)/2]^2*Sin[(d*x)/2])/(24*d) - (15*Sec[c/2]*Sec[c/2 + (d*x)/2]^4*Sin[(d*x)/2])/(4*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]^6*Sin[(d*x)/2])/(3*d) + (523*Sec[c/2 + (d*x)/2]*Tan[c/2])/(24*d) - (15*Sec[c/2 + (d*x)/2]^3*Tan[c/2])/(4*d) + (Sec[c/2 + (d*x)/2]^5*Tan[c/2])/(3*d)))/(a*(1 + Cos[c + d*x]))^(7/2)","C",1
252,1,412,214,6.744493,"\int \frac{\cos ^{\frac{7}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx","Integrate[Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^(7/2),x]","\frac{\cos ^7\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\cos (c+d x)} \left(-\frac{\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{\tan \left(\frac{c}{2}\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{11 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}+\frac{11 \tan \left(\frac{c}{2}\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}-\frac{247 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d}-\frac{247 \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 e^{\frac{1}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \cos ^7\left(\frac{c}{2}+\frac{d x}{2}\right) \left(64 \sinh ^{-1}\left(e^{i (c+d x)}\right)+\frac{177 \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)}{\sqrt{2}}-64 \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{4 \sqrt{2} d \sqrt{1+e^{2 i (c+d x)}} (a (\cos (c+d x)+1))^{7/2}}","\frac{2 \sin ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right)}{a^{7/2} d}-\frac{177 \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{49 \sin (c+d x) \sqrt{\cos (c+d x)}}{64 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}-\frac{17 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{48 a d (a \cos (c+d x)+a)^{5/2}}",1,"((-1/4*I)*E^((I/2)*(c + d*x))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(64*ArcSinh[E^(I*(c + d*x))] + (177*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/Sqrt[2] - 64*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Cos[c/2 + (d*x)/2]^7)/(Sqrt[2]*d*Sqrt[1 + E^((2*I)*(c + d*x))]*(a*(1 + Cos[c + d*x]))^(7/2)) + (Cos[c/2 + (d*x)/2]^7*Sqrt[Cos[c + d*x]]*((-247*Sec[c/2]*Sec[c/2 + (d*x)/2]^2*Sin[(d*x)/2])/(24*d) + (11*Sec[c/2]*Sec[c/2 + (d*x)/2]^4*Sin[(d*x)/2])/(4*d) - (Sec[c/2]*Sec[c/2 + (d*x)/2]^6*Sin[(d*x)/2])/(3*d) - (247*Sec[c/2 + (d*x)/2]*Tan[c/2])/(24*d) + (11*Sec[c/2 + (d*x)/2]^3*Tan[c/2])/(4*d) - (Sec[c/2 + (d*x)/2]^5*Tan[c/2])/(3*d)))/(a*(1 + Cos[c + d*x]))^(7/2)","C",0
253,1,176,177,2.6757822,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx","Integrate[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) \sqrt{a (\cos (c+d x)+1)} \left(15 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right)} \sin ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right)}}\right)+\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \left(8 \tan ^4\left(\frac{1}{2} (c+d x)\right)-26 \tan ^2\left(\frac{1}{2} (c+d x)\right)+33\right)\right)}{24 a^4 d \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right)} (\cos (c+d x)+1)^4}","\frac{5 \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{67 \sin (c+d x) \sqrt{\cos (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}-\frac{13 \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*Sqrt[a*(1 + Cos[c + d*x])]*(15*ArcSin[Sin[(c + d*x)/2]/Sqrt[Cos[(c + d*x)/2]^2]]*Sqrt[Cos[(c + d*x)/2]^2] + Sqrt[2]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sin[(c + d*x)/2]*(33 - 26*Tan[(c + d*x)/2]^2 + 8*Tan[(c + d*x)/2]^4)))/(24*a^4*d*Sqrt[Cos[(c + d*x)/2]^2]*(1 + Cos[c + d*x])^4)","A",0
254,1,148,177,1.93418,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx","Integrate[Cos[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(7/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) \left((135 \cos (c+d x)+140 \cos (2 (c+d x))+17 \cos (3 (c+d x))+140) \sqrt{2-2 \sec (c+d x)}+672 \cos ^6\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right) (-\sec (c+d x))}\right)\right)}{3072 \sqrt{2} a^3 d \sqrt{\cos (c+d x)-1} \sqrt{a (\cos (c+d x)+1)}}","\frac{7 \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 \sin (c+d x) \sqrt{\cos (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{3 \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}",1,"(Sec[(c + d*x)/2]^4*(672*ArcTanh[Sqrt[-(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]]*Cos[(c + d*x)/2]^6 + (140 + 135*Cos[c + d*x] + 140*Cos[2*(c + d*x)] + 17*Cos[3*(c + d*x)])*Sqrt[2 - 2*Sec[c + d*x]])*Tan[(c + d*x)/2])/(3072*Sqrt[2]*a^3*d*Sqrt[-1 + Cos[c + d*x]]*Sqrt[a*(1 + Cos[c + d*x])])","A",0
255,1,149,177,2.8602724,"\int \frac{\sqrt{\cos (c+d x)}}{(a+a \cos (c+d x))^{7/2}} \, dx","Integrate[Sqrt[Cos[c + d*x]]/(a + a*Cos[c + d*x])^(7/2),x]","\frac{\sin \left(\frac{1}{2} (c+d x)\right) \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(4 \cos (c+d x)-5 \cos (2 (c+d x))-156 \cos ^4\left(\frac{1}{2} (c+d x)\right) \cot ^2\left(\frac{1}{2} (c+d x)\right) \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)+73\right)}{192 a^4 d (\cos (c+d x)+1)^4}","\frac{13 \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 \sin (c+d x) \sqrt{\cos (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{5/2}}+\frac{\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]*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Cos[c + d*x])]*(73 + 4*Cos[c + d*x] - 5*Cos[2*(c + d*x)] - 156*ArcTanh[Sqrt[-(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]]*Cos[(c + d*x)/2]^4*Cot[(c + d*x)/2]^2*Sqrt[2 - 2*Sec[c + d*x]])*Sin[(c + d*x)/2])/(192*a^4*d*(1 + Cos[c + d*x])^4)","A",1
256,1,148,177,2.1650509,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+a \cos (c+d x))^{7/2}} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^(7/2)),x]","-\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) \left((1089 \cos (c+d x)+532 \cos (2 (c+d x))+103 \cos (3 (c+d x))+532) \sqrt{2-2 \sec (c+d x)}-6048 \cos ^6\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right) (-\sec (c+d x))}\right)\right)}{3072 \sqrt{2} a^3 d \sqrt{\cos (c+d x)-1} \sqrt{a (\cos (c+d x)+1)}}","\frac{63 \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 \sin (c+d x) \sqrt{\cos (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}",1,"-1/3072*(Sec[(c + d*x)/2]^4*(-6048*ArcTanh[Sqrt[-(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]]*Cos[(c + d*x)/2]^6 + (532 + 1089*Cos[c + d*x] + 532*Cos[2*(c + d*x)] + 103*Cos[3*(c + d*x)])*Sqrt[2 - 2*Sec[c + d*x]])*Tan[(c + d*x)/2])/(Sqrt[2]*a^3*d*Sqrt[-1 + Cos[c + d*x]]*Sqrt[a*(1 + Cos[c + d*x])])","A",0
257,1,559,217,8.440839,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}} \, dx","Integrate[1/(Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^(7/2)),x]","\frac{2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right) \cos ^7\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(\frac{16 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \cos ^8\left(\frac{1}{2} (c+d x)\right) \, _5F_4\left(2,2,2,2,\frac{5}{2};1,1,1,\frac{13}{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)}{3465 \left(2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1\right)}-\frac{\left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2 \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}} \csc ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right) \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}} \left(1144608 \sin ^{14}\left(\frac{c}{2}+\frac{d x}{2}\right)-6712984 \sin ^{12}\left(\frac{c}{2}+\frac{d x}{2}\right)+16548816 \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)-22251094 \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)+17646926 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-8267707 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+2120790 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-229635\right)+105 \left(8752 \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)-26380 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)+27986 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-12908 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+2187\right) \cos ^6\left(\frac{1}{2} (c+d x)\right) \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)\right)}{1680}\right)}{d \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^{3/2} (a (\cos (c+d x)+1))^{7/2}}","-\frac{363 \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 \sin (c+d x)}{192 a^3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{199 \sin (c+d x)}{192 a^2 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{19 \sin (c+d x)}{48 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x)}{6 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^{7/2}}",1,"(2*Cos[c/2 + (d*x)/2]^7*Sec[(c + d*x)/2]^6*Sin[c/2 + (d*x)/2]*((16*Cos[(c + d*x)/2]^8*HypergeometricPFQ[{2, 2, 2, 2, 5/2}, {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]^2)/(3465*(-1 + 2*Sin[c/2 + (d*x)/2]^2)) - (Csc[c/2 + (d*x)/2]^10*(1 - 2*Sin[c/2 + (d*x)/2]^2)^2*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*(105*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Cos[(c + d*x)/2]^6*(2187 - 12908*Sin[c/2 + (d*x)/2]^2 + 27986*Sin[c/2 + (d*x)/2]^4 - 26380*Sin[c/2 + (d*x)/2]^6 + 8752*Sin[c/2 + (d*x)/2]^8) + Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*(-229635 + 2120790*Sin[c/2 + (d*x)/2]^2 - 8267707*Sin[c/2 + (d*x)/2]^4 + 17646926*Sin[c/2 + (d*x)/2]^6 - 22251094*Sin[c/2 + (d*x)/2]^8 + 16548816*Sin[c/2 + (d*x)/2]^10 - 6712984*Sin[c/2 + (d*x)/2]^12 + 1144608*Sin[c/2 + (d*x)/2]^14)))/1680))/(d*(a*(1 + Cos[c + d*x]))^(7/2)*(1 - 2*Sin[c/2 + (d*x)/2]^2)^(3/2))","C",0
258,1,273,257,8.4325005,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^(7/2)),x]","\frac{i e^{-\frac{3}{2} i (c+d x)} \cos ^7\left(\frac{1}{2} (c+d x)\right) \left(3045 \sqrt{2} \left(1+e^{i (c+d x)}\right)^6 \left(1+e^{2 i (c+d x)}\right)^{3/2} \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)-2 \left(8277 e^{i (c+d x)}+14388 e^{2 i (c+d x)}+13108 e^{3 i (c+d x)}+5622 e^{4 i (c+d x)}-5622 e^{5 i (c+d x)}-13108 e^{6 i (c+d x)}-14388 e^{7 i (c+d x)}-8277 e^{8 i (c+d x)}-1887 e^{9 i (c+d x)}+1887\right)\right)}{96 d \left(1+e^{i (c+d x)}\right)^6 \cos ^{\frac{3}{2}}(c+d x) (a (\cos (c+d x)+1))^{7/2}}","\frac{1015 \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{193 \sin (c+d x)}{64 a^3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}-\frac{629 \sin (c+d x)}{64 a^3 d \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{109 \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 \sin (c+d x)}{48 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x)}{6 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}",1,"((I/96)*(-2*(1887 + 8277*E^(I*(c + d*x)) + 14388*E^((2*I)*(c + d*x)) + 13108*E^((3*I)*(c + d*x)) + 5622*E^((4*I)*(c + d*x)) - 5622*E^((5*I)*(c + d*x)) - 13108*E^((6*I)*(c + d*x)) - 14388*E^((7*I)*(c + d*x)) - 8277*E^((8*I)*(c + d*x)) - 1887*E^((9*I)*(c + d*x))) + 3045*Sqrt[2]*(1 + E^(I*(c + d*x)))^6*(1 + E^((2*I)*(c + d*x)))^(3/2)*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])*Cos[(c + d*x)/2]^7)/(d*E^(((3*I)/2)*(c + d*x))*(1 + E^(I*(c + d*x)))^6*Cos[c + d*x]^(3/2)*(a*(1 + Cos[c + d*x]))^(7/2))","C",1
259,1,347,217,6.0401812,"\int \frac{\cos ^{\frac{7}{2}}(c+d x)}{(a+a \cos (c+d x))^{9/2}} \, dx","Integrate[Cos[c + d*x]^(7/2)/(a + a*Cos[c + d*x])^(9/2),x]","\frac{2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right) \cos ^9\left(\frac{c}{2}+\frac{d x}{2}\right) \left(1-\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{9/2} \left(\frac{1}{8} \left(\frac{1}{1-\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}+\frac{7}{6 \left(1-\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{35}{24 \left(1-\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{35}{16 \left(1-\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^4}\right)+\frac{35 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right)} \csc \left(\frac{c}{2}+\frac{d x}{2}\right) \sin ^{-1}\left(\frac{\sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right)}}\right)}{128 \left(1-\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{9/2}}\right)}{d \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right)} (a (\cos (c+d x)+1))^{9/2}}","\frac{35 \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)}{1024 \sqrt{2} a^{9/2} d}+\frac{853 \sin (c+d x) \sqrt{\cos (c+d x)}}{3072 a^3 d (a \cos (c+d x)+a)^{3/2}}-\frac{187 \sin (c+d x) \sqrt{\cos (c+d x)}}{768 a^2 d (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}-\frac{19 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{96 a d (a \cos (c+d x)+a)^{7/2}}",1,"(2*Cos[c/2 + (d*x)/2]^9*Sin[c/2 + (d*x)/2]*(1 - Sec[(c + d*x)/2]^2*Sin[c/2 + (d*x)/2]^2)^(9/2)*((35*ArcSin[Sin[c/2 + (d*x)/2]/Sqrt[Cos[(c + d*x)/2]^2]]*Sqrt[Cos[(c + d*x)/2]^2]*Csc[c/2 + (d*x)/2])/(128*(1 - Sec[(c + d*x)/2]^2*Sin[c/2 + (d*x)/2]^2)^(9/2)) + (35/(16*(1 - Sec[(c + d*x)/2]^2*Sin[c/2 + (d*x)/2]^2)^4) + 35/(24*(1 - Sec[(c + d*x)/2]^2*Sin[c/2 + (d*x)/2]^2)^3) + 7/(6*(1 - Sec[(c + d*x)/2]^2*Sin[c/2 + (d*x)/2]^2)^2) + (1 - Sec[(c + d*x)/2]^2*Sin[c/2 + (d*x)/2]^2)^(-1))/8))/(d*Sqrt[Cos[(c + d*x)/2]^2]*(a*(1 + Cos[c + d*x]))^(9/2))","A",1
260,1,158,217,2.2139515,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{9/2}} \, dx","Integrate[Cos[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(9/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^6\left(\frac{1}{2} (c+d x)\right) \left((2466 \cos (c+d x)+1072 \cos (2 (c+d x))+702 \cos (3 (c+d x))+73 \cos (4 (c+d x))+999) \sqrt{2-2 \sec (c+d x)}+5760 \cos ^8\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right) (-\sec (c+d x))}\right)\right)}{65536 \sqrt{2} a^4 d \sqrt{\cos (c+d x)-1} \sqrt{a (\cos (c+d x)+1)}}","\frac{45 \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)}{1024 \sqrt{2} a^{9/2} d}+\frac{73 \sin (c+d x) \sqrt{\cos (c+d x)}}{1024 a^3 d (a \cos (c+d x)+a)^{3/2}}+\frac{33 \sin (c+d x) \sqrt{\cos (c+d x)}}{256 a^2 d (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}-\frac{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{32 a d (a \cos (c+d x)+a)^{7/2}}",1,"(Sec[(c + d*x)/2]^6*(5760*ArcTanh[Sqrt[-(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]]*Cos[(c + d*x)/2]^8 + (999 + 2466*Cos[c + d*x] + 1072*Cos[2*(c + d*x)] + 702*Cos[3*(c + d*x)] + 73*Cos[4*(c + d*x)])*Sqrt[2 - 2*Sec[c + d*x]])*Tan[(c + d*x)/2])/(65536*Sqrt[2]*a^4*d*Sqrt[-1 + Cos[c + d*x]]*Sqrt[a*(1 + Cos[c + d*x])])","A",0
261,1,30,16,0.0233758,"\int \frac{1}{\sqrt{\cos (x)} \sqrt{1+\cos (x)}} \, dx","Integrate[1/(Sqrt[Cos[x]]*Sqrt[1 + Cos[x]]),x]","\frac{2 \cos \left(\frac{x}{2}\right) \tan ^{-1}\left(\frac{\sin \left(\frac{x}{2}\right)}{\sqrt{\cos (x)}}\right)}{\sqrt{\cos (x)+1}}","\sqrt{2} \sin ^{-1}\left(\frac{\sin (x)}{\cos (x)+1}\right)",1,"(2*ArcTan[Sin[x/2]/Sqrt[Cos[x]]]*Cos[x/2])/Sqrt[1 + Cos[x]]","A",1
262,1,32,41,0.0202273,"\int \frac{1}{\sqrt{\cos (x)} \sqrt{a+a \cos (x)}} \, dx","Integrate[1/(Sqrt[Cos[x]]*Sqrt[a + a*Cos[x]]),x]","\frac{2 \cos \left(\frac{x}{2}\right) \tan ^{-1}\left(\frac{\sin \left(\frac{x}{2}\right)}{\sqrt{\cos (x)}}\right)}{\sqrt{a (\cos (x)+1)}}","\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \sin (x)}{\sqrt{2} \sqrt{\cos (x)} \sqrt{a \cos (x)+a}}\right)}{\sqrt{a}}",1,"(2*ArcTan[Sin[x/2]/Sqrt[Cos[x]]]*Cos[x/2])/Sqrt[a*(1 + Cos[x])]","A",1
263,1,289,129,4.2166351,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^(3/2)*Sqrt[a - a*Cos[c + d*x]],x]","-\frac{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)} \left(2 \sqrt{2} \left(\cos \left(\frac{3}{2} (c+d x)\right)-2 \cos \left(\frac{1}{2} (c+d x)\right)\right) \csc \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x) (\cos (d x)+i \sin (d x))}+3 \sqrt{\cos (c)-i \sin (c)} \left(\cot \left(\frac{1}{2} (c+d x)\right)+i\right) \tanh ^{-1}\left(\frac{e^{i d x}}{\sqrt{\cos (c)-i \sin (c)} \sqrt{e^{2 i d x} (\cos (c)+i \sin (c))-i \sin (c)+\cos (c)}}\right)+3 \sqrt{\cos (c)-i \sin (c)} \left(\cot \left(\frac{1}{2} (c+d x)\right)+i\right) \tanh ^{-1}\left(\frac{\sqrt{e^{2 i d x} (\cos (c)+i \sin (c))-i \sin (c)+\cos (c)}}{\sqrt{\cos (c)-i \sin (c)}}\right)\right)}{8 d \sqrt{i \sin (c) \left(-1+e^{2 i d x}\right)+\cos (c) \left(1+e^{2 i d x}\right)}}","-\frac{a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a-a \cos (c+d x)}}+\frac{3 a \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a-a \cos (c+d x)}}-\frac{3 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{4 d}",1,"-1/8*(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]]*(3*ArcTanh[E^(I*d*x)/(Sqrt[Cos[c] - I*Sin[c]]*Sqrt[Cos[c] + E^((2*I)*d*x)*(Cos[c] + I*Sin[c]) - I*Sin[c]])]*(I + Cot[(c + d*x)/2])*Sqrt[Cos[c] - I*Sin[c]] + 3*ArcTanh[Sqrt[Cos[c] + E^((2*I)*d*x)*(Cos[c] + I*Sin[c]) - I*Sin[c]]/Sqrt[Cos[c] - I*Sin[c]]]*(I + Cot[(c + d*x)/2])*Sqrt[Cos[c] - I*Sin[c]] + 2*Sqrt[2]*(-2*Cos[(c + d*x)/2] + Cos[(3*(c + d*x))/2])*Csc[(c + d*x)/2]*Sqrt[Cos[c + d*x]*(Cos[d*x] + I*Sin[d*x])]))/(d*Sqrt[(1 + E^((2*I)*d*x))*Cos[c] + I*(-1 + E^((2*I)*d*x))*Sin[c]])","C",1
264,1,264,85,0.7510754,"\int \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)} \, dx","Integrate[Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]],x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)} \left(-2 \sqrt{2} \cot \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x) (\cos (d x)+i \sin (d x))}+\sqrt{\cos (c)-i \sin (c)} \left(\cot \left(\frac{1}{2} (c+d x)\right)+i\right) \tanh ^{-1}\left(\frac{e^{i d x}}{\sqrt{\cos (c)-i \sin (c)} \sqrt{e^{2 i d x} (\cos (c)+i \sin (c))-i \sin (c)+\cos (c)}}\right)+\sqrt{\cos (c)-i \sin (c)} \left(\cot \left(\frac{1}{2} (c+d x)\right)+i\right) \tanh ^{-1}\left(\frac{\sqrt{e^{2 i d x} (\cos (c)+i \sin (c))-i \sin (c)+\cos (c)}}{\sqrt{\cos (c)-i \sin (c)}}\right)\right)}{2 d \sqrt{i \sin (c) \left(-1+e^{2 i d x}\right)+\cos (c) \left(1+e^{2 i d x}\right)}}","\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{d}-\frac{a \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a-a \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]]*(ArcTanh[E^(I*d*x)/(Sqrt[Cos[c] - I*Sin[c]]*Sqrt[Cos[c] + E^((2*I)*d*x)*(Cos[c] + I*Sin[c]) - I*Sin[c]])]*(I + Cot[(c + d*x)/2])*Sqrt[Cos[c] - I*Sin[c]] + ArcTanh[Sqrt[Cos[c] + E^((2*I)*d*x)*(Cos[c] + I*Sin[c]) - I*Sin[c]]/Sqrt[Cos[c] - I*Sin[c]]]*(I + Cot[(c + d*x)/2])*Sqrt[Cos[c] - I*Sin[c]] - 2*Sqrt[2]*Cot[(c + d*x)/2]*Sqrt[Cos[c + d*x]*(Cos[d*x] + I*Sin[d*x])]))/(2*d*Sqrt[(1 + E^((2*I)*d*x))*Cos[c] + I*(-1 + E^((2*I)*d*x))*Sin[c]])","C",1
265,1,278,48,0.5388853,"\int \frac{\sqrt{a-a \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx","Integrate[Sqrt[a - a*Cos[c + d*x]]/Sqrt[Cos[c + d*x]],x]","\frac{2 e^{i d x} \left(\cos \left(\frac{c}{2}\right)+i \sin \left(\frac{c}{2}\right)\right) \sqrt{\cos (c)-i \sin (c)} \sqrt{a-a \cos (c+d x)} \sqrt{e^{-i d x} \left(i \sin (c) \left(-1+e^{2 i d x}\right)+\cos (c) \left(1+e^{2 i d x}\right)\right)} \left(\tanh ^{-1}\left(\frac{e^{i d x}}{\sqrt{\cos (c)-i \sin (c)} \sqrt{e^{2 i d x} (\cos (c)+i \sin (c))-i \sin (c)+\cos (c)}}\right)+\tanh ^{-1}\left(\frac{\sqrt{e^{2 i d x} (\cos (c)+i \sin (c))-i \sin (c)+\cos (c)}}{\sqrt{\cos (c)-i \sin (c)}}\right)\right)}{d \left(i \cos \left(\frac{c}{2}\right) \left(-1+e^{i d x}\right)-\sin \left(\frac{c}{2}\right) \left(1+e^{i d x}\right)\right) \sqrt{2 i \sin (c) \left(-1+e^{2 i d x}\right)+2 \cos (c) \left(1+e^{2 i d x}\right)}}","-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{d}",1,"(2*E^(I*d*x)*(ArcTanh[E^(I*d*x)/(Sqrt[Cos[c] - I*Sin[c]]*Sqrt[Cos[c] + E^((2*I)*d*x)*(Cos[c] + I*Sin[c]) - I*Sin[c]])] + ArcTanh[Sqrt[Cos[c] + E^((2*I)*d*x)*(Cos[c] + I*Sin[c]) - I*Sin[c]]/Sqrt[Cos[c] - I*Sin[c]]])*Sqrt[a - a*Cos[c + d*x]]*(Cos[c/2] + I*Sin[c/2])*Sqrt[Cos[c] - I*Sin[c]]*Sqrt[((1 + E^((2*I)*d*x))*Cos[c] + I*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)])/(d*(I*(-1 + E^(I*d*x))*Cos[c/2] - (1 + E^(I*d*x))*Sin[c/2])*Sqrt[2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c]])","C",1
266,1,40,37,0.0466097,"\int \frac{\sqrt{a-a \cos (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[Sqrt[a - a*Cos[c + d*x]]/Cos[c + d*x]^(3/2),x]","\frac{2 \cot \left(\frac{1}{2} (c+d x)\right) \sqrt{a-a \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}","\frac{2 a \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}",1,"(2*Sqrt[a - a*Cos[c + d*x]]*Cot[(c + d*x)/2])/(d*Sqrt[Cos[c + d*x]])","A",1
267,1,52,79,0.1185015,"\int \frac{\sqrt{a-a \cos (c+d x)}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[Sqrt[a - a*Cos[c + d*x]]/Cos[c + d*x]^(5/2),x]","-\frac{2 (2 \cos (c+d x)-1) \cot \left(\frac{1}{2} (c+d x)\right) \sqrt{a-a \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 a \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}-\frac{4 a \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}",1,"(-2*(-1 + 2*Cos[c + d*x])*Sqrt[a - a*Cos[c + d*x]]*Cot[(c + d*x)/2])/(3*d*Cos[c + d*x]^(3/2))","A",1
268,1,62,118,0.1480936,"\int \frac{\sqrt{a-a \cos (c+d x)}}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[Sqrt[a - a*Cos[c + d*x]]/Cos[c + d*x]^(7/2),x]","\frac{2 (-4 \cos (c+d x)+4 \cos (2 (c+d x))+7) \cot \left(\frac{1}{2} (c+d x)\right) \sqrt{a-a \cos (c+d x)}}{15 d \cos ^{\frac{5}{2}}(c+d x)}","-\frac{8 a \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{2 a \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{16 a \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}",1,"(2*Sqrt[a - a*Cos[c + d*x]]*(7 - 4*Cos[c + d*x] + 4*Cos[2*(c + d*x)])*Cot[(c + d*x)/2])/(15*d*Cos[c + d*x]^(5/2))","A",1
269,1,284,114,0.479065,"\int \sqrt{1-\cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x) \, dx","Integrate[Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(3/2),x]","-\frac{\sqrt{-((\cos (c+d x)-1) \cos (c+d x))} \left(2 \sqrt{2} \left(\cos \left(\frac{3}{2} (c+d x)\right)-2 \cos \left(\frac{1}{2} (c+d x)\right)\right) \csc \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x) (\cos (d x)+i \sin (d x))}+3 \sqrt{\cos (c)-i \sin (c)} \left(\cot \left(\frac{1}{2} (c+d x)\right)+i\right) \tanh ^{-1}\left(\frac{e^{i d x}}{\sqrt{\cos (c)-i \sin (c)} \sqrt{e^{2 i d x} (\cos (c)+i \sin (c))-i \sin (c)+\cos (c)}}\right)+3 \sqrt{\cos (c)-i \sin (c)} \left(\cot \left(\frac{1}{2} (c+d x)\right)+i\right) \tanh ^{-1}\left(\frac{\sqrt{e^{2 i d x} (\cos (c)+i \sin (c))-i \sin (c)+\cos (c)}}{\sqrt{\cos (c)-i \sin (c)}}\right)\right)}{8 d \sqrt{i \sin (c) \left(-1+e^{2 i d x}\right)+\cos (c) \left(1+e^{2 i d x}\right)}}","-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{1-\cos (c+d x)}}+\frac{3 \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{1-\cos (c+d x)}}-\frac{3 \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{4 d}",1,"-1/8*(Sqrt[-((-1 + Cos[c + d*x])*Cos[c + d*x])]*(3*ArcTanh[E^(I*d*x)/(Sqrt[Cos[c] - I*Sin[c]]*Sqrt[Cos[c] + E^((2*I)*d*x)*(Cos[c] + I*Sin[c]) - I*Sin[c]])]*(I + Cot[(c + d*x)/2])*Sqrt[Cos[c] - I*Sin[c]] + 3*ArcTanh[Sqrt[Cos[c] + E^((2*I)*d*x)*(Cos[c] + I*Sin[c]) - I*Sin[c]]/Sqrt[Cos[c] - I*Sin[c]]]*(I + Cot[(c + d*x)/2])*Sqrt[Cos[c] - I*Sin[c]] + 2*Sqrt[2]*(-2*Cos[(c + d*x)/2] + Cos[(3*(c + d*x))/2])*Csc[(c + d*x)/2]*Sqrt[Cos[c + d*x]*(Cos[d*x] + I*Sin[d*x])]))/(d*Sqrt[(1 + E^((2*I)*d*x))*Cos[c] + I*(-1 + E^((2*I)*d*x))*Sin[c]])","C",0
270,1,252,72,0.7325546,"\int \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)} \, dx","Integrate[Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]],x]","\frac{\sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right) \cos (c+d x)} \left(-2 \sqrt{2} \cot \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x) (\cos (d x)+i \sin (d x))}+\sqrt{\cos (c)-i \sin (c)} \left(\cot \left(\frac{1}{2} (c+d x)\right)+i\right) \tanh ^{-1}\left(\frac{e^{i d x}}{\sqrt{\cos (c)-i \sin (c)} \sqrt{e^{2 i d x} (\cos (c)+i \sin (c))-i \sin (c)+\cos (c)}}\right)+\sqrt{\cos (c)-i \sin (c)} \left(\cot \left(\frac{1}{2} (c+d x)\right)+i\right) \tanh ^{-1}\left(\frac{\sqrt{e^{2 i d x} (\cos (c)+i \sin (c))-i \sin (c)+\cos (c)}}{\sqrt{\cos (c)-i \sin (c)}}\right)\right)}{2 d \sqrt{\cos (c+d x) (\cos (d x)+i \sin (d x))}}","\frac{\tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{1-\cos (c+d x)}}",1,"((ArcTanh[E^(I*d*x)/(Sqrt[Cos[c] - I*Sin[c]]*Sqrt[Cos[c] + E^((2*I)*d*x)*(Cos[c] + I*Sin[c]) - I*Sin[c]])]*(I + Cot[(c + d*x)/2])*Sqrt[Cos[c] - I*Sin[c]] + ArcTanh[Sqrt[Cos[c] + E^((2*I)*d*x)*(Cos[c] + I*Sin[c]) - I*Sin[c]]/Sqrt[Cos[c] - I*Sin[c]]]*(I + Cot[(c + d*x)/2])*Sqrt[Cos[c] - I*Sin[c]] - 2*Sqrt[2]*Cot[(c + d*x)/2]*Sqrt[Cos[c + d*x]*(Cos[d*x] + I*Sin[d*x])])*Sqrt[Cos[c + d*x]*Sin[(c + d*x)/2]^2])/(2*d*Sqrt[Cos[c + d*x]*(Cos[d*x] + I*Sin[d*x])])","C",1
271,1,277,37,0.5084333,"\int \frac{\sqrt{1-\cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx","Integrate[Sqrt[1 - Cos[c + d*x]]/Sqrt[Cos[c + d*x]],x]","\frac{2 e^{i d x} \left(\cos \left(\frac{c}{2}\right)+i \sin \left(\frac{c}{2}\right)\right) \sqrt{\cos (c)-i \sin (c)} \sqrt{1-\cos (c+d x)} \sqrt{e^{-i d x} \left(i \sin (c) \left(-1+e^{2 i d x}\right)+\cos (c) \left(1+e^{2 i d x}\right)\right)} \left(\tanh ^{-1}\left(\frac{e^{i d x}}{\sqrt{\cos (c)-i \sin (c)} \sqrt{e^{2 i d x} (\cos (c)+i \sin (c))-i \sin (c)+\cos (c)}}\right)+\tanh ^{-1}\left(\frac{\sqrt{e^{2 i d x} (\cos (c)+i \sin (c))-i \sin (c)+\cos (c)}}{\sqrt{\cos (c)-i \sin (c)}}\right)\right)}{d \left(i \cos \left(\frac{c}{2}\right) \left(-1+e^{i d x}\right)-\sin \left(\frac{c}{2}\right) \left(1+e^{i d x}\right)\right) \sqrt{2 i \sin (c) \left(-1+e^{2 i d x}\right)+2 \cos (c) \left(1+e^{2 i d x}\right)}}","-\frac{2 \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}",1,"(2*E^(I*d*x)*(ArcTanh[E^(I*d*x)/(Sqrt[Cos[c] - I*Sin[c]]*Sqrt[Cos[c] + E^((2*I)*d*x)*(Cos[c] + I*Sin[c]) - I*Sin[c]])] + ArcTanh[Sqrt[Cos[c] + E^((2*I)*d*x)*(Cos[c] + I*Sin[c]) - I*Sin[c]]/Sqrt[Cos[c] - I*Sin[c]]])*Sqrt[1 - Cos[c + d*x]]*(Cos[c/2] + I*Sin[c/2])*Sqrt[Cos[c] - I*Sin[c]]*Sqrt[((1 + E^((2*I)*d*x))*Cos[c] + I*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)])/(d*(I*(-1 + E^(I*d*x))*Cos[c/2] - (1 + E^(I*d*x))*Sin[c/2])*Sqrt[2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c]])","C",1
272,1,39,35,0.0398441,"\int \frac{\sqrt{1-\cos (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[Sqrt[1 - Cos[c + d*x]]/Cos[c + d*x]^(3/2),x]","\frac{2 \sqrt{1-\cos (c+d x)} \cot \left(\frac{1}{2} (c+d x)\right)}{d \sqrt{\cos (c+d x)}}","\frac{2 \sin (c+d x)}{d \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}",1,"(2*Sqrt[1 - Cos[c + d*x]]*Cot[(c + d*x)/2])/(d*Sqrt[Cos[c + d*x]])","A",1
273,1,51,75,0.0960969,"\int \frac{\sqrt{1-\cos (c+d x)}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[Sqrt[1 - Cos[c + d*x]]/Cos[c + d*x]^(5/2),x]","-\frac{2 \sqrt{1-\cos (c+d x)} (2 \cos (c+d x)-1) \cot \left(\frac{1}{2} (c+d x)\right)}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 \sin (c+d x)}{3 d \sqrt{1-\cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)}-\frac{4 \sin (c+d x)}{3 d \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}",1,"(-2*Sqrt[1 - Cos[c + d*x]]*(-1 + 2*Cos[c + d*x])*Cot[(c + d*x)/2])/(3*d*Cos[c + d*x]^(3/2))","A",1
274,1,61,112,0.1150192,"\int \frac{\sqrt{1-\cos (c+d x)}}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[Sqrt[1 - Cos[c + d*x]]/Cos[c + d*x]^(7/2),x]","\frac{2 \sqrt{1-\cos (c+d x)} \left(8 \cos ^2(c+d x)-4 \cos (c+d x)+3\right) \cot \left(\frac{1}{2} (c+d x)\right)}{15 d \cos ^{\frac{5}{2}}(c+d x)}","-\frac{8 \sin (c+d x)}{15 d \sqrt{1-\cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x)}{5 d \sqrt{1-\cos (c+d x)} \cos ^{\frac{5}{2}}(c+d x)}+\frac{16 \sin (c+d x)}{15 d \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}",1,"(2*Sqrt[1 - Cos[c + d*x]]*(3 - 4*Cos[c + d*x] + 8*Cos[c + d*x]^2)*Cot[(c + d*x)/2])/(15*d*Cos[c + d*x]^(5/2))","A",1
275,1,256,185,1.1477319,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{\sqrt{a-a \cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]^(5/2)/Sqrt[a - a*Cos[c + d*x]],x]","-\frac{i e^{-2 i (c+d x)} \left(-1+e^{i (c+d x)}\right) \sqrt{\cos (c+d x)} \left(7 \sqrt{2} e^{2 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)-16 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} \left(\sqrt{1+e^{2 i (c+d x)}} \left(2 e^{i (c+d x)}+2 e^{2 i (c+d x)}+e^{3 i (c+d x)}+1\right)+7 e^{2 i (c+d x)} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)\right)}{8 \sqrt{2} d \sqrt{1+e^{2 i (c+d x)}} \sqrt{a-a \cos (c+d x)}}","\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{a-a \cos (c+d x)}}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{a-a \cos (c+d x)}}+\frac{7 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}",1,"((-1/8*I)*(-1 + E^(I*(c + d*x)))*(7*Sqrt[2]*E^((2*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))] - 16*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]*(Sqrt[1 + E^((2*I)*(c + d*x))]*(1 + 2*E^(I*(c + d*x)) + 2*E^((2*I)*(c + d*x)) + E^((3*I)*(c + d*x))) + 7*E^((2*I)*(c + d*x))*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))*Sqrt[Cos[c + d*x]])/(Sqrt[2]*d*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[a - a*Cos[c + d*x]])","C",1
276,1,228,141,0.8911555,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{\sqrt{a-a \cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]^(3/2)/Sqrt[a - a*Cos[c + d*x]],x]","-\frac{i e^{-i (c+d x)} \left(-1+e^{i (c+d x)}\right) \sqrt{\cos (c+d x)} \left(\sqrt{2} e^{i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)-4 e^{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} \left(\sqrt{1+e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)+e^{i (c+d x)} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)\right)}{2 \sqrt{2} d \sqrt{1+e^{2 i (c+d x)}} \sqrt{a-a \cos (c+d x)}}","\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{a-a \cos (c+d x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}",1,"((-1/2*I)*(-1 + E^(I*(c + d*x)))*(Sqrt[2]*E^(I*(c + d*x))*ArcSinh[E^(I*(c + d*x))] - 4*E^(I*(c + d*x))*ArcTanh[(1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] + Sqrt[2]*((1 + E^(I*(c + d*x)))*Sqrt[1 + E^((2*I)*(c + d*x))] + E^(I*(c + d*x))*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))*Sqrt[Cos[c + d*x]])/(Sqrt[2]*d*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[a - a*Cos[c + d*x]])","C",1
277,1,161,107,0.3831948,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{a-a \cos (c+d x)}} \, dx","Integrate[Sqrt[Cos[c + d*x]]/Sqrt[a - a*Cos[c + d*x]],x]","-\frac{i \left(-1+e^{i (c+d x)}\right) \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \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)}{\sqrt{2} d \sqrt{1+e^{2 i (c+d x)}} \sqrt{a-a \cos (c+d x)}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}",1,"((-I)*(-1 + E^(I*(c + d*x)))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(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))]]))/(Sqrt[2]*d*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[a - a*Cos[c + d*x]])","C",1
278,1,118,58,0.3267765,"\int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]]),x]","\frac{i \left(-1+e^{i (c+d x)}\right) \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)}{d \sqrt{1+e^{2 i (c+d x)}} \sqrt{a-a \cos (c+d x)}}","-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}",1,"(I*(-1 + E^(I*(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))])])/(d*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[a - a*Cos[c + d*x]])","C",1
279,1,157,95,0.3847177,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}} \, dx","Integrate[1/(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) \left(2 \sqrt{1+e^{2 i (c+d x)}} \cos \left(\frac{1}{2} (c+d x)\right)-\frac{e^{-\frac{1}{2} 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{2}}\right)}{d \sqrt{1+e^{2 i (c+d x)}} \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}","\frac{2 \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}",1,"(2*(-(((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]*E^((I/2)*(c + d*x)))) + 2*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[(c + d*x)/2])*Sin[(c + d*x)/2])/(d*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[Cos[c + d*x]]*Sqrt[a - a*Cos[c + d*x]])","C",1
280,1,171,135,0.3210556,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*Sqrt[a - a*Cos[c + d*x]]),x]","\frac{2 \sin \left(\frac{1}{2} (c+d x)\right) \left(2 \sqrt{1+e^{2 i (c+d x)}} \cos \left(\frac{1}{2} (c+d x)\right) (\cos (c+d x)+1)-\frac{3 e^{-\frac{3}{2} i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^2 \tanh ^{-1}\left(\frac{1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)}{2 \sqrt{2}}\right)}{3 d \sqrt{1+e^{2 i (c+d x)}} \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}","\frac{2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{2 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}",1,"(2*((-3*(1 + E^((2*I)*(c + d*x)))^2*ArcTanh[(1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/(2*Sqrt[2]*E^(((3*I)/2)*(c + d*x))) + 2*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[(c + d*x)/2]*(1 + Cos[c + d*x]))*Sin[(c + d*x)/2])/(3*d*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^(3/2)*Sqrt[a - a*Cos[c + d*x]])","C",1
281,1,218,173,0.6612399,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}} \, dx","Integrate[1/(Cos[c + d*x]^(7/2)*Sqrt[a - a*Cos[c + d*x]]),x]","\frac{e^{-\frac{5}{2} i (c+d x)} \sin \left(\frac{1}{2} (c+d x)\right) \left(2 \sqrt{1+e^{2 i (c+d x)}} \left(15 e^{i (c+d x)}+40 e^{2 i (c+d x)}+40 e^{3 i (c+d x)}+15 e^{4 i (c+d x)}+13 e^{5 i (c+d x)}+13\right)-15 \sqrt{2} \left(1+e^{2 i (c+d x)}\right)^3 \tanh ^{-1}\left(\frac{1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)\right)}{60 d \sqrt{1+e^{2 i (c+d x)}} \cos ^{\frac{5}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}","\frac{2 \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a-a \cos (c+d x)}}+\frac{26 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a-a \cos (c+d x)}}\right)}{\sqrt{a} d}",1,"((2*Sqrt[1 + E^((2*I)*(c + d*x))]*(13 + 15*E^(I*(c + d*x)) + 40*E^((2*I)*(c + d*x)) + 40*E^((3*I)*(c + d*x)) + 15*E^((4*I)*(c + d*x)) + 13*E^((5*I)*(c + d*x))) - 15*Sqrt[2]*(1 + E^((2*I)*(c + d*x)))^3*ArcTanh[(1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])*Sin[(c + d*x)/2])/(60*d*E^(((5*I)/2)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^(5/2)*Sqrt[a - a*Cos[c + d*x]])","C",1
282,1,255,161,0.1945504,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{\sqrt{1-\cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]^(5/2)/Sqrt[1 - Cos[c + d*x]],x]","-\frac{i e^{-2 i (c+d x)} \left(-1+e^{i (c+d x)}\right) \sqrt{\cos (c+d x)} \left(7 \sqrt{2} e^{2 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)-16 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} \left(\sqrt{1+e^{2 i (c+d x)}} \left(2 e^{i (c+d x)}+2 e^{2 i (c+d x)}+e^{3 i (c+d x)}+1\right)+7 e^{2 i (c+d x)} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)\right)}{8 \sqrt{2} d \sqrt{1+e^{2 i (c+d x)}} \sqrt{1-\cos (c+d x)}}","\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{1-\cos (c+d x)}}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \sqrt{1-\cos (c+d x)}}+\frac{7 \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{4 d}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{2} \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}",1,"((-1/8*I)*(-1 + E^(I*(c + d*x)))*(7*Sqrt[2]*E^((2*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))] - 16*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]*(Sqrt[1 + E^((2*I)*(c + d*x))]*(1 + 2*E^(I*(c + d*x)) + 2*E^((2*I)*(c + d*x)) + E^((3*I)*(c + d*x))) + 7*E^((2*I)*(c + d*x))*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))*Sqrt[Cos[c + d*x]])/(Sqrt[2]*d*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[1 - Cos[c + d*x]])","C",1
283,1,227,118,0.2149825,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{\sqrt{1-\cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]^(3/2)/Sqrt[1 - Cos[c + d*x]],x]","-\frac{i e^{-i (c+d x)} \left(-1+e^{i (c+d x)}\right) \sqrt{\cos (c+d x)} \left(\sqrt{2} e^{i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)-4 e^{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} \left(\sqrt{1+e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)+e^{i (c+d x)} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)\right)}{2 \sqrt{2} d \sqrt{1+e^{2 i (c+d x)}} \sqrt{1-\cos (c+d x)}}","\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{1-\cos (c+d x)}}+\frac{\tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{2} \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}",1,"((-1/2*I)*(-1 + E^(I*(c + d*x)))*(Sqrt[2]*E^(I*(c + d*x))*ArcSinh[E^(I*(c + d*x))] - 4*E^(I*(c + d*x))*ArcTanh[(1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] + Sqrt[2]*((1 + E^(I*(c + d*x)))*Sqrt[1 + E^((2*I)*(c + d*x))] + E^(I*(c + d*x))*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))*Sqrt[Cos[c + d*x]])/(Sqrt[2]*d*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[1 - Cos[c + d*x]])","C",1
284,1,160,85,0.11944,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{1-\cos (c+d x)}} \, dx","Integrate[Sqrt[Cos[c + d*x]]/Sqrt[1 - Cos[c + d*x]],x]","-\frac{i \left(-1+e^{i (c+d x)}\right) \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} \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)}{\sqrt{2} d \sqrt{1+e^{2 i (c+d x)}} \sqrt{1-\cos (c+d x)}}","\frac{2 \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{2} \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}",1,"((-I)*(-1 + E^(I*(c + d*x)))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(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))]]))/(Sqrt[2]*d*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[1 - Cos[c + d*x]])","C",1
285,1,110,47,0.1358343,"\int \frac{1}{\sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}} \, dx","Integrate[1/(Sqrt[1 - Cos[c + d*x]]*Sqrt[Cos[c + d*x]]),x]","\frac{i e^{-i (c+d x)} \left(-1+e^{i (c+d x)}\right) \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} d \sqrt{-((\cos (c+d x)-1) \cos (c+d x))}}","-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{2} \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}",1,"(I*(-1 + 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]*d*E^(I*(c + d*x))*Sqrt[-((-1 + Cos[c + d*x])*Cos[c + d*x])])","C",1
286,1,152,83,0.171634,"\int \frac{1}{\sqrt{1-\cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[1/(Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(3/2)),x]","\frac{2 \sin \left(\frac{1}{2} (c+d x)\right) \left(2 \sqrt{1+e^{2 i (c+d x)}} \cos \left(\frac{1}{2} (c+d x)\right)-\frac{e^{-\frac{1}{2} 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{2}}\right)}{d \sqrt{1+e^{2 i (c+d x)}} \sqrt{-((\cos (c+d x)-1) \cos (c+d x))}}","\frac{2 \sin (c+d x)}{d \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{2} \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}",1,"(2*(-(((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]*E^((I/2)*(c + d*x)))) + 2*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[(c + d*x)/2])*Sin[(c + d*x)/2])/(d*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[-((-1 + Cos[c + d*x])*Cos[c + d*x])])","C",0
287,1,170,122,0.2801813,"\int \frac{1}{\sqrt{1-\cos (c+d x)} \cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[1/(Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(5/2)),x]","\frac{2 \sin \left(\frac{1}{2} (c+d x)\right) \left(2 \sqrt{1+e^{2 i (c+d x)}} \cos \left(\frac{1}{2} (c+d x)\right) (\cos (c+d x)+1)-\frac{3 e^{-\frac{3}{2} i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^2 \tanh ^{-1}\left(\frac{1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)}{2 \sqrt{2}}\right)}{3 d \sqrt{1+e^{2 i (c+d x)}} \sqrt{1-\cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 \sin (c+d x)}{3 d \sqrt{1-\cos (c+d x)} \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x)}{3 d \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{2} \sqrt{1-\cos (c+d x)} \sqrt{\cos (c+d x)}}\right)}{d}",1,"(2*((-3*(1 + E^((2*I)*(c + d*x)))^2*ArcTanh[(1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/(2*Sqrt[2]*E^(((3*I)/2)*(c + d*x))) + 2*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[(c + d*x)/2]*(1 + Cos[c + d*x]))*Sin[(c + d*x)/2])/(3*d*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(3/2))","C",1
288,0,0,78,15.6084431,"\int \cos ^{\frac{4}{3}}(c+d x) \sqrt[3]{a+a \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^(4/3)*(a + a*Cos[c + d*x])^(1/3),x]","\int \cos ^{\frac{4}{3}}(c+d x) \sqrt[3]{a+a \cos (c+d x)} \, dx","\frac{2^{5/6} \sin (c+d x) \sqrt[3]{a \cos (c+d x)+a} F_1\left(\frac{1}{2};-\frac{4}{3},\frac{1}{6};\frac{3}{2};1-\cos (c+d x),\frac{1}{2} (1-\cos (c+d x))\right)}{d (\cos (c+d x)+1)^{5/6}}",1,"Integrate[Cos[c + d*x]^(4/3)*(a + a*Cos[c + d*x])^(1/3), x]","F",-1
289,0,0,79,3.4253806,"\int \cos ^{\frac{4}{3}}(c+d x) (a+a \cos (c+d x))^{2/3} \, dx","Integrate[Cos[c + d*x]^(4/3)*(a + a*Cos[c + d*x])^(2/3),x]","\int \cos ^{\frac{4}{3}}(c+d x) (a+a \cos (c+d x))^{2/3} \, dx","\frac{2 \sqrt[6]{2} \sin (c+d x) (a \cos (c+d x)+a)^{2/3} F_1\left(\frac{1}{2};-\frac{4}{3},-\frac{1}{6};\frac{3}{2};1-\cos (c+d x),\frac{1}{2} (1-\cos (c+d x))\right)}{d (\cos (c+d x)+1)^{7/6}}",1,"Integrate[Cos[c + d*x]^(4/3)*(a + a*Cos[c + d*x])^(2/3), x]","F",-1
290,0,0,79,2.703427,"\int \cos ^{\frac{5}{3}}(c+d x) (a+a \cos (c+d x))^{2/3} \, dx","Integrate[Cos[c + d*x]^(5/3)*(a + a*Cos[c + d*x])^(2/3),x]","\int \cos ^{\frac{5}{3}}(c+d x) (a+a \cos (c+d x))^{2/3} \, dx","\frac{2 \sqrt[6]{2} \sin (c+d x) (a \cos (c+d x)+a)^{2/3} F_1\left(\frac{1}{2};-\frac{5}{3},-\frac{1}{6};\frac{3}{2};1-\cos (c+d x),\frac{1}{2} (1-\cos (c+d x))\right)}{d (\cos (c+d x)+1)^{7/6}}",1,"Integrate[Cos[c + d*x]^(5/3)*(a + a*Cos[c + d*x])^(2/3), x]","F",-1
291,1,268,151,1.6294314,"\int (a+a \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*Sec[c + d*x]^(7/2),x]","\frac{a (\cos (c+d x)+1) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(9 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+9 \left(1+e^{2 i (c+d x)}\right)\right)+\left(1-e^{2 i c}\right) \sqrt{\sec (c+d x)} (9 \csc (c) \cos (d x)+\tan (c+d x) (3 \sec (c+d x)+5))\right)}{15 \left(d-e^{2 i c} d\right)}","\frac{2 a \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{6 a \sin (c+d x) \sqrt{\sec (c+d x)}}{5 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 d}-\frac{6 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",1,"(a*(1 + Cos[c + d*x])*Sec[(c + d*x)/2]^2*((I*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(9*(1 + E^((2*I)*(c + d*x))) + 9*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/E^(I*(c + d*x)) + (1 - E^((2*I)*c))*Sqrt[Sec[c + d*x]]*(9*Cos[d*x]*Csc[c] + (5 + 3*Sec[c + d*x])*Tan[c + d*x])))/(15*(d - d*E^((2*I)*c)))","C",1
292,1,255,123,1.1503573,"\int (a+a \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + a*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) \left(i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(3 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+\left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+3 \left(1+e^{2 i (c+d x)}\right)\right)-\left(-1+e^{2 i c}\right) \sqrt{\sec (c+d x)} (\tan (c+d x)+3 \csc (c) \cos (d x))\right)}{3 \left(d-e^{2 i c} d\right)}","\frac{2 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)}}{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 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)}{d}",1,"(a*(1 + Cos[c + d*x])*Sec[(c + d*x)/2]^2*((I*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(3*(1 + E^((2*I)*(c + d*x))) + 3*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/E^(I*(c + d*x)) - (-1 + E^((2*I)*c))*Sqrt[Sec[c + d*x]]*(3*Cos[d*x]*Csc[c] + Tan[c + d*x])))/(3*(d - d*E^((2*I)*c)))","C",1
293,1,124,97,1.3527127,"\int (a+a \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])*Sec[c + d*x]^(3/2),x]","-\frac{2 i a e^{-i (c+d x)} \left(\sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)-1\right) \sqrt{\sec (c+d x)}}{d}","\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)}}{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)}{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)}{d}",1,"((-2*I)*a*(-1 + Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sqrt[Sec[c + d*x]])/(d*E^(I*(c + d*x)))","C",1
294,1,141,75,1.1115387,"\int (a+a \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]],x]","-\frac{2 i a \left(-2 \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+2 e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+e^{2 i (c+d x)}+1\right)}{d \left(1+e^{2 i (c+d x)}\right) \sqrt{\sec (c+d x)}}","\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)}{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)}{d}",1,"((-2*I)*a*(1 + E^((2*I)*(c + d*x)) - 2*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 2*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*(1 + E^((2*I)*(c + d*x)))*Sqrt[Sec[c + d*x]])","C",1
295,1,140,101,1.2948791,"\int \frac{a+a \cos (c+d x)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(a + a*Cos[c + d*x])/Sqrt[Sec[c + d*x]],x]","\frac{a e^{-2 i c} (\sin (2 c)-i \cos (2 c)) \left(-\frac{12 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}+2 \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right) \sec (c+d x)+2 i \sin (c+d x)+6\right)}{3 d \sqrt{\sec (c+d x)}}","\frac{2 a \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\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 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)}{d}",1,"(a*((-I)*Cos[2*c] + Sin[2*c])*(6 - (12*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] + 2*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]*Sec[c + d*x] + (2*I)*Sin[c + d*x]))/(3*d*E^((2*I)*c)*Sqrt[Sec[c + d*x]])","C",1
296,1,224,127,0.9433858,"\int \frac{a+a \cos (c+d x)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])/Sec[c + d*x]^(3/2),x]","-\frac{i a e^{-3 i (c+d x)} \left(-72 e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+40 e^{3 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)-10 e^{i (c+d x)}+33 e^{2 i (c+d x)}+39 e^{4 i (c+d x)}+10 e^{5 i (c+d x)}+3 e^{6 i (c+d x)}-3\right) (\cos (c+d x)+1) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)}}{120 d}","\frac{2 a \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\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 d}+\frac{6 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",1,"((-1/120*I)*a*(1 + Cos[c + d*x])*(-3 - 10*E^(I*(c + d*x)) + 33*E^((2*I)*(c + d*x)) + 39*E^((4*I)*(c + d*x)) + 10*E^((5*I)*(c + d*x)) + 3*E^((6*I)*(c + d*x)) - 72*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 40*E^((3*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sec[(c + d*x)/2]^2*Sqrt[Sec[c + d*x]])/(d*E^((3*I)*(c + d*x)))","C",1
297,1,198,151,2.1583401,"\int \frac{a+a \cos (c+d x)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])/Sec[c + d*x]^(5/2),x]","\frac{a e^{-4 i (c+d x)} \sqrt{\sec (c+d x)} (\cos (4 (c+d x))+i \sin (4 (c+d x))) \left(504 i e^{-i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-200 i \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+42 \sin (c+d x)+130 \sin (2 (c+d x))+42 \sin (3 (c+d x))+15 \sin (4 (c+d x))-504 i \cos (c+d x)\right)}{420 d}","\frac{2 a \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{10 a \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{10 a \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 \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,"(a*Sqrt[Sec[c + d*x]]*(Cos[4*(c + d*x)] + I*Sin[4*(c + d*x)])*((-504*I)*Cos[c + d*x] + ((504*I)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/E^(I*(c + d*x)) - (200*I)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))] + 42*Sin[c + d*x] + 130*Sin[2*(c + d*x)] + 42*Sin[3*(c + d*x)] + 15*Sin[4*(c + d*x)]))/(420*d*E^((4*I)*(c + d*x)))","C",1
298,1,261,161,1.8716834,"\int (a+a \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2),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)} (24 \csc (c) \cos (d x)+\tan (c+d x) (3 \sec (c+d x)+10))-\frac{2 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(12 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+12 \left(1+e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{30 d}","\frac{2 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{4 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{16 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{16 a^2 \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,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(((-2*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(12*(1 + E^((2*I)*(c + d*x))) + 12*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) + Sqrt[Sec[c + d*x]]*(24*Cos[d*x]*Csc[c] + (10 + 3*Sec[c + d*x])*Tan[c + d*x])))/(30*d)","C",1
299,1,250,131,1.3590269,"\int (a+a \cos (c+d x))^2 \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*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)} (\tan (c+d x)+6 \csc (c) \cos (d x))-\frac{2 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(3 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+2 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+3 \left(1+e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{6 d}","\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{4 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{8 a^2 \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 \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*(((-2*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(3*(1 + E^((2*I)*(c + d*x))) + 3*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 2*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) + Sqrt[Sec[c + d*x]]*(6*Cos[d*x]*Csc[c] + Tan[c + d*x])))/(6*d)","C",1
300,1,48,64,0.1426261,"\int (a+a \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2),x]","\frac{2 a^2 \sqrt{\sec (c+d x)} \left(\sin (c+d x)+2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d}","\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{4 a^2 \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*a^2*Sqrt[Sec[c + d*x]]*(2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + Sin[c + d*x]))/d","A",1
301,1,127,107,1.049862,"\int (a+a \cos (c+d x))^2 \sqrt{\sec (c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]],x]","\frac{a^2 \left(\frac{24 i \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}+2 \left(-4 i \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right) \sec (c+d x)+\sin (c+d x)-6 i\right)\right)}{3 d \sqrt{\sec (c+d x)}}","\frac{2 a^2 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{8 a^2 \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 \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*(((24*I)*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] + 2*(-6*I - (4*I)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]*Sec[c + d*x] + Sin[c + d*x])))/(3*d*Sqrt[Sec[c + d*x]])","C",1
302,1,136,135,1.5132469,"\int \frac{(a+a \cos (c+d x))^2}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(a + a*Cos[c + d*x])^2/Sqrt[Sec[c + d*x]],x]","\frac{a^2 \left(\frac{192 i \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}-40 i \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right) \sec (c+d x)+40 \sin (c+d x)+6 \sin (2 (c+d x))-96 i\right)}{30 d \sqrt{\sec (c+d x)}}","\frac{2 a^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{16 a^2 \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,"(a^2*(-96*I + ((192*I)*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] - (40*I)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]*Sec[c + d*x] + 40*Sin[c + d*x] + 6*Sin[2*(c + d*x)]))/(30*d*Sqrt[Sec[c + d*x]])","C",1
303,1,149,161,1.7096225,"\int \frac{(a+a \cos (c+d x))^2}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^2/Sec[c + d*x]^(3/2),x]","\frac{a^2 \left(\frac{672 i \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}+2 \left(-80 i \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right) \sec (c+d x)+85 \sin (c+d x)+28 \sin (2 (c+d x))+5 \sin (3 (c+d x))-168 i\right)\right)}{140 d \sqrt{\sec (c+d x)}}","\frac{4 a^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a^2 \sin (c+d x)}{7 d \sqrt{\sec (c+d x)}}+\frac{8 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d}+\frac{12 a^2 \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,"(a^2*(((672*I)*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] + 2*(-168*I - (80*I)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]*Sec[c + d*x] + 85*Sin[c + d*x] + 28*Sin[2*(c + d*x)] + 5*Sin[3*(c + d*x)])))/(140*d*Sqrt[Sec[c + d*x]])","C",1
304,1,279,187,2.7545892,"\int (a+a \cos (c+d x))^3 \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(9/2),x]","\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(\sqrt{\sec (c+d x)} \left(294 \csc (c) \cos (d x)+(63 \cos (c+d x)+65 \cos (2 (c+d x))+80) \tan (c+d x) \sec ^2(c+d x)\right)-\frac{2 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(147 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+65 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+147 \left(1+e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{420 d}","\frac{2 a^3 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{6 a^3 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{52 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{28 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{52 a^3 \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{28 a^3 \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,"(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(((-2*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(147*(1 + E^((2*I)*(c + d*x))) + 147*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 65*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) + Sqrt[Sec[c + d*x]]*(294*Cos[d*x]*Csc[c] + (80 + 63*Cos[c + d*x] + 65*Cos[2*(c + d*x)])*Sec[c + d*x]^2*Tan[c + d*x])))/(420*d)","C",1
305,1,259,157,1.9007577,"\int (a+a \cos (c+d x))^3 \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2),x]","\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(\sqrt{\sec (c+d x)} (18 \csc (c) \cos (d x)+\tan (c+d x) (\sec (c+d x)+5))-\frac{2 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(9 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+9 \left(1+e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{20 d}","\frac{2 a^3 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d}+\frac{36 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{36 a^3 \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,"(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(((-2*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(9*(1 + E^((2*I)*(c + d*x))) + 9*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) + Sqrt[Sec[c + d*x]]*(18*Cos[d*x]*Csc[c] + (5 + Sec[c + d*x])*Tan[c + d*x])))/(20*d)","C",1
306,1,157,131,0.9709088,"\int (a+a \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2),x]","-\frac{i a^3 \sec ^{\frac{3}{2}}(c+d x) \left(6 e^{-2 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{3/2} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+20 \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right) \cos (c+d x)+2 i \sin (c+d x)+9 i \sin (2 (c+d x))-6 \cos (2 (c+d x))-6\right)}{3 d}","\frac{2 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{6 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{20 a^3 \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 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"((-1/3*I)*a^3*Sec[c + d*x]^(3/2)*(-6 - 6*Cos[2*(c + d*x)] + (6*(1 + E^((2*I)*(c + d*x)))^(3/2)*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/E^((2*I)*(c + d*x)) + 20*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))] + (2*I)*Sin[c + d*x] + (9*I)*Sin[2*(c + d*x)]))/d","C",1
307,1,135,131,1.307676,"\int (a+a \cos (c+d x))^3 \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2),x]","\frac{a^3 \left(\frac{24 i \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}+2 \left(-10 i \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right) \sec (c+d x)+\sin (c+d x)+3 \tan (c+d x)-6 i\right)\right)}{3 d \sqrt{\sec (c+d x)}}","\frac{2 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a^3 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{20 a^3 \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 \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^3*(((24*I)*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] + 2*(-6*I - (10*I)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]*Sec[c + d*x] + Sin[c + d*x] + 3*Tan[c + d*x])))/(3*d*Sqrt[Sec[c + d*x]])","C",1
308,1,137,131,1.2800291,"\int (a+a \cos (c+d x))^3 \sqrt{\sec (c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]],x]","\frac{a^3 \left(\frac{144 i \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}+2 \left(-20 i \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right) \sec (c+d x)+10 \sin (c+d x)+\sin (2 (c+d x))-36 i\right)\right)}{10 d \sqrt{\sec (c+d x)}}","\frac{2 a^3 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^3 \sin (c+d x)}{d \sqrt{\sec (c+d x)}}+\frac{4 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{36 a^3 \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,"(a^3*(((144*I)*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] + 2*(-36*I - (20*I)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]*Sec[c + d*x] + 10*Sin[c + d*x] + Sin[2*(c + d*x)])))/(10*d*Sqrt[Sec[c + d*x]])","C",1
309,1,146,161,1.8002217,"\int \frac{(a+a \cos (c+d x))^3}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(a + a*Cos[c + d*x])^3/Sqrt[Sec[c + d*x]],x]","\frac{a^3 \left(\frac{4704 i \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}-1040 i \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right) \sec (c+d x)+1070 \sin (c+d x)+252 \sin (2 (c+d x))+30 \sin (3 (c+d x))-2352 i\right)}{420 d \sqrt{\sec (c+d x)}}","\frac{6 a^3 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^3 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{52 a^3 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{52 a^3 \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{28 a^3 \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,"(a^3*(-2352*I + ((4704*I)*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] - (1040*I)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]*Sec[c + d*x] + 1070*Sin[c + d*x] + 252*Sin[2*(c + d*x)] + 30*Sin[3*(c + d*x)]))/(420*d*Sqrt[Sec[c + d*x]])","C",1
310,1,156,187,2.2876584,"\int \frac{(a+a \cos (c+d x))^3}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^3/Sec[c + d*x]^(3/2),x]","\frac{a^3 \left(\frac{22848 i \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}-5280 i \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right) \sec (c+d x)+5820 \sin (c+d x)+2044 \sin (2 (c+d x))+540 \sin (3 (c+d x))+70 \sin (4 (c+d x))-11424 i\right)}{2520 d \sqrt{\sec (c+d x)}}","\frac{68 a^3 \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{6 a^3 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a^3 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{44 a^3 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{44 a^3 \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{68 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}",1,"(a^3*(-11424*I + ((22848*I)*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] - (5280*I)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]*Sec[c + d*x] + 5820*Sin[c + d*x] + 2044*Sin[2*(c + d*x)] + 540*Sin[3*(c + d*x)] + 70*Sin[4*(c + d*x)]))/(2520*d*Sqrt[Sec[c + d*x]])","C",1
311,1,271,187,2.0323406,"\int (a+a \cos (c+d x))^4 \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*Sec[c + d*x]^(9/2),x]","\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \left(\sqrt{\sec (c+d x)} \left(672 \csc (c) \cos (d x)+\tan (c+d x) \left(15 \sec ^2(c+d x)+84 \sec (c+d x)+235\right)\right)-\frac{4 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(168 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+85 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+168 \left(1+e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{840 d}","\frac{2 a^4 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{8 a^4 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{94 a^4 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{64 a^4 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{136 a^4 \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{64 a^4 \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,"(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*(((-4*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(168*(1 + E^((2*I)*(c + d*x))) + 168*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 85*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) + Sqrt[Sec[c + d*x]]*(672*Cos[d*x]*Csc[c] + (235 + 84*Sec[c + d*x] + 15*Sec[c + d*x]^2)*Tan[c + d*x])))/(840*d)","C",0
312,1,278,161,2.4947762,"\int (a+a \cos (c+d x))^4 \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*Sec[c + d*x]^(7/2),x]","\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \left(\sqrt{\sec (c+d x)} (30 \cos (c) \sin (d x)-3 (5 \cos (2 c)-61) \csc (c) \cos (d x)+2 \tan (c+d x) (3 \sec (c+d x)+20))-\frac{8 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(21 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+20 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+21 \left(1+e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{240 d}","\frac{2 a^4 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{8 a^4 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{66 a^4 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{32 a^4 \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{56 a^4 \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,"(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*(((-8*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(21*(1 + E^((2*I)*(c + d*x))) + 21*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 20*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) + Sqrt[Sec[c + d*x]]*(-3*(-61 + 5*Cos[2*c])*Cos[d*x]*Csc[c] + 30*Cos[c]*Sin[d*x] + 2*(20 + 3*Sec[c + d*x])*Tan[c + d*x])))/(240*d)","C",1
313,1,70,118,0.3401637,"\int (a+a \cos (c+d x))^4 \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*Sec[c + d*x]^(5/2),x]","\frac{a^4 \sec ^{\frac{3}{2}}(c+d x) \left(5 \sin (c+d x)+24 \sin (2 (c+d x))+\sin (3 (c+d x))+80 \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{6 d}","\frac{2 a^4 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{8 a^4 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a^4 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{40 a^4 \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,"(a^4*Sec[c + d*x]^(3/2)*(80*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 5*Sin[c + d*x] + 24*Sin[2*(c + d*x)] + Sin[3*(c + d*x)]))/(6*d)","A",1
314,1,150,159,1.4996897,"\int (a+a \cos (c+d x))^4 \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^4*Sec[c + d*x]^(3/2),x]","\frac{a^4 \left(\frac{672 i \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}-320 i \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right) \sec (c+d x)+80 \sin (c+d x)+63 \tan (c+d x)+3 \sin (3 (c+d x)) \sec (c+d x)-336 i\right)}{30 d \sqrt{\sec (c+d x)}}","\frac{2 a^4 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{8 a^4 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{32 a^4 \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{56 a^4 \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,"(a^4*(-336*I + ((672*I)*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] - (320*I)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]*Sec[c + d*x] + 80*Sin[c + d*x] + 3*Sec[c + d*x]*Sin[3*(c + d*x)] + 63*Tan[c + d*x]))/(30*d*Sqrt[Sec[c + d*x]])","C",1
315,1,146,161,1.5599475,"\int (a+a \cos (c+d x))^4 \sqrt{\sec (c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^4*Sqrt[Sec[c + d*x]],x]","\frac{a^4 \left(\frac{10752 i \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}-2720 i \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right) \sec (c+d x)+1910 \sin (c+d x)+336 \sin (2 (c+d x))+30 \sin (3 (c+d x))-5376 i\right)}{420 d \sqrt{\sec (c+d x)}}","\frac{8 a^4 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{94 a^4 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{136 a^4 \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{64 a^4 \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,"(a^4*(-5376*I + ((10752*I)*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] - (2720*I)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]*Sec[c + d*x] + 1910*Sin[c + d*x] + 336*Sin[2*(c + d*x)] + 30*Sin[3*(c + d*x)]))/(420*d*Sqrt[Sec[c + d*x]])","C",1
316,1,156,187,2.1192807,"\int \frac{(a+a \cos (c+d x))^4}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(a + a*Cos[c + d*x])^4/Sqrt[Sec[c + d*x]],x]","\frac{a^4 \left(\frac{51072 i \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}-11520 i \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right) \sec (c+d x)+12240 \sin (c+d x)+3556 \sin (2 (c+d x))+720 \sin (3 (c+d x))+70 \sin (4 (c+d x))-25536 i\right)}{2520 d \sqrt{\sec (c+d x)}}","\frac{122 a^4 \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 a^4 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{32 a^4 \sin (c+d x)}{7 d \sqrt{\sec (c+d x)}}+\frac{32 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d}+\frac{152 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}",1,"(a^4*(-25536*I + ((51072*I)*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] - (11520*I)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]*Sec[c + d*x] + 12240*Sin[c + d*x] + 3556*Sin[2*(c + d*x)] + 720*Sin[3*(c + d*x)] + 70*Sin[4*(c + d*x)]))/(2520*d*Sqrt[Sec[c + d*x]])","C",1
317,1,285,164,3.2869273,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x]),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(-\sqrt{\sec (c+d x)} \left(18 \csc (c) \cos (d x)+\sec (c+d x) \left(\tan \left(\frac{1}{2} (c+d x)\right)-5 \sin \left(\frac{3}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right)\right)\right)+\frac{2 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(9 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+9 \left(1+e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{3 a d (\cos (c+d x)+1)}","-\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{d (a \sec (c+d x)+a)}+\frac{5 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}-\frac{3 \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}+\frac{5 \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 \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*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(9*(1 + E^((2*I)*(c + d*x))) + 9*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) - Sqrt[Sec[c + d*x]]*(18*Cos[d*x]*Csc[c] + Sec[c + d*x]*(-5*Sec[(c + d*x)/2]*Sin[(3*(c + d*x))/2] + Tan[(c + d*x)/2]))))/(3*a*d*(1 + Cos[c + d*x]))","C",1
318,1,256,136,1.9587321,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[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(\frac{\sqrt{\sec (c+d x)} \left(6 \csc (c) \cos (d x)-2 \tan \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{2 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(3 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-\left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+3 \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d}\right)}{a (\cos (c+d x)+1)}","-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d (a \sec (c+d x)+a)}+\frac{3 \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{\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 \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*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(3*(1 + E^((2*I)*(c + d*x))) + 3*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))) + (Sqrt[Sec[c + d*x]]*(6*Cos[d*x]*Csc[c] - 2*Tan[(c + d*x)/2]))/d))/(a*(1 + Cos[c + d*x]))","C",1
319,1,180,110,1.0166992,"\int \frac{\sqrt{\sec (c+d x)}}{a+a \cos (c+d x)} \, dx","Integrate[Sqrt[Sec[c + d*x]]/(a + a*Cos[c + d*x]),x]","-\frac{4 i \left(-\left(\left(1+e^{i (c+d x)}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)\right)+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}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+e^{2 i (c+d x)}+1\right) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)}}{a d \left(1+e^{i (c+d x)}\right)^3}","-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \sec (c+d x)+a)}+\frac{\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{\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,"((-4*I)*Cos[(c + d*x)/2]^2*(1 + E^((2*I)*(c + d*x)) - (1 + E^(I*(c + d*x)))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -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[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sqrt[Sec[c + d*x]])/(a*d*(1 + E^(I*(c + d*x)))^3)","C",1
320,1,181,110,0.9609118,"\int \frac{1}{(a+a \cos (c+d x)) \sqrt{\sec (c+d x)}} \, dx","Integrate[1/((a + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]),x]","-\frac{4 i \left(\left(1+e^{i (c+d x)}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+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}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)-e^{2 i (c+d x)}-1\right) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)}}{a d \left(1+e^{i (c+d x)}\right)^3}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \sec (c+d x)+a)}+\frac{\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{\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,"((-4*I)*Cos[(c + d*x)/2]^2*(-1 - E^((2*I)*(c + d*x)) + (1 + E^(I*(c + d*x)))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -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[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sqrt[Sec[c + d*x]])/(a*d*(1 + E^(I*(c + d*x)))^3)","C",1
321,1,311,112,1.6945167,"\int \frac{1}{(a+a \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[1/((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) \left(\cos \left(\frac{1}{2} (c-d x)\right)+2 \cos \left(\frac{1}{2} (3 c+d x)\right)+2 \cos \left(\frac{1}{2} (c+3 d x)\right)+\cos \left(\frac{1}{2} (5 c+3 d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)}}{2 d}+\frac{2 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(3 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+\left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+3 \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d}\right)}{a (\cos (c+d x)+1)}","-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \sec (c+d x)+a)}-\frac{\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 \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*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(3*(1 + E^((2*I)*(c + d*x))) + 3*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))) - ((Cos[(c - d*x)/2] + 2*Cos[(3*c + d*x)/2] + 2*Cos[(c + 3*d*x)/2] + Cos[(5*c + 3*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]])/(2*d)))/(a*(1 + Cos[c + d*x]))","C",1
322,1,312,140,4.1168905,"\int \frac{1}{(a+a \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[1/((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(2 \sqrt{\sec (c+d x)} \left(\sin (2 c) \cos (2 d x)-6 \cos (c) \sin (d x)+\cos (2 c) \sin (2 d x)+3 (\cos (2 c)+2) \csc (c) \cos (d x)-3 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)-3 \tan \left(\frac{c}{2}\right)\right)-\frac{2 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(9 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+9 \left(1+e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{3 a d (\cos (c+d x)+1)}","\frac{5 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{\sin (c+d x)}{d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)}+\frac{5 \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 \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*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(9*(1 + E^((2*I)*(c + d*x))) + 9*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) + 2*Sqrt[Sec[c + d*x]]*(3*(2 + Cos[2*c])*Cos[d*x]*Csc[c] + Cos[2*d*x]*Sin[2*c] - 3*Sec[c/2]*Sec[(c + d*x)/2]*Sin[(d*x)/2] - 6*Cos[c]*Sin[d*x] + Cos[2*c]*Sin[2*d*x] - 3*Tan[c/2])))/(3*a*d*(1 + Cos[c + d*x]))","C",1
323,1,341,168,2.6931642,"\int \frac{1}{(a+a \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[1/((a + a*Cos[c + d*x])*Sec[c + d*x]^(7/2)),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(-\sqrt{\sec (c+d x)} \left(18 (11 \cos (2 c)+17) \csc (c) \cos (d x)+4 \left(10 \sin (2 c) \cos (2 d x)-3 \sin (3 c) \cos (3 d x)-99 \cos (c) \sin (d x)+10 \cos (2 c) \sin (2 d x)-3 \cos (3 c) \sin (3 d x)-30 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)-30 \tan \left(\frac{c}{2}\right)\right)\right)+\frac{8 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(63 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+25 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+63 \left(1+e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{60 a d (\cos (c+d x)+1)}","-\frac{\sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)}+\frac{7 \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{5 \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{21 \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*(((8*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(63*(1 + E^((2*I)*(c + d*x))) + 63*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 25*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) - Sqrt[Sec[c + d*x]]*(18*(17 + 11*Cos[2*c])*Cos[d*x]*Csc[c] + 4*(10*Cos[2*d*x]*Sin[2*c] - 3*Cos[3*d*x]*Sin[3*c] - 30*Sec[c/2]*Sec[(c + d*x)/2]*Sin[(d*x)/2] - 99*Cos[c]*Sin[d*x] + 10*Cos[2*c]*Sin[2*d*x] - 3*Cos[3*c]*Sin[3*d*x] - 30*Tan[c/2]))))/(60*a*d*(1 + Cos[c + d*x]))","C",1
324,1,287,202,2.4436542,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^2,x]","-\frac{\left(-1+e^{i c}\right) \csc \left(\frac{c}{2}\right) e^{-\frac{1}{2} i (4 c+3 d x)} \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(7 e^{i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{3/2} \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)-37 e^{i (c+d x)}-65 e^{2 i (c+d x)}-82 e^{3 i (c+d x)}-68 e^{4 i (c+d x)}-53 e^{5 i (c+d x)}-21 e^{6 i (c+d x)}+10 i \left(1+e^{2 i (c+d x)}\right) \left(1+e^{i (c+d x)}\right)^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-10\right)}{12 a^2 d \left(1+e^{2 i (c+d x)}\right) (\cos (c+d x)+1)^2}","-\frac{7 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{10 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d}-\frac{7 \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}+\frac{10 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{7 \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{\sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"-1/12*((-1 + E^(I*c))*Cos[(c + d*x)/2]*Csc[c/2]*(-10 - 37*E^(I*(c + d*x)) - 65*E^((2*I)*(c + d*x)) - 82*E^((3*I)*(c + d*x)) - 68*E^((4*I)*(c + d*x)) - 53*E^((5*I)*(c + d*x)) - 21*E^((6*I)*(c + d*x)) + (10*I)*(1 + E^(I*(c + d*x)))^3*(1 + E^((2*I)*(c + d*x)))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 7*E^(I*(c + d*x))*(1 + E^(I*(c + d*x)))^3*(1 + E^((2*I)*(c + d*x)))^(3/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sqrt[Sec[c + d*x]])/(a^2*d*E^((I/2)*(4*c + 3*d*x))*(1 + E^((2*I)*(c + d*x)))*(1 + Cos[c + d*x])^2)","C",1
325,1,252,176,1.3029684,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[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(-4 i 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)+i (12 i \sin (c+d x)+7 i \sin (2 (c+d x))+50 \cos (c+d x)+17 \cos (2 (c+d x))+29)+40 \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)\right)}{6 a^2 d (\cos (c+d x)+1)^2}","-\frac{5 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{4 \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}-\frac{5 \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 \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{\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]]*(((-4*I)*(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)) + 40*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*(29 + 50*Cos[c + d*x] + 17*Cos[2*(c + d*x)] + (12*I)*Sin[c + d*x] + (7*I)*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
326,1,242,149,1.2148892,"\int \frac{\sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^2} \, dx","Integrate[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(-i 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)+i (i \sin (2 (c+d x))+14 \cos (c+d x)+5 \cos (2 (c+d x))+5)+16 \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)\right)}{6 a^2 d (\cos (c+d x)+1)^2}","-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d (\sec (c+d x)+1)}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{\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{\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)*(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)) + 16*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*(5 + 14*Cos[c + d*x] + 5*Cos[2*(c + d*x)] + I*Sin[2*(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
327,1,98,77,0.3670291,"\int \frac{1}{(a+a \cos (c+d x))^2 \sqrt{\sec (c+d x)}} \, dx","Integrate[1/((a + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]),x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{3}{2} (c+d x)\right)+4 \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)\right)}{3 a^2 d (\cos (c+d x)+1)^2}","\frac{\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{\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]]*(4*Cos[(c + d*x)/2]^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2]))/(3*a^2*d*(1 + Cos[c + d*x])^2)","A",1
328,1,239,149,1.3867778,"\int \frac{1}{(a+a \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[1/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)),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 \left(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)+i \sin (2 (c+d x))-10 \cos (c+d x)-7 \cos (2 (c+d x))-7\right)+16 \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{\sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d (\sec (c+d x)+1)}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{\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{\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]]*(16*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*(-7 - 10*Cos[c + d*x] - 7*Cos[2*(c + d*x)] + ((1 + E^(I*(c + d*x)))^3*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^(I*(c + d*x)) + I*Sin[2*(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
329,1,259,152,1.9943169,"\int \frac{1}{(a+a \cos (c+d x))^2 \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[1/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)),x]","-\frac{\sin (c) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) e^{-i d x} \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(2 i e^{-\frac{1}{2} i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\sin \left(\frac{1}{2} (c+d x)\right)+2 \sin \left(\frac{3}{2} (c+d x)\right)+3 \sin \left(\frac{5}{2} (c+d x)\right)-24 i \cos \left(\frac{1}{2} (c+d x)\right)-18 i \cos \left(\frac{3}{2} (c+d x)\right)-6 i \cos \left(\frac{5}{2} (c+d x)\right)+20 \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)\right)}{6 a^2 d (\cos (c+d x)+1)^2}","-\frac{5 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d (\sec (c+d x)+1)}-\frac{5 \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 \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{\sin (c+d x) \sqrt{\sec (c+d x)}}{3 d (a \sec (c+d x)+a)^2}",1,"-1/6*(Cos[(c + d*x)/2]*Csc[c/2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*Sin[c]*(Cos[d*x] + I*Sin[d*x])*((-24*I)*Cos[(c + d*x)/2] - (18*I)*Cos[(3*(c + d*x))/2] - (6*I)*Cos[(5*(c + d*x))/2] + 20*Cos[(c + d*x)/2]^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + ((2*I)*(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/2)*(c + d*x)) + Sin[(c + d*x)/2] + 2*Sin[(3*(c + d*x))/2] + 3*Sin[(5*(c + d*x))/2]))/(a^2*d*E^(I*d*x)*(1 + Cos[c + d*x])^2)","C",1
330,1,257,178,1.8118692,"\int \frac{1}{(a+a \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[1/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(7/2)),x]","\frac{e^{-i d x} \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(7 i e^{-\frac{1}{2} i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+3 \sin \left(\frac{1}{2} (c+d x)\right)+10 \sin \left(\frac{3}{2} (c+d x)\right)+12 \sin \left(\frac{5}{2} (c+d x)\right)+\sin \left(\frac{7}{2} (c+d x)\right)-84 i \cos \left(\frac{1}{2} (c+d x)\right)-63 i \cos \left(\frac{3}{2} (c+d x)\right)-21 i \cos \left(\frac{5}{2} (c+d x)\right)+80 \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)\right)}{6 a^2 d (\cos (c+d x)+1)^2}","\frac{10 \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{7 \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)} (\sec (c+d x)+1)}+\frac{10 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{7 \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{\sin (c+d x)}{3 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*((-84*I)*Cos[(c + d*x)/2] - (63*I)*Cos[(3*(c + d*x))/2] - (21*I)*Cos[(5*(c + d*x))/2] + 80*Cos[(c + d*x)/2]^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + ((7*I)*(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/2)*(c + d*x)) + 3*Sin[(c + d*x)/2] + 10*Sin[(3*(c + d*x))/2] + 12*Sin[(5*(c + d*x))/2] + Sin[(7*(c + d*x))/2]))/(6*a^2*d*E^(I*d*x)*(1 + Cos[c + d*x])^2)","C",1
331,1,271,200,1.8436188,"\int \frac{1}{(a+a \cos (c+d x))^2 \sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[1/((a + a*Cos[c + d*x])^2*Sec[c + d*x]^(9/2)),x]","\frac{e^{-i d x} \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-112 i e^{-\frac{1}{2} i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-34 \sin \left(\frac{1}{2} (c+d x)\right)-148 \sin \left(\frac{3}{2} (c+d x)\right)-168 \sin \left(\frac{5}{2} (c+d x)\right)-11 \sin \left(\frac{7}{2} (c+d x)\right)+3 \sin \left(\frac{9}{2} (c+d x)\right)+1344 i \cos \left(\frac{1}{2} (c+d x)\right)+1008 i \cos \left(\frac{3}{2} (c+d x)\right)+336 i \cos \left(\frac{5}{2} (c+d x)\right)-1200 \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)\right)}{60 a^2 d (\cos (c+d x)+1)^2}","-\frac{3 \sin (c+d x)}{a^2 d \sec ^{\frac{3}{2}}(c+d x) (\sec (c+d x)+1)}+\frac{56 \sin (c+d x)}{15 a^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 \sin (c+d x)}{a^2 d \sqrt{\sec (c+d x)}}-\frac{5 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{56 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{\sin (c+d x)}{3 d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*((1344*I)*Cos[(c + d*x)/2] + (1008*I)*Cos[(3*(c + d*x))/2] + (336*I)*Cos[(5*(c + d*x))/2] - 1200*Cos[(c + d*x)/2]^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - ((112*I)*(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/2)*(c + d*x)) - 34*Sin[(c + d*x)/2] - 148*Sin[(3*(c + d*x))/2] - 168*Sin[(5*(c + d*x))/2] - 11*Sin[(7*(c + d*x))/2] + 3*Sin[(9*(c + d*x))/2]))/(60*a^2*d*E^(I*d*x)*(1 + Cos[c + d*x])^2)","C",1
332,1,363,221,2.3119975,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^3,x]","\frac{2 \cos ^6\left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{32} \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(1284 \cos \left(\frac{1}{2} (c-d x)\right)+921 \cos \left(\frac{1}{2} (3 c+d x)\right)+1243 \cos \left(\frac{1}{2} (c+3 d x)\right)+374 \cos \left(\frac{1}{2} (5 c+3 d x)\right)+670 \cos \left(\frac{1}{2} (3 c+5 d x)\right)+65 \cos \left(\frac{1}{2} (7 c+5 d x)\right)+147 \cos \left(\frac{1}{2} (5 c+7 d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)}-\frac{2 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(147 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-65 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+147 \left(1+e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{15 a^3 d (\cos (c+d x)+1)^3}","-\frac{13 \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 \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{13 \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 \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{\sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{8 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"(2*Cos[(c + d*x)/2]^6*(((-2*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(147*(1 + E^((2*I)*(c + d*x))) + 147*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 65*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) + ((1284*Cos[(c - d*x)/2] + 921*Cos[(3*c + d*x)/2] + 1243*Cos[(c + 3*d*x)/2] + 374*Cos[(5*c + 3*d*x)/2] + 670*Cos[(3*c + 5*d*x)/2] + 65*Cos[(7*c + 5*d*x)/2] + 147*Cos[(5*c + 7*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^5*Sqrt[Sec[c + d*x]])/32))/(15*a^3*d*(1 + Cos[c + d*x])^3)","C",1
333,1,274,195,2.3540493,"\int \frac{\sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^3} \, dx","Integrate[Sqrt[Sec[c + d*x]]/(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(-3 i e^{-2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)^5 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+2 i (2 i \sin (c+d x)+6 i \sin (2 (c+d x))+2 i \sin (3 (c+d x))+69 \cos (c+d x)+34 \cos (2 (c+d x))+7 \cos (3 (c+d x))+34)+160 \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)}{40 a^3 d (\cos (c+d x)+1)^3}","-\frac{9 \sin (c+d x) \sqrt{\sec (c+d x)}}{10 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{9 \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{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 a d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(((-3*I)*(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*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)*(34 + 69*Cos[c + d*x] + 34*Cos[2*(c + d*x)] + 7*Cos[3*(c + d*x)] + (2*I)*Sin[c + d*x] + (6*I)*Sin[2*(c + d*x)] + (2*I)*Sin[3*(c + d*x)]))*(Cos[(c + 3*d*x)/2] + I*Sin[(c + 3*d*x)/2]))/(40*a^3*d*E^(I*d*x)*(1 + Cos[c + d*x])^3)","C",1
334,1,363,195,2.0939055,"\int \frac{1}{(a+a \cos (c+d x))^3 \sqrt{\sec (c+d x)}} \, dx","Integrate[1/((a + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]),x]","\frac{2 \cos ^6\left(\frac{1}{2} (c+d x)\right) \left(-\frac{1}{32} \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(36 \cos \left(\frac{1}{2} (c-d x)\right)+9 \cos \left(\frac{1}{2} (3 c+d x)\right)+7 \cos \left(\frac{1}{2} (c+3 d x)\right)+26 \cos \left(\frac{1}{2} (5 c+3 d x)\right)+10 \cos \left(\frac{1}{2} (3 c+5 d x)\right)+5 \cos \left(\frac{1}{2} (7 c+5 d x)\right)+3 \cos \left(\frac{1}{2} (5 c+7 d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)}+\frac{2 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(3 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+3 \left(1+e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{15 a^3 d (\cos (c+d x)+1)^3}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{\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{\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{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{4 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}",1,"(2*Cos[(c + d*x)/2]^6*(((2*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(3*(1 + E^((2*I)*(c + d*x))) + 3*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) - ((36*Cos[(c - d*x)/2] + 9*Cos[(3*c + d*x)/2] + 7*Cos[(c + 3*d*x)/2] + 26*Cos[(5*c + 3*d*x)/2] + 10*Cos[(3*c + 5*d*x)/2] + 5*Cos[(7*c + 5*d*x)/2] + 3*Cos[(5*c + 7*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^5*Sqrt[Sec[c + d*x]])/32))/(15*a^3*d*(1 + Cos[c + d*x])^3)","C",1
335,1,363,195,2.0033001,"\int \frac{1}{(a+a \cos (c+d x))^3 \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[1/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)),x]","\frac{2 \cos ^6\left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{32} \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(36 \cos \left(\frac{1}{2} (c-d x)\right)+9 \cos \left(\frac{1}{2} (3 c+d x)\right)+17 \cos \left(\frac{1}{2} (c+3 d x)\right)+16 \cos \left(\frac{1}{2} (5 c+3 d x)\right)+20 \cos \left(\frac{1}{2} (3 c+5 d x)\right)-5 \cos \left(\frac{1}{2} (7 c+5 d x)\right)+3 \cos \left(\frac{1}{2} (5 c+7 d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)}-\frac{2 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(3 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+3 \left(1+e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{15 a^3 d (\cos (c+d x)+1)^3}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{\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{\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{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}",1,"(2*Cos[(c + d*x)/2]^6*(((-2*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(3*(1 + E^((2*I)*(c + d*x))) + 3*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) + ((36*Cos[(c - d*x)/2] + 9*Cos[(3*c + d*x)/2] + 17*Cos[(c + 3*d*x)/2] + 16*Cos[(5*c + 3*d*x)/2] + 20*Cos[(3*c + 5*d*x)/2] - 5*Cos[(7*c + 5*d*x)/2] + 3*Cos[(5*c + 7*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^5*Sqrt[Sec[c + d*x]])/32))/(15*a^3*d*(1 + Cos[c + d*x])^3)","C",1
336,1,272,195,3.0429337,"\int \frac{1}{(a+a \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[1/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)),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 \left(3 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)+6 i \sin (c+d x)+8 i \sin (2 (c+d x))+6 i \sin (3 (c+d x))-128 \cos (c+d x)-68 \cos (2 (c+d x))-24 \cos (3 (c+d x))-68\right)+160 \sqrt{\cos (c+d x)} \cos ^5\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)}{40 a^3 d (\cos (c+d x)+1)^3}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{2 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{9 \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{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 a d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(160*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]) + I*(-68 - 128*Cos[c + d*x] - 68*Cos[2*(c + d*x)] - 24*Cos[3*(c + d*x)] + (3*(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)) + (6*I)*Sin[c + d*x] + (8*I)*Sin[2*(c + d*x)] + (6*I)*Sin[3*(c + d*x)]))*(Cos[(c + 3*d*x)/2] + I*Sin[(c + 3*d*x)/2]))/(40*a^3*d*E^(I*d*x)*(1 + Cos[c + d*x])^3)","C",1
337,1,378,195,2.1543676,"\int \frac{1}{(a+a \cos (c+d x))^3 \sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[1/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)),x]","\frac{2 \cos ^6\left(\frac{1}{2} (c+d x)\right) \left(-\frac{1}{32} \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(1134 \cos \left(\frac{1}{2} (c-d x)\right)+1071 \cos \left(\frac{1}{2} (3 c+d x)\right)+923 \cos \left(\frac{1}{2} (c+3 d x)\right)+694 \cos \left(\frac{1}{2} (5 c+3 d x)\right)+470 \cos \left(\frac{1}{2} (3 c+5 d x)\right)+265 \cos \left(\frac{1}{2} (7 c+5 d x)\right)+117 \cos \left(\frac{1}{2} (5 c+7 d x)\right)+30 \cos \left(\frac{1}{2} (9 c+7 d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)}+\frac{2 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(147 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+65 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+147 \left(1+e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{15 a^3 d (\cos (c+d x)+1)^3}","-\frac{13 \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{13 \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 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{8 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{5 d (a \sec (c+d x)+a)^3}",1,"(2*Cos[(c + d*x)/2]^6*(((2*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(147*(1 + E^((2*I)*(c + d*x))) + 147*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 65*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) - ((1134*Cos[(c - d*x)/2] + 1071*Cos[(3*c + d*x)/2] + 923*Cos[(c + 3*d*x)/2] + 694*Cos[(5*c + 3*d*x)/2] + 470*Cos[(3*c + 5*d*x)/2] + 265*Cos[(7*c + 5*d*x)/2] + 117*Cos[(5*c + 7*d*x)/2] + 30*Cos[(9*c + 7*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^5*Sqrt[Sec[c + d*x]])/32))/(15*a^3*d*(1 + Cos[c + d*x])^3)","C",1
338,1,285,221,2.3239092,"\int \frac{1}{(a+a \cos (c+d x))^3 \sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[1/((a + a*Cos[c + d*x])^3*Sec[c + d*x]^(9/2)),x]","\frac{e^{-i d x} \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(119 i e^{-\frac{3}{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)+193 \sin \left(\frac{1}{2} (c+d x)\right)+579 \sin \left(\frac{3}{2} (c+d x)\right)+555 \sin \left(\frac{5}{2} (c+d x)\right)+227 \sin \left(\frac{7}{2} (c+d x)\right)+10 \sin \left(\frac{9}{2} (c+d x)\right)-5355 i \cos \left(\frac{1}{2} (c+d x)\right)-3927 i \cos \left(\frac{3}{2} (c+d x)\right)-1785 i \cos \left(\frac{5}{2} (c+d x)\right)-357 i \cos \left(\frac{7}{2} (c+d x)\right)+5280 \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)\right)}{120 a^3 d (\cos (c+d x)+1)^3}","\frac{11 \sin (c+d x)}{2 a^3 d \sqrt{\sec (c+d x)}}-\frac{119 \sin (c+d x)}{30 d \sqrt{\sec (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}+\frac{11 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{119 \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 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*((-5355*I)*Cos[(c + d*x)/2] - (3927*I)*Cos[(3*(c + d*x))/2] - (1785*I)*Cos[(5*(c + d*x))/2] - (357*I)*Cos[(7*(c + d*x))/2] + 5280*Cos[(c + d*x)/2]^5*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + ((119*I)*(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^(((3*I)/2)*(c + d*x)) + 193*Sin[(c + d*x)/2] + 579*Sin[(3*(c + d*x))/2] + 555*Sin[(5*(c + d*x))/2] + 227*Sin[(7*(c + d*x))/2] + 10*Sin[(9*(c + d*x))/2]))/(120*a^3*d*E^(I*d*x)*(1 + Cos[c + d*x])^3)","C",1
339,1,71,153,0.2106876,"\int \sqrt{a+a \cos (c+d x)} \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(9/2),x]","\frac{2 (18 \cos (c+d x)+4 \cos (2 (c+d x))+4 \cos (3 (c+d x))+9) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)}}{35 d}","\frac{2 a \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}+\frac{12 a \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{32 a \sin (c+d x) \sqrt{\sec (c+d x)}}{35 d \sqrt{a \cos (c+d x)+a}}",1,"(2*Sqrt[a*(1 + Cos[c + d*x])]*(9 + 18*Cos[c + d*x] + 4*Cos[2*(c + d*x)] + 4*Cos[3*(c + d*x)])*Sec[c + d*x]^(7/2)*Tan[(c + d*x)/2])/(35*d)","A",1
340,1,61,115,0.1270128,"\int \sqrt{a+a \cos (c+d x)} \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(7/2),x]","\frac{2 (4 \cos (c+d x)+4 \cos (2 (c+d x))+7) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)}}{15 d}","\frac{2 a \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}+\frac{8 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}",1,"(2*Sqrt[a*(1 + Cos[c + d*x])]*(7 + 4*Cos[c + d*x] + 4*Cos[2*(c + d*x)])*Sec[c + d*x]^(5/2)*Tan[(c + d*x)/2])/(15*d)","A",1
341,1,51,77,0.1021659,"\int \sqrt{a+a \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(5/2),x]","\frac{2 (2 \cos (c+d x)+1) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)}}{3 d}","\frac{2 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{4 a \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}",1,"(2*Sqrt[a*(1 + Cos[c + d*x])]*(1 + 2*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Tan[(c + d*x)/2])/(3*d)","A",1
342,1,39,36,0.0680843,"\int \sqrt{a+a \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2),x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)}}{d}","\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}",1,"(2*Sqrt[a*(1 + Cos[c + d*x])]*Sqrt[Sec[c + d*x]]*Tan[(c + d*x)/2])/d","A",1
343,1,70,57,0.0865744,"\int \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)} \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \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)}}{d}","\frac{2 \sqrt{a} \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[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]])/d","A",1
344,1,97,92,0.1285341,"\int \frac{\sqrt{a+a \cos (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx","Integrate[Sqrt[a + a*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} \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)}\right)}{2 d}","\frac{\sqrt{a} \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 \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]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*Sin[(c + d*x)/2]))/(2*d)","A",1
345,1,111,136,0.2631857,"\int \frac{\sqrt{a+a \cos (c+d x)}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + a*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} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{3}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}\right)}{8 d}","\frac{a \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{3 \sqrt{a} \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{3 a \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(3*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(2*Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(8*d)","A",1
346,1,72,161,0.2795803,"\int (a+a \cos (c+d x))^{3/2} \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2),x]","\frac{2 a (117 \cos (c+d x)+26 \cos (2 (c+d x))+26 \cos (3 (c+d x))+41) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)}}{105 d}","\frac{2 a^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}+\frac{26 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{104 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{208 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \cos (c+d x)+a}}",1,"(2*a*Sqrt[a*(1 + Cos[c + d*x])]*(41 + 117*Cos[c + d*x] + 26*Cos[2*(c + d*x)] + 26*Cos[3*(c + d*x)])*Sec[c + d*x]^(7/2)*Tan[(c + d*x)/2])/(105*d)","A",1
347,1,62,121,0.1887144,"\int (a+a \cos (c+d x))^{3/2} \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2),x]","\frac{2 a (3 \cos (c+d x)+3 \cos (2 (c+d x))+4) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)}}{5 d}","\frac{2 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}+\frac{6 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}+\frac{12 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d \sqrt{a \cos (c+d x)+a}}",1,"(2*a*Sqrt[a*(1 + Cos[c + d*x])]*(4 + 3*Cos[c + d*x] + 3*Cos[2*(c + d*x)])*Sec[c + d*x]^(5/2)*Tan[(c + d*x)/2])/(5*d)","A",1
348,1,52,81,0.1375904,"\int (a+a \cos (c+d x))^{3/2} \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2),x]","\frac{2 a (5 \cos (c+d x)+1) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)}}{3 d}","\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{10 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}",1,"(2*a*Sqrt[a*(1 + Cos[c + d*x])]*(1 + 5*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Tan[(c + d*x)/2])/(3*d)","A",1
349,1,85,96,0.1799571,"\int (a+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\sqrt{2} \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^{3/2} \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 \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*Sin[(c + d*x)/2]))/d","A",1
350,1,99,95,0.152099,"\int (a+a \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]],x]","\frac{a \sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(3 \sqrt{2} \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)}\right)}{2 d}","\frac{3 a^{3/2} \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 \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(a*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(3*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*Sin[(c + d*x)/2]))/(2*d)","A",1
351,1,111,140,0.2504185,"\int \frac{(a+a \cos (c+d x))^{3/2}}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)/Sqrt[Sec[c + d*x]],x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(-5 \sin \left(\frac{1}{2} (c+d x)\right)+6 \sin \left(\frac{3}{2} (c+d x)\right)+\sin \left(\frac{5}{2} (c+d x)\right)+7 \sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}\right)}{8 d}","\frac{7 a^{3/2} \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 \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{7 a^2 \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(7*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] - 5*Sin[(c + d*x)/2] + 6*Sin[(3*(c + d*x))/2] + Sin[(5*(c + d*x))/2]))/(8*d)","A",1
352,1,126,180,0.5457923,"\int \frac{(a+a \cos (c+d x))^{3/2}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)/Sec[c + d*x]^(3/2),x]","\frac{a \sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(33 \sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \left(26 \sin \left(\frac{1}{2} (c+d x)\right)+9 \sin \left(\frac{3}{2} (c+d x)\right)+2 \sin \left(\frac{5}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}\right)}{48 d}","\frac{11 a^{3/2} \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{11 a^2 \sin (c+d x)}{12 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \sin (c+d x)}{3 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{11 a^2 \sin (c+d x)}{8 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"(a*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(33*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(26*Sin[(c + d*x)/2] + 9*Sin[(3*(c + d*x))/2] + 2*Sin[(5*(c + d*x))/2])))/(48*d)","A",1
353,1,84,201,5.3826052,"\int (a+a \cos (c+d x))^{5/2} \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(11/2),x]","\frac{a^2 (698 \cos (c+d x)+803 \cos (2 (c+d x))+146 \cos (3 (c+d x))+146 \cos (4 (c+d x))+727) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)}}{315 d}","\frac{38 a^3 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d \sqrt{a \cos (c+d x)+a}}+\frac{146 a^3 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d \sqrt{a \cos (c+d x)+a}}+\frac{584 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{1168 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{9 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(727 + 698*Cos[c + d*x] + 803*Cos[2*(c + d*x)] + 146*Cos[3*(c + d*x)] + 146*Cos[4*(c + d*x)])*Sec[c + d*x]^(9/2)*Tan[(c + d*x)/2])/(315*d)","A",1
354,1,74,161,5.358301,"\int (a+a \cos (c+d x))^{5/2} \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(9/2),x]","\frac{a^2 (93 \cos (c+d x)+23 \cos (2 (c+d x))+23 \cos (3 (c+d x))+29) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)}}{21 d}","\frac{6 a^3 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{7 d \sqrt{a \cos (c+d x)+a}}+\frac{46 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d \sqrt{a \cos (c+d x)+a}}+\frac{92 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{21 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{7 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(29 + 93*Cos[c + d*x] + 23*Cos[2*(c + d*x)] + 23*Cos[3*(c + d*x)])*Sec[c + d*x]^(7/2)*Tan[(c + d*x)/2])/(21*d)","A",1
355,1,64,121,0.2829094,"\int (a+a \cos (c+d x))^{5/2} \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2),x]","\frac{a^2 (28 \cos (c+d x)+43 \cos (2 (c+d x))+49) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a (\cos (c+d x)+1)}}{15 d}","\frac{22 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{86 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{5 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*(49 + 28*Cos[c + d*x] + 43*Cos[2*(c + d*x)])*Sec[c + d*x]^(5/2)*Tan[(c + d*x)/2])/(15*d)","A",1
356,1,404,138,6.3154801,"\int (a+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2),x]","\frac{\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)} \csc ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right) (a (\cos (c+d x)+1))^{5/2} \left(256 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \cos ^4\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(\frac{3}{2},2,\frac{7}{2};1,\frac{9}{2};2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)+512 \left(\sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-3 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+2\right) \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \, _2F_1\left(\frac{3}{2},\frac{7}{2};\frac{9}{2};2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)+\frac{21 \sqrt{2} \sin ^{-1}\left(\sqrt{2} \sqrt{\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)-10 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+15\right)}{\sqrt{\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}}-14 \sqrt{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)} \left(12 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-31 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+30 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+45\right)\right)}{672 d}","\frac{2 a^{5/2} \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{14 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}",1,"((a*(1 + Cos[c + d*x]))^(5/2)*Csc[c/2 + (d*x)/2]^3*Sec[c/2 + (d*x)/2]^5*Sqrt[(1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]*(256*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{3/2, 2, 7/2}, {1, 9/2}, 2*Sin[c/2 + (d*x)/2]^2]*Sin[c/2 + (d*x)/2]^6 + 512*Hypergeometric2F1[3/2, 7/2, 9/2, 2*Sin[c/2 + (d*x)/2]^2]*Sin[c/2 + (d*x)/2]^6*(2 - 3*Sin[c/2 + (d*x)/2]^2 + Sin[c/2 + (d*x)/2]^4) + (21*Sqrt[2]*ArcSin[Sqrt[2]*Sqrt[Sin[c/2 + (d*x)/2]^2]]*(15 - 10*Sin[c/2 + (d*x)/2]^2 + 3*Sin[c/2 + (d*x)/2]^4))/Sqrt[Sin[c/2 + (d*x)/2]^2] - 14*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]*(45 + 30*Sin[c/2 + (d*x)/2]^2 - 31*Sin[c/2 + (d*x)/2]^4 + 12*Sin[c/2 + (d*x)/2]^6)))/(672*d)","C",0
357,1,202,134,3.00312,"\int (a+a \cos (c+d x))^{5/2} \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} (a (\cos (c+d x)+1))^{5/2} \left(6 \sin ^4(c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(\frac{3}{2},2,\frac{5}{2};1,\frac{9}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)+24 \sin ^2(c+d x) (\cos (c+d x)+3) \, _2F_1\left(\frac{3}{2},\frac{5}{2};\frac{9}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)+7 (28 \cos (c+d x)+3 \cos (2 (c+d x))+89) \, _2F_1\left(\frac{1}{2},\frac{3}{2};\frac{7}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}{420 d}","\frac{5 a^{5/2} \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 \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{d}",1,"(Sqrt[Cos[c + d*x]]*(a*(1 + Cos[c + d*x]))^(5/2)*Sec[(c + d*x)/2]^4*Sqrt[Sec[c + d*x]]*(7*(89 + 28*Cos[c + d*x] + 3*Cos[2*(c + d*x)])*Hypergeometric2F1[1/2, 3/2, 7/2, 2*Sin[(c + d*x)/2]^2] + 24*(3 + Cos[c + d*x])*Hypergeometric2F1[3/2, 5/2, 9/2, 2*Sin[(c + d*x)/2]^2]*Sin[c + d*x]^2 + 6*Csc[(c + d*x)/2]^2*HypergeometricPFQ[{3/2, 2, 5/2}, {1, 9/2}, 2*Sin[(c + d*x)/2]^2]*Sin[c + d*x]^4)*Tan[(c + d*x)/2])/(420*d)","C",0
358,1,202,140,3.014055,"\int (a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} (a (\cos (c+d x)+1))^{5/2} \left(2 \sin ^4(c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(\frac{3}{2},\frac{3}{2},2;1,\frac{9}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)+8 \sin ^2(c+d x) (\cos (c+d x)+3) \, _2F_1\left(\frac{3}{2},\frac{3}{2};\frac{9}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)+7 (28 \cos (c+d x)+3 \cos (2 (c+d x))+89) \, _2F_1\left(\frac{1}{2},\frac{1}{2};\frac{7}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}{420 d}","\frac{19 a^{5/2} \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{9 a^3 \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{2 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(a*(1 + Cos[c + d*x]))^(5/2)*Sec[(c + d*x)/2]^4*Sqrt[Sec[c + d*x]]*(7*(89 + 28*Cos[c + d*x] + 3*Cos[2*(c + d*x)])*Hypergeometric2F1[1/2, 1/2, 7/2, 2*Sin[(c + d*x)/2]^2] + 8*(3 + Cos[c + d*x])*Hypergeometric2F1[3/2, 3/2, 9/2, 2*Sin[(c + d*x)/2]^2]*Sin[c + d*x]^2 + 2*Csc[(c + d*x)/2]^2*HypergeometricPFQ[{3/2, 3/2, 2}, {1, 9/2}, 2*Sin[(c + d*x)/2]^2]*Sin[c + d*x]^4)*Tan[(c + d*x)/2])/(420*d)","C",0
359,1,202,180,3.1150871,"\int \frac{(a+a \cos (c+d x))^{5/2}}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} (a (\cos (c+d x)+1))^{5/2} \left(-2 \sin ^4(c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(\frac{1}{2},\frac{3}{2},2;1,\frac{9}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)-8 \sin ^2(c+d x) (\cos (c+d x)+3) \, _2F_1\left(\frac{1}{2},\frac{3}{2};\frac{9}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)+7 (28 \cos (c+d x)+3 \cos (2 (c+d x))+89) \, _2F_1\left(-\frac{1}{2},\frac{1}{2};\frac{7}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}{420 d}","\frac{25 a^{5/2} \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{13 a^3 \sin (c+d x)}{12 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{25 a^3 \sin (c+d x)}{8 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Cos[c + d*x]]*(a*(1 + Cos[c + d*x]))^(5/2)*Sec[(c + d*x)/2]^4*Sqrt[Sec[c + d*x]]*(7*(89 + 28*Cos[c + d*x] + 3*Cos[2*(c + d*x)])*Hypergeometric2F1[-1/2, 1/2, 7/2, 2*Sin[(c + d*x)/2]^2] - 8*(3 + Cos[c + d*x])*Hypergeometric2F1[1/2, 3/2, 9/2, 2*Sin[(c + d*x)/2]^2]*Sin[c + d*x]^2 - 2*Csc[(c + d*x)/2]^2*HypergeometricPFQ[{1/2, 3/2, 2}, {1, 9/2}, 2*Sin[(c + d*x)/2]^2]*Sin[c + d*x]^4)*Tan[(c + d*x)/2])/(420*d)","C",0
360,1,202,220,3.0977022,"\int \frac{(a+a \cos (c+d x))^{5/2}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2)/Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} (a (\cos (c+d x)+1))^{5/2} \left(-6 \sin ^4(c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(-\frac{1}{2},\frac{3}{2},2;1,\frac{9}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)-24 \sin ^2(c+d x) (\cos (c+d x)+3) \, _2F_1\left(-\frac{1}{2},\frac{3}{2};\frac{9}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)+7 (28 \cos (c+d x)+3 \cos (2 (c+d x))+89) \, _2F_1\left(-\frac{3}{2},\frac{1}{2};\frac{7}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}{420 d}","\frac{163 a^{5/2} \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{163 a^3 \sin (c+d x)}{96 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{17 a^3 \sin (c+d x)}{24 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+a}}+\frac{163 a^3 \sin (c+d x)}{64 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(Sqrt[Cos[c + d*x]]*(a*(1 + Cos[c + d*x]))^(5/2)*Sec[(c + d*x)/2]^4*Sqrt[Sec[c + d*x]]*(7*(89 + 28*Cos[c + d*x] + 3*Cos[2*(c + d*x)])*Hypergeometric2F1[-3/2, 1/2, 7/2, 2*Sin[(c + d*x)/2]^2] - 24*(3 + Cos[c + d*x])*Hypergeometric2F1[-1/2, 3/2, 9/2, 2*Sin[(c + d*x)/2]^2]*Sin[c + d*x]^2 - 6*Csc[(c + d*x)/2]^2*HypergeometricPFQ[{-1/2, 3/2, 2}, {1, 9/2}, 2*Sin[(c + d*x)/2]^2]*Sin[c + d*x]^4)*Tan[(c + d*x)/2])/(420*d)","C",0
361,1,1540,154,7.8132675,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{\sqrt{1+\cos (c+d x)}} \, dx","Integrate[Sec[c + d*x]^(7/2)/Sqrt[1 + Cos[c + d*x]],x]","-\frac{2 \cot \left(\frac{c}{2}+\frac{d x}{2}\right) \csc ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{1}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}\right)^{7/2} \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)}{675 d \sqrt{\cos (c+d x)+1} \left(2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1\right)}","\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d \sqrt{\cos (c+d x)+1}}-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{\cos (c+d x)+1}}-\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}+\frac{26 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{\cos (c+d x)+1}}",1,"(-2*Cot[c/2 + (d*x)/2]*Csc[c/2 + (d*x)/2]^6*((1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1))^(7/2)*(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[1 + Cos[c + d*x]]*(-1 + 2*Sin[c/2 + (d*x)/2]^2))","C",0
362,1,473,118,6.605879,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{\sqrt{1+\cos (c+d x)}} \, dx","Integrate[Sec[c + d*x]^(5/2)/Sqrt[1 + Cos[c + d*x]],x]","-\frac{2 \left(\frac{1}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}\right)^{7/2} \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)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\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)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right)+7 \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3 \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}} \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(\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}} \left(7 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-3\right)+\left(3-6 \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)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right)\right)\right)}{63 d \sqrt{\cos (c+d x)+1}}","\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d \sqrt{\cos (c+d x)+1}}+\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}-\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{\cos (c+d x)+1}}",1,"(-2*Cot[c/2 + (d*x)/2]*Csc[c/2 + (d*x)/2]^4*((1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1))^(7/2)*(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*(1 - 2*Sin[c/2 + (d*x)/2]^2)^3*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*(15 - 20*Sin[c/2 + (d*x)/2]^2 + 8*Sin[c/2 + (d*x)/2]^4)*(ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*(3 - 6*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 + 7*Sin[c/2 + (d*x)/2]^2))))/(63*d*Sqrt[1 + Cos[c + d*x]])","C",0
363,1,178,82,1.953193,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{\sqrt{1+\cos (c+d x)}} \, dx","Integrate[Sec[c + d*x]^(3/2)/Sqrt[1 + 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) \sec ^{\frac{3}{2}}(c+d x) \left(\frac{1}{2} \cos (c+d x) (\cos (c+d x)+2) \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)-\frac{1}{10} \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)}{d \sqrt{\cos (c+d x)+1}}","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{\cos (c+d x)+1}}-\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}",1,"(2*Cos[(c + d*x)/2]*Sec[c + d*x]^(3/2)*Sin[(c + d*x)/2]*((Cos[c + d*x]*(2 + Cos[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]]))/2 - (Hypergeometric2F1[2, 5/2, 7/2, -(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]*Sin[c + d*x]*Tan[c + d*x])/10))/(d*Sqrt[1 + Cos[c + d*x]])","C",0
364,1,68,47,0.1070445,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{1+\cos (c+d x)}} \, dx","Integrate[Sqrt[Sec[c + d*x]]/Sqrt[1 + Cos[c + d*x]],x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)}{d}","\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}",1,"(2*ArcTan[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]]*Cos[(c + d*x)/2]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[Sec[c + d*x]])/d","A",1
365,1,171,94,0.5991996,"\int \frac{1}{\sqrt{1+\cos (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Integrate[1/(Sqrt[1 + Cos[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","\frac{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)}} \cos \left(\frac{1}{2} (c+d x)\right) \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)}{d \sqrt{\cos (c+d x)+1}}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{\cos (c+d x)+1}}\right)}{d}-\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}",1,"(I*Sqrt[2]*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 + d*x)/2])/(d*E^((I/2)*(c + d*x))*Sqrt[1 + Cos[c + d*x]])","C",1
366,1,257,125,0.877494,"\int \frac{1}{\sqrt{1+\cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[1/(Sqrt[1 + Cos[c + d*x]]*Sec[c + d*x]^(3/2)),x]","\frac{i e^{-2 i (c+d x)} \left(1+e^{i (c+d x)}\right) \sqrt{\sec (c+d x)} \left(-e^{i (c+d x)}+e^{2 i (c+d x)}-e^{3 i (c+d x)}+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} 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)-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)+1\right)}{4 d \sqrt{\cos (c+d x)+1}}","\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)}{d}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left(\frac{\sin (c+d x)}{\sqrt{\cos (c+d x)+1}}\right)}{d}+\frac{\sin (c+d x)}{d \sqrt{\cos (c+d x)+1} \sqrt{\sec (c+d x)}}",1,"((I/4)*(1 + E^(I*(c + d*x)))*(1 - E^(I*(c + d*x)) + E^((2*I)*(c + d*x)) - E^((3*I)*(c + d*x)) + E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))] + 2*Sqrt[2]*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))])] - 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[1 + Cos[c + d*x]])","C",1
367,1,1542,189,7.7605058,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[Sec[c + d*x]^(7/2)/Sqrt[a + a*Cos[c + d*x]],x]","-\frac{2 \cot \left(\frac{c}{2}+\frac{d x}{2}\right) \csc ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{1}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}\right)^{7/2} \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)}{675 d \sqrt{a (\cos (c+d x)+1)} \left(2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1\right)}","\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d \sqrt{a \cos (c+d x)+a}}-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{26 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} \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*Cot[c/2 + (d*x)/2]*Csc[c/2 + (d*x)/2]^6*((1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1))^(7/2)*(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))","C",0
368,1,475,151,6.632104,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[Sec[c + d*x]^(5/2)/Sqrt[a + a*Cos[c + d*x]],x]","-\frac{2 \left(\frac{1}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}\right)^{7/2} \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)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\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)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}\right)+7 \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3 \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}} \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(\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}} \left(7 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-3\right)+\left(3-6 \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)}{2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1}}\right)\right)\right)}{63 d \sqrt{a (\cos (c+d x)+1)}}","\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} \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*Cot[c/2 + (d*x)/2]*Csc[c/2 + (d*x)/2]^4*((1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1))^(7/2)*(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*(1 - 2*Sin[c/2 + (d*x)/2]^2)^3*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*(15 - 20*Sin[c/2 + (d*x)/2]^2 + 8*Sin[c/2 + (d*x)/2]^4)*(ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*(3 - 6*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 + 7*Sin[c/2 + (d*x)/2]^2))))/(63*d*Sqrt[a*(1 + Cos[c + d*x])])","C",0
369,1,180,113,1.872772,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[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) \sec ^{\frac{3}{2}}(c+d x) \left(\frac{1}{2} \cos (c+d x) (\cos (c+d x)+2) \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)-\frac{1}{10} \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)}{d \sqrt{a (\cos (c+d x)+1)}}","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}-\frac{\sqrt{2} \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]*Sec[c + d*x]^(3/2)*Sin[(c + d*x)/2]*((Cos[c + d*x]*(2 + Cos[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]]))/2 - (Hypergeometric2F1[2, 5/2, 7/2, -(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]*Sin[c + d*x]*Tan[c + d*x])/10))/(d*Sqrt[a*(1 + Cos[c + d*x])])","C",0
370,1,71,56,0.0948729,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[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)} \tan ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)}{d \sqrt{a (\cos (c+d x)+1)}}","\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(2*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
371,1,173,105,0.2535233,"\int \frac{1}{\sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Integrate[1/(Sqrt[a + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","\frac{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)}} \cos \left(\frac{1}{2} (c+d x)\right) \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)}{d \sqrt{a (\cos (c+d x)+1)}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(I*Sqrt[2]*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 + d*x)/2])/(d*E^((I/2)*(c + d*x))*Sqrt[a*(1 + Cos[c + d*x])])","C",1
372,1,259,168,0.44961,"\int \frac{1}{\sqrt{a+a \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[1/(Sqrt[a + a*Cos[c + d*x]]*Sec[c + d*x]^(3/2)),x]","\frac{i e^{-2 i (c+d x)} \left(1+e^{i (c+d x)}\right) \sqrt{\sec (c+d x)} \left(-e^{i (c+d x)}+e^{2 i (c+d x)}-e^{3 i (c+d x)}+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} 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)-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)+1\right)}{4 d \sqrt{a (\cos (c+d x)+1)}}","-\frac{\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{\sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{\sqrt{2} \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,"((I/4)*(1 + E^(I*(c + d*x)))*(1 - E^(I*(c + d*x)) + E^((2*I)*(c + d*x)) - E^((3*I)*(c + d*x)) + E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))] + 2*Sqrt[2]*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))])] - 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
373,-1,0,197,0,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(3/2),x]","\text{\$Aborted}","\frac{11 \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 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 a d \sqrt{a \cos (c+d x)+a}}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac{19 \sin (c+d x) \sqrt{\sec (c+d x)}}{6 a d \sqrt{a \cos (c+d x)+a}}",1,"$Aborted","F",-1
374,1,458,157,6.5077021,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(3/2),x]","\frac{2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{1}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}\right)^{3/2} \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(\frac{4 \sin ^2\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{5}{2};1,\frac{9}{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)}{70 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-35}-\frac{1}{6} \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2 \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}} \csc ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \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}} \left(124 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-350 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+298 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-75\right)-3 \left(34 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-100 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+91 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-25\right) \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)\right)\right)}{d (a (\cos (c+d x)+1))^{3/2}}","-\frac{7 \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 \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \sqrt{a \cos (c+d x)+a}}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(2*Cos[c/2 + (d*x)/2]^3*Sec[(c + d*x)/2]^2*Sin[c/2 + (d*x)/2]*((1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1))^(3/2)*((4*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 5/2}, {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]^2)/(-35 + 70*Sin[c/2 + (d*x)/2]^2) - (Csc[c/2 + (d*x)/2]^6*(1 - 2*Sin[c/2 + (d*x)/2]^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)]]*(-25 + 91*Sin[c/2 + (d*x)/2]^2 - 100*Sin[c/2 + (d*x)/2]^4 + 34*Sin[c/2 + (d*x)/2]^6) + Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*(-75 + 298*Sin[c/2 + (d*x)/2]^2 - 350*Sin[c/2 + (d*x)/2]^4 + 124*Sin[c/2 + (d*x)/2]^6)))/6))/(d*(a*(1 + Cos[c + d*x]))^(3/2))","C",0
375,1,99,117,0.5226698,"\int \frac{\sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[Sqrt[Sec[c + d*x]]/(a + a*Cos[c + d*x])^(3/2),x]","-\frac{\tan \left(\frac{1}{2} (c+d x)\right) \left(3 \cot ^2\left(\frac{1}{2} (c+d x)\right) \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)+2\right)}{4 a d \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)}}","\frac{3 \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{\sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"-1/4*((2 + 3*ArcTanh[Sqrt[-(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]]*Cot[(c + d*x)/2]^2*Sqrt[2 - 2*Sec[c + d*x]])*Tan[(c + d*x)/2])/(a*d*Sqrt[a*(1 + Cos[c + d*x])]*Sqrt[Sec[c + d*x]])","A",1
376,1,140,117,0.455887,"\int \frac{1}{(a+a \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}} \, dx","Integrate[1/((a + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]),x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\sec (c+d x)} \left(\sqrt{\cos (c+d x)+1} \sin ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right)}}\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}}\right)}{2 a d \sqrt{a (\cos (c+d x)+1)}}","\frac{\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{\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]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[Sec[c + d*x]]*(ArcSin[Sin[(c + d*x)/2]/Sqrt[Cos[(c + d*x)/2]^2]]*Sqrt[1 + Cos[c + d*x]] + 2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sin[(c + d*x)/2]))/(2*a*d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
377,1,316,174,6.5410783,"\int \frac{1}{(a+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[1/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)),x]","\frac{\cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\sec (c+d x)} \left(-\frac{2 \sin \left(\frac{c}{2}\right) \cos \left(\frac{d x}{2}\right)}{d}-\frac{2 \cos \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)}{d}+\frac{\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{\tan \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}\right)}{(a (\cos (c+d x)+1))^{3/2}}-\frac{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)}} \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \left(2 \sinh ^{-1}\left(e^{i (c+d x)}\right)+\frac{5 \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)}{\sqrt{2}}-2 \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{d (a (\cos (c+d x)+1))^{3/2}}","\frac{2 \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 \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{\sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"((-I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(2*ArcSinh[E^(I*(c + d*x))] + (5*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/Sqrt[2] - 2*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*Cos[(d*x)/2]*Sin[c/2])/d - (2*Cos[c/2]*Sin[(d*x)/2])/d + (Sec[c/2]*Sec[c/2 + (d*x)/2]^2*Sin[(d*x)/2])/d + (Sec[c/2 + (d*x)/2]*Tan[c/2])/d))/(a*(1 + Cos[c + d*x]))^(3/2)","C",1
378,1,316,214,6.5695507,"\int \frac{1}{(a+a \cos (c+d x))^{3/2} \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[1/((a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)),x]","\frac{\cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\sec (c+d x)} \left(\frac{2 \sin \left(\frac{3 c}{2}\right) \cos \left(\frac{3 d x}{2}\right)}{d}+\frac{2 \cos \left(\frac{3 c}{2}\right) \sin \left(\frac{3 d x}{2}\right)}{d}-\frac{\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{\tan \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}\right)}{(a (\cos (c+d x)+1))^{3/2}}+\frac{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)}} \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \left(\sinh ^{-1}\left(e^{i (c+d x)}\right)+\frac{3 \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)}{\sqrt{2}}-\tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{d (a (\cos (c+d x)+1))^{3/2}}","-\frac{3 \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{9 \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{\sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}+\frac{3 \sin (c+d x)}{2 a d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}",1,"((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))]*(ArcSinh[E^(I*(c + d*x))] + (3*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/Sqrt[2] - 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*Cos[(3*d*x)/2]*Sin[(3*c)/2])/d - (Sec[c/2]*Sec[c/2 + (d*x)/2]^2*Sin[(d*x)/2])/d + (2*Cos[(3*c)/2]*Sin[(3*d*x)/2])/d - (Sec[c/2 + (d*x)/2]*Tan[c/2])/d))/(a*(1 + Cos[c + d*x]))^(3/2)","C",1
379,1,641,237,8.1101522,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(5/2),x]","-\frac{\left(\frac{1}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}\right)^{7/2} \cot ^5\left(\frac{c}{2}+\frac{d x}{2}\right) \csc ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(640 \sin ^{12}\left(\frac{c}{2}+\frac{d x}{2}\right) \cos ^8\left(\frac{1}{2} (c+d x)\right) \, _5F_4\left(2,2,2,2,\frac{7}{2};1,1,1,\frac{13}{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)-1280 \left(5 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-6\right) \sin ^{12}\left(\frac{c}{2}+\frac{d x}{2}\right) \cos ^6\left(\frac{1}{2} (c+d x)\right) \, _4F_3\left(2,2,2,\frac{7}{2};1,1,\frac{13}{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)+33 \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3 \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}} \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}} \left(4344400 \sin ^{14}\left(\frac{c}{2}+\frac{d x}{2}\right)-26448512 \sin ^{12}\left(\frac{c}{2}+\frac{d x}{2}\right)+68243596 \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)-96281836 \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)+79946462 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-38990350 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+10333785 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1148175\right)-105 \left(33208 \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)-140732 \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)+234156 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-188110 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+72902 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-10935\right) \cos ^4\left(\frac{1}{2} (c+d x)\right) \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)\right)\right)}{41580 d (a (\cos (c+d x)+1))^{5/2}}","\frac{163 \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 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{48 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{299 \sin (c+d x) \sqrt{\sec (c+d x)}}{48 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{17 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"-1/41580*(Cot[c/2 + (d*x)/2]^5*Csc[c/2 + (d*x)/2]^4*Sec[(c + d*x)/2]^4*((1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1))^(7/2)*(640*Cos[(c + d*x)/2]^8*HypergeometricPFQ[{2, 2, 2, 2, 7/2}, {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 - 1280*Cos[(c + d*x)/2]^6*HypergeometricPFQ[{2, 2, 2, 7/2}, {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) + 33*(1 - 2*Sin[c/2 + (d*x)/2]^2)^3*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*(-105*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Cos[(c + d*x)/2]^4*(-10935 + 72902*Sin[c/2 + (d*x)/2]^2 - 188110*Sin[c/2 + (d*x)/2]^4 + 234156*Sin[c/2 + (d*x)/2]^6 - 140732*Sin[c/2 + (d*x)/2]^8 + 33208*Sin[c/2 + (d*x)/2]^10) + Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*(-1148175 + 10333785*Sin[c/2 + (d*x)/2]^2 - 38990350*Sin[c/2 + (d*x)/2]^4 + 79946462*Sin[c/2 + (d*x)/2]^6 - 96281836*Sin[c/2 + (d*x)/2]^8 + 68243596*Sin[c/2 + (d*x)/2]^10 - 26448512*Sin[c/2 + (d*x)/2]^12 + 4344400*Sin[c/2 + (d*x)/2]^14))))/(d*(a*(1 + Cos[c + d*x]))^(5/2))","C",0
380,1,508,197,6.8148039,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(5/2),x]","\frac{2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{1}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}\right)^{3/2} \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(\frac{8 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \cos ^6\left(\frac{1}{2} (c+d x)\right) \, _4F_3\left(2,2,2,\frac{5}{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)}{315 \left(2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1\right)}+\frac{1}{120} \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2 \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}} \csc ^8\left(\frac{c}{2}+\frac{d x}{2}\right) \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}} \left(15344 \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)-66122 \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)+109737 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-87764 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+33980 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-5145\right)-15 \left(824 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-2021 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+1465 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-343\right) \cos ^4\left(\frac{1}{2} (c+d x)\right) \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)\right)\right)}{d (a (\cos (c+d x)+1))^{5/2}}","-\frac{75 \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 \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a^2 d \sqrt{a \cos (c+d x)+a}}-\frac{13 \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a d (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(2*Cos[c/2 + (d*x)/2]^5*Sec[(c + d*x)/2]^4*Sin[c/2 + (d*x)/2]*((1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1))^(3/2)*((8*Cos[(c + d*x)/2]^6*HypergeometricPFQ[{2, 2, 2, 5/2}, {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]^2)/(315*(-1 + 2*Sin[c/2 + (d*x)/2]^2)) + (Csc[c/2 + (d*x)/2]^8*(1 - 2*Sin[c/2 + (d*x)/2]^2)^2*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*(-15*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Cos[(c + d*x)/2]^4*(-343 + 1465*Sin[c/2 + (d*x)/2]^2 - 2021*Sin[c/2 + (d*x)/2]^4 + 824*Sin[c/2 + (d*x)/2]^6) + Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*(-5145 + 33980*Sin[c/2 + (d*x)/2]^2 - 87764*Sin[c/2 + (d*x)/2]^4 + 109737*Sin[c/2 + (d*x)/2]^6 - 66122*Sin[c/2 + (d*x)/2]^8 + 15344*Sin[c/2 + (d*x)/2]^10)))/120))/(d*(a*(1 + Cos[c + d*x]))^(5/2))","C",0
381,1,131,157,0.9394719,"\int \frac{\sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[Sqrt[Sec[c + d*x]]/(a + a*Cos[c + d*x])^(5/2),x]","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(76 \tanh ^{-1}\left(\sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right) (-\sec (c+d x))}\right)-\cos (c+d x) (9 \cos (c+d x)+13) \sec ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{2-2 \sec (c+d x)}\right)}{64 \sqrt{2} a^2 d \sqrt{1-\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)}}","\frac{19 \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 \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}",1,"((76*ArcTanh[Sqrt[-(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]] - Cos[c + d*x]*(13 + 9*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Sqrt[2 - 2*Sec[c + d*x]])*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(64*Sqrt[2]*a^2*d*Sqrt[a*(1 + Cos[c + d*x])]*Sqrt[1 - Sec[c + d*x]])","A",1
382,1,122,157,0.689483,"\int \frac{1}{(a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)}} \, dx","Integrate[1/((a + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]),x]","\frac{-2 \tan ^3\left(\frac{1}{2} (c+d x)\right)+48 \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)-5 \cot \left(\frac{1}{2} (c+d x)\right) \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)}{32 a^2 d \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)}}","\frac{5 \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{\sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{\sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}",1,"(-5*ArcTanh[Sqrt[-(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]]*Cot[(c + d*x)/2]*Sqrt[2 - 2*Sec[c + d*x]] + 48*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 2*Tan[(c + d*x)/2]^3)/(32*a^2*d*Sqrt[a*(1 + Cos[c + d*x])]*Sqrt[Sec[c + d*x]])","A",0
383,1,164,157,0.6952225,"\int \frac{1}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[1/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)),x]","\frac{\sqrt{\cos (c+d x)} (\cos (c+d x)+1)^{3/2} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(6 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)+1} \sin ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right)}}\right)-\left(\sin \left(\frac{1}{2} (c+d x)\right)-7 \sin \left(\frac{3}{2} (c+d x)\right)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}}\right)}{32 d (a (\cos (c+d x)+1))^{5/2}}","\frac{3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{7 \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}",1,"(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])^(3/2)*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(6*ArcSin[Sin[(c + d*x)/2]/Sqrt[Cos[(c + d*x)/2]^2]]*Cos[(c + d*x)/2]^2*Sqrt[1 + Cos[c + d*x]] - Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*(Sin[(c + d*x)/2] - 7*Sin[(3*(c + d*x))/2])))/(32*d*(a*(1 + Cos[c + d*x]))^(5/2))","A",1
384,1,373,214,2.2476596,"\int \frac{1}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[1/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)),x]","\frac{e^{-\frac{1}{2} i (c+d x)} \left(\frac{1}{16} i e^{-2 i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \cos \left(\frac{1}{2} (c+d x)\right) \left(\sqrt{2} \left(-7 e^{i (c+d x)}-8 e^{2 i (c+d x)}+8 e^{3 i (c+d x)}+7 e^{4 i (c+d x)}+15 e^{5 i (c+d x)}+16 \sqrt{1+e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)^4 \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)-15\right)-43 \left(1+e^{i (c+d x)}\right)^4 \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)-16 i \sqrt{2} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^5\left(\frac{1}{2} (c+d x)\right) \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{4 d (a (\cos (c+d x)+1))^{5/2}}","\frac{2 \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 \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{\sin (c+d x)}{4 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}-\frac{11 \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}",1,"(((I/16)*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(-43*(1 + E^(I*(c + d*x)))^4*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]*(-15 - 7*E^(I*(c + d*x)) - 8*E^((2*I)*(c + d*x)) + 8*E^((3*I)*(c + d*x)) + 7*E^((4*I)*(c + d*x)) + 15*E^((5*I)*(c + d*x)) + 16*(1 + E^(I*(c + d*x)))^4*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))*Cos[(c + d*x)/2])/E^((2*I)*(c + d*x)) - (16*I)*Sqrt[2]*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))]*Cos[(c + d*x)/2]^5)/(4*d*E^((I/2)*(c + d*x))*(a*(1 + Cos[c + d*x]))^(5/2))","C",1
385,1,412,254,3.2325439,"\int \frac{1}{(a+a \cos (c+d x))^{5/2} \sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[1/((a + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2)),x]","\frac{e^{-\frac{1}{2} i (c+d x)} \left(40 i \sqrt{2} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^5\left(\frac{1}{2} (c+d x)\right) \sinh ^{-1}\left(e^{i (c+d x)}\right)+115 i \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^5\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)-\frac{1}{16} i e^{-3 i (c+d x)} \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(-47 e^{i (c+d x)}-39 e^{2 i (c+d x)}-16 e^{3 i (c+d x)}+16 e^{4 i (c+d x)}+39 e^{5 i (c+d x)}+47 e^{6 i (c+d x)}+8 e^{7 i (c+d x)}+40 e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)^4 \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)-8\right)\right)}{4 d (a (\cos (c+d x)+1))^{5/2}}","-\frac{5 \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{115 \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{35 \sin (c+d x)}{16 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{15 \sin (c+d x)}{16 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}",1,"((40*I)*Sqrt[2]*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))]*Cos[(c + d*x)/2]^5 + (115*I)*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 + d*x)/2]^5 - ((I/16)*(-8 - 47*E^(I*(c + d*x)) - 39*E^((2*I)*(c + d*x)) - 16*E^((3*I)*(c + d*x)) + 16*E^((4*I)*(c + d*x)) + 39*E^((5*I)*(c + d*x)) + 47*E^((6*I)*(c + d*x)) + 8*E^((7*I)*(c + d*x)) + 40*E^(I*(c + d*x))*(1 + E^(I*(c + d*x)))^4*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Cos[(c + d*x)/2]*Sqrt[Sec[c + d*x]])/E^((3*I)*(c + d*x)))/(4*d*E^((I/2)*(c + d*x))*(a*(1 + Cos[c + d*x]))^(5/2))","C",1
386,1,696,277,8.396483,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx","Integrate[Sec[c + d*x]^(5/2)/(a + a*Cos[c + d*x])^(7/2),x]","\frac{\left(\frac{1}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}\right)^{7/2} \cot ^7\left(\frac{c}{2}+\frac{d x}{2}\right) \csc ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(-7680 \sin ^{14}\left(\frac{c}{2}+\frac{d x}{2}\right) \cos ^{10}\left(\frac{1}{2} (c+d x)\right) \, _6F_5\left(2,2,2,2,2,\frac{7}{2};1,1,1,1,\frac{15}{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)+19200 \left(6 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-7\right) \sin ^{14}\left(\frac{c}{2}+\frac{d x}{2}\right) \cos ^8\left(\frac{1}{2} (c+d x)\right) \, _5F_4\left(2,2,2,2,\frac{7}{2};1,1,1,\frac{15}{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)+143 \left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3 \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}} \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}} \left(704274992 \sin ^{18}\left(\frac{c}{2}+\frac{d x}{2}\right)-5410719584 \sin ^{16}\left(\frac{c}{2}+\frac{d x}{2}\right)+18305254212 \sin ^{14}\left(\frac{c}{2}+\frac{d x}{2}\right)-35736693140 \sin ^{12}\left(\frac{c}{2}+\frac{d x}{2}\right)+44313222590 \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)-36160322412 \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)+19406027859 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-6601900452 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+1291549455 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-110685960\right)+315 \left(1793816 \sin ^{12}\left(\frac{c}{2}+\frac{d x}{2}\right)-8670660 \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)+17139064 \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)-17629526 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)+9953934 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-2928877 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+351384\right) \cos ^6\left(\frac{1}{2} (c+d x)\right) \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)\right)\right)}{3243240 d (a (\cos (c+d x)+1))^{7/2}}","\frac{1015 \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{193 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{64 a^3 d \sqrt{a \cos (c+d x)+a}}-\frac{629 \sin (c+d x) \sqrt{\sec (c+d x)}}{64 a^3 d \sqrt{a \cos (c+d x)+a}}-\frac{109 \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 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{48 a d (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}",1,"(Cot[c/2 + (d*x)/2]^7*Csc[c/2 + (d*x)/2]^4*Sec[(c + d*x)/2]^6*((1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1))^(7/2)*(-7680*Cos[(c + d*x)/2]^10*HypergeometricPFQ[{2, 2, 2, 2, 2, 7/2}, {1, 1, 1, 1, 15/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14 + 19200*Cos[(c + d*x)/2]^8*HypergeometricPFQ[{2, 2, 2, 2, 7/2}, {1, 1, 1, 15/2}, Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*Sin[c/2 + (d*x)/2]^14*(-7 + 6*Sin[c/2 + (d*x)/2]^2) + 143*(1 - 2*Sin[c/2 + (d*x)/2]^2)^3*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*(315*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Cos[(c + d*x)/2]^6*(351384 - 2928877*Sin[c/2 + (d*x)/2]^2 + 9953934*Sin[c/2 + (d*x)/2]^4 - 17629526*Sin[c/2 + (d*x)/2]^6 + 17139064*Sin[c/2 + (d*x)/2]^8 - 8670660*Sin[c/2 + (d*x)/2]^10 + 1793816*Sin[c/2 + (d*x)/2]^12) + Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*(-110685960 + 1291549455*Sin[c/2 + (d*x)/2]^2 - 6601900452*Sin[c/2 + (d*x)/2]^4 + 19406027859*Sin[c/2 + (d*x)/2]^6 - 36160322412*Sin[c/2 + (d*x)/2]^8 + 44313222590*Sin[c/2 + (d*x)/2]^10 - 35736693140*Sin[c/2 + (d*x)/2]^12 + 18305254212*Sin[c/2 + (d*x)/2]^14 - 5410719584*Sin[c/2 + (d*x)/2]^16 + 704274992*Sin[c/2 + (d*x)/2]^18))))/(3243240*d*(a*(1 + Cos[c + d*x]))^(7/2))","C",0
387,1,561,237,6.9238377,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx","Integrate[Sec[c + d*x]^(3/2)/(a + a*Cos[c + d*x])^(7/2),x]","\frac{2 \sin \left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{1}{1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}\right)^{3/2} \cos ^7\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(\frac{16 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \cos ^8\left(\frac{1}{2} (c+d x)\right) \, _5F_4\left(2,2,2,2,\frac{5}{2};1,1,1,\frac{13}{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)}{3465 \left(2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-1\right)}-\frac{\left(1-2 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2 \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}} \csc ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right) \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}} \left(1144608 \sin ^{14}\left(\frac{c}{2}+\frac{d x}{2}\right)-6712984 \sin ^{12}\left(\frac{c}{2}+\frac{d x}{2}\right)+16548816 \sin ^{10}\left(\frac{c}{2}+\frac{d x}{2}\right)-22251094 \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)+17646926 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-8267707 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)+2120790 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-229635\right)+105 \left(8752 \sin ^8\left(\frac{c}{2}+\frac{d x}{2}\right)-26380 \sin ^6\left(\frac{c}{2}+\frac{d x}{2}\right)+27986 \sin ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-12908 \sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+2187\right) \cos ^6\left(\frac{1}{2} (c+d x)\right) \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)\right)}{1680}\right)}{d (a (\cos (c+d x)+1))^{7/2}}","-\frac{363 \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 \sin (c+d x) \sqrt{\sec (c+d x)}}{192 a^3 d \sqrt{a \cos (c+d x)+a}}-\frac{199 \sin (c+d x) \sqrt{\sec (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac{19 \sin (c+d x) \sqrt{\sec (c+d x)}}{48 a d (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}}",1,"(2*Cos[c/2 + (d*x)/2]^7*Sec[(c + d*x)/2]^6*Sin[c/2 + (d*x)/2]*((1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1))^(3/2)*((16*Cos[(c + d*x)/2]^8*HypergeometricPFQ[{2, 2, 2, 2, 5/2}, {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]^2)/(3465*(-1 + 2*Sin[c/2 + (d*x)/2]^2)) - (Csc[c/2 + (d*x)/2]^10*(1 - 2*Sin[c/2 + (d*x)/2]^2)^2*Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*(105*ArcTanh[Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]]*Cos[(c + d*x)/2]^6*(2187 - 12908*Sin[c/2 + (d*x)/2]^2 + 27986*Sin[c/2 + (d*x)/2]^4 - 26380*Sin[c/2 + (d*x)/2]^6 + 8752*Sin[c/2 + (d*x)/2]^8) + Sqrt[Sin[c/2 + (d*x)/2]^2/(-1 + 2*Sin[c/2 + (d*x)/2]^2)]*(-229635 + 2120790*Sin[c/2 + (d*x)/2]^2 - 8267707*Sin[c/2 + (d*x)/2]^4 + 17646926*Sin[c/2 + (d*x)/2]^6 - 22251094*Sin[c/2 + (d*x)/2]^8 + 16548816*Sin[c/2 + (d*x)/2]^10 - 6712984*Sin[c/2 + (d*x)/2]^12 + 1144608*Sin[c/2 + (d*x)/2]^14)))/1680))/(d*(a*(1 + Cos[c + d*x]))^(7/2))","C",0
388,1,153,197,3.7748075,"\int \frac{\sqrt{\sec (c+d x)}}{(a+a \cos (c+d x))^{7/2}} \, dx","Integrate[Sqrt[Sec[c + d*x]]/(a + a*Cos[c + d*x])^(7/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(6048 \cos ^6\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \tanh ^{-1}\left(\sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right) (-\sec (c+d x))}\right)-2 (532 \cos (c+d x)+103 \cos (2 (c+d x))+493) \sqrt{2-2 \sec (c+d x)}\right)}{3072 \sqrt{2} a^3 d \sqrt{-((\sec (c+d x)-1) \sec (c+d x))} \sqrt{a (\cos (c+d x)+1)}}","\frac{63 \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 \sin (c+d x)}{192 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{5 \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x)}{6 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{7/2}}",1,"(Sec[(c + d*x)/2]^4*(-2*(493 + 532*Cos[c + d*x] + 103*Cos[2*(c + d*x)])*Sqrt[2 - 2*Sec[c + d*x]] + 6048*ArcTanh[Sqrt[-(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]]*Cos[(c + d*x)/2]^6*Sec[c + d*x])*Tan[(c + d*x)/2])/(3072*Sqrt[2]*a^3*d*Sqrt[a*(1 + Cos[c + d*x])]*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])])","A",1
389,1,125,197,1.1796067,"\int \frac{1}{(a+a \cos (c+d x))^{7/2} \sqrt{\sec (c+d x)}} \, dx","Integrate[1/((a + a*Cos[c + d*x])^(7/2)*Sqrt[Sec[c + d*x]]),x]","\frac{\sin (c+d x) (4 \cos (c+d x)-5 \cos (2 (c+d x))+73) \sec ^6\left(\frac{1}{2} (c+d x)\right)-312 \cot \left(\frac{1}{2} (c+d x)\right) \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)}{3072 a^3 d \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)}}","\frac{13 \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 \sin (c+d x)}{192 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{\sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}+\frac{\sin (c+d x)}{6 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{7/2}}",1,"(-312*ArcTanh[Sqrt[-(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]]*Cot[(c + d*x)/2]*Sqrt[2 - 2*Sec[c + d*x]] + (73 + 4*Cos[c + d*x] - 5*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^6*Sin[c + d*x])/(3072*a^3*d*Sqrt[a*(1 + Cos[c + d*x])]*Sqrt[Sec[c + d*x]])","A",1
390,1,153,197,3.6869285,"\int \frac{1}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[1/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(3/2)),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(2 (140 \cos (c+d x)+17 \cos (2 (c+d x))+59) \sqrt{2-2 \sec (c+d x)}+672 \cos ^6\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \tanh ^{-1}\left(\sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right) (-\sec (c+d x))}\right)\right)}{3072 \sqrt{2} a^3 d \sqrt{-((\sec (c+d x)-1) \sec (c+d x))} \sqrt{a (\cos (c+d x)+1)}}","\frac{7 \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 \sin (c+d x)}{192 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{3 \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x)}{6 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{7/2}}",1,"(Sec[(c + d*x)/2]^4*(2*(59 + 140*Cos[c + d*x] + 17*Cos[2*(c + d*x)])*Sqrt[2 - 2*Sec[c + d*x]] + 672*ArcTanh[Sqrt[-(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]]*Cos[(c + d*x)/2]^6*Sec[c + d*x])*Tan[(c + d*x)/2])/(3072*Sqrt[2]*a^3*d*Sqrt[a*(1 + Cos[c + d*x])]*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])])","A",0
391,1,196,197,4.3891514,"\int \frac{1}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[1/((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) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(15 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right)} \sin ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right)}}\right)+\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \left(8 \tan ^4\left(\frac{1}{2} (c+d x)\right)-26 \tan ^2\left(\frac{1}{2} (c+d x)\right)+33\right)\right)}{24 a^4 d \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right)} (\cos (c+d x)+1)^4}","\frac{5 \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{67 \sin (c+d x)}{192 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{6 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}-\frac{13 \sin (c+d x)}{48 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}",1,"(Cos[(c + d*x)/2]^7*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Cos[c + d*x])]*Sqrt[Sec[c + d*x]]*(15*ArcSin[Sin[(c + d*x)/2]/Sqrt[Cos[(c + d*x)/2]^2]]*Sqrt[Cos[(c + d*x)/2]^2] + Sqrt[2]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sin[(c + d*x)/2]*(33 - 26*Tan[(c + d*x)/2]^2 + 8*Tan[(c + d*x)/2]^4)))/(24*a^4*d*Sqrt[Cos[(c + d*x)/2]^2]*(1 + Cos[c + d*x])^4)","A",1
392,1,454,254,6.794334,"\int \frac{1}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[1/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(7/2)),x]","\frac{\cos ^7\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{\sec (c+d x)} \left(-\frac{247 \sin \left(\frac{c}{2}\right) \cos \left(\frac{d x}{2}\right)}{12 d}-\frac{247 \cos \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)}{12 d}+\frac{\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{\tan \left(\frac{c}{2}\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{41 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{12 d}-\frac{41 \tan \left(\frac{c}{2}\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{12 d}+\frac{379 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d}+\frac{379 \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 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) \left(64 \sinh ^{-1}\left(e^{i (c+d x)}\right)+\frac{177 \tanh ^{-1}\left(\frac{1-e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)}{\sqrt{2}}-64 \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{4 \sqrt{2} d (a (\cos (c+d x)+1))^{7/2}}","\frac{2 \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 \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{49 \sin (c+d x)}{64 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{17 \sin (c+d x)}{48 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x)}{6 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}",1,"((-1/4*I)*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(64*ArcSinh[E^(I*(c + d*x))] + (177*ArcTanh[(1 - E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/Sqrt[2] - 64*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Cos[c/2 + (d*x)/2]^7)/(Sqrt[2]*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*Cos[(d*x)/2]*Sin[c/2])/(12*d) - (247*Cos[c/2]*Sin[(d*x)/2])/(12*d) + (379*Sec[c/2]*Sec[c/2 + (d*x)/2]^2*Sin[(d*x)/2])/(24*d) - (41*Sec[c/2]*Sec[c/2 + (d*x)/2]^4*Sin[(d*x)/2])/(12*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]^6*Sin[(d*x)/2])/(3*d) + (379*Sec[c/2 + (d*x)/2]*Tan[c/2])/(24*d) - (41*Sec[c/2 + (d*x)/2]^3*Tan[c/2])/(12*d) + (Sec[c/2 + (d*x)/2]^5*Tan[c/2])/(3*d)))/(a*(1 + Cos[c + d*x]))^(7/2)","C",0
393,1,460,294,3.4835852,"\int \frac{1}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[1/((a + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(9/2)),x]","\frac{e^{-\frac{1}{2} i (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left(672 i \sqrt{2} \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{1}{2} (c+d x)\right) \sinh ^{-1}\left(e^{i (c+d x)}\right)-\frac{1}{64} i e^{-4 i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \cos \left(\frac{1}{2} (c+d x)\right) \left(\sqrt{2} \left(-1003 e^{i (c+d x)}-2169 e^{2 i (c+d x)}-2297 e^{3 i (c+d x)}-779 e^{4 i (c+d x)}+779 e^{5 i (c+d x)}+2297 e^{6 i (c+d x)}+2169 e^{7 i (c+d x)}+1003 e^{8 i (c+d x)}+96 e^{9 i (c+d x)}+672 e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)^6 \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)-96\right)-1911 e^{i (c+d x)} \left(1+e^{i (c+d x)}\right)^6 \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)\right)}{24 a^4 d (\cos (c+d x)+1)^4}","-\frac{7 \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{637 \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{189 \sin (c+d x)}{64 a^3 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{259 \sin (c+d x)}{192 a^2 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac{7 \sin (c+d x)}{16 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x)}{6 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}",1,"((((-1/64*I)*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(-1911*E^(I*(c + d*x))*(1 + E^(I*(c + d*x)))^6*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]*(-96 - 1003*E^(I*(c + d*x)) - 2169*E^((2*I)*(c + d*x)) - 2297*E^((3*I)*(c + d*x)) - 779*E^((4*I)*(c + d*x)) + 779*E^((5*I)*(c + d*x)) + 2297*E^((6*I)*(c + d*x)) + 2169*E^((7*I)*(c + d*x)) + 1003*E^((8*I)*(c + d*x)) + 96*E^((9*I)*(c + d*x)) + 672*E^(I*(c + d*x))*(1 + E^(I*(c + d*x)))^6*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))*Cos[(c + d*x)/2])/E^((4*I)*(c + d*x)) + (672*I)*Sqrt[2]*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))]*Cos[(c + d*x)/2]^7)*Sqrt[a*(1 + Cos[c + d*x])])/(24*a^4*d*E^((I/2)*(c + d*x))*(1 + Cos[c + d*x])^4)","C",1
394,1,163,237,4.3515369,"\int \frac{1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[1/((a + a*Cos[c + d*x])^(9/2)*Sec[c + d*x]^(5/2)),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(2 (999 \cos (c+d x)+702 \cos (2 (c+d x))+73 \cos (3 (c+d x))+882) \sqrt{2-2 \sec (c+d x)}+5760 \cos ^8\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \tanh ^{-1}\left(\sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right) (-\sec (c+d x))}\right)\right)}{65536 \sqrt{2} a^4 d \sqrt{-((\sec (c+d x)-1) \sec (c+d x))} \sqrt{a (\cos (c+d x)+1)}}","\frac{45 \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)}{1024 \sqrt{2} a^{9/2} d}+\frac{73 \sin (c+d x)}{1024 a^3 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}+\frac{33 \sin (c+d x)}{256 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac{\sin (c+d x)}{8 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{9/2}}-\frac{5 \sin (c+d x)}{32 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{7/2}}",1,"(Sec[(c + d*x)/2]^6*(2*(882 + 999*Cos[c + d*x] + 702*Cos[2*(c + d*x)] + 73*Cos[3*(c + d*x)])*Sqrt[2 - 2*Sec[c + d*x]] + 5760*ArcTanh[Sqrt[-(Sec[c + d*x]*Sin[(c + d*x)/2]^2)]]*Cos[(c + d*x)/2]^8*Sec[c + d*x])*Tan[(c + d*x)/2])/(65536*Sqrt[2]*a^4*d*Sqrt[a*(1 + Cos[c + d*x])]*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])])","A",0
395,1,395,237,6.0951646,"\int \frac{1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[1/((a + a*Cos[c + d*x])^(9/2)*Sec[c + d*x]^(7/2)),x]","\frac{2 \sin \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)} \cos ^9\left(\frac{c}{2}+\frac{d x}{2}\right) \left(1-\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{9/2} \left(\frac{1}{8} \left(\frac{1}{1-\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}+\frac{7}{6 \left(1-\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{35}{24 \left(1-\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{35}{16 \left(1-\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^4}\right)+\frac{35 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right)} \csc \left(\frac{c}{2}+\frac{d x}{2}\right) \sin ^{-1}\left(\frac{\sin \left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right)}}\right)}{128 \left(1-\sin ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{9/2}}\right)}{d \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right)} (a (\cos (c+d x)+1))^{9/2}}","\frac{35 \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)}{1024 \sqrt{2} a^{9/2} d}+\frac{853 \sin (c+d x)}{3072 a^3 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac{187 \sin (c+d x)}{768 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac{19 \sin (c+d x)}{96 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}-\frac{\sin (c+d x)}{8 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^{9/2}}",1,"(2*Cos[c/2 + (d*x)/2]^9*Sin[c/2 + (d*x)/2]*Sqrt[(1 - 2*Sin[c/2 + (d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[c/2 + (d*x)/2]^2]*(1 - Sec[(c + d*x)/2]^2*Sin[c/2 + (d*x)/2]^2)^(9/2)*((35*ArcSin[Sin[c/2 + (d*x)/2]/Sqrt[Cos[(c + d*x)/2]^2]]*Sqrt[Cos[(c + d*x)/2]^2]*Csc[c/2 + (d*x)/2])/(128*(1 - Sec[(c + d*x)/2]^2*Sin[c/2 + (d*x)/2]^2)^(9/2)) + (35/(16*(1 - Sec[(c + d*x)/2]^2*Sin[c/2 + (d*x)/2]^2)^4) + 35/(24*(1 - Sec[(c + d*x)/2]^2*Sin[c/2 + (d*x)/2]^2)^3) + 7/(6*(1 - Sec[(c + d*x)/2]^2*Sin[c/2 + (d*x)/2]^2)^2) + (1 - Sec[(c + d*x)/2]^2*Sin[c/2 + (d*x)/2]^2)^(-1))/8))/(d*Sqrt[Cos[(c + d*x)/2]^2]*(a*(1 + Cos[c + d*x]))^(9/2))","A",1
396,1,51,38,0.1171811,"\int (a+a \cos (c+d x))^{3/2} \sec ^{\frac{5}{4}}(c+d x) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/4),x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt[4]{\sec (c+d x)} (a (\cos (c+d x)+1))^{3/2}}{d}","\frac{4 a^2 \sin (c+d x) \sqrt[4]{\sec (c+d x)}}{d \sqrt{a \cos (c+d x)+a}}",1,"(2*(a*(1 + Cos[c + d*x]))^(3/2)*Sec[(c + d*x)/2]^2*Sec[c + d*x]^(1/4)*Tan[(c + d*x)/2])/d","A",1
397,0,0,302,3.0584739,"\int \cos ^m(c+d x) (a+a \cos (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^m*(a + a*Cos[c + d*x])^4,x]","\int \cos ^m(c+d x) (a+a \cos (c+d x))^4 \, dx","-\frac{a^4 \left(8 m^2+40 m+35\right) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) (m+4) \sqrt{\sin ^2(c+d x)}}-\frac{4 a^4 (2 m+5) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{a^4 \left(4 m^2+29 m+55\right) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2) (m+3) (m+4)}+\frac{2 (m+5) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right) \cos ^{m+1}(c+d x)}{d (m+3) (m+4)}+\frac{\sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2 \cos ^{m+1}(c+d x)}{d (m+4)}",1,"Integrate[Cos[c + d*x]^m*(a + a*Cos[c + d*x])^4, x]","F",-1
398,0,0,232,1.271415,"\int \cos ^m(c+d x) (a+a \cos (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^m*(a + a*Cos[c + d*x])^3,x]","\int \cos ^m(c+d x) (a+a \cos (c+d x))^3 \, dx","-\frac{a^3 (4 m+5) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) \sqrt{\sin ^2(c+d x)}}-\frac{a^3 (4 m+11) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{a^3 (2 m+7) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2) (m+3)}+\frac{\sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right) \cos ^{m+1}(c+d x)}{d (m+3)}",1,"Integrate[Cos[c + d*x]^m*(a + a*Cos[c + d*x])^3, x]","F",-1
399,0,0,173,0.5253883,"\int \cos ^m(c+d x) (a+a \cos (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^m*(a + a*Cos[c + d*x])^2,x]","\int \cos ^m(c+d x) (a+a \cos (c+d x))^2 \, dx","-\frac{a^2 (2 m+3) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) \sqrt{\sin ^2(c+d x)}}-\frac{2 a^2 \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) \sqrt{\sin ^2(c+d x)}}+\frac{a^2 \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2)}",1,"Integrate[Cos[c + d*x]^m*(a + a*Cos[c + d*x])^2, x]","F",-1
400,1,208,131,1.0212477,"\int \cos ^m(c+d x) (a+a \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^m*(a + a*Cos[c + d*x]),x]","\frac{i a 2^{-m-2} \left(e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)\right)^{m+1} (\cos (c+d x)+1) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left((m-1) m \, _2F_1\left(1,\frac{m+1}{2};\frac{1-m}{2};-e^{2 i (c+d x)}\right)+(m+1) e^{i (c+d x)} \left(2 (m-1) \, _2F_1\left(1,\frac{m+2}{2};1-\frac{m}{2};-e^{2 i (c+d x)}\right)+m e^{i (c+d x)} \, _2F_1\left(1,\frac{m+3}{2};\frac{3-m}{2};-e^{2 i (c+d x)}\right)\right)\right)}{d (m-1) m (m+1)}","-\frac{a \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) \sqrt{\sin ^2(c+d x)}}-\frac{a \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) \sqrt{\sin ^2(c+d x)}}",1,"(I*2^(-2 - m)*a*((1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x)))^(1 + m)*(1 + Cos[c + d*x])*((-1 + m)*m*Hypergeometric2F1[1, (1 + m)/2, (1 - m)/2, -E^((2*I)*(c + d*x))] + E^(I*(c + d*x))*(1 + m)*(2*(-1 + m)*Hypergeometric2F1[1, (2 + m)/2, 1 - m/2, -E^((2*I)*(c + d*x))] + E^(I*(c + d*x))*m*Hypergeometric2F1[1, (3 + m)/2, (3 - m)/2, -E^((2*I)*(c + d*x))]))*Sec[(c + d*x)/2]^2)/(d*(-1 + m)*m*(1 + m))","C",0
401,0,0,156,0.8660949,"\int \frac{\cos ^m(c+d x)}{a+a \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^m/(a + a*Cos[c + d*x]),x]","\int \frac{\cos ^m(c+d x)}{a+a \cos (c+d x)} \, dx","\frac{m \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{a d (m+1) \sqrt{\sin ^2(c+d x)}}-\frac{\sin (c+d x) \cos ^m(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m}{2};\frac{m+2}{2};\cos ^2(c+d x)\right)}{a d \sqrt{\sin ^2(c+d x)}}+\frac{\sin (c+d x) \cos ^m(c+d x)}{d (a \cos (c+d x)+a)}",1,"Integrate[Cos[c + d*x]^m/(a + a*Cos[c + d*x]), x]","F",-1
402,0,0,229,1.1130162,"\int \frac{\cos ^m(c+d x)}{(a+a \cos (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^m/(a + a*Cos[c + d*x])^2,x]","\int \frac{\cos ^m(c+d x)}{(a+a \cos (c+d x))^2} \, dx","\frac{(1-2 m) m \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{3 a^2 d (m+1) \sqrt{\sin ^2(c+d x)}}-\frac{2 (1-m) (m+1) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{3 a^2 d (m+2) \sqrt{\sin ^2(c+d x)}}-\frac{2 (1-m) \sin (c+d x) \cos ^{m+1}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}-\frac{\sin (c+d x) \cos ^{m+1}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"Integrate[Cos[c + d*x]^m/(a + a*Cos[c + d*x])^2, x]","F",-1
403,1,135,150,0.1977408,"\int \cos ^7(c+d x) (a+b \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^7*(a + b*Cos[c + d*x]),x]","-\frac{a \sin ^7(c+d x)}{7 d}+\frac{3 a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x)}{d}+\frac{a \sin (c+d x)}{d}+\frac{35 b (c+d x)}{128 d}+\frac{7 b \sin (2 (c+d x))}{32 d}+\frac{7 b \sin (4 (c+d x))}{128 d}+\frac{b \sin (6 (c+d x))}{96 d}+\frac{b \sin (8 (c+d x))}{1024 d}","-\frac{a \sin ^7(c+d x)}{7 d}+\frac{3 a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x)}{d}+\frac{a \sin (c+d x)}{d}+\frac{b \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{7 b \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{35 b \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{35 b \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35 b x}{128}",1,"(35*b*(c + d*x))/(128*d) + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/d + (3*a*Sin[c + d*x]^5)/(5*d) - (a*Sin[c + d*x]^7)/(7*d) + (7*b*Sin[2*(c + d*x)])/(32*d) + (7*b*Sin[4*(c + d*x)])/(128*d) + (b*Sin[6*(c + d*x)])/(96*d) + (b*Sin[8*(c + d*x)])/(1024*d)","A",1
404,1,89,128,0.1864159,"\int \cos ^6(c+d x) (a+b \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^6*(a + b*Cos[c + d*x]),x]","\frac{35 a (45 \sin (2 (c+d x))+9 \sin (4 (c+d x))+\sin (6 (c+d x))+60 c+60 d x)-960 b \sin ^7(c+d x)+4032 b \sin ^5(c+d x)-6720 b \sin ^3(c+d x)+6720 b \sin (c+d x)}{6720 d}","\frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16}-\frac{b \sin ^7(c+d x)}{7 d}+\frac{3 b \sin ^5(c+d x)}{5 d}-\frac{b \sin ^3(c+d x)}{d}+\frac{b \sin (c+d x)}{d}",1,"(6720*b*Sin[c + d*x] - 6720*b*Sin[c + d*x]^3 + 4032*b*Sin[c + d*x]^5 - 960*b*Sin[c + d*x]^7 + 35*a*(60*c + 60*d*x + 45*Sin[2*(c + d*x)] + 9*Sin[4*(c + d*x)] + Sin[6*(c + d*x)]))/(6720*d)","A",1
405,1,78,114,0.0992359,"\int \cos ^5(c+d x) (a+b \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*(a + b*Cos[c + d*x]),x]","\frac{192 a \sin ^5(c+d x)-640 a \sin ^3(c+d x)+960 a \sin (c+d x)+5 b (45 \sin (2 (c+d x))+9 \sin (4 (c+d x))+\sin (6 (c+d x))+60 c+60 d x)}{960 d}","\frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{b \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 b \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 b \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 b x}{16}",1,"(960*a*Sin[c + d*x] - 640*a*Sin[c + d*x]^3 + 192*a*Sin[c + d*x]^5 + 5*b*(60*c + 60*d*x + 45*Sin[2*(c + d*x)] + 9*Sin[4*(c + d*x)] + Sin[6*(c + d*x)]))/(960*d)","A",1
406,1,89,92,0.1302598,"\int \cos ^4(c+d x) (a+b \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Cos[c + d*x]),x]","\frac{3 a (c+d x)}{8 d}+\frac{a \sin (2 (c+d x))}{4 d}+\frac{a \sin (4 (c+d x))}{32 d}+\frac{b \sin ^5(c+d x)}{5 d}-\frac{2 b \sin ^3(c+d x)}{3 d}+\frac{b \sin (c+d x)}{d}","\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a x}{8}+\frac{b \sin ^5(c+d x)}{5 d}-\frac{2 b \sin ^3(c+d x)}{3 d}+\frac{b \sin (c+d x)}{d}",1,"(3*a*(c + d*x))/(8*d) + (b*Sin[c + d*x])/d - (2*b*Sin[c + d*x]^3)/(3*d) + (b*Sin[c + d*x]^5)/(5*d) + (a*Sin[2*(c + d*x)])/(4*d) + (a*Sin[4*(c + d*x)])/(32*d)","A",1
407,1,73,76,0.0936748,"\int \cos ^3(c+d x) (a+b \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Cos[c + d*x]),x]","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{3 b (c+d x)}{8 d}+\frac{b \sin (2 (c+d x))}{4 d}+\frac{b \sin (4 (c+d x))}{32 d}","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{b \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 b \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 b x}{8}",1,"(3*b*(c + d*x))/(8*d) + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d) + (b*Sin[2*(c + d*x)])/(4*d) + (b*Sin[4*(c + d*x)])/(32*d)","A",1
408,1,57,54,0.0642208,"\int \cos ^2(c+d x) (a+b \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x]),x]","\frac{a (c+d x)}{2 d}+\frac{a \sin (2 (c+d x))}{4 d}-\frac{b \sin ^3(c+d x)}{3 d}+\frac{b \sin (c+d x)}{d}","\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}-\frac{b \sin ^3(c+d x)}{3 d}+\frac{b \sin (c+d x)}{d}",1,"(a*(c + d*x))/(2*d) + (b*Sin[c + d*x])/d - (b*Sin[c + d*x]^3)/(3*d) + (a*Sin[2*(c + d*x)])/(4*d)","A",1
409,1,35,38,0.0617822,"\int \cos (c+d x) (a+b \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x]),x]","\frac{4 a \sin (c+d x)+b (2 (c+d x)+\sin (2 (c+d x)))}{4 d}","\frac{a \sin (c+d x)}{d}+\frac{b \sin (c+d x) \cos (c+d x)}{2 d}+\frac{b x}{2}",1,"(4*a*Sin[c + d*x] + b*(2*(c + d*x) + Sin[2*(c + d*x)]))/(4*d)","A",1
410,1,26,15,0.0063685,"\int (a+b \cos (c+d x)) \, dx","Integrate[a + b*Cos[c + d*x],x]","a x+\frac{b \sin (c) \cos (d x)}{d}+\frac{b \cos (c) \sin (d x)}{d}","a x+\frac{b \sin (c+d x)}{d}",1,"a*x + (b*Cos[d*x]*Sin[c])/d + (b*Cos[c]*Sin[d*x])/d","A",1
411,1,16,16,0.0061955,"\int (a+b \cos (c+d x)) \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*Sec[c + d*x],x]","\frac{a \tanh ^{-1}(\sin (c+d x))}{d}+b x","\frac{a \tanh ^{-1}(\sin (c+d x))}{d}+b x",1,"b*x + (a*ArcTanh[Sin[c + d*x]])/d","A",1
412,1,24,24,0.0079099,"\int (a+b \cos (c+d x)) \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{a \tan (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}","\frac{a \tan (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"(b*ArcTanh[Sin[c + d*x]])/d + (a*Tan[c + d*x])/d","A",1
413,1,47,47,0.0146778,"\int (a+b \cos (c+d x)) \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d}+\frac{b \tan (c+d x)}{d}","\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d}+\frac{b \tan (c+d x)}{d}",1,"(a*ArcTanh[Sin[c + d*x]])/(2*d) + (b*Tan[c + d*x])/d + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
414,1,60,63,0.1692993,"\int (a+b \cos (c+d x)) \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{a \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b \tan (c+d x) \sec (c+d x)}{2 d}","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b \tan (c+d x) \sec (c+d x)}{2 d}",1,"(b*ArcTanh[Sin[c + d*x]])/(2*d) + (b*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d","A",1
415,1,76,85,0.2298124,"\int (a+b \cos (c+d x)) \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*Sec[c + d*x]^5,x]","\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)}{8 d}+\frac{b \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}","\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b \tan ^3(c+d x)}{3 d}+\frac{b \tan (c+d x)}{d}",1,"(a*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (3*a*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x]))/(8*d) + (b*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d","A",1
416,1,88,101,0.3335854,"\int (a+b \cos (c+d x)) \sec ^6(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*Sec[c + d*x]^6,x]","\frac{a \left(\frac{1}{5} \tan ^5(c+d x)+\frac{2}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}+\frac{b \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 b \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)}{8 d}","\frac{a \tan ^5(c+d x)}{5 d}+\frac{2 a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{3 b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 b \tan (c+d x) \sec (c+d x)}{8 d}",1,"(b*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (3*b*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x]))/(8*d) + (a*(Tan[c + d*x] + (2*Tan[c + d*x]^3)/3 + Tan[c + d*x]^5/5))/d","A",1
417,1,123,150,0.3094499,"\int \cos ^4(c+d x) (a+b \cos (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^4*(a + b*Cos[c + d*x])^2,x]","\frac{5 \left(\left(48 a^2+45 b^2\right) \sin (2 (c+d x))+\left(6 a^2+9 b^2\right) \sin (4 (c+d x))+72 a^2 c+72 a^2 d x+b^2 \sin (6 (c+d x))+60 b^2 c+60 b^2 d x\right)+384 a b \sin ^5(c+d x)-1280 a b \sin ^3(c+d x)+1920 a b \sin (c+d x)}{960 d}","\frac{\left(6 a^2+5 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{\left(6 a^2+5 b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left(6 a^2+5 b^2\right)+\frac{2 a b \sin ^5(c+d x)}{5 d}-\frac{4 a b \sin ^3(c+d x)}{3 d}+\frac{2 a b \sin (c+d x)}{d}+\frac{b^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}",1,"(1920*a*b*Sin[c + d*x] - 1280*a*b*Sin[c + d*x]^3 + 384*a*b*Sin[c + d*x]^5 + 5*(72*a^2*c + 60*b^2*c + 72*a^2*d*x + 60*b^2*d*x + (48*a^2 + 45*b^2)*Sin[2*(c + d*x)] + (6*a^2 + 9*b^2)*Sin[4*(c + d*x)] + b^2*Sin[6*(c + d*x)]))/(960*d)","A",1
418,1,85,111,0.1385501,"\int \cos ^3(c+d x) (a+b \cos (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^3*(a + b*Cos[c + d*x])^2,x]","\frac{-80 \left(a^2+2 b^2\right) \sin ^3(c+d x)+240 \left(a^2+b^2\right) \sin (c+d x)+15 a b (12 (c+d x)+8 \sin (2 (c+d x))+\sin (4 (c+d x)))+48 b^2 \sin ^5(c+d x)}{240 d}","-\frac{\left(a^2+2 b^2\right) \sin ^3(c+d x)}{3 d}+\frac{\left(a^2+b^2\right) \sin (c+d x)}{d}+\frac{a b \sin (c+d x) \cos ^3(c+d x)}{2 d}+\frac{3 a b \sin (c+d x) \cos (c+d x)}{4 d}+\frac{3 a b x}{4}+\frac{b^2 \sin ^5(c+d x)}{5 d}",1,"(240*(a^2 + b^2)*Sin[c + d*x] - 80*(a^2 + 2*b^2)*Sin[c + d*x]^3 + 48*b^2*Sin[c + d*x]^5 + 15*a*b*(12*(c + d*x) + 8*Sin[2*(c + d*x)] + Sin[4*(c + d*x)]))/(240*d)","A",1
419,1,86,101,0.1606563,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^2,x]","\frac{24 \left(a^2+b^2\right) \sin (2 (c+d x))+48 a^2 c+48 a^2 d x-64 a b \sin ^3(c+d x)+192 a b \sin (c+d x)+3 b^2 \sin (4 (c+d x))+36 b^2 c+36 b^2 d x}{96 d}","\frac{\left(4 a^2+3 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(4 a^2+3 b^2\right)-\frac{2 a b \sin ^3(c+d x)}{3 d}+\frac{2 a b \sin (c+d x)}{d}+\frac{b^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"(48*a^2*c + 36*b^2*c + 48*a^2*d*x + 36*b^2*d*x + 192*a*b*Sin[c + d*x] - 64*a*b*Sin[c + d*x]^3 + 24*(a^2 + b^2)*Sin[2*(c + d*x)] + 3*b^2*Sin[4*(c + d*x)])/(96*d)","A",1
420,1,59,71,0.1515659,"\int \cos (c+d x) (a+b \cos (c+d x))^2 \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^2,x]","\frac{3 \left(4 a^2+3 b^2\right) \sin (c+d x)+b (12 a (c+d x)+6 a \sin (2 (c+d x))+b \sin (3 (c+d x)))}{12 d}","\frac{2 \left(a^2+b^2\right) \sin (c+d x)}{3 d}+\frac{\sin (c+d x) (a+b \cos (c+d x))^2}{3 d}+\frac{a b \sin (c+d x) \cos (c+d x)}{3 d}+a b x",1,"(3*(4*a^2 + 3*b^2)*Sin[c + d*x] + b*(12*a*(c + d*x) + 6*a*Sin[2*(c + d*x)] + b*Sin[3*(c + d*x)]))/(12*d)","A",1
421,1,46,50,0.0751437,"\int (a+b \cos (c+d x))^2 \, dx","Integrate[(a + b*Cos[c + d*x])^2,x]","\frac{2 \left(2 a^2+b^2\right) (c+d x)+8 a b \sin (c+d x)+b^2 \sin (2 (c+d x))}{4 d}","\frac{1}{2} x \left(2 a^2+b^2\right)+\frac{2 a b \sin (c+d x)}{d}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{2 d}",1,"(2*(2*a^2 + b^2)*(c + d*x) + 8*a*b*Sin[c + d*x] + b^2*Sin[2*(c + d*x)])/(4*d)","A",1
422,1,46,33,0.0134906,"\int (a+b \cos (c+d x))^2 \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*Sec[c + d*x],x]","\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+2 a b x+\frac{b^2 \sin (c) \cos (d x)}{d}+\frac{b^2 \cos (c) \sin (d x)}{d}","\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+2 a b x+\frac{b^2 \sin (c+d x)}{d}",1,"2*a*b*x + (a^2*ArcTanh[Sin[c + d*x]])/d + (b^2*Cos[d*x]*Sin[c])/d + (b^2*Cos[c]*Sin[d*x])/d","A",1
423,1,32,33,0.0839573,"\int (a+b \cos (c+d x))^2 \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2,x]","\frac{a^2 \tan (c+d x)+2 a b \tanh ^{-1}(\sin (c+d x))+b^2 d x}{d}","\frac{a^2 \tan (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+b^2 x",1,"(b^2*d*x + 2*a*b*ArcTanh[Sin[c + d*x]] + a^2*Tan[c + d*x])/d","A",1
424,1,67,59,0.0138074,"\int (a+b \cos (c+d x))^2 \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*Sec[c + d*x]^3,x]","\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{2 d}+\frac{2 a b \tan (c+d x)}{d}+\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{d}","\frac{\left(a^2+2 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{2 d}+\frac{2 a b \tan (c+d x)}{d}",1,"(a^2*ArcTanh[Sin[c + d*x]])/(2*d) + (b^2*ArcTanh[Sin[c + d*x]])/d + (2*a*b*Tan[c + d*x])/d + (a^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
425,1,71,80,0.2238875,"\int (a+b \cos (c+d x))^2 \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4,x]","\frac{a^2 \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}+\frac{a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a b \tan (c+d x) \sec (c+d x)}{d}+\frac{b^2 \tan (c+d x)}{d}","\frac{\left(2 a^2+3 b^2\right) \tan (c+d x)}{3 d}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a b \tan (c+d x) \sec (c+d x)}{d}",1,"(a*b*ArcTanh[Sin[c + d*x]])/d + (b^2*Tan[c + d*x])/d + (a*b*Sec[c + d*x]*Tan[c + d*x])/d + (a^2*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d","A",1
426,1,82,110,0.2778435,"\int (a+b \cos (c+d x))^2 \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*Sec[c + d*x]^5,x]","\frac{3 \left(3 a^2+4 b^2\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(3 \left(3 a^2+4 b^2\right) \sec (c+d x)+6 a^2 \sec ^3(c+d x)+16 a b \left(\tan ^2(c+d x)+3\right)\right)}{24 d}","\frac{\left(3 a^2+4 b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(3 a^2+4 b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^2 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{2 a b \tan ^3(c+d x)}{3 d}+\frac{2 a b \tan (c+d x)}{d}",1,"(3*(3*a^2 + 4*b^2)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(3*(3*a^2 + 4*b^2)*Sec[c + d*x] + 6*a^2*Sec[c + d*x]^3 + 16*a*b*(3 + Tan[c + d*x]^2)))/(24*d)","A",1
427,1,118,135,0.5561802,"\int (a+b \cos (c+d x))^2 \sec ^6(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*Sec[c + d*x]^6,x]","\frac{a^2 \left(\frac{1}{5} \tan ^5(c+d x)+\frac{2}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}+\frac{a b \tan (c+d x) \sec ^3(c+d x)}{2 d}+\frac{3 a b \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)}{4 d}+\frac{b^2 \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}","\frac{\left(4 a^2+5 b^2\right) \tan ^3(c+d x)}{15 d}+\frac{\left(4 a^2+5 b^2\right) \tan (c+d x)}{5 d}+\frac{a^2 \tan (c+d x) \sec ^4(c+d x)}{5 d}+\frac{3 a b \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a b \tan (c+d x) \sec ^3(c+d x)}{2 d}+\frac{3 a b \tan (c+d x) \sec (c+d x)}{4 d}",1,"(a*b*Sec[c + d*x]^3*Tan[c + d*x])/(2*d) + (3*a*b*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x]))/(4*d) + (b^2*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d + (a^2*(Tan[c + d*x] + (2*Tan[c + d*x]^3)/3 + Tan[c + d*x]^5/5))/d","A",1
428,1,159,170,0.325368,"\int \cos ^3(c+d x) (a+b \cos (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^3*(a + b*Cos[c + d*x])^3,x]","\frac{80 a^3 \sin (3 (c+d x))+45 \left(16 a^2 b+5 b^3\right) \sin (2 (c+d x))+360 a \left(2 a^2+5 b^2\right) \sin (c+d x)+90 a^2 b \sin (4 (c+d x))+1080 a^2 b c+1080 a^2 b d x+300 a b^2 \sin (3 (c+d x))+36 a b^2 \sin (5 (c+d x))+45 b^3 \sin (4 (c+d x))+5 b^3 \sin (6 (c+d x))+300 b^3 c+300 b^3 d x}{960 d}","-\frac{a \left(a^2+6 b^2\right) \sin ^3(c+d x)}{3 d}+\frac{a \left(a^2+3 b^2\right) \sin (c+d x)}{d}+\frac{b \left(18 a^2+5 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{b \left(18 a^2+5 b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{9}{8} a^2 b x+\frac{3 a b^2 \sin ^5(c+d x)}{5 d}+\frac{b^3 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 b^3 x}{16}",1,"(1080*a^2*b*c + 300*b^3*c + 1080*a^2*b*d*x + 300*b^3*d*x + 360*a*(2*a^2 + 5*b^2)*Sin[c + d*x] + 45*(16*a^2*b + 5*b^3)*Sin[2*(c + d*x)] + 80*a^3*Sin[3*(c + d*x)] + 300*a*b^2*Sin[3*(c + d*x)] + 90*a^2*b*Sin[4*(c + d*x)] + 45*b^3*Sin[4*(c + d*x)] + 36*a*b^2*Sin[5*(c + d*x)] + 5*b^3*Sin[6*(c + d*x)])/(960*d)","A",1
429,1,130,180,0.3257793,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^3,x]","\frac{120 \left(a^3+3 a b^2\right) \sin (2 (c+d x))+240 a^3 c+240 a^3 d x+60 b \left(18 a^2+5 b^2\right) \sin (c+d x)+120 a^2 b \sin (3 (c+d x))+45 a b^2 \sin (4 (c+d x))+540 a b^2 c+540 a b^2 d x+50 b^3 \sin (3 (c+d x))+6 b^3 \sin (5 (c+d x))}{480 d}","-\frac{\left(3 a^2-16 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^2}{60 b d}-\frac{a \left(6 a^2-71 b^2\right) \sin (c+d x) \cos (c+d x)}{120 d}+\frac{1}{8} a x \left(4 a^2+9 b^2\right)-\frac{\left(3 a^4-52 a^2 b^2-16 b^4\right) \sin (c+d x)}{30 b d}+\frac{\sin (c+d x) (a+b \cos (c+d x))^4}{5 b d}-\frac{a \sin (c+d x) (a+b \cos (c+d x))^3}{20 b d}",1,"(240*a^3*c + 540*a*b^2*c + 240*a^3*d*x + 540*a*b^2*d*x + 60*b*(18*a^2 + 5*b^2)*Sin[c + d*x] + 120*(a^3 + 3*a*b^2)*Sin[2*(c + d*x)] + 120*a^2*b*Sin[3*(c + d*x)] + 50*b^3*Sin[3*(c + d*x)] + 45*a*b^2*Sin[4*(c + d*x)] + 6*b^3*Sin[5*(c + d*x)])/(480*d)","A",1
430,1,100,121,0.2692643,"\int \cos (c+d x) (a+b \cos (c+d x))^3 \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^3,x]","\frac{8 a \left(4 a^2+9 b^2\right) \sin (c+d x)+b \left(8 \left(3 a^2+b^2\right) \sin (2 (c+d x))+48 a^2 c+48 a^2 d x+8 a b \sin (3 (c+d x))+b^2 \sin (4 (c+d x))+12 b^2 c+12 b^2 d x\right)}{32 d}","\frac{a \left(a^2+4 b^2\right) \sin (c+d x)}{2 d}+\frac{b \left(2 a^2+3 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3}{8} b x \left(4 a^2+b^2\right)+\frac{\sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac{a \sin (c+d x) (a+b \cos (c+d x))^2}{4 d}",1,"(8*a*(4*a^2 + 9*b^2)*Sin[c + d*x] + b*(48*a^2*c + 12*b^2*c + 48*a^2*d*x + 12*b^2*d*x + 8*(3*a^2 + b^2)*Sin[2*(c + d*x)] + 8*a*b*Sin[3*(c + d*x)] + b^2*Sin[4*(c + d*x)]))/(32*d)","A",1
431,1,80,76,0.1247661,"\int (a+b \cos (c+d x))^3 \, dx","Integrate[(a + b*Cos[c + d*x])^3,x]","\frac{12 a^3 c+12 a^3 d x+9 b \left(4 a^2+b^2\right) \sin (c+d x)+9 a b^2 \sin (2 (c+d x))+18 a b^2 c+18 a b^2 d x+b^3 \sin (3 (c+d x))}{12 d}","a^3 x+\frac{b \left(3 a^2+b^2\right) \sin (c+d x)}{d}+\frac{3 a b^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3}{2} a b^2 x-\frac{b^3 \sin ^3(c+d x)}{3 d}",1,"(12*a^3*c + 18*a*b^2*c + 12*a^3*d*x + 18*a*b^2*d*x + 9*b*(4*a^2 + b^2)*Sin[c + d*x] + 9*a*b^2*Sin[2*(c + d*x)] + b^3*Sin[3*(c + d*x)])/(12*d)","A",1
432,1,105,73,0.1370613,"\int (a+b \cos (c+d x))^3 \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*Sec[c + d*x],x]","\frac{-4 a^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 a^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 b \left(6 a^2+b^2\right) (c+d x)+12 a b^2 \sin (c+d x)+b^3 \sin (2 (c+d x))}{4 d}","\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} b x \left(6 a^2+b^2\right)+\frac{5 a b^2 \sin (c+d x)}{2 d}+\frac{b^2 \sin (c+d x) (a+b \cos (c+d x))}{2 d}",1,"(2*b*(6*a^2 + b^2)*(c + d*x) - 4*a^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*a^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 12*a*b^2*Sin[c + d*x] + b^3*Sin[2*(c + d*x)])/(4*d)","A",1
433,1,88,68,0.3481555,"\int (a+b \cos (c+d x))^3 \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^2,x]","\frac{a^3 \tan (c+d x)+3 a b \left(-a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+b c+b d x\right)+b^3 \sin (c+d x)}{d}","-\frac{b \left(a^2-b^2\right) \sin (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \tan (c+d x) (a+b \cos (c+d x))}{d}+3 a b^2 x",1,"(3*a*b*(b*c + b*d*x - a*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + a*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + b^3*Sin[c + d*x] + a^3*Tan[c + d*x])/d","A",1
434,1,55,79,0.1765014,"\int (a+b \cos (c+d x))^3 \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^3,x]","\frac{a \left(a^2+6 b^2\right) \tanh ^{-1}(\sin (c+d x))+a^2 \tan (c+d x) (a \sec (c+d x)+6 b)+2 b^3 d x}{2 d}","\frac{a \left(a^2+6 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{5 a^2 b \tan (c+d x)}{2 d}+\frac{a^2 \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))}{2 d}+b^3 x",1,"(2*b^3*d*x + a*(a^2 + 6*b^2)*ArcTanh[Sin[c + d*x]] + a^2*(6*b + a*Sec[c + d*x])*Tan[c + d*x])/(2*d)","A",1
435,1,70,109,0.2563357,"\int (a+b \cos (c+d x))^3 \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^4,x]","\frac{\left(9 a^2 b+6 b^3\right) \tanh ^{-1}(\sin (c+d x))+a \tan (c+d x) \left(2 a^2 \tan ^2(c+d x)+6 a^2+9 a b \sec (c+d x)+18 b^2\right)}{6 d}","\frac{a \left(2 a^2+9 b^2\right) \tan (c+d x)}{3 d}+\frac{b \left(3 a^2+2 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{7 a^2 b \tan (c+d x) \sec (c+d x)}{6 d}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))}{3 d}",1,"((9*a^2*b + 6*b^3)*ArcTanh[Sin[c + d*x]] + a*Tan[c + d*x]*(6*a^2 + 18*b^2 + 9*a*b*Sec[c + d*x] + 2*a^2*Tan[c + d*x]^2))/(6*d)","A",1
436,1,90,133,0.4195767,"\int (a+b \cos (c+d x))^3 \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^5,x]","\frac{3 a \left(a^2+4 b^2\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(2 a^3 \sec ^3(c+d x)+8 b \left(a^2 \tan ^2(c+d x)+3 a^2+b^2\right)+3 a \left(a^2+4 b^2\right) \sec (c+d x)\right)}{8 d}","\frac{b \left(2 a^2+b^2\right) \tan (c+d x)}{d}+\frac{3 a \left(a^2+4 b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a \left(a^2+4 b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{3 a^2 b \tan (c+d x) \sec ^2(c+d x)}{4 d}+\frac{a^2 \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))}{4 d}",1,"(3*a*(a^2 + 4*b^2)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(3*a*(a^2 + 4*b^2)*Sec[c + d*x] + 2*a^3*Sec[c + d*x]^3 + 8*b*(3*a^2 + b^2 + a^2*Tan[c + d*x]^2)))/(8*d)","A",1
437,1,120,169,0.8993135,"\int (a+b \cos (c+d x))^3 \sec ^6(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^6,x]","\frac{15 b \left(9 a^2+4 b^2\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(8 a \left(5 \left(2 a^2+3 b^2\right) \tan ^2(c+d x)+15 \left(a^2+3 b^2\right)+3 a^2 \tan ^4(c+d x)\right)+15 b \left(9 a^2+4 b^2\right) \sec (c+d x)+90 a^2 b \sec ^3(c+d x)\right)}{120 d}","\frac{a \left(4 a^2+15 b^2\right) \tan ^3(c+d x)}{15 d}+\frac{a \left(4 a^2+15 b^2\right) \tan (c+d x)}{5 d}+\frac{b \left(9 a^2+4 b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \left(9 a^2+4 b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{11 a^2 b \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac{a^2 \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))}{5 d}",1,"(15*b*(9*a^2 + 4*b^2)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*b*(9*a^2 + 4*b^2)*Sec[c + d*x] + 90*a^2*b*Sec[c + d*x]^3 + 8*a*(15*(a^2 + 3*b^2) + 5*(2*a^2 + 3*b^2)*Tan[c + d*x]^2 + 3*a^2*Tan[c + d*x]^4)))/(120*d)","A",1
438,1,181,247,0.4113711,"\int \cos ^3(c+d x) (a+b \cos (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^3*(a + b*Cos[c + d*x])^4,x]","\frac{1680 a b \left(6 a^2+5 b^2\right) (c+d x)+21 b^2 \left(24 a^2+7 b^2\right) \sin (5 (c+d x))+420 a b \left(16 a^2+15 b^2\right) \sin (2 (c+d x))+420 a b \left(2 a^2+3 b^2\right) \sin (4 (c+d x))+105 \left(48 a^4+240 a^2 b^2+35 b^4\right) \sin (c+d x)+35 \left(16 a^4+120 a^2 b^2+21 b^4\right) \sin (3 (c+d x))+140 a b^3 \sin (6 (c+d x))+15 b^4 \sin (7 (c+d x))}{6720 d}","\frac{b^2 \left(37 a^2+6 b^2\right) \sin (c+d x) \cos ^4(c+d x)}{35 d}+\frac{a b \left(6 a^2+5 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{6 d}+\frac{a b \left(6 a^2+5 b^2\right) \sin (c+d x) \cos (c+d x)}{4 d}+\frac{1}{4} a b x \left(6 a^2+5 b^2\right)-\frac{\left(35 a^4+168 a^2 b^2+24 b^4\right) \sin ^3(c+d x)}{105 d}+\frac{\left(35 a^4+168 a^2 b^2+24 b^4\right) \sin (c+d x)}{35 d}+\frac{8 a b^3 \sin (c+d x) \cos ^5(c+d x)}{21 d}+\frac{b^2 \sin (c+d x) \cos ^4(c+d x) (a+b \cos (c+d x))^2}{7 d}",1,"(1680*a*b*(6*a^2 + 5*b^2)*(c + d*x) + 105*(48*a^4 + 240*a^2*b^2 + 35*b^4)*Sin[c + d*x] + 420*a*b*(16*a^2 + 15*b^2)*Sin[2*(c + d*x)] + 35*(16*a^4 + 120*a^2*b^2 + 21*b^4)*Sin[3*(c + d*x)] + 420*a*b*(2*a^2 + 3*b^2)*Sin[4*(c + d*x)] + 21*b^2*(24*a^2 + 7*b^2)*Sin[5*(c + d*x)] + 140*a*b^3*Sin[6*(c + d*x)] + 15*b^4*Sin[7*(c + d*x)])/(6720*d)","A",1
439,1,156,235,0.4365728,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^4,x]","\frac{45 b^2 \left(4 a^2+b^2\right) \sin (4 (c+d x))+480 a b \left(6 a^2+5 b^2\right) \sin (c+d x)+80 a b \left(4 a^2+5 b^2\right) \sin (3 (c+d x))+60 \left(8 a^4+36 a^2 b^2+5 b^4\right) (c+d x)+15 \left(16 a^4+96 a^2 b^2+15 b^4\right) \sin (2 (c+d x))+48 a b^3 \sin (5 (c+d x))+5 b^4 \sin (6 (c+d x))}{960 d}","-\frac{\left(4 a^2-25 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^3}{120 b d}-\frac{a \left(4 a^2-53 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^2}{120 b d}-\frac{a \left(4 a^4-121 a^2 b^2-128 b^4\right) \sin (c+d x)}{60 b d}-\frac{\left(8 a^4-178 a^2 b^2-75 b^4\right) \sin (c+d x) \cos (c+d x)}{240 d}+\frac{1}{16} x \left(8 a^4+36 a^2 b^2+5 b^4\right)+\frac{\sin (c+d x) (a+b \cos (c+d x))^5}{6 b d}-\frac{a \sin (c+d x) (a+b \cos (c+d x))^4}{30 b d}",1,"(60*(8*a^4 + 36*a^2*b^2 + 5*b^4)*(c + d*x) + 480*a*b*(6*a^2 + 5*b^2)*Sin[c + d*x] + 15*(16*a^4 + 96*a^2*b^2 + 15*b^4)*Sin[2*(c + d*x)] + 80*a*b*(4*a^2 + 5*b^2)*Sin[3*(c + d*x)] + 45*b^2*(4*a^2 + b^2)*Sin[4*(c + d*x)] + 48*a*b^3*Sin[5*(c + d*x)] + 5*b^4*Sin[6*(c + d*x)])/(960*d)","A",1
440,1,133,170,0.4856159,"\int \cos (c+d x) (a+b \cos (c+d x))^4 \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^4,x]","\frac{30 \left(8 a^4+36 a^2 b^2+5 b^4\right) \sin (c+d x)+b \left(480 a^3 c+480 a^3 d x+5 \left(24 a^2 b+5 b^3\right) \sin (3 (c+d x))+240 a \left(a^2+b^2\right) \sin (2 (c+d x))+30 a b^2 \sin (4 (c+d x))+360 a b^2 c+360 a b^2 d x+3 b^3 \sin (5 (c+d x))\right)}{240 d}","\frac{\left(3 a^2+4 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^2}{15 d}+\frac{a b \left(6 a^2+29 b^2\right) \sin (c+d x) \cos (c+d x)}{30 d}+\frac{1}{2} a b x \left(4 a^2+3 b^2\right)+\frac{2 \left(3 a^4+28 a^2 b^2+4 b^4\right) \sin (c+d x)}{15 d}+\frac{\sin (c+d x) (a+b \cos (c+d x))^4}{5 d}+\frac{a \sin (c+d x) (a+b \cos (c+d x))^3}{5 d}",1,"(30*(8*a^4 + 36*a^2*b^2 + 5*b^4)*Sin[c + d*x] + b*(480*a^3*c + 360*a*b^2*c + 480*a^3*d*x + 360*a*b^2*d*x + 240*a*(a^2 + b^2)*Sin[2*(c + d*x)] + 5*(24*a^2*b + 5*b^3)*Sin[3*(c + d*x)] + 30*a*b^2*Sin[4*(c + d*x)] + 3*b^3*Sin[5*(c + d*x)]))/(240*d)","A",1
441,1,104,137,0.2051399,"\int (a+b \cos (c+d x))^4 \, dx","Integrate[(a + b*Cos[c + d*x])^4,x]","\frac{24 b^2 \left(6 a^2+b^2\right) \sin (2 (c+d x))+96 a b \left(4 a^2+3 b^2\right) \sin (c+d x)+12 \left(8 a^4+24 a^2 b^2+3 b^4\right) (c+d x)+32 a b^3 \sin (3 (c+d x))+3 b^4 \sin (4 (c+d x))}{96 d}","\frac{a b \left(19 a^2+16 b^2\right) \sin (c+d x)}{6 d}+\frac{b^2 \left(26 a^2+9 b^2\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left(8 a^4+24 a^2 b^2+3 b^4\right)+\frac{b \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac{7 a b \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}",1,"(12*(8*a^4 + 24*a^2*b^2 + 3*b^4)*(c + d*x) + 96*a*b*(4*a^2 + 3*b^2)*Sin[c + d*x] + 24*b^2*(6*a^2 + b^2)*Sin[2*(c + d*x)] + 32*a*b^3*Sin[3*(c + d*x)] + 3*b^4*Sin[4*(c + d*x)])/(96*d)","A",1
442,1,128,107,0.1555359,"\int (a+b \cos (c+d x))^4 \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*Sec[c + d*x],x]","\frac{-12 a^4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 a^4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+24 a b \left(2 a^2+b^2\right) (c+d x)+9 b^2 \left(8 a^2+b^2\right) \sin (c+d x)+12 a b^3 \sin (2 (c+d x))+b^4 \sin (3 (c+d x))}{12 d}","\frac{a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \left(17 a^2+2 b^2\right) \sin (c+d x)}{3 d}+2 a b x \left(2 a^2+b^2\right)+\frac{4 a b^3 \sin (c+d x) \cos (c+d x)}{3 d}+\frac{b^2 \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}",1,"(24*a*b*(2*a^2 + b^2)*(c + d*x) - 12*a^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*a^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 9*b^2*(8*a^2 + b^2)*Sin[c + d*x] + 12*a*b^3*Sin[2*(c + d*x)] + b^4*Sin[3*(c + d*x)])/(12*d)","A",1
443,1,119,114,0.6584212,"\int (a+b \cos (c+d x))^4 \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*Sec[c + d*x]^2,x]","\frac{4 a^4 \tan (c+d x)+2 b \left(-8 a^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+8 a^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+b \left(12 a^2+b^2\right) (c+d x)\right)+16 a b^3 \sin (c+d x)+b^4 \sin (2 (c+d x))}{4 d}","\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{2 a b \left(a^2-2 b^2\right) \sin (c+d x)}{d}-\frac{b^2 \left(2 a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} b^2 x \left(12 a^2+b^2\right)+\frac{a^2 \tan (c+d x) (a+b \cos (c+d x))^2}{d}",1,"(2*b*(b*(12*a^2 + b^2)*(c + d*x) - 8*a^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 8*a^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 16*a*b^3*Sin[c + d*x] + b^4*Sin[2*(c + d*x)] + 4*a^4*Tan[c + d*x])/(4*d)","A",1
444,1,174,108,2.4413452,"\int (a+b \cos (c+d x))^4 \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*Sec[c + d*x]^3,x]","\frac{16 a^3 b \tan (c+d x)+a \left(\frac{a^3}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{a^3}{\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+12 b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 a \left(a^2+12 b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+16 b^3 c+16 b^3 d x\right)+4 b^4 \sin (c+d x)}{4 d}","\frac{3 a^3 b \tan (c+d x)}{d}-\frac{b^2 \left(a^2-2 b^2\right) \sin (c+d x)}{2 d}+\frac{a^2 \left(a^2+12 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{2 d}+4 a b^3 x",1,"(a*(16*b^3*c + 16*b^3*d*x - 2*a*(a^2 + 12*b^2)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*a*(a^2 + 12*b^2)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + a^3/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 - a^3/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + 4*b^4*Sin[c + d*x] + 16*a^3*b*Tan[c + d*x])/(4*d)","A",1
445,1,77,115,0.3867882,"\int (a+b \cos (c+d x))^4 \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*Sec[c + d*x]^4,x]","\frac{a^4 \tan ^3(c+d x)+6 a b \left(a^2+2 b^2\right) \tanh ^{-1}(\sin (c+d x))+3 a^2 \tan (c+d x) \left(a^2+2 a b \sec (c+d x)+6 b^2\right)+3 b^4 d x}{3 d}","\frac{4 a^3 b \tan (c+d x) \sec (c+d x)}{3 d}+\frac{a^2 \left(2 a^2+17 b^2\right) \tan (c+d x)}{3 d}+\frac{2 a b \left(a^2+2 b^2\right) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{3 d}+b^4 x",1,"(3*b^4*d*x + 6*a*b*(a^2 + 2*b^2)*ArcTanh[Sin[c + d*x]] + 3*a^2*(a^2 + 6*b^2 + 2*a*b*Sec[c + d*x])*Tan[c + d*x] + a^4*Tan[c + d*x]^3)/(3*d)","A",1
446,1,101,154,0.4869982,"\int (a+b \cos (c+d x))^4 \sec ^5(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*Sec[c + d*x]^5,x]","\frac{3 \left(3 a^4+24 a^2 b^2+8 b^4\right) \tanh ^{-1}(\sin (c+d x))+a \tan (c+d x) \left(6 a^3 \sec ^3(c+d x)+32 b \left(3 \left(a^2+b^2\right)+a^2 \tan ^2(c+d x)\right)+9 a \left(a^2+8 b^2\right) \sec (c+d x)\right)}{24 d}","\frac{5 a^3 b \tan (c+d x) \sec ^2(c+d x)}{6 d}+\frac{4 a b \left(2 a^2+3 b^2\right) \tan (c+d x)}{3 d}+\frac{a^2 \left(3 a^2+22 b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^2 \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{4 d}+\frac{\left(3 a^4+24 a^2 b^2+8 b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 d}",1,"(3*(3*a^4 + 24*a^2*b^2 + 8*b^4)*ArcTanh[Sin[c + d*x]] + a*Tan[c + d*x]*(9*a*(a^2 + 8*b^2)*Sec[c + d*x] + 6*a^3*Sec[c + d*x]^3 + 32*b*(3*(a^2 + b^2) + a^2*Tan[c + d*x]^2)))/(24*d)","A",1
447,1,125,188,0.7457655,"\int (a+b \cos (c+d x))^4 \sec ^6(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*Sec[c + d*x]^6,x]","\frac{15 a b \left(3 a^2+4 b^2\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(6 a^4 \tan ^4(c+d x)+30 a^3 b \sec ^3(c+d x)+20 a^2 \left(a^2+3 b^2\right) \tan ^2(c+d x)+15 a b \left(3 a^2+4 b^2\right) \sec (c+d x)+30 \left(a^4+6 a^2 b^2+b^4\right)\right)}{30 d}","\frac{3 a^3 b \tan (c+d x) \sec ^3(c+d x)}{5 d}+\frac{a b \left(3 a^2+4 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 \left(4 a^2+27 b^2\right) \tan (c+d x) \sec ^2(c+d x)}{15 d}+\frac{a b \left(3 a^2+4 b^2\right) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a^2 \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}+\frac{\left(8 a^4+60 a^2 b^2+15 b^4\right) \tan (c+d x)}{15 d}",1,"(15*a*b*(3*a^2 + 4*b^2)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(30*(a^4 + 6*a^2*b^2 + b^4) + 15*a*b*(3*a^2 + 4*b^2)*Sec[c + d*x] + 30*a^3*b*Sec[c + d*x]^3 + 20*a^2*(a^2 + 3*b^2)*Tan[c + d*x]^2 + 6*a^4*Tan[c + d*x]^4))/(30*d)","A",1
448,1,154,222,1.007814,"\int (a+b \cos (c+d x))^4 \sec ^7(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^4*Sec[c + d*x]^7,x]","\frac{15 \left(5 a^4+36 a^2 b^2+8 b^4\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(40 a^4 \sec ^5(c+d x)+64 a b \left(5 \left(2 a^2+b^2\right) \tan ^2(c+d x)+15 \left(a^2+b^2\right)+3 a^2 \tan ^4(c+d x)\right)+10 a^2 \left(5 a^2+36 b^2\right) \sec ^3(c+d x)+15 \left(5 a^4+36 a^2 b^2+8 b^4\right) \sec (c+d x)\right)}{240 d}","\frac{7 a^3 b \tan (c+d x) \sec ^4(c+d x)}{15 d}+\frac{4 a b \left(4 a^2+5 b^2\right) \tan ^3(c+d x)}{15 d}+\frac{4 a b \left(4 a^2+5 b^2\right) \tan (c+d x)}{5 d}+\frac{a^2 \left(5 a^2+32 b^2\right) \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{a^2 \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^2}{6 d}+\frac{\left(5 a^4+36 a^2 b^2+8 b^4\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\left(5 a^4+36 a^2 b^2+8 b^4\right) \tan (c+d x) \sec (c+d x)}{16 d}",1,"(15*(5*a^4 + 36*a^2*b^2 + 8*b^4)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*(5*a^4 + 36*a^2*b^2 + 8*b^4)*Sec[c + d*x] + 10*a^2*(5*a^2 + 36*b^2)*Sec[c + d*x]^3 + 40*a^4*Sec[c + d*x]^5 + 64*a*b*(15*(a^2 + b^2) + 5*(2*a^2 + b^2)*Tan[c + d*x]^2 + 3*a^2*Tan[c + d*x]^4)))/(240*d)","A",1
449,1,153,193,0.6382135,"\int \frac{\cos ^5(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^5/(a + b*Cos[c + d*x]),x]","\frac{-24 a b \left(4 a^2+3 b^2\right) \sin (c+d x)+24 b^2 \left(a^2+b^2\right) \sin (2 (c+d x))+\frac{192 a^5 \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}}+12 \left(8 a^4+4 a^2 b^2+3 b^4\right) (c+d x)-8 a b^3 \sin (3 (c+d x))+3 b^4 \sin (4 (c+d x))}{96 b^5 d}","-\frac{2 a^5 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d \sqrt{a-b} \sqrt{a+b}}-\frac{a \left(3 a^2+2 b^2\right) \sin (c+d x)}{3 b^4 d}+\frac{\left(4 a^2+3 b^2\right) \sin (c+d x) \cos (c+d x)}{8 b^3 d}+\frac{x \left(8 a^4+4 a^2 b^2+3 b^4\right)}{8 b^5}-\frac{a \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 b d}",1,"(12*(8*a^4 + 4*a^2*b^2 + 3*b^4)*(c + d*x) + (192*a^5*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] - 24*a*b*(4*a^2 + 3*b^2)*Sin[c + d*x] + 24*b^2*(a^2 + b^2)*Sin[2*(c + d*x)] - 8*a*b^3*Sin[3*(c + d*x)] + 3*b^4*Sin[4*(c + d*x)])/(96*b^5*d)","A",1
450,1,122,148,0.3306319,"\int \frac{\cos ^4(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^4/(a + b*Cos[c + d*x]),x]","\frac{-6 a \left(2 a^2+b^2\right) (c+d x)+3 b \left(4 a^2+3 b^2\right) \sin (c+d x)-\frac{24 a^4 \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}-3 a b^2 \sin (2 (c+d x))+b^3 \sin (3 (c+d x))}{12 b^4 d}","\frac{2 a^4 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d \sqrt{a-b} \sqrt{a+b}}-\frac{a x \left(2 a^2+b^2\right)}{2 b^4}+\frac{\left(3 a^2+2 b^2\right) \sin (c+d x)}{3 b^3 d}-\frac{a \sin (c+d x) \cos (c+d x)}{2 b^2 d}+\frac{\sin (c+d x) \cos ^2(c+d x)}{3 b d}",1,"(-6*a*(2*a^2 + b^2)*(c + d*x) - (24*a^4*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + 3*b*(4*a^2 + 3*b^2)*Sin[c + d*x] - 3*a*b^2*Sin[2*(c + d*x)] + b^3*Sin[3*(c + d*x)])/(12*b^4*d)","A",1
451,1,97,110,0.2355999,"\int \frac{\cos ^3(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^3/(a + b*Cos[c + d*x]),x]","\frac{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))}{4 b^3 d}","-\frac{2 a^3 \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\right)}{2 b^3}-\frac{a \sin (c+d x)}{b^2 d}+\frac{\sin (c+d x) \cos (c+d x)}{2 b d}",1,"(2*(2*a^2 + b^2)*(c + d*x) + (8*a^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
452,1,72,76,0.1417052,"\int \frac{\cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Cos[c + d*x]),x]","\frac{-\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)}{b^2 d}","\frac{2 a^2 \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 x}{b^2}+\frac{\sin (c+d x)}{b d}",1,"(-(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
453,1,58,59,0.0912781,"\int \frac{\cos (c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]/(a + b*Cos[c + d*x]),x]","\frac{\frac{2 a \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+c+d x}{b d}","\frac{x}{b}-\frac{2 a \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,"(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
454,1,48,49,0.0359231,"\int \frac{1}{a+b \cos (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^(-1),x]","-\frac{2 \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 \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*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(Sqrt[-a^2 + b^2]*d)","A",1
455,1,102,68,0.0820308,"\int \frac{\sec (c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[Sec[c + d*x]/(a + b*Cos[c + d*x]),x]","\frac{\frac{2 b \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a d}","\frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{2 b \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a 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] - 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
456,1,115,85,0.3779409,"\int \frac{\sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Cos[c + d*x]),x]","\frac{-\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)}{a^2 d}","\frac{2 b^2 \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 \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{\tan (c+d x)}{a d}",1,"((-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
457,1,238,119,1.0574882,"\int \frac{\sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[Sec[c + d*x]^3/(a + b*Cos[c + d*x]),x]","\frac{\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)}{4 a^3 d}","-\frac{2 b^3 \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 \tan (c+d x)}{a^2 d}+\frac{\left(a^2+2 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{\tan (c+d x) \sec (c+d x)}{2 a d}",1,"((8*b^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
458,1,258,157,2.4629855,"\int \frac{\sec ^4(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[Sec[c + d*x]^4/(a + b*Cos[c + d*x]),x]","\frac{\frac{1}{2} \sec ^3(c+d x) \left(4 a \sin (c+d x) \left(\left(2 a^2+3 b^2\right) \cos (2 (c+d x))+4 a^2-3 a b \cos (c+d x)+3 b^2\right)+9 b \left(a^2+2 b^2\right) \cos (c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+3 b \left(a^2+2 b^2\right) \cos (3 (c+d x)) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)-\frac{24 b^4 \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}}}{12 a^4 d}","\frac{2 b^4 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d \sqrt{a-b} \sqrt{a+b}}-\frac{b \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{b \left(a^2+2 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{\left(2 a^2+3 b^2\right) \tan (c+d x)}{3 a^3 d}+\frac{\tan (c+d x) \sec ^2(c+d x)}{3 a d}",1,"((-24*b^4*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (Sec[c + d*x]^3*(9*b*(a^2 + 2*b^2)*Cos[c + d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 3*b*(a^2 + 2*b^2)*Cos[3*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 4*a*(4*a^2 + 3*b^2 - 3*a*b*Cos[c + d*x] + (2*a^2 + 3*b^2)*Cos[2*(c + d*x)])*Sin[c + d*x]))/2)/(12*a^4*d)","A",1
459,1,176,266,0.9024641,"\int \frac{\cos ^5(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^5/(a + b*Cos[c + d*x])^2,x]","\frac{\frac{12 a^5 b \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}+9 b \left(4 a^2+b^2\right) \sin (c+d x)+\frac{24 a^4 \left(4 a^2-5 b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}-6 a b^2 \sin (2 (c+d x))-12 a (2 a-i b) (2 a+i b) (c+d x)+b^3 \sin (3 (c+d x))}{12 b^5 d}","-\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\left(4 a^2-b^2\right) \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d \left(a^2-b^2\right)}-\frac{a x \left(4 a^2+b^2\right)}{b^5}-\frac{a \left(2 a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{b^3 d \left(a^2-b^2\right)}+\frac{2 a^4 \left(4 a^2-5 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left(12 a^4-7 a^2 b^2-2 b^4\right) \sin (c+d x)}{3 b^4 d \left(a^2-b^2\right)}",1,"(-12*a*(2*a - I*b)*(2*a + I*b)*(c + d*x) + (24*a^4*(4*a^2 - 5*b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + 9*b*(4*a^2 + b^2)*Sin[c + d*x] + (12*a^5*b*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])) - 6*a*b^2*Sin[2*(c + d*x)] + b^3*Sin[3*(c + d*x)])/(12*b^5*d)","C",1
460,1,144,166,0.783373,"\int \frac{\cos ^4(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^4/(a + b*Cos[c + d*x])^2,x]","\frac{-\frac{4 a^4 b \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}+2 \left(6 a^2+b^2\right) (c+d x)-\frac{8 a^3 \left(3 a^2-4 b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}-8 a b \sin (c+d x)+b^2 \sin (2 (c+d x))}{4 b^4 d}","\frac{x \left(6 a^2+b^2\right)}{2 b^4}-\frac{a^4 \sin (c+d x)}{b^3 d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{2 a^3 \left(3 a^2-4 b^2\right) \tanh ^{-1}\left(\frac{(a-b) \sin (c+d x)}{\sqrt{b^2-a^2} (\cos (c+d x)+1)}\right)}{b^4 d \left(b^2-a^2\right)^{3/2}}-\frac{2 a \sin (c+d x)}{b^3 d}+\frac{\sin (c+d x) \cos (c+d x)}{2 b^2 d}",1,"(2*(6*a^2 + b^2)*(c + d*x) - (8*a^3*(3*a^2 - 4*b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) - 8*a*b*Sin[c + d*x] - (4*a^4*b*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])) + b^2*Sin[2*(c + d*x)])/(4*b^4*d)","A",1
461,1,113,155,0.7339626,"\int \frac{\cos ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^3/(a + b*Cos[c + d*x])^2,x]","\frac{\sin (c+d x) \left(\frac{a^3 b}{(a-b) (a+b) (a+b \cos (c+d x))}+b\right)+\frac{2 a^2 \left(2 a^2-3 b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}-2 a (c+d x)}{b^3 d}","\frac{\left(2 a^2-b^2\right) \sin (c+d x)}{b^2 d \left(a^2-b^2\right)}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{2 a^2 \left(2 a^2-3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{2 a x}{b^3}",1,"(-2*a*(c + d*x) + (2*a^2*(2*a^2 - 3*b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + (b + (a^3*b)/((a - b)*(a + b)*(a + b*Cos[c + d*x])))*Sin[c + d*x])/(b^3*d)","A",1
462,1,103,108,0.4057079,"\int \frac{\cos ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Cos[c + d*x])^2,x]","\frac{-\frac{2 a \left(a^2-2 b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}-\frac{a^2 b \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}+c+d x}{b^2 d}","-\frac{2 a \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{a^2 \sin (c+d x)}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{x}{b^2}",1,"(c + d*x - (2*a*(a^2 - 2*b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) - (a^2*b*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])))/(b^2*d)","A",1
463,1,83,85,0.2380064,"\int \frac{\cos (c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[Cos[c + d*x]/(a + b*Cos[c + d*x])^2,x]","\frac{\frac{a \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d 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)}{\left(b^2-a^2\right)^{3/2}}}{d}","\frac{a \sin (c+d x)}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{2 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}}",1,"((-2*b*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + (a*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])))/d","A",1
464,1,84,86,0.1875502,"\int \frac{1}{(a+b \cos (c+d x))^2} \, dx","Integrate[(a + b*Cos[c + d*x])^(-2),x]","\frac{\frac{2 a \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}-\frac{b \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}}{d}","\frac{2 a \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{3/2} (a+b)^{3/2}}-\frac{b \sin (c+d x)}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"((2*a*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) - (b*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])))/d","A",1
465,1,146,118,0.3714315,"\int \frac{\sec (c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[Sec[c + d*x]/(a + b*Cos[c + d*x])^2,x]","\frac{\frac{2 b \left(b^2-2 a^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}+\frac{a b^2 \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^2 d}","-\frac{2 b \left(2 a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{\tanh ^{-1}(\sin (c+d x))}{a^2 d}",1,"((2*b*(-2*a^2 + b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) - Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a*b^2*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])))/(a^2*d)","A",1
466,1,163,155,0.966438,"\int \frac{\sec ^2(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Cos[c + d*x])^2,x]","\frac{-\frac{2 b^2 \left(2 b^2-3 a^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}-\frac{a b^3 \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}+a \tan (c+d x)+2 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^3 d}","-\frac{2 b \tanh ^{-1}(\sin (c+d x))}{a^3 d}+\frac{\left(a^2-2 b^2\right) \tan (c+d x)}{a^2 d \left(a^2-b^2\right)}+\frac{b^2 \tan (c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{2 b^2 \left(3 a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{3/2} (a+b)^{3/2}}",1,"((-2*b^2*(-3*a^2 + 2*b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + 2*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 2*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - (a*b^3*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])) + a*Tan[c + d*x])/(a^3*d)","A",1
467,1,285,217,5.5217047,"\int \frac{\sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^3/(a + b*Cos[c + d*x])^2,x]","\frac{\frac{8 b^3 \left(3 b^2-4 a^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}+\frac{a^2}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{a^2}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-2 a^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 a^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{4 a b^4 \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}-8 a b \tan (c+d x)-12 b^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 a^4 d}","\frac{\left(a^2-3 b^2\right) \tan (c+d x) \sec (c+d x)}{2 a^2 d \left(a^2-b^2\right)}+\frac{b^2 \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+6 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{2 b^3 \left(4 a^2-3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{b \left(2 a^2-3 b^2\right) \tan (c+d x)}{a^3 d \left(a^2-b^2\right)}",1,"((8*b^3*(-4*a^2 + 3*b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) - 2*a^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 12*b^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*a^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 12*b^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + a^2/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 - a^2/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*a*b^4*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])) - 8*a*b*Tan[c + d*x])/(4*a^4*d)","A",1
468,1,499,270,6.1539525,"\int \frac{\sec ^4(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^4/(a + b*Cos[c + d*x])^2,x]","-\frac{b^5 \sin (c+d x)}{a^4 d (a-b) (a+b) (a+b \cos (c+d x))}+\frac{a-6 b}{12 a^3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{6 b-a}{12 a^3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{6 a^2 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{6 a^2 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{\left(a^2 b+4 b^3\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a^5 d}+\frac{\left(-a^2 b-4 b^3\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^5 d}-\frac{2 b^4 \left(5 a^2-4 b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{a^5 d \left(a^2-b^2\right) \sqrt{b^2-a^2}}+\frac{2 a^2 \sin \left(\frac{1}{2} (c+d x)\right)+9 b^2 \sin \left(\frac{1}{2} (c+d x)\right)}{3 a^4 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{2 a^2 \sin \left(\frac{1}{2} (c+d x)\right)+9 b^2 \sin \left(\frac{1}{2} (c+d x)\right)}{3 a^4 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\frac{\left(a^2-4 b^2\right) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d \left(a^2-b^2\right)}+\frac{b^2 \tan (c+d x) \sec ^2(c+d x)}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{b \left(a^2+4 b^2\right) \tanh ^{-1}(\sin (c+d x))}{a^5 d}+\frac{2 b^4 \left(5 a^2-4 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left(2 a^4+7 a^2 b^2-12 b^4\right) \tan (c+d x)}{3 a^4 d \left(a^2-b^2\right)}-\frac{b \left(a^2-2 b^2\right) \tan (c+d x) \sec (c+d x)}{a^3 d \left(a^2-b^2\right)}",1,"(-2*b^4*(5*a^2 - 4*b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(a^5*(a^2 - b^2)*Sqrt[-a^2 + b^2]*d) + ((a^2*b + 4*b^3)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(a^5*d) + ((-(a^2*b) - 4*b^3)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(a^5*d) + (a - 6*b)/(12*a^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + Sin[(c + d*x)/2]/(6*a^2*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + Sin[(c + d*x)/2]/(6*a^2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (-a + 6*b)/(12*a^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (2*a^2*Sin[(c + d*x)/2] + 9*b^2*Sin[(c + d*x)/2])/(3*a^4*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (2*a^2*Sin[(c + d*x)/2] + 9*b^2*Sin[(c + d*x)/2])/(3*a^4*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) - (b^5*Sin[c + d*x])/(a^4*(a - b)*(a + b)*d*(a + b*Cos[c + d*x]))","A",1
469,1,199,300,2.0192785,"\int \frac{\cos ^5(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^5/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{2 a^5 b \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}+2 \left(12 a^2+b^2\right) (c+d x)+\frac{2 a^4 b \left(10 b^2-7 a^2\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}+\frac{4 a^3 \left(12 a^4-29 a^2 b^2+20 b^4\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}-12 a b \sin (c+d x)+b^2 \sin (2 (c+d x))}{4 b^5 d}","-\frac{a^2 \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{a^2 \left(4 a^2-7 b^2\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{x \left(12 a^2+b^2\right)}{2 b^5}-\frac{3 a \left(4 a^4-7 a^2 b^2+2 b^4\right) \sin (c+d x)}{2 b^4 d \left(a^2-b^2\right)^2}+\frac{\left(6 a^4-10 a^2 b^2+b^4\right) \sin (c+d x) \cos (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2}-\frac{a^3 \left(12 a^4-29 a^2 b^2+20 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{5/2} (a+b)^{5/2}}",1,"(2*(12*a^2 + b^2)*(c + d*x) + (4*a^3*(12*a^4 - 29*a^2*b^2 + 20*b^4)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) - 12*a*b*Sin[c + d*x] + (2*a^5*b*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^2) + (2*a^4*b*(-7*a^2 + 10*b^2)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])) + b^2*Sin[2*(c + d*x)])/(4*b^5*d)","A",1
470,1,177,221,1.4842034,"\int \frac{\cos ^4(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^4/(a + b*Cos[c + d*x])^3,x]","\frac{-\frac{a^4 b \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}-\frac{6 a^2 \left(2 a^4-5 a^2 b^2+4 b^4\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}+\frac{a^3 b \left(5 a^2-8 b^2\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}-6 a (c+d x)+2 b \sin (c+d x)}{2 b^4 d}","-\frac{a^2 \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^2-2 b^2\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right)}+\frac{3 a^2 \left(2 a^4-5 a^2 b^2+4 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{3 a^3 \left(a^2-2 b^2\right) \sin (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{3 a x}{b^4}",1,"(-6*a*(c + d*x) - (6*a^2*(2*a^4 - 5*a^2*b^2 + 4*b^4)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + 2*b*Sin[c + d*x] - (a^4*b*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^2) + (a^3*b*(5*a^2 - 8*b^2)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])))/(2*b^4*d)","A",1
471,1,149,179,1.1245783,"\int \frac{\cos ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^3/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{2 a \left(2 a^4-5 a^2 b^2+6 b^4\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}-\frac{a^2 b \sin (c+d x) \left(2 a^3+3 b \left(a^2-2 b^2\right) \cos (c+d x)-5 a b^2\right)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))^2}+2 (c+d x)}{2 b^3 d}","-\frac{a^2 \left(2 a^2-5 b^2\right) \sin (c+d x)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{a \left(2 a^4-5 a^2 b^2+6 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{x}{b^3}",1,"(2*(c + d*x) + (2*a*(2*a^4 - 5*a^2*b^2 + 6*b^4)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) - (a^2*b*(2*a^3 - 5*a*b^2 + 3*b*(a^2 - 2*b^2)*Cos[c + d*x])*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])^2))/(2*b^3*d)","A",1
472,1,115,149,0.5515116,"\int \frac{\cos ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{a \sin (c+d x) \left(\left(a^2-4 b^2\right) \cos (c+d x)-3 a b\right)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))^2}-\frac{2 \left(a^2+2 b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}}{2 d}","\frac{\left(a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{a^2 \sin (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{a \left(a^2-4 b^2\right) \sin (c+d x)}{2 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}",1,"((-2*(a^2 + 2*b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (a*(-3*a*b + (a^2 - 4*b^2)*Cos[c + d*x])*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])^2))/(2*d)","A",1
473,1,115,134,0.3838623,"\int \frac{\cos (c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[Cos[c + d*x]/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{\sin (c+d x) \left(b \left(a^2+2 b^2\right) \cos (c+d x)+a \left(2 a^2+b^2\right)\right)}{(a+b \cos (c+d x))^2}+\frac{6 a b \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}}{2 d (a-b)^2 (a+b)^2}","\frac{\left(a^2+2 b^2\right) \sin (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{a \sin (c+d x)}{2 d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{3 a b \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}",1,"((6*a*b*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + ((a*(2*a^2 + b^2) + b*(a^2 + 2*b^2)*Cos[c + d*x])*Sin[c + d*x])/(a + b*Cos[c + d*x])^2)/(2*(a - b)^2*(a + b)^2*d)","A",1
474,1,113,133,0.3994735,"\int \frac{1}{(a+b \cos (c+d x))^3} \, dx","Integrate[(a + b*Cos[c + d*x])^(-3),x]","\frac{\frac{b \sin (c+d x) \left(-4 a^2-3 a b \cos (c+d x)+b^2\right)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))^2}-\frac{2 \left(2 a^2+b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}}{2 d}","\frac{\left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{3 a b \sin (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{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 + b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (b*(-4*a^2 + b^2 - 3*a*b*Cos[c + d*x])*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])^2))/(2*d)","A",1
475,1,192,182,1.1397466,"\int \frac{\sec (c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[Sec[c + d*x]/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{2 b \left(6 a^4-5 a^2 b^2+2 b^4\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}+\frac{a b^2 \sin (c+d x) \left(6 a^3+b \left(5 a^2-2 b^2\right) \cos (c+d x)-3 a b^2\right)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))^2}-2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^3 d}","\frac{\tanh ^{-1}(\sin (c+d x))}{a^3 d}+\frac{b^2 \left(5 a^2-2 b^2\right) \sin (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{b^2 \sin (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{b \left(6 a^4-5 a^2 b^2+2 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}",1,"((2*b*(6*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) - 2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a*b^2*(6*a^3 - 3*a*b^2 + b*(5*a^2 - 2*b^2)*Cos[c + d*x])*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])^2))/(2*a^3*d)","A",1
476,1,205,232,4.2111171,"\int \frac{\sec ^2(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Cos[c + d*x])^3,x]","-\frac{\frac{6 b^2 \left(4 a^4-5 a^2 b^2+2 b^4\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}+\frac{a b^3 \sin (c+d x) \left(8 a^3+b \left(7 a^2-4 b^2\right) \cos (c+d x)-5 a b^2\right)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))^2}-2 a \tan (c+d x)-6 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 b \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{3 b \tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac{3 b^2 \left(2 a^2-b^2\right) \tan (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{b^2 \tan (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{3 b^2 \left(4 a^4-5 a^2 b^2+2 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{\left(2 a^4-11 a^2 b^2+6 b^4\right) \tan (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}",1,"-1/2*((6*b^2*(4*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) - 6*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a*b^3*(8*a^3 - 5*a*b^2 + b*(7*a^2 - 4*b^2)*Cos[c + d*x])*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])^2) - 2*a*Tan[c + d*x])/(a^4*d)","A",1
477,1,427,305,6.1829838,"\int \frac{\sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^3/(a + b*Cos[c + d*x])^3,x]","-\frac{3 b \sin \left(\frac{1}{2} (c+d x)\right)}{a^4 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{3 b \sin \left(\frac{1}{2} (c+d x)\right)}{a^4 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{b^4 \sin (c+d x)}{2 a^3 d (a-b) (a+b) (a+b \cos (c+d x))^2}+\frac{1}{4 a^3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{1}{4 a^3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{\left(-a^2-12 b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a^5 d}+\frac{\left(a^2+12 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^5 d}+\frac{3 \left(3 a^2 b^4 \sin (c+d x)-2 b^6 \sin (c+d x)\right)}{2 a^4 d (a-b)^2 (a+b)^2 (a+b \cos (c+d x))}+\frac{b^3 \left(20 a^4-29 a^2 b^2+12 b^4\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{a^5 d \left(a^2-b^2\right)^2 \sqrt{b^2-a^2}}","\frac{b^2 \left(7 a^2-4 b^2\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{b^2 \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+12 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 a^5 d}-\frac{3 b \left(2 a^4-7 a^2 b^2+4 b^4\right) \tan (c+d x)}{2 a^4 d \left(a^2-b^2\right)^2}-\frac{b^3 \left(20 a^4-29 a^2 b^2+12 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{\left(a^4-10 a^2 b^2+6 b^4\right) \tan (c+d x) \sec (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}",1,"(b^3*(20*a^4 - 29*a^2*b^2 + 12*b^4)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(a^5*(a^2 - b^2)^2*Sqrt[-a^2 + b^2]*d) + ((-a^2 - 12*b^2)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(2*a^5*d) + ((a^2 + 12*b^2)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(2*a^5*d) + 1/(4*a^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) - (3*b*Sin[(c + d*x)/2])/(a^4*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - 1/(4*a^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) - (3*b*Sin[(c + d*x)/2])/(a^4*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (b^4*Sin[c + d*x])/(2*a^3*(a - b)*(a + b)*d*(a + b*Cos[c + d*x])^2) + (3*(3*a^2*b^4*Sin[c + d*x] - 2*b^6*Sin[c + d*x]))/(2*a^4*(a - b)^2*(a + b)^2*d*(a + b*Cos[c + d*x]))","A",1
478,1,240,307,5.7814739,"\int \frac{\cos ^5(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Integrate[Cos[c + d*x]^5/(a + b*Cos[c + d*x])^4,x]","\frac{\frac{2 a^5 b \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^3}+\frac{5 a^4 b \left(3 b^2-2 a^2\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))^2}+\frac{6 a^2 \left(8 a^6-28 a^4 b^2+35 a^2 b^4-20 b^6\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{7/2}}+\frac{a^3 b \left(26 a^4-71 a^2 b^2+60 b^4\right) \sin (c+d x)}{(a-b)^3 (a+b)^3 (a+b \cos (c+d x))}-24 a (c+d x)+6 b \sin (c+d x)}{6 b^5 d}","-\frac{a^2 \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^2 \left(4 a^2-9 b^2\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^4-23 a^2 b^2+6 b^4\right) \sin (c+d x)}{6 b^4 d \left(a^2-b^2\right)^2}+\frac{a^2 \left(8 a^6-28 a^4 b^2+35 a^2 b^4-20 b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{a^3 \left(4 a^4-11 a^2 b^2+12 b^4\right) \sin (c+d x)}{2 b^4 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{4 a x}{b^5}",1,"(-24*a*(c + d*x) + (6*a^2*(8*a^6 - 28*a^4*b^2 + 35*a^2*b^4 - 20*b^6)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) + 6*b*Sin[c + d*x] + (2*a^5*b*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^3) + (5*a^4*b*(-2*a^2 + 3*b^2)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])^2) + (a^3*b*(26*a^4 - 71*a^2*b^2 + 60*b^4)*Sin[c + d*x])/((a - b)^3*(a + b)^3*(a + b*Cos[c + d*x])))/(6*b^5*d)","A",1
479,1,227,250,2.7330018,"\int \frac{\cos ^4(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Integrate[Cos[c + d*x]^4/(a + b*Cos[c + d*x])^4,x]","\frac{-\frac{2 a^4 b \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^3}+\frac{a^2 b \left(-11 a^4+32 a^2 b^2-36 b^4\right) \sin (c+d x)}{(a-b)^3 (a+b)^3 (a+b \cos (c+d x))}+\frac{a^3 b \left(7 a^2-12 b^2\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))^2}-\frac{6 a \left(2 a^6-7 a^4 b^2+8 a^2 b^4-8 b^6\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{7/2}}+6 (c+d x)}{6 b^4 d}","-\frac{a^2 \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(9 a^4-28 a^2 b^2+34 b^4\right) \sin (c+d x)}{6 b^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{a^3 \left(3 a^2-8 b^2\right) \sin (c+d x)}{6 b^3 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{a \left(2 a^6-7 a^4 b^2+8 a^2 b^4-8 b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{x}{b^4}",1,"(6*(c + d*x) - (6*a*(2*a^6 - 7*a^4*b^2 + 8*a^2*b^4 - 8*b^6)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) - (2*a^4*b*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^3) + (a^3*b*(7*a^2 - 12*b^2)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])^2) + (a^2*b*(-11*a^4 + 32*a^2*b^2 - 36*b^4)*Sin[c + d*x])/((a - b)^3*(a + b)^3*(a + b*Cos[c + d*x])))/(6*b^4*d)","A",1
480,1,158,222,1.2337254,"\int \frac{\cos ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Integrate[Cos[c + d*x]^3/(a + b*Cos[c + d*x])^4,x]","\frac{\frac{a \sin (c+d x) \left(4 a^4+3 a b \left(a^2+9 b^2\right) \cos (c+d x)+11 a^2 b^2+\left(2 a^4-5 a^2 b^2+18 b^4\right) \cos ^2(c+d x)\right)}{(a-b)^3 (a+b)^3 (a+b \cos (c+d x))^3}-\frac{6 b \left(3 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{7/2}}}{6 d}","-\frac{b \left(3 a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{a^2 \left(2 a^2-7 b^2\right) \sin (c+d x)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{a \left(2 a^4-5 a^2 b^2+18 b^4\right) \sin (c+d x)}{6 b^2 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}",1,"((-6*b*(3*a^2 + 2*b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) + (a*(4*a^4 + 11*a^2*b^2 + 3*a*b*(a^2 + 9*b^2)*Cos[c + d*x] + (2*a^4 - 5*a^2*b^2 + 18*b^4)*Cos[c + d*x]^2)*Sin[c + d*x])/((a - b)^3*(a + b)^3*(a + b*Cos[c + d*x])^3))/(6*d)","A",1
481,1,162,206,1.1951229,"\int \frac{\cos ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Cos[c + d*x])^4,x]","\frac{\frac{6 a \left(a^2+4 b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{7/2}}+\frac{\sin (c+d x) \left(-13 a^4 b-2 a^2 b^3+b \left(a^4-10 a^2 b^2-6 b^4\right) \cos ^2(c+d x)+3 a \left(a^4-9 a^2 b^2-2 b^4\right) \cos (c+d x)\right)}{(a-b)^3 (a+b)^3 (a+b \cos (c+d x))^3}}{6 d}","\frac{a \left(a^2+4 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{a^2 \sin (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{a \left(a^2-6 b^2\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^4-10 a^2 b^2-6 b^4\right) \sin (c+d x)}{6 b d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}",1,"((6*a*(a^2 + 4*b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) + ((-13*a^4*b - 2*a^2*b^3 + 3*a*(a^4 - 9*a^2*b^2 - 2*b^4)*Cos[c + d*x] + b*(a^4 - 10*a^2*b^2 - 6*b^4)*Cos[c + d*x]^2)*Sin[c + d*x])/((a - b)^3*(a + b)^3*(a + b*Cos[c + d*x])^3))/(6*d)","A",1
482,1,164,192,1.0603827,"\int \frac{\cos (c+d x)}{(a+b \cos (c+d x))^4} \, dx","Integrate[Cos[c + d*x]/(a + b*Cos[c + d*x])^4,x]","\frac{\frac{\sin (c+d x) \left(6 a^5+10 a^3 b^2+a b^2 \left(2 a^2+13 b^2\right) \cos ^2(c+d x)-3 b \left(-2 a^4-9 a^2 b^2+b^4\right) \cos (c+d x)-a b^4\right)}{(a-b)^3 (a+b)^3 (a+b \cos (c+d x))^3}-\frac{6 b \left(4 a^2+b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{7/2}}}{6 d}","-\frac{b \left(4 a^2+b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac{a \left(2 a^2+13 b^2\right) \sin (c+d x)}{6 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{\left(2 a^2+3 b^2\right) \sin (c+d x)}{6 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{a \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}",1,"((-6*b*(4*a^2 + b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) + ((6*a^5 + 10*a^3*b^2 - a*b^4 - 3*b*(-2*a^4 - 9*a^2*b^2 + b^4)*Cos[c + d*x] + a*b^2*(2*a^2 + 13*b^2)*Cos[c + d*x]^2)*Sin[c + d*x])/((a - b)^3*(a + b)^3*(a + b*Cos[c + d*x])^3))/(6*d)","A",1
483,1,159,184,0.9171883,"\int \frac{1}{(a+b \cos (c+d x))^4} \, dx","Integrate[(a + b*Cos[c + d*x])^(-4),x]","\frac{\frac{6 a \left(2 a^2+3 b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{7/2}}-\frac{b \sin (c+d x) \left(18 a^4+b^2 \left(11 a^2+4 b^2\right) \cos ^2(c+d x)+3 a b \left(9 a^2+b^2\right) \cos (c+d x)-5 a^2 b^2+2 b^4\right)}{(a-b)^3 (a+b)^3 (a+b \cos (c+d x))^3}}{6 d}","\frac{a \left(2 a^2+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{b \left(11 a^2+4 b^2\right) \sin (c+d x)}{6 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{5 a b \sin (c+d x)}{6 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{b \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}",1,"((6*a*(2*a^2 + 3*b^2)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) - (b*(18*a^4 - 5*a^2*b^2 + 2*b^4 + 3*a*b*(9*a^2 + b^2)*Cos[c + d*x] + b^2*(11*a^2 + 4*b^2)*Cos[c + d*x]^2)*Sin[c + d*x])/((a - b)^3*(a + b)^3*(a + b*Cos[c + d*x])^3))/(6*d)","A",1
484,1,274,251,3.0728983,"\int \frac{\sec (c+d x)}{(a+b \cos (c+d x))^4} \, dx","Integrate[Sec[c + d*x]/(a + b*Cos[c + d*x])^4,x]","\frac{\frac{2 a^3 b^2 \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^3}+\frac{a^2 b^2 \left(8 a^2-3 b^2\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))^2}+\frac{a b^2 \left(26 a^4-17 a^2 b^2+6 b^4\right) \sin (c+d x)}{(a-b)^3 (a+b)^3 (a+b \cos (c+d x))}+\frac{6 b \left(-8 a^6+8 a^4 b^2-7 a^2 b^4+2 b^6\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{7/2}}-6 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{6 a^4 d}","\frac{\tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac{b^2 \left(8 a^2-3 b^2\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^2 \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}-\frac{b \left(8 a^6-8 a^4 b^2+7 a^2 b^4-2 b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{b^2 \left(26 a^4-17 a^2 b^2+6 b^4\right) \sin (c+d x)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}",1,"((6*b*(-8*a^6 + 8*a^4*b^2 - 7*a^2*b^4 + 2*b^6)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) - 6*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*a^3*b^2*Sin[c + d*x])/((a - b)*(a + b)*(a + b*Cos[c + d*x])^3) + (a^2*b^2*(8*a^2 - 3*b^2)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Cos[c + d*x])^2) + (a*b^2*(26*a^4 - 17*a^2*b^2 + 6*b^4)*Sin[c + d*x])/((a - b)^3*(a + b)^3*(a + b*Cos[c + d*x])))/(6*a^4*d)","A",1
485,1,416,308,6.2261634,"\int \frac{\sec ^2(c+d x)}{(a+b \cos (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Cos[c + d*x])^4,x]","\frac{4 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a^5 d}-\frac{4 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^5 d}+\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{a^4 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{a^4 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{b^3 \sin (c+d x)}{3 a^2 d (a-b) (a+b) (a+b \cos (c+d x))^3}+\frac{-47 a^4 b^3 \sin (c+d x)+50 a^2 b^5 \sin (c+d x)-18 b^7 \sin (c+d x)}{6 a^4 d (a-b)^3 (a+b)^3 (a+b \cos (c+d x))}+\frac{6 b^5 \sin (c+d x)-11 a^2 b^3 \sin (c+d x)}{6 a^3 d (a-b)^2 (a+b)^2 (a+b \cos (c+d x))^2}-\frac{b^2 \left(20 a^6-35 a^4 b^2+28 a^2 b^4-8 b^6\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{a^5 d \left(a^2-b^2\right)^3 \sqrt{b^2-a^2}}","-\frac{4 b \tanh ^{-1}(\sin (c+d x))}{a^5 d}+\frac{b^2 \left(9 a^2-4 b^2\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^2 \tan (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^3}+\frac{\left(6 a^6-65 a^4 b^2+68 a^2 b^4-24 b^6\right) \tan (c+d x)}{6 a^4 d \left(a^2-b^2\right)^3}+\frac{b^2 \left(12 a^4-11 a^2 b^2+4 b^4\right) \tan (c+d x)}{2 a^3 d \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{b^2 \left(20 a^6-35 a^4 b^2+28 a^2 b^4-8 b^6\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{7/2} (a+b)^{7/2}}",1,"-((b^2*(20*a^6 - 35*a^4*b^2 + 28*a^2*b^4 - 8*b^6)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(a^5*(a^2 - b^2)^3*Sqrt[-a^2 + b^2]*d)) + (4*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(a^5*d) - (4*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(a^5*d) + Sin[(c + d*x)/2]/(a^4*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + Sin[(c + d*x)/2]/(a^4*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) - (b^3*Sin[c + d*x])/(3*a^2*(a - b)*(a + b)*d*(a + b*Cos[c + d*x])^3) + (-11*a^2*b^3*Sin[c + d*x] + 6*b^5*Sin[c + d*x])/(6*a^3*(a - b)^2*(a + b)^2*d*(a + b*Cos[c + d*x])^2) + (-47*a^4*b^3*Sin[c + d*x] + 50*a^2*b^5*Sin[c + d*x] - 18*b^7*Sin[c + d*x])/(6*a^4*(a - b)^3*(a + b)^3*d*(a + b*Cos[c + d*x]))","A",1
486,1,214,264,1.1413946,"\int \cos ^3(c+d x) \sqrt{a+b \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]],x]","\frac{-4 \left(8 a^4+17 a^2 b^2-25 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+b \sin (c+d x) \left(-16 a^3+\left(145 b^3-4 a^2 b\right) \cos (c+d x)+36 a b^2 \cos (2 (c+d x))+136 a b^2+15 b^3 \cos (3 (c+d x))\right)+4 a \left(8 a^3+8 a^2 b+19 a b^2+19 b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{210 b^3 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(8 a^2+25 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b^2 d}+\frac{2 a \left(8 a^2+19 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 \left(8 a^4+17 a^2 b^2-25 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{8 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b^2 d}+\frac{2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{3/2}}{7 b d}",1,"(4*a*(8*a^3 + 8*a^2*b + 19*a*b^2 + 19*b^3)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 4*(8*a^4 + 17*a^2*b^2 - 25*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + b*(-16*a^3 + 136*a*b^2 + (-4*a^2*b + 145*b^3)*Cos[c + d*x] + 36*a*b^2*Cos[2*(c + d*x)] + 15*b^3*Cos[3*(c + d*x)])*Sin[c + d*x])/(210*b^3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
487,1,180,207,0.8607438,"\int \cos ^2(c+d x) \sqrt{a+b \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]],x]","\frac{b \sin (c+d x) \left(2 a^2+8 a b \cos (c+d x)+3 b^2 \cos (2 (c+d x))+3 b^2\right)+4 a \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-2 \left(2 a^3+2 a^2 b-9 a b^2-9 b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{a+b \cos (c+d x)}}","\frac{4 a \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(2 a^2-9 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 b d}-\frac{4 a \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b d}",1,"(-2*(2*a^3 + 2*a^2*b - 9*a*b^2 - 9*b^3)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + 4*a*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + b*(2*a^2 + 3*b^2 + 8*a*b*Cos[c + d*x] + 3*b^2*Cos[2*(c + d*x)])*Sin[c + d*x])/(15*b^2*d*Sqrt[a + b*Cos[c + d*x]])","A",1
488,1,137,162,0.5547029,"\int \cos (c+d x) \sqrt{a+b \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]],x]","\frac{-2 \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+2 b \sin (c+d x) (a+b \cos (c+d x))+2 a (a+b) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{2 a \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*a*(a + b)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 2*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + 2*b*(a + b*Cos[c + d*x])*Sin[c + d*x])/(3*b*d*Sqrt[a + b*Cos[c + d*x]])","A",1
489,1,57,57,0.0594098,"\int \sqrt{a+b \cos (c+d x)} \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","\frac{2 \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])","A",1
490,1,81,118,2.1801537,"\int \sqrt{a+b \cos (c+d x)} \sec (c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x],x]","\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+a \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)}{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)}}+\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)}}",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
491,1,307,197,7.3279534,"\int \sqrt{a+b \cos (c+d x)} \sec ^2(c+d x) \, dx","Integrate[Sqrt[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)}+\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)}{\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 b \sqrt{-\frac{1}{a+b}}}}{4 d}","\frac{\tan (c+d x) \sqrt{a+b \cos (c+d x)}}{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{\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{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)])/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*b*Sqrt[-(a + b)^(-1)]) + 4*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*d)","C",1
492,1,515,262,6.4949476,"\int \sqrt{a+b \cos (c+d x)} \sec ^3(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^3,x]","\frac{\sqrt{a+b \cos (c+d x)} \left(\frac{b \tan (c+d x)}{4 a}+\frac{1}{2} \tan (c+d x) \sec (c+d x)\right)}{d}+\frac{\frac{2 \left(8 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{2 i b^2 \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{b \cos (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left(2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right)}+\frac{8 a 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 d}","\frac{\left(4 a^2-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{b \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a d}+\frac{3 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{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{\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 - 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)*b^2*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2)))/(16*a*d) + (Sqrt[a + b*Cos[c + d*x]]*((b*Tan[c + d*x])/(4*a) + (Sec[c + d*x]*Tan[c + d*x])/2))/d","C",1
493,1,262,314,1.3570672,"\int \cos ^3(c+d x) (a+b \cos (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^3*(a + b*Cos[c + d*x])^(3/2),x]","\frac{-8 a \left(8 a^4+31 a^2 b^2-39 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+b \sin (c+d x) \left(-32 a^4+\left(1606 a b^3-8 a^3 b\right) \cos (c+d x)+916 a^2 b^2+4 \left(53 a^2 b^2+84 b^4\right) \cos (2 (c+d x))+170 a b^3 \cos (3 (c+d x))+35 b^4 \cos (4 (c+d x))+301 b^4\right)+8 \left(8 a^5+8 a^4 b+33 a^3 b^2+33 a^2 b^3+147 a b^4+147 b^5\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)}{1260 b^3 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(8 a^2+49 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b^2 d}+\frac{2 a \left(8 a^2+39 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b^2 d}-\frac{2 a \left(8 a^4+31 a^2 b^2-39 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)}{315 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(8 a^4+33 a^2 b^2+147 b^4\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{315 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b^2 d}+\frac{2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}",1,"(8*(8*a^5 + 8*a^4*b + 33*a^3*b^2 + 33*a^2*b^3 + 147*a*b^4 + 147*b^5)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 8*a*(8*a^4 + 31*a^2*b^2 - 39*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + b*(-32*a^4 + 916*a^2*b^2 + 301*b^4 + (-8*a^3*b + 1606*a*b^3)*Cos[c + d*x] + 4*(53*a^2*b^2 + 84*b^4)*Cos[2*(c + d*x)] + 170*a*b^3*Cos[3*(c + d*x)] + 35*b^4*Cos[4*(c + d*x)])*Sin[c + d*x])/(1260*b^3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
494,1,214,258,1.1537519,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(3/2),x]","\frac{4 \left(6 a^4-31 a^2 b^2+25 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+b \sin (c+d x) \left(12 a^3+b \left(108 a^2+145 b^2\right) \cos (c+d x)+78 a b^2 \cos (2 (c+d x))+178 a b^2+15 b^3 \cos (3 (c+d x))\right)-8 a \left(3 a^3+3 a^2 b-41 a b^2-41 b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{210 b^2 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(6 a^2-25 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 b d}-\frac{4 a \left(3 a^2-41 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \left(6 a^4-31 a^2 b^2+25 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}-\frac{4 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 b d}",1,"(-8*a*(3*a^3 + 3*a^2*b - 41*a*b^2 - 41*b^3)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + 4*(6*a^4 - 31*a^2*b^2 + 25*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + b*(12*a^3 + 178*a*b^2 + b*(108*a^2 + 145*b^2)*Cos[c + d*x] + 78*a*b^2*Cos[2*(c + d*x)] + 15*b^3*Cos[3*(c + d*x)])*Sin[c + d*x])/(210*b^2*d*Sqrt[a + b*Cos[c + d*x]])","A",1
495,1,174,199,0.7816659,"\int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^(3/2),x]","\frac{b \sin (c+d x) \left(4 a^2+6 a b \cos (c+d x)+b^2 \cos (2 (c+d x))+b^2\right)-2 a \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+2 \left(a^3+a^2 b+3 a b^2+3 b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{5 b d \sqrt{a+b \cos (c+d x)}}","-\frac{2 a \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{5 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2+3 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{5 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}+\frac{2 a \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}",1,"(2*(a^3 + a^2*b + 3*a*b^2 + 3*b^3)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 2*a*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + b*(4*a^2 + b^2 + 6*a*b*Cos[c + d*x] + b^2*Cos[2*(c + d*x)])*Sin[c + d*x])/(5*b*d*Sqrt[a + b*Cos[c + d*x]])","A",1
496,1,134,157,0.5460918,"\int (a+b \cos (c+d x))^{3/2} \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2),x]","\frac{-2 \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+2 b \sin (c+d x) (a+b \cos (c+d x))+8 a (a+b) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{8 a \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(8*a*(a + b)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 2*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + 2*b*(a + b*Cos[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
497,1,107,179,2.2990322,"\int (a+b \cos (c+d x))^{3/2} \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x],x]","\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b (a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+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)\right)}{d \sqrt{a+b \cos (c+d x)}}","\frac{2 a^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}+\frac{2 a 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{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)]*(b*(a + 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
498,1,363,209,11.1769631,"\int (a+b \cos (c+d x))^{3/2} \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^2,x]","\frac{4 a \tan (c+d x) \sqrt{a+b \cos (c+d x)}+b \left(-\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)}{b^2 \sqrt{-\frac{1}{a+b}}}+\frac{8 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{10 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)}{\sqrt{a+b \cos (c+d x)}}\right)}{4 d}","\frac{\left(a^2+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)}{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}}}+\frac{3 a b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}",1,"(b*((8*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]] + (10*a*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)])))/(b^2*Sqrt[-(a + b)^(-1)])) + 4*a*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*d)","C",1
499,1,508,255,6.4526091,"\int (a+b \cos (c+d x))^{3/2} \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3,x]","\frac{\sqrt{a+b \cos (c+d x)} \left(\frac{1}{2} a \tan (c+d x) \sec (c+d x)+\frac{5}{4} b \tan (c+d x)\right)}{d}+\frac{\frac{2 \left(8 a^2+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{10 i b^2 \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{b \cos (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left(2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right)}+\frac{8 a 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+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)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{5 b \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{7 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{5 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 \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 + 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]] + ((10*I)*b^2*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2)))/(16*d) + (Sqrt[a + b*Cos[c + d*x]]*((5*b*Tan[c + d*x])/4 + (a*Sec[c + d*x]*Tan[c + d*x])/2))/d","C",1
500,1,268,371,1.2057636,"\int \cos ^3(c+d x) (a+b \cos (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^3*(a + b*Cos[c + d*x])^(5/2),x]","\frac{16 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \left(b \left(2 a^4 b+663 a^2 b^3+135 b^5\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+a \left(8 a^4+51 a^2 b^2+741 b^4\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 \left(\left(64 a^4-3732 a^2 b^2-2610 b^4\right) \sin (c+d x)-b \left(4 \left(6 a^3+619 a b^2\right) \sin (2 (c+d x))+b \left(\left(452 a^2+513 b^2\right) \sin (3 (c+d x))+7 b (46 a \sin (4 (c+d x))+9 b \sin (5 (c+d x)))\right)\right)\right) (a+b \cos (c+d x))}{5544 b^3 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(8 a^2+81 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{693 b^2 d}+\frac{2 a \left(8 a^2+67 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{693 b^2 d}+\frac{2 \left(8 a^4+57 a^2 b^2+135 b^4\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{693 b^2 d}+\frac{2 a \left(8 a^4+51 a^2 b^2+741 b^4\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{693 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 \left(8 a^6+49 a^4 b^2+78 a^2 b^4-135 b^6\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{693 b^3 d \sqrt{a+b \cos (c+d x)}}-\frac{8 a \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{99 b^2 d}+\frac{2 \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}",1,"(16*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b*(2*a^4*b + 663*a^2*b^3 + 135*b^5)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + a*(8*a^4 + 51*a^2*b^2 + 741*b^4)*((a + b)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) - b*(a + b*Cos[c + d*x])*((64*a^4 - 3732*a^2*b^2 - 2610*b^4)*Sin[c + d*x] - b*(4*(6*a^3 + 619*a*b^2)*Sin[2*(c + d*x)] + b*((452*a^2 + 513*b^2)*Sin[3*(c + d*x)] + 7*b*(46*a*Sin[4*(c + d*x)] + 9*b*Sin[5*(c + d*x)])))))/(5544*b^3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
501,1,263,308,1.3784537,"\int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2),x]","\frac{16 a \left(5 a^4-62 a^2 b^2+57 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+b \sin (c+d x) \left(40 a^4+4 a b \left(160 a^2+619 b^2\right) \cos (c+d x)+1984 a^2 b^2+8 \left(85 a^2 b^2+42 b^4\right) \cos (2 (c+d x))+260 a b^3 \cos (3 (c+d x))+35 b^4 \cos (4 (c+d x))+301 b^4\right)-8 \left(10 a^5+10 a^4 b-279 a^3 b^2-279 a^2 b^3-147 a b^4-147 b^5\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)}{1260 b^2 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(10 a^2-49 b^2\right) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{315 b d}-\frac{4 a \left(5 a^2-57 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 b d}+\frac{4 a \left(5 a^4-62 a^2 b^2+57 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)}{315 b^2 d \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(10 a^4-279 a^2 b^2-147 b^4\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 \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}-\frac{4 a \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{63 b d}",1,"(-8*(10*a^5 + 10*a^4*b - 279*a^3*b^2 - 279*a^2*b^3 - 147*a*b^4 - 147*b^5)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + 16*a*(5*a^4 - 62*a^2*b^2 + 57*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + b*(40*a^4 + 1984*a^2*b^2 + 301*b^4 + 4*a*b*(160*a^2 + 619*b^2)*Cos[c + d*x] + 8*(85*a^2*b^2 + 42*b^4)*Cos[2*(c + d*x)] + 260*a*b^3*Cos[3*(c + d*x)] + 35*b^4*Cos[4*(c + d*x)])*Sin[c + d*x])/(1260*b^2*d*Sqrt[a + b*Cos[c + d*x]])","A",1
502,1,214,249,0.8826176,"\int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2),x]","\frac{-4 \left(3 a^4+2 a^2 b^2-5 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+b \sin (c+d x) \left(36 a^3+b \left(72 a^2+29 b^2\right) \cos (c+d x)+24 a b^2 \cos (2 (c+d x))+44 a b^2+3 b^3 \cos (3 (c+d x))\right)+4 a \left(3 a^3+3 a^2 b+29 a b^2+29 b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{42 b d \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(3 a^2+5 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{21 d}+\frac{2 a \left(3 a^2+29 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{21 b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 \left(3 a^4+2 a^2 b^2-5 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)}{21 b d \sqrt{a+b \cos (c+d x)}}+\frac{2 \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}+\frac{2 a \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{7 d}",1,"(4*a*(3*a^3 + 3*a^2*b + 29*a*b^2 + 29*b^3)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 4*(3*a^4 + 2*a^2*b^2 - 5*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + b*(36*a^3 + 44*a*b^2 + b*(72*a^2 + 29*b^2)*Cos[c + d*x] + 24*a*b^2*Cos[2*(c + d*x)] + 3*b^3*Cos[3*(c + d*x)])*Sin[c + d*x])/(42*b*d*Sqrt[a + b*Cos[c + d*x]])","A",1
503,1,177,197,0.7741526,"\int (a+b \cos (c+d x))^{5/2} \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2),x]","\frac{b \sin (c+d x) \left(22 a^2+28 a b \cos (c+d x)+3 b^2 \cos (2 (c+d x))+3 b^2\right)-16 a \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+2 \left(23 a^3+23 a^2 b+9 a b^2+9 b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{a+b \cos (c+d x)}}","-\frac{16 a \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(23 a^2+9 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}+\frac{16 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}",1,"(2*(23*a^3 + 23*a^2*b + 9*a*b^2 + 9*b^3)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 16*a*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + b*(22*a^2 + 3*b^2 + 28*a*b*Cos[c + d*x] + 3*b^2*Cos[2*(c + d*x)])*Sin[c + d*x])/(15*d*Sqrt[a + b*Cos[c + d*x]])","A",1
504,1,379,222,1.7613538,"\int (a+b \cos (c+d x))^{5/2} \sec (c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x],x]","\frac{\frac{4 b \left(9 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)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 a \left(6 a^2+7 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 b^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}+\frac{14 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)}{\sqrt{-\frac{1}{a+b}}}}{6 d}","\frac{2 a^3 \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 \left(2 a^2+b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 b^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{14 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}}}",1,"((4*b*(9*a^2 + b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*a*(6*a^2 + 7*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]] + ((14*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)])))/Sqrt[-(a + b)^(-1)] + 4*b^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(6*d)","C",1
505,1,390,222,2.1940314,"\int (a+b \cos (c+d x))^{5/2} \sec ^2(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^2,x]","\frac{\frac{2 b \left(9 a^2+2 b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 i \left(a^2-2 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}}}+4 a^2 \tan (c+d x) \sqrt{a+b \cos (c+d x)}+\frac{24 a b^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{4 d}","\frac{a \left(a^2+4 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{\left(a^2-2 b^2\right) \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 \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{d}+\frac{5 a^2 b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}",1,"((24*a*b^2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*b*(9*a^2 + 2*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)*(a^2 - 2*b^2)*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^2*Sqrt[a + b*Cos[c + d*x]]*Tan[c + d*x])/(4*d)","C",1
506,1,395,270,2.6461077,"\int (a+b \cos (c+d x))^{5/2} \sec ^3(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^3,x]","\frac{\frac{4 b \left(a^2+4 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{a \left(8 a^2+21 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)}}+2 a \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)} (2 a+9 b \cos (c+d x))-\frac{9 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)}{\sqrt{-\frac{1}{a+b}}}}{8 d}","\frac{b \left(11 a^2+8 b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{a \left(4 a^2+15 b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 d \sqrt{a+b \cos (c+d x)}}+\frac{a^2 \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d}+\frac{9 a b \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{9 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}}}",1,"((4*b*(a^2 + 4*b^2)*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]] + (a*(8*a^2 + 21*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]] - ((9*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)])))/Sqrt[-(a + b)^(-1)] + 2*a*Sqrt[a + b*Cos[c + d*x]]*(2*a + 9*b*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(8*d)","C",1
507,1,434,323,4.1226548,"\int (a+b \cos (c+d x))^{5/2} \sec ^4(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^4,x]","\frac{\frac{2 b \left(104 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)}}+4 \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)} \left(\left(8 a^2+\frac{33 b^2}{2}\right) \sin (2 (c+d x))+8 a^2 \tan (c+d x)+26 a b \sin (c+d x)\right)-\frac{2 i \left(16 a^2+33 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(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{104 a b^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}}{96 d}","\frac{\left(16 a^2+33 b^2\right) \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{a \left(16 a^2+59 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+33 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{5 b \left(4 a^2+b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{8 d \sqrt{a+b \cos (c+d x)}}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{13 a b \tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{12 d}",1,"((104*a*b^2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*b*(104*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)*(16*a^2 + 33*b^2)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Cos[c + d*x]))/(a - b))]*Csc[c + d*x]*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b*Sqrt[-(a + b)^(-1)]) + 4*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*(26*a*b*Sin[c + d*x] + (8*a^2 + (33*b^2)/2)*Sin[2*(c + d*x)] + 8*a^2*Tan[c + d*x]))/(96*d)","C",1
508,1,211,246,1.0979775,"\int (a+b \cos (c+d x))^{7/2} \, dx","Integrate[(a + b*Cos[c + d*x])^(7/2),x]","\frac{-4 \left(71 a^4-46 a^2 b^2-25 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+b \sin (c+d x) \left(488 a^3+b \left(752 a^2+145 b^2\right) \cos (c+d x)+162 a b^2 \cos (2 (c+d x))+262 a b^2+15 b^3 \cos (3 (c+d x))\right)+64 a \left(11 a^3+11 a^2 b+13 a b^2+13 b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{210 d \sqrt{a+b \cos (c+d x)}}","\frac{2 b \left(71 a^2+25 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 d}+\frac{32 a \left(11 a^2+13 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 \left(71 a^4-46 a^2 b^2-25 b^4\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{105 d \sqrt{a+b \cos (c+d x)}}+\frac{2 b \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}+\frac{24 a b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{35 d}",1,"(64*a*(11*a^3 + 11*a^2*b + 13*a*b^2 + 13*b^3)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 4*(71*a^4 - 46*a^2*b^2 - 25*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + b*(488*a^3 + 262*a*b^2 + b*(752*a^2 + 145*b^2)*Cos[c + d*x] + 162*a*b^2*Cos[2*(c + d*x)] + 15*b^3*Cos[3*(c + d*x)])*Sin[c + d*x])/(210*d*Sqrt[a + b*Cos[c + d*x]])","A",1
509,1,92,138,0.2450929,"\int \cos ^3(c+d x) \sqrt{3+4 \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^3*Sqrt[3 + 4*Cos[c + d*x]],x]","\frac{59 \sqrt{7} F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)+141 \sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)+(212 \sin (c+d x)+9 \sin (2 (c+d x))+30 \sin (3 (c+d x))) \sqrt{4 \cos (c+d x)+3}}{420 d}","\frac{59 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{60 \sqrt{7} d}+\frac{47 E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{20 \sqrt{7} d}+\frac{\sin (c+d x) \cos (c+d x) (4 \cos (c+d x)+3)^{3/2}}{14 d}-\frac{3 \sin (c+d x) (4 \cos (c+d x)+3)^{3/2}}{70 d}+\frac{59 \sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{105 d}",1,"(141*Sqrt[7]*EllipticE[(c + d*x)/2, 8/7] + 59*Sqrt[7]*EllipticF[(c + d*x)/2, 8/7] + Sqrt[3 + 4*Cos[c + d*x]]*(212*Sin[c + d*x] + 9*Sin[2*(c + d*x)] + 30*Sin[3*(c + d*x)]))/(420*d)","A",1
510,1,81,105,0.1401448,"\int \cos ^2(c+d x) \sqrt{3+4 \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^2*Sqrt[3 + 4*Cos[c + d*x]],x]","\frac{-\sqrt{7} F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)+21 \sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)+2 (\sin (c+d x)+2 \sin (2 (c+d x))) \sqrt{4 \cos (c+d x)+3}}{20 d}","-\frac{\sqrt{7} F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{20 d}+\frac{21 \sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{20 d}+\frac{\sin (c+d x) (4 \cos (c+d x)+3)^{3/2}}{10 d}-\frac{\sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{5 d}",1,"(21*Sqrt[7]*EllipticE[(c + d*x)/2, 8/7] - Sqrt[7]*EllipticF[(c + d*x)/2, 8/7] + 2*Sqrt[3 + 4*Cos[c + d*x]]*(Sin[c + d*x] + 2*Sin[2*(c + d*x)]))/(20*d)","A",1
511,1,69,78,0.0630851,"\int \cos (c+d x) \sqrt{3+4 \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]*Sqrt[3 + 4*Cos[c + d*x]],x]","\frac{\sqrt{7} F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)+3 \sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)+4 \sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{6 d}","\frac{\sqrt{7} F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{6 d}+\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{2 d}+\frac{2 \sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{3 d}",1,"(3*Sqrt[7]*EllipticE[(c + d*x)/2, 8/7] + Sqrt[7]*EllipticF[(c + d*x)/2, 8/7] + 4*Sqrt[3 + 4*Cos[c + d*x]]*Sin[c + d*x])/(6*d)","A",1
512,1,23,23,0.0241355,"\int \sqrt{3+4 \cos (c+d x)} \, dx","Integrate[Sqrt[3 + 4*Cos[c + d*x]],x]","\frac{2 \sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{d}","\frac{2 \sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{d}",1,"(2*Sqrt[7]*EllipticE[(c + d*x)/2, 8/7])/d","A",1
513,1,41,48,0.0510011,"\int \sqrt{3+4 \cos (c+d x)} \sec (c+d x) \, dx","Integrate[Sqrt[3 + 4*Cos[c + d*x]]*Sec[c + d*x],x]","\frac{8 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)+6 \Pi \left(2;\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}","\frac{8 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}+\frac{6 \Pi \left(2;\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}",1,"(8*EllipticF[(c + d*x)/2, 8/7] + 6*EllipticPi[2, (c + d*x)/2, 8/7])/(Sqrt[7]*d)","A",1
514,1,157,95,1.121119,"\int \sqrt{3+4 \cos (c+d x)} \sec ^2(c+d x) \, dx","Integrate[Sqrt[3 + 4*Cos[c + d*x]]*Sec[c + d*x]^2,x]","\frac{6 \sqrt{7} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{8}{7}\right)+21 \sqrt{4 \cos (c+d x)+3} \tan (c+d x)+\frac{i \sqrt{7} \sin (c+d x) \left(-12 F\left(i \sinh ^{-1}\left(\sqrt{4 \cos (c+d x)+3}\right)|-\frac{1}{7}\right)+21 E\left(i \sinh ^{-1}\left(\sqrt{4 \cos (c+d x)+3}\right)|-\frac{1}{7}\right)-8 \Pi \left(-\frac{1}{3};i \sinh ^{-1}\left(\sqrt{4 \cos (c+d x)+3}\right)|-\frac{1}{7}\right)\right)}{\sqrt{\sin ^2(c+d x)}}}{21 d}","\frac{3 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{d}+\frac{4 \Pi \left(2;\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}+\frac{\sqrt{4 \cos (c+d x)+3} \tan (c+d x)}{d}",1,"(6*Sqrt[7]*EllipticPi[2, (c + d*x)/2, 8/7] + (I*Sqrt[7]*(21*EllipticE[I*ArcSinh[Sqrt[3 + 4*Cos[c + d*x]]], -1/7] - 12*EllipticF[I*ArcSinh[Sqrt[3 + 4*Cos[c + d*x]]], -1/7] - 8*EllipticPi[-1/3, I*ArcSinh[Sqrt[3 + 4*Cos[c + d*x]]], -1/7])*Sin[c + d*x])/Sqrt[Sin[c + d*x]^2] + 21*Sqrt[3 + 4*Cos[c + d*x]]*Tan[c + d*x])/(21*d)","C",1
515,1,194,135,1.2675806,"\int \sqrt{3+4 \cos (c+d x)} \sec ^3(c+d x) \, dx","Integrate[Sqrt[3 + 4*Cos[c + d*x]]*Sec[c + d*x]^3,x]","\frac{\frac{12 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7}}+\frac{6 \Pi \left(2;\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7}}+(2 \cos (c+d x)+3) \sqrt{4 \cos (c+d x)+3} \tan (c+d x) \sec (c+d x)+\frac{2 i \sin (c+d x) \left(-12 F\left(i \sinh ^{-1}\left(\sqrt{4 \cos (c+d x)+3}\right)|-\frac{1}{7}\right)+21 E\left(i \sinh ^{-1}\left(\sqrt{4 \cos (c+d x)+3}\right)|-\frac{1}{7}\right)-8 \Pi \left(-\frac{1}{3};i \sinh ^{-1}\left(\sqrt{4 \cos (c+d x)+3}\right)|-\frac{1}{7}\right)\right)}{3 \sqrt{7} \sqrt{\sin ^2(c+d x)}}}{6 d}","\frac{3 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{3 d}+\frac{5 \Pi \left(2;\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{3 \sqrt{7} d}+\frac{\sqrt{4 \cos (c+d x)+3} \tan (c+d x)}{3 d}+\frac{\sqrt{4 \cos (c+d x)+3} \tan (c+d x) \sec (c+d x)}{2 d}",1,"((12*EllipticF[(c + d*x)/2, 8/7])/Sqrt[7] + (6*EllipticPi[2, (c + d*x)/2, 8/7])/Sqrt[7] + (((2*I)/3)*(21*EllipticE[I*ArcSinh[Sqrt[3 + 4*Cos[c + d*x]]], -1/7] - 12*EllipticF[I*ArcSinh[Sqrt[3 + 4*Cos[c + d*x]]], -1/7] - 8*EllipticPi[-1/3, I*ArcSinh[Sqrt[3 + 4*Cos[c + d*x]]], -1/7])*Sin[c + d*x])/(Sqrt[7]*Sqrt[Sin[c + d*x]^2]) + (3 + 2*Cos[c + d*x])*Sqrt[3 + 4*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(6*d)","C",1
516,1,114,140,0.2081354,"\int \sqrt{3-4 \cos (c+d x)} \cos ^3(c+d x) \, dx","Integrate[Sqrt[3 - 4*Cos[c + d*x]]*Cos[c + d*x]^3,x]","\frac{654 \sin (c+d x)-511 \sin (2 (c+d x))+108 \sin (3 (c+d x))-60 \sin (4 (c+d x))-413 \sqrt{4 \cos (c+d x)-3} F\left(\left.\frac{1}{2} (c+d x)\right|8\right)+141 \sqrt{4 \cos (c+d x)-3} E\left(\left.\frac{1}{2} (c+d x)\right|8\right)}{420 d \sqrt{3-4 \cos (c+d x)}}","-\frac{59 F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{60 \sqrt{7} d}-\frac{47 E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{20 \sqrt{7} d}-\frac{\sin (c+d x) \cos (c+d x) (3-4 \cos (c+d x))^{3/2}}{14 d}-\frac{3 \sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{70 d}+\frac{59 \sin (c+d x) \sqrt{3-4 \cos (c+d x)}}{105 d}",1,"(141*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 8] - 413*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 8] + 654*Sin[c + d*x] - 511*Sin[2*(c + d*x)] + 108*Sin[3*(c + d*x)] - 60*Sin[4*(c + d*x)])/(420*d*Sqrt[3 - 4*Cos[c + d*x]])","A",1
517,1,104,107,0.2070753,"\int \sqrt{3-4 \cos (c+d x)} \cos ^2(c+d x) \, dx","Integrate[Sqrt[3 - 4*Cos[c + d*x]]*Cos[c + d*x]^2,x]","-\frac{14 \sin (c+d x)-16 \sin (2 (c+d x))+8 \sin (3 (c+d x))+7 \sqrt{4 \cos (c+d x)-3} F\left(\left.\frac{1}{2} (c+d x)\right|8\right)+21 \sqrt{4 \cos (c+d x)-3} E\left(\left.\frac{1}{2} (c+d x)\right|8\right)}{20 d \sqrt{3-4 \cos (c+d x)}}","-\frac{\sqrt{7} F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{20 d}+\frac{21 \sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{20 d}-\frac{\sin (c+d x) (3-4 \cos (c+d x))^{3/2}}{10 d}+\frac{\sin (c+d x) \sqrt{3-4 \cos (c+d x)}}{5 d}",1,"-1/20*(21*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 8] + 7*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 8] + 14*Sin[c + d*x] - 16*Sin[2*(c + d*x)] + 8*Sin[3*(c + d*x)])/(d*Sqrt[3 - 4*Cos[c + d*x]])","A",1
518,1,94,80,0.0965061,"\int \sqrt{3-4 \cos (c+d x)} \cos (c+d x) \, dx","Integrate[Sqrt[3 - 4*Cos[c + d*x]]*Cos[c + d*x],x]","\frac{12 \sin (c+d x)-8 \sin (2 (c+d x))-7 \sqrt{4 \cos (c+d x)-3} F\left(\left.\frac{1}{2} (c+d x)\right|8\right)+3 \sqrt{4 \cos (c+d x)-3} E\left(\left.\frac{1}{2} (c+d x)\right|8\right)}{6 d \sqrt{3-4 \cos (c+d x)}}","-\frac{\sqrt{7} F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{6 d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{2 d}+\frac{2 \sin (c+d x) \sqrt{3-4 \cos (c+d x)}}{3 d}",1,"(3*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 8] - 7*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 8] + 12*Sin[c + d*x] - 8*Sin[2*(c + d*x)])/(6*d*Sqrt[3 - 4*Cos[c + d*x]])","A",1
519,1,44,24,0.0313599,"\int \sqrt{3-4 \cos (c+d x)} \, dx","Integrate[Sqrt[3 - 4*Cos[c + d*x]],x]","-\frac{2 \sqrt{4 \cos (c+d x)-3} E\left(\left.\frac{1}{2} (c+d x)\right|8\right)}{d \sqrt{3-4 \cos (c+d x)}}","\frac{2 \sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{d}",1,"(-2*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 8])/(d*Sqrt[3 - 4*Cos[c + d*x]])","A",1
520,1,61,50,0.0617052,"\int \sqrt{3-4 \cos (c+d x)} \sec (c+d x) \, dx","Integrate[Sqrt[3 - 4*Cos[c + d*x]]*Sec[c + d*x],x]","\frac{2 \sqrt{4 \cos (c+d x)-3} \left(3 \Pi \left(2;\left.\frac{1}{2} (c+d x)\right|8\right)-4 F\left(\left.\frac{1}{2} (c+d x)\right|8\right)\right)}{d \sqrt{3-4 \cos (c+d x)}}","-\frac{8 F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{\sqrt{7} d}-\frac{6 \Pi \left(2;\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{\sqrt{7} d}",1,"(2*Sqrt[-3 + 4*Cos[c + d*x]]*(-4*EllipticF[(c + d*x)/2, 8] + 3*EllipticPi[2, (c + d*x)/2, 8]))/(d*Sqrt[3 - 4*Cos[c + d*x]])","A",1
521,1,178,98,1.4671591,"\int \sqrt{3-4 \cos (c+d x)} \sec ^2(c+d x) \, dx","Integrate[Sqrt[3 - 4*Cos[c + d*x]]*Sec[c + d*x]^2,x]","\frac{21 \sqrt{3-4 \cos (c+d x)} \tan (c+d x)-\frac{42 \sqrt{4 \cos (c+d x)-3} \Pi \left(2;\left.\frac{1}{2} (c+d x)\right|8\right)}{\sqrt{3-4 \cos (c+d x)}}-\frac{i \sqrt{7} \sin (c+d x) \left(-12 F\left(i \sinh ^{-1}\left(\sqrt{3-4 \cos (c+d x)}\right)|-\frac{1}{7}\right)+21 E\left(i \sinh ^{-1}\left(\sqrt{3-4 \cos (c+d x)}\right)|-\frac{1}{7}\right)-8 \Pi \left(-\frac{1}{3};i \sinh ^{-1}\left(\sqrt{3-4 \cos (c+d x)}\right)|-\frac{1}{7}\right)\right)}{\sqrt{\sin ^2(c+d x)}}}{21 d}","\frac{3 F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{\sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{d}+\frac{4 \Pi \left(2;\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{\sqrt{7} d}+\frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x)}{d}",1,"((-42*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticPi[2, (c + d*x)/2, 8])/Sqrt[3 - 4*Cos[c + d*x]] - (I*Sqrt[7]*(21*EllipticE[I*ArcSinh[Sqrt[3 - 4*Cos[c + d*x]]], -1/7] - 12*EllipticF[I*ArcSinh[Sqrt[3 - 4*Cos[c + d*x]]], -1/7] - 8*EllipticPi[-1/3, I*ArcSinh[Sqrt[3 - 4*Cos[c + d*x]]], -1/7])*Sin[c + d*x])/Sqrt[Sin[c + d*x]^2] + 21*Sqrt[3 - 4*Cos[c + d*x]]*Tan[c + d*x])/(21*d)","C",1
522,1,237,138,1.8839461,"\int \sqrt{3-4 \cos (c+d x)} \sec ^3(c+d x) \, dx","Integrate[Sqrt[3 - 4*Cos[c + d*x]]*Sec[c + d*x]^3,x]","\frac{-\frac{12 \sqrt{4 \cos (c+d x)-3} F\left(\left.\frac{1}{2} (c+d x)\right|8\right)}{\sqrt{3-4 \cos (c+d x)}}+\frac{6 \sqrt{4 \cos (c+d x)-3} \Pi \left(2;\left.\frac{1}{2} (c+d x)\right|8\right)}{\sqrt{3-4 \cos (c+d x)}}-\sqrt{3-4 \cos (c+d x)} (2 \cos (c+d x)-3) \tan (c+d x) \sec (c+d x)+\frac{2 i \sin (c+d x) \left(-12 F\left(i \sinh ^{-1}\left(\sqrt{3-4 \cos (c+d x)}\right)|-\frac{1}{7}\right)+21 E\left(i \sinh ^{-1}\left(\sqrt{3-4 \cos (c+d x)}\right)|-\frac{1}{7}\right)-8 \Pi \left(-\frac{1}{3};i \sinh ^{-1}\left(\sqrt{3-4 \cos (c+d x)}\right)|-\frac{1}{7}\right)\right)}{3 \sqrt{7} \sqrt{\sin ^2(c+d x)}}}{6 d}","-\frac{3 F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{\sqrt{7} d}+\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{3 d}-\frac{5 \Pi \left(2;\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{3 \sqrt{7} d}-\frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x)}{3 d}+\frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x) \sec (c+d x)}{2 d}",1,"((-12*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 8])/Sqrt[3 - 4*Cos[c + d*x]] + (6*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticPi[2, (c + d*x)/2, 8])/Sqrt[3 - 4*Cos[c + d*x]] + (((2*I)/3)*(21*EllipticE[I*ArcSinh[Sqrt[3 - 4*Cos[c + d*x]]], -1/7] - 12*EllipticF[I*ArcSinh[Sqrt[3 - 4*Cos[c + d*x]]], -1/7] - 8*EllipticPi[-1/3, I*ArcSinh[Sqrt[3 - 4*Cos[c + d*x]]], -1/7])*Sin[c + d*x])/(Sqrt[7]*Sqrt[Sin[c + d*x]^2]) - Sqrt[3 - 4*Cos[c + d*x]]*(-3 + 2*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(6*d)","C",1
523,1,182,215,0.913565,"\int \frac{\cos ^3(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]^3/Sqrt[a + b*Cos[c + d*x]],x]","\frac{b \sin (c+d x) \left(-8 a^2-2 a b \cos (c+d x)+3 b^2 \cos (2 (c+d x))+3 b^2\right)-2 a \left(8 a^2+7 b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+2 \left(8 a^3+8 a^2 b+9 a b^2+9 b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 a \left(8 a^2+7 b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(8 a^2+9 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 b^3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{8 a \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^2 d}+\frac{2 \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b d}",1,"(2*(8*a^3 + 8*a^2*b + 9*a*b^2 + 9*b^3)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 2*a*(8*a^2 + 7*b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + b*(-8*a^2 + 3*b^2 - 2*a*b*Cos[c + d*x] + 3*b^2*Cos[2*(c + d*x)])*Sin[c + d*x])/(15*b^3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
524,1,137,165,0.615715,"\int \frac{\cos ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]^2/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \left(2 a^2+b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+2 b \sin (c+d x) (a+b \cos (c+d x))-4 a (a+b) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(2 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^2 d \sqrt{a+b \cos (c+d x)}}-\frac{4 a \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b d}",1,"(-4*a*(a + b)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + 2*(2*a^2 + b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + 2*b*(a + b*Cos[c + d*x])*Sin[c + d*x])/(3*b^2*d*Sqrt[a + b*Cos[c + d*x]])","A",1
525,1,86,122,2.416077,"\int \frac{\cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[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) 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 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 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)}{b d \sqrt{a+b \cos (c+d x)}}",1,"(2*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*d*Sqrt[a + b*Cos[c + d*x]])","A",1
526,1,57,57,0.0453484,"\int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[1/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}","\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}",1,"(2*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
527,1,58,58,0.0865138,"\int \frac{\sec (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[Sec[c + d*x]/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \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 \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*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
528,1,310,206,8.4996646,"\int \frac{\sec ^2(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[Sec[c + d*x]^2/Sqrt[a + b*Cos[c + d*x]],x]","\frac{4 \tan (c+d x) \sqrt{a+b \cos (c+d x)}-\frac{6 b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \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{\tan (c+d x) \sqrt{a+b \cos (c+d x)}}{a d}+\frac{\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{\sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{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,"((-6*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)*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]]*Tan[c + d*x])/(4*a*d)","C",1
529,1,518,268,6.3966629,"\int \frac{\sec ^3(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[Sec[c + d*x]^3/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\sqrt{a+b \cos (c+d x)} \left(\frac{\tan (c+d x) \sec (c+d x)}{2 a}-\frac{3 b \tan (c+d x)}{4 a^2}\right)}{d}+\frac{\frac{2 \left(8 a^2+9 b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{6 i b^2 \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{b-b \cos (c+d x)}{a+b}} \sqrt{-\frac{b \cos (c+d x)+b}{a-b}} \left(2 a (a-b) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)+b \left(2 a F\left(i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)-b \Pi \left(\frac{a+b}{a};i \sinh ^{-1}\left(\sqrt{-\frac{1}{a+b}} \sqrt{a+b \cos (c+d x)}\right)|\frac{a+b}{a-b}\right)\right)\right)}{a \sqrt{-\frac{1}{a+b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{-\frac{a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left(2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right)}+\frac{8 a 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+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)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{3 b \tan (c+d x) \sqrt{a+b \cos (c+d x)}}{4 a^2 d}+\frac{3 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{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{\tan (c+d x) \sec (c+d x) \sqrt{a+b \cos (c+d x)}}{2 a 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 + 9*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]] - ((6*I)*b^2*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[-((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d*x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x]) + 2*(a + b*Cos[c + d*x])^2)))/(16*a^2*d) + (Sqrt[a + b*Cos[c + d*x]]*((-3*b*Tan[c + d*x])/(4*a^2) + (Sec[c + d*x]*Tan[c + d*x])/(2*a)))/d","C",1
530,1,242,326,1.3133349,"\int \frac{\cos ^4(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^4/(a + b*Cos[c + d*x])^(3/2),x]","\frac{-b \sin (c+d x) \left(16 a^4+4 a b \left(a^2-b^2\right) \cos (c+d x)-7 a^2 b^2+\left(b^4-a^2 b^2\right) \cos (2 (c+d x))+b^4\right)-8 a \left(4 a^4-3 a^2 b^2-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)+2 \left(16 a^5+16 a^4 b-8 a^3 b^2-8 a^2 b^3-3 a b^4-3 b^5\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)}{5 b^4 d (a-b) (a+b) \sqrt{a+b \cos (c+d x)}}","-\frac{2 a^2 \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^2\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{8 a \left(4 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)}{5 b^4 d \sqrt{a+b \cos (c+d x)}}-\frac{2 a \left(8 a^2-3 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 b^3 d \left(a^2-b^2\right)}+\frac{2 \left(16 a^4-8 a^2 b^2-3 b^4\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{5 b^4 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*(16*a^5 + 16*a^4*b - 8*a^3*b^2 - 8*a^2*b^3 - 3*a*b^4 - 3*b^5)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 8*a*(4*a^4 - 3*a^2*b^2 - b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - b*(16*a^4 - 7*a^2*b^2 + b^4 + 4*a*b*(a^2 - b^2)*Cos[c + d*x] + (-(a^2*b^2) + b^4)*Cos[2*(c + d*x)])*Sin[c + d*x])/(5*(a - b)*b^4*(a + b)*d*Sqrt[a + b*Cos[c + d*x]])","A",1
531,1,197,257,0.8811771,"\int \frac{\cos ^3(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^3/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 \left(8 a^4-7 a^2 b^2-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)-2 b \sin (c+d x) \left(-4 a^3+\left(b^3-a^2 b\right) \cos (c+d x)+a b^2\right)-2 a \left(8 a^3+8 a^2 b-5 a b^2-5 b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d (a-b) (a+b) \sqrt{a+b \cos (c+d x)}}","-\frac{2 a^2 \sin (c+d x) \cos (c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(4 a^2-b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^2 d \left(a^2-b^2\right)}+\frac{2 \left(8 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^3 d \sqrt{a+b \cos (c+d x)}}-\frac{2 a \left(8 a^2-5 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 b^3 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(-2*a*(8*a^3 + 8*a^2*b - 5*a*b^2 - 5*b^3)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + 2*(8*a^4 - 7*a^2*b^2 - b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - 2*b*(-4*a^3 + a*b^2 + (-(a^2*b) + b^3)*Cos[c + d*x])*Sin[c + d*x])/(3*(a - b)*b^3*(a + b)*d*Sqrt[a + b*Cos[c + d*x]])","A",1
532,1,159,186,0.6996084,"\int \frac{\cos ^2(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 \left(2 a^3+2 a^2 b-a b^2-b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-2 a \left(2 \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+a b \sin (c+d x)\right)}{b^2 d (a-b) (a+b) \sqrt{a+b \cos (c+d x)}}","-\frac{2 a^2 \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^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{4 a \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*(2*a^3 + 2*a^2*b - a*b^2 - b^3)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 2*a*(2*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*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
533,1,137,170,0.5275159,"\int \frac{\cos (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+2 a b \sin (c+d x)-2 a (a+b) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{b d (a-b) (a+b) \sqrt{a+b \cos (c+d x)}}","\frac{2 a \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 a \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 \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*(a + b)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + 2*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + 2*a*b*Sin[c + d*x])/((a - b)*b*(a + b)*d*Sqrt[a + b*Cos[c + d*x]])","A",1
534,1,83,106,0.2062151,"\int \frac{1}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(a + b*Cos[c + d*x])^(-3/2),x]","\frac{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)}{d (a-b) (a+b) \sqrt{a+b \cos (c+d x)}}","\frac{2 \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 b \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}",1,"(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
535,1,402,176,5.1912176,"\int \frac{\sec (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\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)}}{2 a d}","\frac{2 b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) \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)}{a d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \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,"(-(((-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
536,1,441,277,4.146873,"\int \frac{\sec ^2(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\frac{4 \tan (c+d x) \left(a^3+b \left(a^2-3 b^2\right) \cos (c+d x)-a b^2\right)}{\left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{b \left(\frac{2 \left(7 a^2-9 b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \left(a^2-3 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^2 \sqrt{-\frac{1}{a+b}}}-\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)}}\right)}{(a-b) (a+b)}}{4 a^2 d}","\frac{b \left(a^2-3 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^2-3 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{3 b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{a^2 d \sqrt{a+b \cos (c+d x)}}+\frac{\tan (c+d x)}{a d \sqrt{a+b \cos (c+d x)}}+\frac{\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,"(-((b*((-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*(7*a^2 - 9*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)*(a^2 - 3*b^2)*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Cos[c + d*x]))/(a - b))]*Csc[c + d*x]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)])))/(a*b^2*Sqrt[-(a + b)^(-1)])))/((a - b)*(a + b))) + (4*(a^3 - a*b^2 + b*(a^2 - 3*b^2)*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
537,1,597,345,6.4826397,"\int \frac{\sec ^3(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^3/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\sqrt{a+b \cos (c+d x)} \left(-\frac{7 b \tan (c+d x)}{4 a^3}+\frac{\tan (c+d x) \sec (c+d x)}{2 a^2}+\frac{2 b^4 \sin (c+d x)}{a^3 \left(a^2-b^2\right) (a+b \cos (c+d x))}\right)}{d}-\frac{\frac{2 \left(4 a^3 b-20 a b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}-\frac{2 i \left(7 a^2 b^2-15 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+29 a^2 b^2-45 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 b \tan (c+d x)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{5 b \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^2 d \sqrt{a+b \cos (c+d x)}}-\frac{b^2 \left(7 a^2-15 b^2\right) \sin (c+d x)}{4 a^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{b \left(7 a^2-15 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{4 a^3 d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\left(4 a^2+15 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{\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*b - 20*a*b^3)*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 + 29*a^2*b^2 - 45*b^4)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(7*a^2*b^2 - 15*b^4)*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]]*((2*b^4*Sin[c + d*x])/(a^3*(a^2 - b^2)*(a + b*Cos[c + d*x])) - (7*b*Tan[c + d*x])/(4*a^3) + (Sec[c + d*x]*Tan[c + d*x])/(2*a^2)))/d","C",1
538,1,272,436,1.9906662,"\int \frac{\cos ^5(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^5/(a + b*Cos[c + d*x])^(5/2),x]","\frac{b \left(\frac{10 a^5 \sin (c+d x)}{a^2-b^2}-\frac{10 a^4 \left(11 a^2-15 b^2\right) \sin (c+d x) (a+b \cos (c+d x))}{\left(a^2-b^2\right)^2}-28 a \sin (c+d x) (a+b \cos (c+d x))^2+3 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(\left(128 a^6-212 a^4 b^2+55 a^2 b^4+9 b^6\right) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+a \left(-128 a^5+128 a^4 b+116 a^3 b^2-116 a^2 b^3+17 a b^4-17 b^5\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)}{(a-b)^2}}{15 b^5 d (a+b \cos (c+d x))^{3/2}}","-\frac{2 a^2 \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{8 a^2 \left(2 a^2-3 b^2\right) \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{4 a \left(32 a^4-49 a^2 b^2+7 b^4\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^4 d \left(a^2-b^2\right)^2}-\frac{2 a \left(128 a^4-116 a^2 b^2-17 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)}{15 b^5 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(48 a^4-71 a^2 b^2+3 b^4\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \cos (c+d x)}}{15 b^3 d \left(a^2-b^2\right)^2}+\frac{2 \left(128 a^6-212 a^4 b^2+55 a^2 b^4+9 b^6\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)*((128*a^6 - 212*a^4*b^2 + 55*a^2*b^4 + 9*b^6)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + a*(-128*a^5 + 128*a^4*b + 116*a^3*b^2 - 116*a^2*b^3 + 17*a*b^4 - 17*b^5)*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/(a - b)^2 + b*((10*a^5*Sin[c + d*x])/(a^2 - b^2) - (10*a^4*(11*a^2 - 15*b^2)*(a + b*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2 - 28*a*(a + b*Cos[c + d*x])^2*Sin[c + d*x] + 3*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
539,1,237,345,1.641905,"\int \frac{\cos ^4(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^4/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(\frac{b \sin (c+d x) \left(16 a^6-25 a^4 b^2+\left(b^3-a^2 b\right)^2 \cos (2 (c+d x))+4 a b \left(5 a^4-8 a^2 b^2+b^4\right) \cos (c+d x)+b^6\right)}{2 \left(a^2-b^2\right)^2}+\frac{\left(\frac{a+b \cos (c+d x)}{a+b}\right)^{3/2} \left(\left(16 a^5-16 a^4 b-16 a^3 b^2+16 a^2 b^3-a b^4+b^5\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-4 \left(4 a^5-7 a^3 b^2+2 a b^4\right) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)}{(a-b)^2}\right)}{3 b^4 d (a+b \cos (c+d x))^{3/2}}","-\frac{2 a^2 \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^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left(a^2-b^2\right)}+\frac{2 \left(16 a^4-16 a^2 b^2-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)}{3 b^4 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{8 a \left(4 a^4-7 a^2 b^2+2 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 b^4 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{4 a^3 \left(3 a^2-5 b^2\right) \sin (c+d x)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}",1,"(2*((((a + b*Cos[c + d*x])/(a + b))^(3/2)*(-4*(4*a^5 - 7*a^3*b^2 + 2*a*b^4)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + (16*a^5 - 16*a^4*b - 16*a^3*b^2 + 16*a^2*b^3 - a*b^4 + b^5)*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/(a - b)^2 + (b*(16*a^6 - 25*a^4*b^2 + b^6 + 4*a*b*(5*a^4 - 8*a^2*b^2 + b^4)*Cos[c + d*x] + (-(a^2*b) + b^3)^2*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
540,1,188,281,1.2728574,"\int \frac{\cos ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^3/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(\frac{a^2 b \sin (c+d x) \left(-4 a^3+\left(9 b^3-5 a^2 b\right) \cos (c+d x)+8 a b^2\right)}{\left(a^2-b^2\right)^2}+\frac{\left(\frac{a+b \cos (c+d x)}{a+b}\right)^{3/2} \left(\left(8 a^4-15 a^2 b^2+3 b^4\right) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+a \left(-8 a^3+8 a^2 b+9 a b^2-9 b^3\right) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)}{(a-b)^2}\right)}{3 b^3 d (a+b \cos (c+d x))^{3/2}}","-\frac{8 a^2 \left(a^2-2 b^2\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 a^2 \sin (c+d x) \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(8 a^2-9 b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(8 a^4-15 a^2 b^2+3 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 b^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*((((a + b*Cos[c + d*x])/(a + b))^(3/2)*((8*a^4 - 15*a^2*b^2 + 3*b^4)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + a*(-8*a^3 + 8*a^2*b + 9*a*b^2 - 9*b^3)*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/(a - b)^2 + (a^2*b*(-4*a^3 + 8*a*b^2 + (-5*a^2*b + 9*b^3)*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
541,1,175,263,1.1502732,"\int \frac{\cos ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(\frac{a b \sin (c+d x) \left(a^3+2 b \left(a^2-3 b^2\right) \cos (c+d x)-5 a b^2\right)}{\left(a^2-b^2\right)^2}-\frac{\left(\frac{a+b \cos (c+d x)}{a+b}\right)^{3/2} \left(2 \left(a^3-3 a b^2\right) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+\left(-2 a^3+2 a^2 b+3 a b^2-3 b^3\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^2 \sin (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{4 a \left(a^2-3 b^2\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^2-3 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^2 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{4 a \left(a^2-3 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*(-((((a + b*Cos[c + d*x])/(a + b))^(3/2)*(2*(a^3 - 3*a*b^2)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + (-2*a^3 + 2*a^2*b + 3*a*b^2 - 3*b^3)*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/(a - b)^2) + (a*b*(a^3 - 5*a*b^2 + 2*b*(a^2 - 3*b^2)*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
542,1,154,243,1.0026107,"\int \frac{\cos (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]/(a + b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(\frac{\sin (c+d x) \left(b \left(a^2+3 b^2\right) \cos (c+d x)+2 a \left(a^2+b^2\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(a^2+3 b^2\right) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+a (b-a) 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+3 b^2\right) \sin (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 a \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\frac{2 a \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 b d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 \left(a^2+3 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 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)*((a^2 + 3*b^2)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + a*(-a + b)*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/((a - b)^2*b)) + ((2*a*(a^2 + b^2) + b*(a^2 + 3*b^2)*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2))/(3*d*(a + b*Cos[c + d*x])^(3/2))","A",1
543,1,158,221,0.9336555,"\int \frac{1}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(a + b*Cos[c + d*x])^(-5/2),x]","\frac{2 b \sin (c+d x) \left(-5 a^2-4 a b \cos (c+d x)+b^2\right)-2 (a-b) (a+b)^2 \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{3/2} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+8 a (a+b)^2 \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{3/2} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d (a-b)^2 (a+b)^2 (a+b \cos (c+d x))^{3/2}}","-\frac{8 a b \sin (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 b \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{8 a \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(8*a*(a + b)^2*((a + b*Cos[c + d*x])/(a + b))^(3/2)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 2*(a - b)*(a + b)^2*((a + b*Cos[c + d*x])/(a + b))^(3/2)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + 2*b*(-5*a^2 + b^2 - 4*a*b*Cos[c + d*x])*Sin[c + d*x])/(3*(a - b)^2*(a + b)^2*d*(a + b*Cos[c + d*x])^(3/2))","A",1
544,1,464,320,4.8006507,"\int \frac{\sec (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]/(a + b*Cos[c + d*x])^(5/2),x]","\frac{\frac{4 b^2 \sin (c+d x) \left(8 a^3+b \left(7 a^2-3 b^2\right) \cos (c+d x)-4 a b^2\right)}{\left(a^3-a b^2\right)^2 (a+b \cos (c+d x))^{3/2}}+\frac{-\frac{8 a b \left(3 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)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 i \left(3 b^2-7 a^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(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}}}+\frac{2 \left(6 a^4-19 a^2 b^2+9 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)}}}{a^2 (a-b)^2 (a+b)^2}}{6 d}","\frac{2 b^2 \left(7 a^2-3 b^2\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 b^2 \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}+\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)}{3 a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}-\frac{2 b \left(7 a^2-3 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 \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)}}",1,"(((-8*a*b*(3*a^2 - b^2)*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 - 19*a^2*b^2 + 9*b^4)*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 + 3*b^2)*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^2*(a - b)^2*(a + b)^2) + (4*b^2*(8*a^3 - 4*a*b^2 + b*(7*a^2 - 3*b^2)*Cos[c + d*x])*Sin[c + d*x])/((a^3 - a*b^2)^2*(a + b*Cos[c + d*x])^(3/2)))/(6*d)","C",1
545,1,638,380,6.5785062,"\int \frac{\sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{a+b \cos (c+d x)} \left(\frac{\tan (c+d x)}{a^3}-\frac{2 b^3 \sin (c+d x)}{3 a^2 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{4 \left(5 a^2 b^3 \sin (c+d x)-3 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{b \left(\frac{2 \left(20 a b^3-36 a^3 b\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \cos (c+d x)}}+\frac{2 \left(33 a^4-86 a^2 b^2+45 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 i \left(3 a^4-26 a^2 b^2+15 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)}\right)}{12 a^3 d (b-a)^2 (a+b)^2}","-\frac{5 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^2-5 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^2-5 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^4-26 a^2 b^2+15 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^4-26 a^2 b^2+15 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{\tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}",1,"-1/12*(b*((2*(-36*a^3*b + 20*a*b^3)*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 - 86*a^2*b^2 + 45*b^4)*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 - 26*a^2*b^2 + 15*b^4)*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)^2*(a + b)^2*d) + (Sqrt[a + b*Cos[c + d*x]]*((-2*b^3*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) - (4*(5*a^2*b^3*Sin[c + d*x] - 3*b^5*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) + Tan[c + d*x]/a^3))/d","C",1
546,1,189,282,1.4179677,"\int \frac{1}{(a+b \cos (c+d x))^{7/2}} \, dx","Integrate[(a + b*Cos[c + d*x])^(-7/2),x]","\frac{2 \left(\frac{\left(\frac{a+b \cos (c+d x)}{a+b}\right)^{5/2} \left(\left(23 a^2+9 b^2\right) E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+8 a (b-a) F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)\right)}{(a-b)^3}+\frac{b \sin (c+d x) \left(34 a^4+b^2 \left(23 a^2+9 b^2\right) \cos ^2(c+d x)+2 a b \left(27 a^2+5 b^2\right) \cos (c+d x)-5 a^2 b^2+3 b^4\right)}{\left(b^2-a^2\right)^3}\right)}{15 d (a+b \cos (c+d x))^{5/2}}","-\frac{2 b \left(23 a^2+9 b^2\right) \sin (c+d x)}{15 d \left(a^2-b^2\right)^3 \sqrt{a+b \cos (c+d x)}}-\frac{16 a b \sin (c+d x)}{15 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^{3/2}}-\frac{2 b \sin (c+d x)}{5 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{5/2}}-\frac{16 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)}{15 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(23 a^2+9 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \left(a^2-b^2\right)^3 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*((((a + b*Cos[c + d*x])/(a + b))^(5/2)*((23*a^2 + 9*b^2)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + 8*a*(-a + b)*EllipticF[(c + d*x)/2, (2*b)/(a + b)]))/(a - b)^3 + (b*(34*a^4 - 5*a^2*b^2 + 3*b^4 + 2*a*b*(27*a^2 + 5*b^2)*Cos[c + d*x] + b^2*(23*a^2 + 9*b^2)*Cos[c + d*x]^2)*Sin[c + d*x])/(-a^2 + b^2)^3))/(15*d*(a + b*Cos[c + d*x])^(5/2))","A",1
547,1,81,111,0.1838002,"\int \frac{\cos ^3(c+d x)}{\sqrt{3+4 \cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]^3/Sqrt[3 + 4*Cos[c + d*x]],x]","\frac{-23 \sqrt{7} F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)+63 \sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)+7 (\sin (2 (c+d x))-2 \sin (c+d x)) \sqrt{4 \cos (c+d x)+3}}{140 d}","-\frac{23 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{20 \sqrt{7} d}+\frac{9 \sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{20 d}+\frac{\sin (c+d x) \cos (c+d x) \sqrt{4 \cos (c+d x)+3}}{10 d}-\frac{\sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{10 d}",1,"(63*Sqrt[7]*EllipticE[(c + d*x)/2, 8/7] - 23*Sqrt[7]*EllipticF[(c + d*x)/2, 8/7] + 7*Sqrt[3 + 4*Cos[c + d*x]]*(-2*Sin[c + d*x] + Sin[2*(c + d*x)]))/(140*d)","A",1
548,1,70,78,0.0829057,"\int \frac{\cos ^2(c+d x)}{\sqrt{3+4 \cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]^2/Sqrt[3 + 4*Cos[c + d*x]],x]","\frac{17 \sqrt{7} F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)-21 \sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)+14 \sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{84 d}","\frac{17 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{12 \sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{4 d}+\frac{\sin (c+d x) \sqrt{4 \cos (c+d x)+3}}{6 d}",1,"(-21*Sqrt[7]*EllipticE[(c + d*x)/2, 8/7] + 17*Sqrt[7]*EllipticF[(c + d*x)/2, 8/7] + 14*Sqrt[3 + 4*Cos[c + d*x]]*Sin[c + d*x])/(84*d)","A",1
549,1,43,51,0.0597909,"\int \frac{\cos (c+d x)}{\sqrt{3+4 \cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]/Sqrt[3 + 4*Cos[c + d*x]],x]","\frac{7 E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)-3 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{2 \sqrt{7} d}","\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{2 d}-\frac{3 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{2 \sqrt{7} d}",1,"(7*EllipticE[(c + d*x)/2, 8/7] - 3*EllipticF[(c + d*x)/2, 8/7])/(2*Sqrt[7]*d)","A",1
550,1,23,23,0.0299911,"\int \frac{1}{\sqrt{3+4 \cos (c+d x)}} \, dx","Integrate[1/Sqrt[3 + 4*Cos[c + d*x]],x]","\frac{2 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}","\frac{2 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}",1,"(2*EllipticF[(c + d*x)/2, 8/7])/(Sqrt[7]*d)","A",1
551,1,24,24,0.0533925,"\int \frac{\sec (c+d x)}{\sqrt{3+4 \cos (c+d x)}} \, dx","Integrate[Sec[c + d*x]/Sqrt[3 + 4*Cos[c + d*x]],x]","\frac{2 \Pi \left(2;\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}","\frac{2 \Pi \left(2;\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}",1,"(2*EllipticPi[2, (c + d*x)/2, 8/7])/(Sqrt[7]*d)","A",1
552,1,158,101,1.1763667,"\int \frac{\sec ^2(c+d x)}{\sqrt{3+4 \cos (c+d x)}} \, dx","Integrate[Sec[c + d*x]^2/Sqrt[3 + 4*Cos[c + d*x]],x]","\frac{-\frac{6 \Pi \left(2;\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7}}+\sqrt{4 \cos (c+d x)+3} \tan (c+d x)+\frac{i \sin (c+d x) \left(-12 F\left(i \sinh ^{-1}\left(\sqrt{4 \cos (c+d x)+3}\right)|-\frac{1}{7}\right)+21 E\left(i \sinh ^{-1}\left(\sqrt{4 \cos (c+d x)+3}\right)|-\frac{1}{7}\right)-8 \Pi \left(-\frac{1}{3};i \sinh ^{-1}\left(\sqrt{4 \cos (c+d x)+3}\right)|-\frac{1}{7}\right)\right)}{3 \sqrt{7} \sqrt{\sin ^2(c+d x)}}}{3 d}","\frac{F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{3 d}-\frac{4 \Pi \left(2;\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{3 \sqrt{7} d}+\frac{\sqrt{4 \cos (c+d x)+3} \tan (c+d x)}{3 d}",1,"((-6*EllipticPi[2, (c + d*x)/2, 8/7])/Sqrt[7] + ((I/3)*(21*EllipticE[I*ArcSinh[Sqrt[3 + 4*Cos[c + d*x]]], -1/7] - 12*EllipticF[I*ArcSinh[Sqrt[3 + 4*Cos[c + d*x]]], -1/7] - 8*EllipticPi[-1/3, I*ArcSinh[Sqrt[3 + 4*Cos[c + d*x]]], -1/7])*Sin[c + d*x])/(Sqrt[7]*Sqrt[Sin[c + d*x]^2]) + Sqrt[3 + 4*Cos[c + d*x]]*Tan[c + d*x])/(3*d)","C",1
553,1,195,137,1.3225899,"\int \frac{\sec ^3(c+d x)}{\sqrt{3+4 \cos (c+d x)}} \, dx","Integrate[Sec[c + d*x]^3/Sqrt[3 + 4*Cos[c + d*x]],x]","\frac{\frac{4 F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7}}+\frac{18 \Pi \left(2;\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{\sqrt{7}}-(2 \cos (c+d x)-1) \sqrt{4 \cos (c+d x)+3} \tan (c+d x) \sec (c+d x)-\frac{2 i \sin (c+d x) \left(-12 F\left(i \sinh ^{-1}\left(\sqrt{4 \cos (c+d x)+3}\right)|-\frac{1}{7}\right)+21 E\left(i \sinh ^{-1}\left(\sqrt{4 \cos (c+d x)+3}\right)|-\frac{1}{7}\right)-8 \Pi \left(-\frac{1}{3};i \sinh ^{-1}\left(\sqrt{4 \cos (c+d x)+3}\right)|-\frac{1}{7}\right)\right)}{3 \sqrt{7} \sqrt{\sin ^2(c+d x)}}}{6 d}","-\frac{F\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{3 \sqrt{7} d}+\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{3 d}+\frac{\sqrt{7} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{8}{7}\right)}{3 d}-\frac{\sqrt{4 \cos (c+d x)+3} \tan (c+d x)}{3 d}+\frac{\sqrt{4 \cos (c+d x)+3} \tan (c+d x) \sec (c+d x)}{6 d}",1,"((4*EllipticF[(c + d*x)/2, 8/7])/Sqrt[7] + (18*EllipticPi[2, (c + d*x)/2, 8/7])/Sqrt[7] - (((2*I)/3)*(21*EllipticE[I*ArcSinh[Sqrt[3 + 4*Cos[c + d*x]]], -1/7] - 12*EllipticF[I*ArcSinh[Sqrt[3 + 4*Cos[c + d*x]]], -1/7] - 8*EllipticPi[-1/3, I*ArcSinh[Sqrt[3 + 4*Cos[c + d*x]]], -1/7])*Sin[c + d*x])/(Sqrt[7]*Sqrt[Sin[c + d*x]^2]) - (-1 + 2*Cos[c + d*x])*Sqrt[3 + 4*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(6*d)","C",1
554,1,102,113,0.1702914,"\int \frac{\cos ^3(c+d x)}{\sqrt{3-4 \cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]^3/Sqrt[3 - 4*Cos[c + d*x]],x]","\frac{-4 \sin (c+d x)+\sin (2 (c+d x))+2 \sin (3 (c+d x))+23 \sqrt{4 \cos (c+d x)-3} F\left(\left.\frac{1}{2} (c+d x)\right|8\right)+9 \sqrt{4 \cos (c+d x)-3} E\left(\left.\frac{1}{2} (c+d x)\right|8\right)}{20 d \sqrt{3-4 \cos (c+d x)}}","\frac{23 F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{20 \sqrt{7} d}-\frac{9 \sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{20 d}-\frac{\sin (c+d x) \sqrt{3-4 \cos (c+d x)} \cos (c+d x)}{10 d}-\frac{\sin (c+d x) \sqrt{3-4 \cos (c+d x)}}{10 d}",1,"(9*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 8] + 23*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 8] - 4*Sin[c + d*x] + Sin[2*(c + d*x)] + 2*Sin[3*(c + d*x)])/(20*d*Sqrt[3 - 4*Cos[c + d*x]])","A",1
555,1,94,80,0.1251433,"\int \frac{\cos ^2(c+d x)}{\sqrt{3-4 \cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]^2/Sqrt[3 - 4*Cos[c + d*x]],x]","\frac{-6 \sin (c+d x)+4 \sin (2 (c+d x))+17 \sqrt{4 \cos (c+d x)-3} F\left(\left.\frac{1}{2} (c+d x)\right|8\right)+3 \sqrt{4 \cos (c+d x)-3} E\left(\left.\frac{1}{2} (c+d x)\right|8\right)}{12 d \sqrt{3-4 \cos (c+d x)}}","\frac{17 F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{12 \sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{4 d}-\frac{\sin (c+d x) \sqrt{3-4 \cos (c+d x)}}{6 d}",1,"(3*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 8] + 17*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 8] - 6*Sin[c + d*x] + 4*Sin[2*(c + d*x)])/(12*d*Sqrt[3 - 4*Cos[c + d*x]])","A",1
556,1,60,53,0.073993,"\int \frac{\cos (c+d x)}{\sqrt{3-4 \cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]/Sqrt[3 - 4*Cos[c + d*x]],x]","\frac{\sqrt{4 \cos (c+d x)-3} \left(3 F\left(\left.\frac{1}{2} (c+d x)\right|8\right)+E\left(\left.\frac{1}{2} (c+d x)\right|8\right)\right)}{2 d \sqrt{3-4 \cos (c+d x)}}","\frac{3 F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{2 \sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{2 d}",1,"(Sqrt[-3 + 4*Cos[c + d*x]]*(EllipticE[(c + d*x)/2, 8] + 3*EllipticF[(c + d*x)/2, 8]))/(2*d*Sqrt[3 - 4*Cos[c + d*x]])","A",1
557,1,44,24,0.0384427,"\int \frac{1}{\sqrt{3-4 \cos (c+d x)}} \, dx","Integrate[1/Sqrt[3 - 4*Cos[c + d*x]],x]","\frac{2 \sqrt{4 \cos (c+d x)-3} F\left(\left.\frac{1}{2} (c+d x)\right|8\right)}{d \sqrt{3-4 \cos (c+d x)}}","\frac{2 F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{\sqrt{7} d}",1,"(2*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 8])/(d*Sqrt[3 - 4*Cos[c + d*x]])","A",1
558,1,45,25,0.0649223,"\int \frac{\sec (c+d x)}{\sqrt{3-4 \cos (c+d x)}} \, dx","Integrate[Sec[c + d*x]/Sqrt[3 - 4*Cos[c + d*x]],x]","\frac{2 \sqrt{4 \cos (c+d x)-3} \Pi \left(2;\left.\frac{1}{2} (c+d x)\right|8\right)}{d \sqrt{3-4 \cos (c+d x)}}","-\frac{2 \Pi \left(2;\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{\sqrt{7} d}",1,"(2*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticPi[2, (c + d*x)/2, 8])/(d*Sqrt[3 - 4*Cos[c + d*x]])","A",1
559,1,179,104,1.5166283,"\int \frac{\sec ^2(c+d x)}{\sqrt{3-4 \cos (c+d x)}} \, dx","Integrate[Sec[c + d*x]^2/Sqrt[3 - 4*Cos[c + d*x]],x]","\frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x)+\frac{6 \sqrt{4 \cos (c+d x)-3} \Pi \left(2;\left.\frac{1}{2} (c+d x)\right|8\right)}{\sqrt{3-4 \cos (c+d x)}}-\frac{i \sin (c+d x) \left(-12 F\left(i \sinh ^{-1}\left(\sqrt{3-4 \cos (c+d x)}\right)|-\frac{1}{7}\right)+21 E\left(i \sinh ^{-1}\left(\sqrt{3-4 \cos (c+d x)}\right)|-\frac{1}{7}\right)-8 \Pi \left(-\frac{1}{3};i \sinh ^{-1}\left(\sqrt{3-4 \cos (c+d x)}\right)|-\frac{1}{7}\right)\right)}{3 \sqrt{7} \sqrt{\sin ^2(c+d x)}}}{3 d}","\frac{F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{\sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{3 d}-\frac{4 \Pi \left(2;\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{3 \sqrt{7} d}+\frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x)}{3 d}",1,"((6*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticPi[2, (c + d*x)/2, 8])/Sqrt[3 - 4*Cos[c + d*x]] - ((I/3)*(21*EllipticE[I*ArcSinh[Sqrt[3 - 4*Cos[c + d*x]]], -1/7] - 12*EllipticF[I*ArcSinh[Sqrt[3 - 4*Cos[c + d*x]]], -1/7] - 8*EllipticPi[-1/3, I*ArcSinh[Sqrt[3 - 4*Cos[c + d*x]]], -1/7])*Sin[c + d*x])/(Sqrt[7]*Sqrt[Sin[c + d*x]^2]) + Sqrt[3 - 4*Cos[c + d*x]]*Tan[c + d*x])/(3*d)","C",1
560,1,236,140,1.9006895,"\int \frac{\sec ^3(c+d x)}{\sqrt{3-4 \cos (c+d x)}} \, dx","Integrate[Sec[c + d*x]^3/Sqrt[3 - 4*Cos[c + d*x]],x]","\frac{-\frac{4 \sqrt{4 \cos (c+d x)-3} F\left(\left.\frac{1}{2} (c+d x)\right|8\right)}{\sqrt{3-4 \cos (c+d x)}}+\frac{18 \sqrt{4 \cos (c+d x)-3} \Pi \left(2;\left.\frac{1}{2} (c+d x)\right|8\right)}{\sqrt{3-4 \cos (c+d x)}}+\sqrt{3-4 \cos (c+d x)} (2 \cos (c+d x)+1) \tan (c+d x) \sec (c+d x)-\frac{2 i \sin (c+d x) \left(-12 F\left(i \sinh ^{-1}\left(\sqrt{3-4 \cos (c+d x)}\right)|-\frac{1}{7}\right)+21 E\left(i \sinh ^{-1}\left(\sqrt{3-4 \cos (c+d x)}\right)|-\frac{1}{7}\right)-8 \Pi \left(-\frac{1}{3};i \sinh ^{-1}\left(\sqrt{3-4 \cos (c+d x)}\right)|-\frac{1}{7}\right)\right)}{3 \sqrt{7} \sqrt{\sin ^2(c+d x)}}}{6 d}","\frac{F\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{3 \sqrt{7} d}-\frac{\sqrt{7} E\left(\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{3 d}-\frac{\sqrt{7} \Pi \left(2;\frac{1}{2} (c+d x+\pi )|\frac{8}{7}\right)}{3 d}+\frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x)}{3 d}+\frac{\sqrt{3-4 \cos (c+d x)} \tan (c+d x) \sec (c+d x)}{6 d}",1,"((-4*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 8])/Sqrt[3 - 4*Cos[c + d*x]] + (18*Sqrt[-3 + 4*Cos[c + d*x]]*EllipticPi[2, (c + d*x)/2, 8])/Sqrt[3 - 4*Cos[c + d*x]] - (((2*I)/3)*(21*EllipticE[I*ArcSinh[Sqrt[3 - 4*Cos[c + d*x]]], -1/7] - 12*EllipticF[I*ArcSinh[Sqrt[3 - 4*Cos[c + d*x]]], -1/7] - 8*EllipticPi[-1/3, I*ArcSinh[Sqrt[3 - 4*Cos[c + d*x]]], -1/7])*Sin[c + d*x])/(Sqrt[7]*Sqrt[Sin[c + d*x]^2]) + Sqrt[3 - 4*Cos[c + d*x]]*(1 + 2*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(6*d)","C",1
561,1,77,111,0.5275043,"\int \cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]),x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} (42 A \cos (c+d x)+15 B \cos (2 (c+d x))+65 B)+126 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+50 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{105 d}","\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{10 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{10 B \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}",1,"(126*A*EllipticE[(c + d*x)/2, 2] + 50*B*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(65*B + 42*A*Cos[c + d*x] + 15*B*Cos[2*(c + d*x)])*Sin[c + d*x])/(105*d)","A",1
562,1,66,87,0.2448534,"\int \cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{2 \left(\sin (c+d x) \sqrt{\cos (c+d x)} (5 A+3 B \cos (c+d x))+5 A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+9 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d}","\frac{2 A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)}}{3 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,"(2*(9*B*EllipticE[(c + d*x)/2, 2] + 5*A*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(5*A + 3*B*Cos[c + d*x])*Sin[c + d*x]))/(15*d)","A",1
563,1,53,61,0.1144443,"\int \sqrt{\cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{2 \left(3 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+B \left(F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)}\right)\right)}{3 d}","\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(2*(3*A*EllipticE[(c + d*x)/2, 2] + B*(EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*Sin[c + d*x])))/(3*d)","A",1
564,1,35,35,0.0684557,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/Sqrt[Cos[c + d*x]],x]","\frac{2 A F\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}","\frac{2 A F\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 + (2*A*EllipticF[(c + d*x)/2, 2])/d","A",1
565,1,51,57,0.1453585,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x])/Cos[c + d*x]^(3/2),x]","\frac{2 \left(-A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{A \sin (c+d x)}{\sqrt{\cos (c+d x)}}+B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d}","-\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*(-(A*EllipticE[(c + d*x)/2, 2]) + B*EllipticF[(c + d*x)/2, 2] + (A*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/d","A",1
566,1,65,83,0.4242966,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x])/Cos[c + d*x]^(5/2),x]","\frac{\frac{2 \sin (c+d x) (A+3 B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)}+2 A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}","\frac{2 A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(-6*B*EllipticE[(c + d*x)/2, 2] + 2*A*EllipticF[(c + d*x)/2, 2] + (2*(A + 3*B*Cos[c + d*x])*Sin[c + d*x])/Cos[c + d*x]^(3/2))/(3*d)","A",1
567,1,95,111,0.3179824,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x])/Cos[c + d*x]^(7/2),x]","\frac{9 A \sin (2 (c+d x))+6 A \tan (c+d x)-18 A \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+10 B \sin (c+d x)+10 B \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{6 A \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-18*A*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 10*B*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 10*B*Sin[c + d*x] + 9*A*Sin[2*(c + d*x)] + 6*A*Tan[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",1
568,1,113,160,0.8027027,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2,x]","\frac{84 \left(9 a^2+7 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(7 \left(36 a^2+43 b^2\right) \cos (c+d x)+5 b (36 a \cos (2 (c+d x))+156 a+7 b \cos (3 (c+d x)))\right)+600 a b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{630 d}","\frac{2 \left(9 a^2+7 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(9 a^2+7 b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{20 a b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a b \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{20 a b \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b^2 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}",1,"(84*(9*a^2 + 7*b^2)*EllipticE[(c + d*x)/2, 2] + 600*a*b*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*(36*a^2 + 43*b^2)*Cos[c + d*x] + 5*b*(156*a + 36*a*Cos[2*(c + d*x)] + 7*b*Cos[3*(c + d*x)]))*Sin[c + d*x])/(630*d)","A",1
569,1,98,135,0.6330888,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2,x]","\frac{10 \left(7 a^2+5 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} \left(70 a^2+84 a b \cos (c+d x)+15 b^2 \cos (2 (c+d x))+65 b^2\right)+252 a b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{105 d}","\frac{2 \left(7 a^2+5 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(7 a^2+5 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{12 a b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a b \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 b^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}",1,"(252*a*b*EllipticE[(c + d*x)/2, 2] + 10*(7*a^2 + 5*b^2)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(70*a^2 + 65*b^2 + 84*a*b*Cos[c + d*x] + 15*b^2*Cos[2*(c + d*x)])*Sin[c + d*x])/(105*d)","A",1
570,1,79,101,0.3215653,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2 \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2,x]","\frac{6 \left(5 a^2+3 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+20 a b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 b \sin (c+d x) \sqrt{\cos (c+d x)} (10 a+3 b \cos (c+d x))}{15 d}","\frac{2 \left(5 a^2+3 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a b \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(6*(5*a^2 + 3*b^2)*EllipticE[(c + d*x)/2, 2] + 20*a*b*EllipticF[(c + d*x)/2, 2] + 2*b*Sqrt[Cos[c + d*x]]*(10*a + 3*b*Cos[c + d*x])*Sin[c + d*x])/(15*d)","A",1
571,1,64,72,0.1649618,"\int \frac{(a+b \cos (c+d x))^2}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(a + b*Cos[c + d*x])^2/Sqrt[Cos[c + d*x]],x]","\frac{2 \left(\left(3 a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 a b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b^2 \sin (c+d x) \sqrt{\cos (c+d x)}\right)}{3 d}","\frac{2 \left(3 a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(2*(6*a*b*EllipticE[(c + d*x)/2, 2] + (3*a^2 + b^2)*EllipticF[(c + d*x)/2, 2] + b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x]))/(3*d)","A",1
572,1,62,68,0.2993188,"\int \frac{(a+b \cos (c+d x))^2}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^2/Cos[c + d*x]^(3/2),x]","\frac{2 \left(\left(b^2-a^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+a \left(\frac{a \sin (c+d x)}{\sqrt{\cos (c+d x)}}+2 b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)\right)}{d}","-\frac{2 \left(a^2-b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{4 a b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*((-a^2 + b^2)*EllipticE[(c + d*x)/2, 2] + a*(2*b*EllipticF[(c + d*x)/2, 2] + (a*Sin[c + d*x])/Sqrt[Cos[c + d*x]])))/d","A",1
573,1,73,95,0.6197443,"\int \frac{(a+b \cos (c+d x))^2}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^2/Cos[c + d*x]^(5/2),x]","\frac{2 \left(\left(a^2+3 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 a b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{a \sin (c+d x) (a+6 b \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)}\right)}{3 d}","\frac{2 \left(a^2+3 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{4 a b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{4 a b \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(2*(-6*a*b*EllipticE[(c + d*x)/2, 2] + (a^2 + 3*b^2)*EllipticF[(c + d*x)/2, 2] + (a*(a + 6*b*Cos[c + d*x])*Sin[c + d*x])/Cos[c + d*x]^(3/2)))/(3*d)","A",1
574,1,124,135,0.3970482,"\int \frac{(a+b \cos (c+d x))^2}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^2/Cos[c + d*x]^(7/2),x]","\frac{-6 \left(3 a^2+5 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 \sin (2 (c+d x))+6 a^2 \tan (c+d x)+20 a b \sin (c+d x)+20 a b \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+15 b^2 \sin (2 (c+d x))}{15 d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{2 \left(3 a^2+5 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(3 a^2+5 b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a b \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-6*(3*a^2 + 5*b^2)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 20*a*b*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 20*a*b*Sin[c + d*x] + 9*a^2*Sin[2*(c + d*x)] + 15*b^2*Sin[2*(c + d*x)] + 6*a^2*Tan[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",1
575,1,137,194,1.0297702,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3,x]","\frac{60 \left(7 a^3+15 a b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+84 \left(27 a^2 b+7 b^3\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(84 a^3+54 a b^2 \cos (2 (c+d x))+234 a b^2+7 b^3 \cos (3 (c+d x))\right)+7 b \left(108 a^2+43 b^2\right) \cos (c+d x)\right)}{630 d}","\frac{2 a \left(7 a^2+15 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 b \left(27 a^2+7 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 b \left(27 a^2+7 b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 a \left(7 a^2+15 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 b^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d}+\frac{40 a b^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}",1,"(84*(27*a^2*b + 7*b^3)*EllipticE[(c + d*x)/2, 2] + 60*(7*a^3 + 15*a*b^2)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*b*(108*a^2 + 43*b^2)*Cos[c + d*x] + 5*(84*a^3 + 234*a*b^2 + 54*a*b^2*Cos[2*(c + d*x)] + 7*b^3*Cos[3*(c + d*x)]))*Sin[c + d*x])/(630*d)","A",1
576,1,110,159,0.7924519,"\int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3 \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3,x]","\frac{42 \left(5 a^3+9 a b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+10 \left(21 a^2 b+5 b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b \sin (c+d x) \sqrt{\cos (c+d x)} \left(210 a^2+126 a b \cos (c+d x)+15 b^2 \cos (2 (c+d x))+65 b^2\right)}{105 d}","\frac{2 b \left(21 a^2+5 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a \left(5 a^2+9 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b \left(21 a^2+5 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{32 a b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}{7 d}",1,"(42*(5*a^3 + 9*a*b^2)*EllipticE[(c + d*x)/2, 2] + 10*(21*a^2*b + 5*b^3)*EllipticF[(c + d*x)/2, 2] + b*Sqrt[Cos[c + d*x]]*(210*a^2 + 65*b^2 + 126*a*b*Cos[c + d*x] + 15*b^2*Cos[2*(c + d*x)])*Sin[c + d*x])/(105*d)","A",1
577,1,84,116,0.400646,"\int \frac{(a+b \cos (c+d x))^3}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(a + b*Cos[c + d*x])^3/Sqrt[Cos[c + d*x]],x]","\frac{2 \left(3 \left(5 a^2 b+b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+5 a \left(a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b^2 \sin (c+d x) \sqrt{\cos (c+d x)} (5 a+b \cos (c+d x))\right)}{5 d}","\frac{2 a \left(a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{6 b \left(5 a^2+b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b^2 \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}{5 d}+\frac{8 a b^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{5 d}",1,"(2*(3*(5*a^2*b + b^3)*EllipticE[(c + d*x)/2, 2] + 5*a*(a^2 + b^2)*EllipticF[(c + d*x)/2, 2] + b^2*Sqrt[Cos[c + d*x]]*(5*a + b*Cos[c + d*x])*Sin[c + d*x]))/(5*d)","A",1
578,1,86,124,0.5596522,"\int \frac{(a+b \cos (c+d x))^3}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^3/Cos[c + d*x]^(3/2),x]","\frac{2 \left(\frac{\sin (c+d x) \left(3 a^3+b^3 \cos (c+d x)\right)}{\sqrt{\cos (c+d x)}}-3 \left(a^3-3 a b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\left(9 a^2 b+b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d}","\frac{2 b \left(9 a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a \left(a^2-3 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 b \left(3 a^2-b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a^2 \sin (c+d x) (a+b \cos (c+d x))}{d \sqrt{\cos (c+d x)}}",1,"(2*(-3*(a^3 - 3*a*b^2)*EllipticE[(c + d*x)/2, 2] + (9*a^2*b + b^3)*EllipticF[(c + d*x)/2, 2] + ((3*a^3 + b^3*Cos[c + d*x])*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(3*d)","A",1
579,1,85,120,1.3287652,"\int \frac{(a+b \cos (c+d x))^3}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^3/Cos[c + d*x]^(5/2),x]","\frac{2 \left(\left(3 b^3-9 a^2 b\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+a \left(\left(a^2+9 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{a \sin (c+d x) (a+9 b \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)}\right)\right)}{3 d}","\frac{2 a \left(a^2+9 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 b \left(3 a^2-b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 \sin (c+d x) (a+b \cos (c+d x))}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{16 a^2 b \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}",1,"(2*((-9*a^2*b + 3*b^3)*EllipticE[(c + d*x)/2, 2] + a*((a^2 + 9*b^2)*EllipticF[(c + d*x)/2, 2] + (a*(a + 9*b*Cos[c + d*x])*Sin[c + d*x])/Cos[c + d*x]^(3/2))))/(3*d)","A",1
580,1,125,149,0.957948,"\int \frac{(a+b \cos (c+d x))^3}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^3/Cos[c + d*x]^(7/2),x]","\frac{3 \left(a^3+5 a b^2\right) \sin (2 (c+d x))+2 a^3 \tan (c+d x)+10 b \left(a^2+b^2\right) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 a \left(a^2+5 b^2\right) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+10 a^2 b \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 b \left(a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{6 a \left(a^2+5 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{6 a \left(a^2+5 b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 \sin (c+d x) (a+b \cos (c+d x))}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{8 a^2 b \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-6*a*(a^2 + 5*b^2)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 10*b*(a^2 + b^2)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 10*a^2*b*Sin[c + d*x] + 3*(a^3 + 5*a*b^2)*Sin[2*(c + d*x)] + 2*a^3*Tan[c + d*x])/(5*d*Cos[c + d*x]^(3/2))","A",1
581,1,177,194,0.8127798,"\int \frac{(a+b \cos (c+d x))^3}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^3/Cos[c + d*x]^(9/2),x]","\frac{25 a^3 \sin (2 (c+d x))+30 a^3 \tan (c+d x)+10 a \left(5 a^2+21 b^2\right) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-42 b \left(9 a^2+5 b^2\right) \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+126 a^2 b \sin (c+d x)+378 a^2 b \sin (c+d x) \cos ^2(c+d x)+105 a b^2 \sin (2 (c+d x))+210 b^3 \sin (c+d x) \cos ^2(c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 a \left(5 a^2+21 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 b \left(9 a^2+5 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \left(5 a^2+21 b^2\right) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left(9 a^2+5 b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 \sin (c+d x) (a+b \cos (c+d x))}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{32 a^2 b \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-42*b*(9*a^2 + 5*b^2)*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 10*a*(5*a^2 + 21*b^2)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 126*a^2*b*Sin[c + d*x] + 378*a^2*b*Cos[c + d*x]^2*Sin[c + d*x] + 210*b^3*Cos[c + d*x]^2*Sin[c + d*x] + 25*a^3*Sin[2*(c + d*x)] + 105*a*b^2*Sin[2*(c + d*x)] + 30*a^3*Tan[c + d*x])/(105*d*Cos[c + d*x]^(5/2))","A",1
582,1,158,112,2.0441528,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^(5/2)/(a + b*Cos[c + d*x]),x]","\frac{-\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)}}{6 b d}","-\frac{2 a^3 \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^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^3 d}-\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d}",1,"(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
583,1,81,75,0.3037507,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^(3/2)/(a + b*Cos[c + d*x]),x]","-\frac{2 \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 \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 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(-2*(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
584,1,48,53,0.0756469,"\int \frac{\sqrt{\cos (c+d x)}}{a+b \cos (c+d x)} \, dx","Integrate[Sqrt[Cos[c + d*x]]/(a + b*Cos[c + d*x]),x]","\frac{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}}{b d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}-\frac{2 a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a+b)}",1,"(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
585,1,29,29,0.0797553,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])),x]","\frac{2 \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{d (a+b)}","\frac{2 \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{d (a+b)}",1,"(2*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/((a + b)*d)","A",1
586,1,195,77,3.1110935,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Integrate[1/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])),x]","-\frac{\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)}}}{2 a d}","-\frac{2 b \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a d (a+b)}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{2 \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}",1,"-1/2*((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
587,1,210,128,4.5849162,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])),x]","\frac{\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)}{6 a^2 d}","\frac{2 b^2 \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 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{2 b \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{2 \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}",1,"((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
588,1,266,245,1.9571342,"\int \frac{\cos ^{\frac{7}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^(7/2)/(a + b*Cos[c + d*x])^2,x]","\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)} \left(\frac{3 a^3}{\left(a^2-b^2\right) (a+b \cos (c+d x))}+2\right)-\frac{\frac{2 \left(5 a^3-8 a b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{8 \left(2 a^2+b^2\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{6 \left(5 a^2-4 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)}{b^2 \sqrt{\sin ^2(c+d x)}}}{(a-b) (a+b)}}{12 b^2 d}","-\frac{a^2 \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^2-2 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \left(a^2-b^2\right)}-\frac{a \left(5 a^2-4 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}+\frac{\left(15 a^4-16 a^2 b^2-2 b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d \left(a^2-b^2\right)}-\frac{a^3 \left(5 a^2-7 b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d (a-b) (a+b)^2}",1,"(4*Sqrt[Cos[c + d*x]]*(2 + (3*a^3)/((a^2 - b^2)*(a + b*Cos[c + d*x])))*Sin[c + d*x] - ((2*(5*a^3 - 8*a*b^2)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(2*a^2 + b^2)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (6*(5*a^2 - 4*b^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])/(b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b)))/(12*b^2*d)","A",1
589,1,251,185,1.8490424,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^2,x]","\frac{\frac{4 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{\left(b^2-a^2\right) (a+b \cos (c+d x))}+\frac{\frac{2 \left(a^2-2 b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{2 \left(3 a^2-2 b^2\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}+4 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)}{(a-b) (a+b)}}{4 b d}","\frac{\left(3 a^2-2 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}-\frac{a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{a \left(3 a^2-4 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}+\frac{a^2 \left(3 a^2-5 b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}",1,"((4*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((-a^2 + b^2)*(a + b*Cos[c + d*x])) + ((2*(a^2 - 2*b^2)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + 4*a*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)) + (2*(3*a^2 - 2*b^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^2*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b)))/(4*b*d)","A",1
590,1,194,163,3.4021608,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^2,x]","\frac{\frac{4 a \sin (c+d x) \sqrt{\cos (c+d x)}}{\left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\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)}{b^2 \sqrt{\sin ^2(c+d x)}}-\frac{10 a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+8 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{(a-b) (a+b)}}{4 d}","\frac{\left(a^2-2 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}-\frac{a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}-\frac{a \left(a^2-3 b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a-b) (a+b)^2}+\frac{a \sin (c+d x) \sqrt{\cos (c+d x)}}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"((4*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*(a + b*Cos[c + d*x])) - (8*EllipticF[(c + d*x)/2, 2] - (10*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (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])/(b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b)))/(4*d)","A",1
591,1,229,148,3.70047,"\int \frac{\sqrt{\cos (c+d x)}}{(a+b \cos (c+d x))^2} \, dx","Integrate[Sqrt[Cos[c + d*x]]/(a + b*Cos[c + d*x])^2,x]","\frac{\frac{4 b \sin (c+d x) \sqrt{\cos (c+d x)}}{\left(b^2-a^2\right) (a+b \cos (c+d x))}-\frac{2 \left(\frac{\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{b^2 \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+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)\right)}{b (b-a) (a+b)}}{4 d}","\frac{a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \left(a^2-b^2\right)}-\frac{\left(a^2+b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a-b) (a+b)^2}-\frac{b \sin (c+d x) \sqrt{\cos (c+d x)}}{d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"((4*b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((-a^2 + b^2)*(a + b*Cos[c + d*x])) - (2*(-((b^2*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)) + ((-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])))/(b*(-a + b)*(a + b)))/(4*d)","A",1
592,1,238,157,3.5116381,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^2} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2),x]","\frac{\frac{4 b^2 \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^2-3 b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}-\frac{2 \sin (c+d x) \left(\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)}}+8 a \left(\frac{a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}-F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{(a-b) (a+b)}}{4 a d}","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \left(a^2-b^2\right)}-\frac{b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \left(a^2-b^2\right)}+\frac{\left(3 a^2-b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a d (a-b) (a+b)^2}+\frac{b^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{a d \left(a^2-b^2\right) (a+b \cos (c+d x))}",1,"((4*b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*(a + b*Cos[c + d*x])) + ((2*(4*a^2 - 3*b^2)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + 8*a*(-EllipticF[(c + d*x)/2, 2] + (a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)) - (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*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b)))/(4*a*d)","A",1
593,1,278,217,3.1353946,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx","Integrate[1/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2),x]","\frac{4 \sqrt{\cos (c+d x)} \left(\frac{b^3 \sin (c+d x)}{\left(b^2-a^2\right) (a+b \cos (c+d x))}+2 \tan (c+d x)\right)-\frac{\frac{\left(8 a b^2-4 a^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(9 b^3-10 a^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(2 a^2-3 b^2\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}}{(b-a) (a+b)}}{4 a^2 d}","\frac{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^2-3 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{b \left(5 a^2-3 b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a-b) (a+b)^2}+\frac{b^2 \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(2 a^2-3 b^2\right) \sin (c+d x)}{a^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}",1,"(-(((2*(-10*a^2*b + 9*b^3)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + ((-4*a^3 + 8*a*b^2)*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)))/b - (2*(2*a^2 - 3*b^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 + b)*(a + b))) + 4*Sqrt[Cos[c + d*x]]*((b^3*Sin[c + d*x])/((-a^2 + b^2)*(a + b*Cos[c + d*x])) + 2*Tan[c + d*x]))/(4*a^2*d)","A",1
594,1,294,281,3.571839,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^2),x]","\frac{4 \sqrt{\cos (c+d x)} \left(\frac{3 b^4 \sin (c+d x)}{\left(a^2-b^2\right) (a+b \cos (c+d x))}+2 \tan (c+d x) (a \sec (c+d x)-6 b)\right)+\frac{\frac{8 \left(7 a^3-10 a b^2\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{6 \left(4 a^2-5 b^2\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a \sqrt{\sin ^2(c+d x)}}+\frac{2 \left(4 a^4+44 a^2 b^2-45 b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{(a-b) (a+b)}}{12 a^3 d}","\frac{\left(2 a^2-5 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^2 \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^2-5 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{b \left(4 a^2-5 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \left(a^2-b^2\right)}+\frac{b^2 \left(7 a^2-5 b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}-\frac{b \left(4 a^2-5 b^2\right) \sin (c+d x)}{a^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}",1,"(((2*(4*a^4 + 44*a^2*b^2 - 45*b^4)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(7*a^3 - 10*a*b^2)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (6*(4*a^2 - 5*b^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*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b)) + 4*Sqrt[Cos[c + d*x]]*((3*b^4*Sin[c + d*x])/((a^2 - b^2)*(a + b*Cos[c + d*x])) + 2*(-6*b + a*Sec[c + d*x])*Tan[c + d*x]))/(12*a^3*d)","A",1
595,1,354,346,3.2304548,"\int \frac{\cos ^{\frac{9}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^(9/2)/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)} \left(35 a^6-57 a^4 b^2+4 \left(b^3-a^2 b\right)^2 \cos (2 (c+d x))+a b \left(49 a^4-83 a^2 b^2+16 b^4\right) \cos (c+d x)+4 b^6\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{\frac{2 \left(35 a^5-73 a^3 b^2+56 a b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{16 \left(7 a^4-14 a^2 b^2-2 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)}{a+b}+\frac{6 \left(35 a^4-65 a^2 b^2+24 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)}{b^2 \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{48 b^3 d}","-\frac{a^2 \left(7 a^2-13 b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{a \left(35 a^4-65 a^2 b^2+24 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d \left(a^2-b^2\right)^2}+\frac{\left(35 a^4-61 a^2 b^2+8 b^4\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{12 b^3 d \left(a^2-b^2\right)^2}+\frac{\left(105 a^6-223 a^4 b^2+128 a^2 b^4+8 b^6\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 b^5 d \left(a^2-b^2\right)^2}-\frac{a^3 \left(35 a^4-86 a^2 b^2+63 b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^5 d (a-b)^2 (a+b)^3}",1,"((4*Sqrt[Cos[c + d*x]]*(35*a^6 - 57*a^4*b^2 + 4*b^6 + a*b*(49*a^4 - 83*a^2*b^2 + 16*b^4)*Cos[c + d*x] + 4*(-(a^2*b) + b^3)^2*Cos[2*(c + d*x)])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) - ((2*(35*a^5 - 73*a^3*b^2 + 56*a*b^4)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (16*(7*a^4 - 14*a^2*b^2 - 2*b^4)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (6*(35*a^4 - 65*a^2*b^2 + 24*b^4)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(48*b^3*d)","A",1
596,1,309,282,3.0068869,"\int \frac{\cos ^{\frac{7}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^(7/2)/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{\frac{8 \left(a^3-4 a b^2\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{\left(5 a^4-7 a^2 b^2+8 b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{\left(15 a^4-29 a^2 b^2+8 b^4\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}-\frac{2 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(5 a^3+b \left(7 a^2-13 b^2\right) \cos (c+d x)-11 a b^2\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}}{8 b^2 d}","-\frac{a^2 \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^2 \left(5 a^2-11 b^2\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{3 a \left(5 a^4-11 a^2 b^2+8 b^4\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^2 \left(15 a^4-38 a^2 b^2+35 b^4\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}+\frac{\left(15 a^4-29 a^2 b^2+8 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d \left(a^2-b^2\right)^2}",1,"((-2*a^2*Sqrt[Cos[c + d*x]]*(5*a^3 - 11*a*b^2 + b*(7*a^2 - 13*b^2)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) + (((5*a^4 - 7*a^2*b^2 + 8*b^4)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(a^3 - 4*a*b^2)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + ((15*a^4 - 29*a^2*b^2 + 8*b^4)*(-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
597,1,284,264,2.1712785,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{4 a \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3+3 b \left(a^2-3 b^2\right) \cos (c+d x)-7 a b^2\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{\frac{2 \left(a^3+5 a b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}-\frac{16 \left(a^2+2 b^2\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{6 \left(a^2-3 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)}{b^2 \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{16 b d}","-\frac{3 a \left(a^2-3 b^2\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{a^2 \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{3 a \left(a^2-3 b^2\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^4-5 a^2 b^2+8 b^4\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{3 a \left(a^4-2 a^2 b^2+5 b^4\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,"((4*a*Sqrt[Cos[c + d*x]]*(a^3 - 7*a*b^2 + 3*b*(a^2 - 3*b^2)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) - ((2*(a^3 + 5*a*b^2)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) - (16*(a^2 + 2*b^2)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (6*(a^2 - 3*b^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])/(b^2*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(16*b*d)","A",1
598,1,272,244,2.0687183,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)} \left(b \left(a^2+5 b^2\right) \cos (c+d x)+3 a \left(a^2+b^2\right)\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{-\frac{2 \left(5 a^2+b^2\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{2 \left(a^2+5 b^2\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b^2 \sqrt{\sin ^2(c+d x)}}+24 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)}{(a-b)^2 (a+b)^2}}{16 d}","\frac{a \left(a^2-7 b^2\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^2+5 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b d \left(a^2-b^2\right)^2}+\frac{\left(a^2+5 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{a \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^4-10 a^2 b^2-3 b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d (a-b)^2 (a+b)^3}",1,"((4*Sqrt[Cos[c + d*x]]*(3*a*(a^2 + b^2) + b*(a^2 + 5*b^2)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) - ((-2*(5*a^2 + b^2)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + 24*a*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)) + (2*(a^2 + 5*b^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^2*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(16*d)","A",1
599,1,291,250,3.0669636,"\int \frac{\sqrt{\cos (c+d x)}}{(a+b \cos (c+d x))^3} \, dx","Integrate[Sqrt[Cos[c + d*x]]/(a + b*Cos[c + d*x])^3,x]","\frac{\frac{\frac{2 \left(3 b^3-9 a^2 b\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{8 a \left(2 a^2+b^2\right) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 a \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{b}+\frac{2 \left(5 a^2+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)}}}{(a-b)^2 (a+b)^2}-\frac{4 b \sin (c+d x) \sqrt{\cos (c+d x)} \left(7 a^3+b \left(5 a^2+b^2\right) \cos (c+d x)-a b^2\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}}{16 a d}","\frac{3 \left(a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b d \left(a^2-b^2\right)^2}+\frac{\left(5 a^2+b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a d \left(a^2-b^2\right)^2}-\frac{b \left(5 a^2+b^2\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{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(3 a^4+10 a^2 b^2-b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b d (a-b)^2 (a+b)^3}",1,"((-4*b*Sqrt[Cos[c + d*x]]*(7*a^3 - a*b^2 + b*(5*a^2 + b^2)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) + ((2*(-9*a^2*b + 3*b^3)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*a*(2*a^2 + b^2)*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)))/b + (2*(5*a^2 + b^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 - b)^2*(a + b)^2))/(16*a*d)","A",1
600,1,301,261,2.9985553,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^3} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3),x]","\frac{\frac{4 b^2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(11 a^3+\left(9 a^2 b-3 b^3\right) \cos (c+d x)-5 a b^2\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+\frac{\frac{16 \left(a b^2-4 a^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{6 \left(3 a^2-b^2\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a \sqrt{\sin ^2(c+d x)}}+\frac{2 \left(16 a^4-19 a^2 b^2+9 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}}{16 a^2 d}","-\frac{\left(7 a^2-b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a d \left(a^2-b^2\right)^2}-\frac{3 b \left(3 a^2-b^2\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{3 b^2 \left(3 a^2-b^2\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{b^2 \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{3 \left(5 a^4-2 a^2 b^2+b^4\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d (a-b)^2 (a+b)^3}",1,"((4*b^2*Sqrt[Cos[c + d*x]]*(11*a^3 - 5*a*b^2 + (9*a^2*b - 3*b^3)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) + ((2*(16*a^4 - 19*a^2*b^2 + 9*b^4)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (16*(-4*a^3 + a*b^2)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) - (6*(3*a^2 - b^2)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(16*a^2*d)","A",1
601,1,334,328,3.6210439,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx","Integrate[1/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3),x]","\frac{4 \sqrt{\cos (c+d x)} \left(\frac{b^3 \sin (c+d x) \left(-15 a^3+\left(7 b^3-13 a^2 b\right) \cos (c+d x)+9 a b^2\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+8 \tan (c+d x)\right)-\frac{\frac{2 \left(56 a^4 b-95 a^2 b^3+45 b^5\right) \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{8 a \left(2 a^4-10 a^2 b^2+5 b^4\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(8 a^4-29 a^2 b^2+15 b^4\right) \sin (c+d x) \left(\left(b^2-2 a^2\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 a (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{16 a^3 d}","\frac{b \left(11 a^2-5 b^2\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^2 \left(11 a^2-5 b^2\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{b^2 \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(8 a^4-29 a^2 b^2+15 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{b \left(35 a^4-38 a^2 b^2+15 b^4\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}+\frac{\left(8 a^4-29 a^2 b^2+15 b^4\right) \sin (c+d x)}{4 a^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}",1,"(-(((2*(56*a^4*b - 95*a^2*b^3 + 45*b^5)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*a*(2*a^4 - 10*a^2*b^2 + 5*b^4)*(2*EllipticF[(c + d*x)/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)))/b + (2*(8*a^4 - 29*a^2*b^2 + 15*b^4)*(-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)) + 4*Sqrt[Cos[c + d*x]]*((b^3*(-15*a^3 + 9*a*b^2 + (-13*a^2*b + 7*b^3)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) + 8*Tan[c + d*x]))/(16*a^3*d)","A",1
602,1,349,395,5.956896,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^3),x]","\frac{4 \sqrt{\cos (c+d x)} \left(\frac{3 b^4 \sin (c+d x) \left(19 a^3+b \left(17 a^2-11 b^2\right) \cos (c+d x)-13 a b^2\right)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}+8 \tan (c+d x) (a \sec (c+d x)-9 b)\right)+\frac{\frac{16 \left(20 a^5-64 a^3 b^2+35 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)}{a+b}+\frac{6 \left(24 a^4-65 a^2 b^2+35 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 \sqrt{\sin ^2(c+d x)}}+\frac{2 \left(16 a^6+328 a^4 b^2-641 a^2 b^4+315 b^6\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}}{48 a^4 d}","\frac{b^2 \left(13 a^2-7 b^2\right) \sin (c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))}+\frac{b^2 \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(24 a^4-65 a^2 b^2+35 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{b^2 \left(63 a^4-86 a^2 b^2+35 b^4\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{b \left(24 a^4-65 a^2 b^2+35 b^4\right) \sin (c+d x)}{4 a^4 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}+\frac{\left(8 a^4-61 a^2 b^2+35 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^4-61 a^2 b^2+35 b^4\right) \sin (c+d x)}{12 a^3 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x)}",1,"(((2*(16*a^6 + 328*a^4*b^2 - 641*a^2*b^4 + 315*b^6)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (16*(20*a^5 - 64*a^3*b^2 + 35*a*b^4)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (6*(24*a^4 - 65*a^2*b^2 + 35*b^4)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2) + 4*Sqrt[Cos[c + d*x]]*((3*b^4*(19*a^3 - 13*a*b^2 + b*(17*a^2 - 11*b^2)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) + 8*(-9*b + a*Sec[c + d*x])*Tan[c + d*x]))/(48*a^4*d)","A",1
603,1,1152,438,17.5945019,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]],x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{2 d}+\frac{-\frac{12 a^2 \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \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 \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 a \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^2-4 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}+\frac{a \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} (a+2 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} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + ((-12*a^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]]) - 16*a*b*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*a*((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
604,1,314,371,7.0648037,"\int \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \, dx","Integrate[Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]],x]","\frac{\sqrt{\cos (c+d x)} \left(-\frac{4 a \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)}{\sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}}}+\frac{2 (a+b) \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)}{\sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}}}+\frac{4 a \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)}{\sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}}}+2 a \tan \left(\frac{1}{2} (c+d x)\right)-b \tan \left(\frac{1}{2} (c+d x)\right)+b \sin \left(\frac{3}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right)\right)}{2 d \sqrt{a+b \cos (c+d x)}}","\frac{\sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}-\frac{(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{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}} \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,"(Sqrt[Cos[c + d*x]]*((2*(a + b)*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (4*a*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (4*a*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + b*Sec[(c + d*x)/2]*Sin[(3*(c + d*x))/2] + 2*a*Tan[(c + d*x)/2] - b*Tan[(c + d*x)/2]))/(2*d*Sqrt[a + b*Cos[c + d*x]])","A",1
605,1,137,135,1.2752344,"\int \frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]/Sqrt[Cos[c + d*x]],x]","\frac{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{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{a+b \cos (c+d x)}}","-\frac{2 \csc (c+d x) \sqrt{\frac{a (1-\cos (c+d x))}{a+b \cos (c+d x)}} \sqrt{\frac{a (\cos (c+d x)+1)}{a+b \cos (c+d x)}} (a+b \cos (c+d x)) \Pi \left(\frac{b}{a+b};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{\cos (c+d x)}}{\sqrt{a+b \cos (c+d x)}}\right)|-\frac{a-b}{a+b}\right)}{d \sqrt{a+b}}",1,"(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[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[a + b*Cos[c + d*x]])","A",1
606,1,203,229,3.4563357,"\int \frac{\sqrt{a+b \cos (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]/Cos[c + d*x]^(3/2),x]","-\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} \left(-\sin (c+d x) \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}}-\sqrt{\cos (c+d x)} \sqrt{\cos (c+d x)+1} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+\sqrt{\cos (c+d x)} \sqrt{\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)}{d \sqrt{\cos (c+d x)} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}}}","\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\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[a + b*Cos[c + d*x]]*Sec[(c + d*x)/2]^2*(Sqrt[Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - Sqrt[Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sin[c + d*x]))/(d*Sqrt[Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]))","A",1
607,1,247,271,7.86796,"\int \frac{\sqrt{a+b \cos (c+d x)}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]/Cos[c + d*x]^(5/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \left(2 a^2+2 a (a+2 b) \cos (c+d x)+b (a+b) \cos (2 (c+d x))+a b+b^2\right)+2 a (a+b) \sqrt{\cos (c+d x)+1} \cos ^{\frac{3}{2}}(c+d x) \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-2 b (a+b) \sqrt{\cos (c+d x)+1} \cos ^{\frac{3}{2}}(c+d x) \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)}{3 a d \cos ^{\frac{3}{2}}(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)}{3 a^2 d}+\frac{2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\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}",1,"(-2*b*(a + b)*Cos[c + d*x]^(3/2)*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*a*(a + b)*Cos[c + d*x]^(3/2)*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)] + (2*a^2 + a*b + b^2 + 2*a*(a + 2*b)*Cos[c + d*x] + b*(a + b)*Cos[2*(c + d*x)])*Tan[(c + d*x)/2])/(3*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]])","A",0
608,1,453,329,13.292812,"\int \frac{\sqrt{a+b \cos (c+d x)}}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[Sqrt[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 \sec (c+d x) \left(9 a^2 \sin (c+d x)-2 b^2 \sin (c+d x)\right)}{15 a^2}+\frac{2 b \tan (c+d x) \sec (c+d x)}{15 a}+\frac{2}{5} \tan (c+d x) \sec ^2(c+d x)\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(-\left(\left(9 a^2-2 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 \left(9 a^2+7 a b-2 b^2\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-2 \left(9 a^3+9 a^2 b-2 a b^2-2 b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{15 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} (9 a+2 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^2-2 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 b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d \cos ^{\frac{5}{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*(9*a^3 + 9*a^2*b - 2*a*b^2 - 2*b^3)*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*(9*a^2 + 7*a*b - 2*b^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)] - (9*a^2 - 2*b^2)*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*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]*(9*a^2*Sin[c + d*x] - 2*b^2*Sin[c + d*x]))/(15*a^2) + (2*b*Sec[c + d*x]*Tan[c + d*x])/(15*a) + (2*Sec[c + d*x]^2*Tan[c + d*x])/5))/d","A",1
609,1,1304,389,6.2327472,"\int \frac{\sqrt{a+b \cos (c+d x)}}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]/Cos[c + d*x]^(9/2),x]","\frac{-\frac{4 a \left(25 a^4-17 b^2 a^2-8 b^4\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-19 b a^3-8 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 b^4-19 a^2 b^2\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{105 a^3 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{7} \tan (c+d x) \sec ^3(c+d x)+\frac{2 \left(25 a^2 \sin (c+d x)-4 b^2 \sin (c+d x)\right) \sec ^2(c+d x)}{105 a^2}+\frac{2 b \tan (c+d x) \sec ^2(c+d x)}{35 a}+\frac{2 \left(8 \sin (c+d x) b^3+19 a^2 \sin (c+d x) b\right) \sec (c+d x)}{105 a^3}\right)}{d}","\frac{2 \left(25 a^2-4 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 b (a-b) \sqrt{a+b} \left(19 a^2+8 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^4 d}+\frac{2 (a-b) \sqrt{a+b} \left(25 a^2+6 a b+8 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 b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 \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 - 17*a^2*b^2 - 8*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]]) - 4*a*(-19*a^3*b - 8*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]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-19*a^2*b^2 - 8*b^4)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(105*a^3*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^2*(25*a^2*Sin[c + d*x] - 4*b^2*Sin[c + d*x]))/(105*a^2) + (2*Sec[c + d*x]*(19*a^2*b*Sin[c + d*x] + 8*b^3*Sin[c + d*x]))/(105*a^3) + (2*b*Sec[c + d*x]^2*Tan[c + d*x])/(35*a) + (2*Sec[c + d*x]^3*Tan[c + d*x])/7))/d","C",1
610,1,1189,508,19.5903427,"\int \cos ^{\frac{3}{2}}(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])^(3/2),x]","\frac{-\frac{4 a \left(17 a^2+16 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)}}-208 a^2 b \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(3 a^2+16 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)}{48 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{7}{12} a \sin (c+d x)+\frac{1}{6} b \sin (2 (c+d x))\right)}{d}","\frac{\left(3 a^2+16 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 b d \sqrt{\cos (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(3 a^2+16 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)}{24 a b d}+\frac{a \sqrt{a+b} \left(a^2-12 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^2 d}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac{a \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{\sqrt{a+b} (a+2 b) (3 a+8 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 d}",1,"((-4*a*(17*a^2 + 16*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]]) - 208*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*(3*a^2 + 16*b^2)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(48*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((7*a*Sin[c + d*x])/12 + (b*Sin[2*(c + d*x)])/6))/d","C",0
611,1,437,433,12.2743521,"\int \sqrt{\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])^(3/2),x]","\frac{\sqrt{\cos (c+d x)} \left(\frac{-4 \left(4 a^2-a b+2 b^2\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)+12 a^2 \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)+10 a^2 \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \tan \left(\frac{1}{2} (c+d x)\right)+16 b^2 \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)-5 a b \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \tan \left(\frac{1}{2} (c+d x)\right)+5 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)+10 a (a+b) \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)}{\sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}}}+4 b \sin (c+d x) (a+b \cos (c+d x))\right)}{8 d \sqrt{a+b \cos (c+d x)}}","-\frac{\sqrt{a+b} \left(3 a^2+4 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d}+\frac{\sin (c+d x) (a+b \cos (c+d x))^{3/2}}{2 d \sqrt{\cos (c+d x)}}+\frac{3 a \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} (5 a+2 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{5 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}",1,"(Sqrt[Cos[c + d*x]]*(4*b*(a + b*Cos[c + d*x])*Sin[c + d*x] + (10*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)] - 4*(4*a^2 - a*b + 2*b^2)*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 12*a^2*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 16*b^2*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 5*a*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*Sin[(3*(c + d*x))/2] + 10*a^2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Tan[(c + d*x)/2] - 5*a*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Tan[(c + d*x)/2])/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]))/(8*d*Sqrt[a + b*Cos[c + d*x]])","A",1
612,1,339,375,7.5745847,"\int \frac{(a+b \cos (c+d x))^{3/2}}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)/Sqrt[Cos[c + d*x]],x]","-\frac{\sqrt{\cos (c+d x)} \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(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))+4 a (a-2 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 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)+12 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)}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{d \left(\tan ^4\left(\frac{1}{2} (c+d x)\right)-1\right) \sqrt{a+b \cos (c+d x)}}","\frac{b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\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{b (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d}-\frac{3 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}} \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,"-((Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]^2*(2*b*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 4*a*(a - 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)] + 12*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]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + b*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(d*Sqrt[a + b*Cos[c + d*x]]*(-1 + Tan[(c + d*x)/2]^4)))","A",1
613,1,357,337,13.1271896,"\int \frac{(a+b \cos (c+d x))^{3/2}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(3/2),x]","\frac{\cos (c+d x) \left(\frac{2 \left(a^2+2 a b-b^2\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)}{\sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}}}-2 a^2 \tan \left(\frac{1}{2} (c+d x)\right)+\frac{4 b^2 \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)}{\sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}}}+a b \tan \left(\frac{1}{2} (c+d x)\right)-a b \sin \left(\frac{3}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right)-\frac{2 a (a+b) \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)}{\sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}}}\right)+2 a \sin (c+d x) (a+b \cos (c+d x))}{d \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}","-\frac{2 (a-2 b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{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*Cos[c + d*x])*Sin[c + d*x] + Cos[c + d*x]*((-2*a*(a + b)*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (2*(a^2 + 2*a*b - b^2)*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (4*b^2*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)])/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - a*b*Sec[(c + d*x)/2]*Sin[(3*(c + d*x))/2] - 2*a^2*Tan[(c + d*x)/2] + a*b*Tan[(c + d*x)/2]))/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])","A",1
614,1,256,277,4.9799013,"\int \frac{(a+b \cos (c+d x))^{3/2}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(5/2),x]","\frac{2 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right)} \left(\left(a^2+4 a b+3 b^2\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)-4 b \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} (a+b \cos (c+d x))-4 b (a+b) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)+\frac{2 \sin (c+d x) (a+b \cos (c+d x)) (a+4 b \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)}}{3 d \sqrt{a+b \cos (c+d x)}}","\frac{2 a \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-3 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}} F\left(\sin ^{-1}\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{8 b (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d}",1,"((2*(a + b*Cos[c + d*x])*(a + 4*b*Cos[c + d*x])*Sin[c + d*x])/Cos[c + d*x]^(3/2) + 2*Sqrt[Cos[(c + d*x)/2]^2]*(-4*b*(a + b)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (a^2 + 4*a*b + 3*b^2)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - 4*b*(a + b*Cos[c + d*x])*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Tan[(c + d*x)/2]))/(3*d*Sqrt[a + b*Cos[c + d*x]])","A",0
615,1,443,325,13.521815,"\int \frac{(a+b \cos (c+d x))^{3/2}}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(7/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \sec (c+d x) \left(3 a^2 \sin (c+d x)+b^2 \sin (c+d x)\right)}{5 a}+\frac{2}{5} a \tan (c+d x) \sec ^2(c+d x)+\frac{4}{5} b \tan (c+d x) \sec (c+d x)\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(-\left(\left(3 a^2+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 \left(3 a^2+4 a b+b^2\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-2 \left(3 a^3+3 a^2 b+a b^2+b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{5 a 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} \left(3 a^2+b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^2 d}+\frac{4 b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 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)}-\frac{2 (a-b) (3 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}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a d}",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*(3*a^3 + 3*a^2*b + a*b^2 + b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(3*a^2 + 4*a*b + b^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)] - (3*a^2 + b^2)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(5*a*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]*(3*a^2*Sin[c + d*x] + b^2*Sin[c + d*x]))/(5*a) + (4*b*Sec[c + d*x]*Tan[c + d*x])/5 + (2*a*Sec[c + d*x]^2*Tan[c + d*x])/5))/d","A",1
616,1,1302,387,6.2609315,"\int \frac{(a+b \cos (c+d x))^{3/2}}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(9/2),x]","\frac{-\frac{4 a \left(25 a^4-31 b^2 a^2+6 b^4\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(6 a b^3-82 a^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(6 b^4-82 a^2 b^2\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{105 a^2 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{7} a \tan (c+d x) \sec ^3(c+d x)+\frac{2 \left(25 \sin (c+d x) a^2+3 b^2 \sin (c+d x)\right) \sec ^2(c+d x)}{105 a}+\frac{16}{35} b \tan (c+d x) \sec ^2(c+d x)+\frac{4 \left(41 a^2 b \sin (c+d x)-3 b^3 \sin (c+d x)\right) \sec (c+d x)}{105 a^2}\right)}{d}","\frac{2 \left(25 a^2+3 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{105 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(25 a^2-57 a b-6 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d}+\frac{4 b (a-b) \sqrt{a+b} \left(41 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)}{105 a^3 d}+\frac{16 b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"((-4*a*(25*a^4 - 31*a^2*b^2 + 6*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]]) - 4*a*(-82*a^3*b + 6*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]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-82*a^2*b^2 + 6*b^4)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(105*a^2*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^2*(25*a^2*Sin[c + d*x] + 3*b^2*Sin[c + d*x]))/(105*a) + (4*Sec[c + d*x]*(41*a^2*b*Sin[c + d*x] - 3*b^3*Sin[c + d*x]))/(105*a^2) + (16*b*Sec[c + d*x]^2*Tan[c + d*x])/35 + (2*a*Sec[c + d*x]^3*Tan[c + d*x])/7))/d","C",1
617,1,1368,454,6.295508,"\int \frac{(a+b \cos (c+d x))^{3/2}}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)/Cos[c + d*x]^(11/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{9} a \tan (c+d x) \sec ^4(c+d x)+\frac{2 \left(49 \sin (c+d x) a^2+3 b^2 \sin (c+d x)\right) \sec ^3(c+d x)}{315 a}+\frac{20}{63} b \tan (c+d x) \sec ^3(c+d x)+\frac{8 \left(22 a^2 b \sin (c+d x)-b^3 \sin (c+d x)\right) \sec ^2(c+d x)}{315 a^2}+\frac{2 \left(147 \sin (c+d x) a^4+33 b^2 \sin (c+d x) a^2+8 b^4 \sin (c+d x)\right) \sec (c+d x)}{315 a^3}\right)}{d}-\frac{-\frac{4 a \left(8 b^5+31 a^2 b^3-39 a^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(147 a^5+33 b^2 a^3+8 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 b^5+33 a^2 b^3+147 a^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)}{315 a^3 d}","\frac{8 b \left(22 a^2-b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(49 a^2+3 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(147 a^4+33 a^2 b^2+8 b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{315 a^4 d}-\frac{2 (a-b) \sqrt{a+b} \left(147 a^3-39 a^2 b-6 a b^2-8 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{20 b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 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*b + 31*a^2*b^3 + 8*b^5)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-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 + 33*a^3*b^2 + 8*a*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*(147*a^4*b + 33*a^2*b^3 + 8*b^5)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a^3*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^3*(49*a^2*Sin[c + d*x] + 3*b^2*Sin[c + d*x]))/(315*a) + (8*Sec[c + d*x]^2*(22*a^2*b*Sin[c + d*x] - b^3*Sin[c + d*x]))/(315*a^2) + (2*Sec[c + d*x]*(147*a^4*Sin[c + d*x] + 33*a^2*b^2*Sin[c + d*x] + 8*b^4*Sin[c + d*x]))/(315*a^3) + (20*b*Sec[c + d*x]^3*Tan[c + d*x])/63 + (2*a*Sec[c + d*x]^4*Tan[c + d*x])/9))/d","C",0
618,1,1203,506,19.0630279,"\int \sqrt{\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])^(5/2),x]","\frac{-\frac{4 a \left(16 b^3+59 a^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^3+76 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(16 b^3+33 a^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}{6} \sin (2 (c+d x)) b^2+\frac{13}{12} a \sin (c+d x) b\right)}{d}","\frac{\left(33 a^2+16 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a+b} \left(33 a^2+26 a b+16 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)}{24 d}-\frac{(a-b) \sqrt{a+b} \left(33 a^2+16 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)}{24 a d}-\frac{5 a \sqrt{a+b} \left(a^2+4 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b d}+\frac{b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{13 a b \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{12 d}",1,"((-4*a*(59*a^2*b + 16*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*(48*a^3 + 76*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]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[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*(33*a^2*b + 16*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]])))/(48*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((13*a*b*Sin[c + d*x])/12 + (b^2*Sin[2*(c + d*x)])/6))/d","C",0
619,1,329,443,6.6173506,"\int \frac{(a+b \cos (c+d x))^{5/2}}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)/Sqrt[Cos[c + d*x]],x]","\frac{\sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right)} \left(2 b \left(15 a^2+4 b^2\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))}{a+b}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+2 \left(4 a^3-12 a^2 b+a b^2-2 b^3\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)+9 a b \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} (a+b \cos (c+d x))+9 a b (a+b) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)+2 b^2 \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))}{4 d \sqrt{a+b \cos (c+d x)}}","\frac{\sqrt{a+b} \left(8 a^2+9 a b+2 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}-\frac{\sqrt{a+b} \left(15 a^2+4 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d}+\frac{b^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}{2 d}+\frac{9 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\cos (c+d x)}}-\frac{9 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)}{4 d}",1,"(2*b^2*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])*Sin[c + d*x] + Sqrt[Cos[(c + d*x)/2]^2]*(9*a*b*(a + b)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 2*(4*a^3 - 12*a^2*b + a*b^2 - 2*b^3)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 2*b*(15*a^2 + 4*b^2)*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 9*a*b*(a + b*Cos[c + d*x])*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Tan[(c + d*x)/2]))/(4*d*Sqrt[a + b*Cos[c + d*x]])","A",0
620,1,1185,445,18.189362,"\int \frac{(a+b \cos (c+d x))^{5/2}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(3/2),x]","\frac{2 \sqrt{a+b \cos (c+d x)} \sin (c+d x) a^2}{d \sqrt{\cos (c+d x)}}+\frac{\frac{4 a \left(-b^3-4 a^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^3-6 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^2 b-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)}{2 d}","-\frac{\left(2 a^2-b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \left(2 a^2-6 a b-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)}{d}+\frac{(a-b) \sqrt{a+b} \left(2 a^2-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{2 a^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{5 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)}{d}",1,"(2*a^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + ((4*a*(-4*a^2*b - 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^3 - 6*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]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[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^2*b - 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]])))/(2*d)","C",0
621,1,328,392,7.0441073,"\int \frac{(a+b \cos (c+d x))^{5/2}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(5/2),x]","\frac{2 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right)} \left(\left(a^3+7 a^2 b+9 a b^2-3 b^3\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)+6 b^3 \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))}{a+b}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-7 a b \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} (a+b \cos (c+d x))-7 a b (a+b) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)+\frac{2 a \sin (c+d x) (a+b \cos (c+d x)) (a+7 b \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)}}{3 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \sqrt{a+b} \left(a^2-7 a b+9 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 d}+\frac{2 a^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 b^2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d}+\frac{14 b (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 d}",1,"((2*a*(a + b*Cos[c + d*x])*(a + 7*b*Cos[c + d*x])*Sin[c + d*x])/Cos[c + d*x]^(3/2) + 2*Sqrt[Cos[(c + d*x)/2]^2]*(-7*a*b*(a + b)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (a^3 + 7*a^2*b + 9*a*b^2 - 3*b^3)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 6*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - 7*a*b*(a + b*Cos[c + d*x])*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Tan[(c + d*x)/2]))/(3*d*Sqrt[a + b*Cos[c + d*x]])","A",0
622,1,427,338,11.796414,"\int \frac{(a+b \cos (c+d x))^{5/2}}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(7/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{15} \sec (c+d x) \left(9 a^2 \sin (c+d x)+23 b^2 \sin (c+d x)\right)+\frac{2}{5} a^2 \tan (c+d x) \sec ^2(c+d x)+\frac{22}{15} a b \tan (c+d x) \sec (c+d x)\right)}{d}-\frac{4 \cos ^2\left(\frac{1}{2} (c+d x)\right)^{5/2} \left(\frac{\cos (c+d x)}{\cos (c+d x)+1}\right)^{3/2} \sqrt{\cos (c+d x)+1} \left(\left(9 a^2+23 b^2\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))-\left(9 a^3+17 a^2 b+23 a b^2+15 b^3\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)+\left(9 a^3+9 a^2 b+23 a b^2+23 b^3\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)\right)}{15 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}","-\frac{2 (a-b) \sqrt{a+b} \left(9 a^2-8 a b+15 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 d}+\frac{2 (a-b) \sqrt{a+b} \left(9 a^2+23 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 a^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{22 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 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]]*((9*a^3 + 9*a^2*b + 23*a*b^2 + 23*b^3)*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)] - (9*a^3 + 17*a^2*b + 23*a*b^2 + 15*b^3)*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)] + (9*a^2 + 23*b^2)*(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]))/(15*d*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]*(9*a^2*Sin[c + d*x] + 23*b^2*Sin[c + d*x]))/15 + (22*a*b*Sec[c + d*x]*Tan[c + d*x])/15 + (2*a^2*Sec[c + d*x]^2*Tan[c + d*x])/5))/d","A",0
623,1,1302,387,6.2971977,"\int \frac{(a+b \cos (c+d x))^{5/2}}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(9/2),x]","\frac{-\frac{4 a \left(5 a^4-2 b^2 a^2-3 b^4\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-29 b a^3-3 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(-3 b^4-29 a^2 b^2\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{21 a d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{7} a^2 \tan (c+d x) \sec ^3(c+d x)+\frac{2}{21} \left(5 \sin (c+d x) a^2+9 b^2 \sin (c+d x)\right) \sec ^2(c+d x)+\frac{6}{7} a b \tan (c+d x) \sec ^2(c+d x)+\frac{2 \left(3 \sin (c+d x) b^3+29 a^2 \sin (c+d x) b\right) \sec (c+d x)}{21 a}\right)}{d}","\frac{2 \left(5 a^2+9 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (a-b) \sqrt{a+b} \left(5 a^2-24 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)}{21 a d}+\frac{2 b (a-b) \sqrt{a+b} \left(29 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)}{21 a^2 d}+\frac{2 a^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{6 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{7 d \cos ^{\frac{5}{2}}(c+d x)}",1,"((-4*a*(5*a^4 - 2*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]]) - 4*a*(-29*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]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-29*a^2*b^2 - 3*b^4)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(21*a*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^2*(5*a^2*Sin[c + d*x] + 9*b^2*Sin[c + d*x]))/21 + (2*Sec[c + d*x]*(29*a^2*b*Sin[c + d*x] + 3*b^3*Sin[c + d*x]))/(21*a) + (6*a*b*Sec[c + d*x]^2*Tan[c + d*x])/7 + (2*a^2*Sec[c + d*x]^3*Tan[c + d*x])/7))/d","C",1
624,1,1368,454,6.307596,"\int \frac{(a+b \cos (c+d x))^{5/2}}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(11/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{9} a^2 \tan (c+d x) \sec ^4(c+d x)+\frac{2}{315} \left(49 \sin (c+d x) a^2+75 b^2 \sin (c+d x)\right) \sec ^3(c+d x)+\frac{38}{63} a b \tan (c+d x) \sec ^3(c+d x)+\frac{2 \left(5 \sin (c+d x) b^3+163 a^2 \sin (c+d x) b\right) \sec ^2(c+d x)}{315 a}+\frac{2 \left(147 \sin (c+d x) a^4+279 b^2 \sin (c+d x) a^2-10 b^4 \sin (c+d x)\right) \sec (c+d x)}{315 a^2}\right)}{d}-\frac{-\frac{4 a \left(-10 b^5+124 a^2 b^3-114 a^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(147 a^5+279 b^2 a^3-10 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 b^5+279 a^2 b^3+147 a^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)}{315 a^2 d}","\frac{2 \left(49 a^2+75 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 b \left(163 a^2+5 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{9 d \cos ^{\frac{9}{2}}(c+d x)}-\frac{2 (a-b) \sqrt{a+b} \left(147 a^3-114 a^2 b+165 a b^2+10 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+279 a^2 b^2-10 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{38 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{63 d \cos ^{\frac{7}{2}}(c+d x)}",1,"-1/315*((-4*a*(-114*a^4*b + 124*a^2*b^3 - 10*b^5)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-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 + 279*a^3*b^2 - 10*a*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*(147*a^4*b + 279*a^2*b^3 - 10*b^5)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a^2*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^3*(49*a^2*Sin[c + d*x] + 75*b^2*Sin[c + d*x]))/315 + (2*Sec[c + d*x]^2*(163*a^2*b*Sin[c + d*x] + 5*b^3*Sin[c + d*x]))/(315*a) + (2*Sec[c + d*x]*(147*a^4*Sin[c + d*x] + 279*a^2*b^2*Sin[c + d*x] - 10*b^4*Sin[c + d*x]))/(315*a^2) + (38*a*b*Sec[c + d*x]^3*Tan[c + d*x])/63 + (2*a^2*Sec[c + d*x]^4*Tan[c + d*x])/9))/d","C",1
625,1,1431,522,6.3474617,"\int \frac{(a+b \cos (c+d x))^{5/2}}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)/Cos[c + d*x]^(13/2),x]","\frac{-\frac{4 a \left(135 a^6-78 b^2 a^4-49 b^4 a^2-8 b^6\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(-741 b a^5-51 b^3 a^3-8 b^5 a\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(-8 b^6-51 a^2 b^4-741 a^4 b^2\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{693 a^3 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2}{11} a^2 \tan (c+d x) \sec ^5(c+d x)+\frac{2}{693} \left(81 \sin (c+d x) a^2+113 b^2 \sin (c+d x)\right) \sec ^4(c+d x)+\frac{46}{99} a b \tan (c+d x) \sec ^4(c+d x)+\frac{2 \left(3 \sin (c+d x) b^3+229 a^2 \sin (c+d x) b\right) \sec ^3(c+d x)}{693 a}+\frac{2 \left(135 \sin (c+d x) a^4+205 b^2 \sin (c+d x) a^2-4 b^4 \sin (c+d x)\right) \sec ^2(c+d x)}{693 a^2}+\frac{2 \left(8 \sin (c+d x) b^5+51 a^2 \sin (c+d x) b^3+741 a^4 \sin (c+d x) b\right) \sec (c+d x)}{693 a^3}\right)}{d}","\frac{2 \left(81 a^2+113 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 b \left(229 a^2+3 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{693 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{11 d \cos ^{\frac{11}{2}}(c+d x)}+\frac{2 \left(135 a^4+205 a^2 b^2-4 b^4\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{693 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b (a-b) \sqrt{a+b} \left(741 a^4+51 a^2 b^2+8 b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{693 a^4 d}+\frac{2 (a-b) \sqrt{a+b} \left(135 a^4-606 a^3 b+57 a^2 b^2+6 a b^3+8 b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{693 a^3 d}+\frac{46 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{99 d \cos ^{\frac{9}{2}}(c+d x)}",1,"((-4*a*(135*a^6 - 78*a^4*b^2 - 49*a^2*b^4 - 8*b^6)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-741*a^5*b - 51*a^3*b^3 - 8*a*b^5)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-741*a^4*b^2 - 51*a^2*b^4 - 8*b^6)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(693*a^3*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*Sec[c + d*x]^4*(81*a^2*Sin[c + d*x] + 113*b^2*Sin[c + d*x]))/693 + (2*Sec[c + d*x]^3*(229*a^2*b*Sin[c + d*x] + 3*b^3*Sin[c + d*x]))/(693*a) + (2*Sec[c + d*x]^2*(135*a^4*Sin[c + d*x] + 205*a^2*b^2*Sin[c + d*x] - 4*b^4*Sin[c + d*x]))/(693*a^2) + (2*Sec[c + d*x]*(741*a^4*b*Sin[c + d*x] + 51*a^2*b^3*Sin[c + d*x] + 8*b^5*Sin[c + d*x]))/(693*a^3) + (46*a*b*Sec[c + d*x]^4*Tan[c + d*x])/99 + (2*a^2*Sec[c + d*x]^5*Tan[c + d*x])/11))/d","C",0
626,1,479,379,4.4559877,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]^(3/2)/Sqrt[a + b*Cos[c + d*x]],x]","\frac{\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 \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{\sin (c+d x) \sqrt{a+b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}+\frac{\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{(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,"(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
627,1,130,116,0.9238696,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[Sqrt[Cos[c + d*x]]/Sqrt[a + b*Cos[c + d*x]],x]","-\frac{2 \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 \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[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
628,1,170,109,1.4027269,"\int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]),x]","-\frac{4 (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 \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)*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
629,1,211,224,4.9565954,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Integrate[1/(Cos[c + d*x]^(3/2)*Sqrt[a + b*Cos[c + d*x]]),x]","\frac{2 \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 (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} \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 + 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
630,1,371,274,13.9682724,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Integrate[1/(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 \tan (c+d x) \sec (c+d x)}{3 a}-\frac{4 b \tan (c+d x)}{3 a^2}\right)}{d}+\frac{16 \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(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))+a (a-2 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 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^2 d \cos ^{\frac{3}{2}}(c+d x) (\cos (c+d x)+1)^{3/2} \sqrt{a+b \cos (c+d x)}}","-\frac{4 b (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d}+\frac{2 \sqrt{a+b} (a+2 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 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 a d \cos ^{\frac{3}{2}}(c+d x)}",1,"(16*(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*b*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*(a - 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)] + 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]]*((-4*b*Tan[c + d*x])/(3*a^2) + (2*Sec[c + d*x]*Tan[c + d*x])/(3*a)))/d","A",1
631,1,1201,465,6.2231214,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 \sqrt{\cos (c+d x)} \sin (c+d x) a^2}{b \left(b^2-a^2\right) d \sqrt{a+b \cos (c+d x)}}+\frac{-\frac{4 a \left(a^2-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)}}-8 a^2 b \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(3 a^2-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)}{2 (a-b) b (a+b) d}","-\frac{2 a^2 \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\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{\left(3 a^2-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 b^2 d \sqrt{a+b}}+\frac{3 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}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d}+\frac{(3 a+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*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(-a^2 + b^2)*d*Sqrt[a + b*Cos[c + d*x]]) + ((-4*a*(a^2 - 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]]) - 8*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*(3*a^2 - b^2)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(2*(a - b)*b*(a + b)*d)","C",1
632,1,985,387,17.7341222,"\int \frac{\cos ^{\frac{3}{2}}(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])^(3/2),x]","\frac{2 a \sqrt{\cos (c+d x)} \sin (c+d x)}{\left(a^2-b^2\right) d \sqrt{a+b \cos (c+d x)}}-\frac{2 a \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 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)}{(a-b) (a+b) d}","-\frac{2 a^2 \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 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^2 d}-\frac{2 \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 \sqrt{a+b}}+\frac{2 \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)}{b d \sqrt{a+b}}",1,"(2*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - (-4*a*b*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*a*((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
633,1,196,266,4.8173225,"\int \frac{\sqrt{\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])^(3/2),x]","\frac{2 \left((a-b) \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right)-(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)+(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)\right)}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}","\frac{2 a \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 \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}}-\frac{2 \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 \sqrt{a+b}}",1,"(2*((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)] - (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)] + (a - b)*Sqrt[Cos[c + d*x]]*Tan[(c + d*x)/2]))/((a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",1
634,1,202,267,5.793165,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{2 \left(b (b-a) \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right)+a (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)-b (a+b) \sqrt{\cos (c+d x)+1} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}","-\frac{2 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 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 \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*(-(b*(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)]) + 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)] + b*(-a + b)*Sqrt[Cos[c + d*x]]*Tan[(c + d*x)/2]))/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",1
635,1,1233,285,6.2839755,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Integrate[1/(Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{-\frac{4 a \left(2 a^2 b-2 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^3-2 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(a^2 b-2 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)}{a^2 (b-a) (a+b) d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \tan (c+d x)}{a^2}-\frac{2 b^3 \sin (c+d x)}{a^2 \left(a^2-b^2\right) (a+b \cos (c+d x))}\right)}{d}","\frac{2 b^2 \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+2 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^2-2 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*b - 2*b^3)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(a^3 - 2*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]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(a^2*b - 2*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^2*(-a + b)*(a + b)*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((-2*b^3*Sin[c + d*x])/(a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])) + (2*Tan[c + d*x])/a^2))/d","C",1
636,1,1269,357,6.3585063,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{-\frac{4 a \left(a^4+7 b^2 a^2-8 b^4\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(5 a^3 b-8 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(5 a^2 b^2-8 b^4\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{3 a^3 (a-b) (a+b) d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \sin (c+d x) b^4}{a^3 \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{10 \tan (c+d x) b}{3 a^3}+\frac{2 \sec (c+d x) \tan (c+d x)}{3 a^2}\right)}{d}","\frac{2 (a+2 b) (a+4 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 b^2 \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(a^2-4 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 \left(5 a^2-8 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^4 d \sqrt{a+b}}",1,"((-4*a*(a^4 + 7*a^2*b^2 - 8*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]]) - 4*a*(5*a^3*b - 8*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]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(5*a^2*b^2 - 8*b^4)*((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*b^4*Sin[c + d*x])/(a^3*(a^2 - b^2)*(a + b*Cos[c + d*x])) - (10*b*Tan[c + d*x])/(3*a^3) + (2*Sec[c + d*x]*Tan[c + d*x])/(3*a^2)))/d","C",0
637,1,1314,433,6.3836309,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}} \, dx","Integrate[1/(Cos[c + d*x]^(7/2)*(a + b*Cos[c + d*x])^(3/2)),x]","\frac{\left(a^2+4 b^2\right) \left(-\frac{4 a \left(4 a^2 b-4 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^3-4 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 a^2 b-4 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)\right)}{5 a^4 (b-a) (a+b) d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(-\frac{2 \sin (c+d x) b^5}{a^4 \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{6 \sec (c+d x) \tan (c+d x) b}{5 a^3}+\frac{2 \sec (c+d x) \left(3 \sin (c+d x) a^2+11 b^2 \sin (c+d x)\right)}{5 a^4}+\frac{2 \sec ^2(c+d x) \tan (c+d x)}{5 a^2}\right)}{d}","\frac{2 b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(a^2-6 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 a^2 d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 (3 a+4 b) \left(a^2+4 b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^4 d \sqrt{a+b}}-\frac{2 b \left(3 a^2-8 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{5 a^3 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(3 a^4+8 a^2 b^2-16 b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^5 d \sqrt{a+b}}",1,"((a^2 + 4*b^2)*((-4*a*(4*a^2*b - 4*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*(3*a^3 - 4*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]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(3*a^2*b - 4*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]]))))/(5*a^4*(-a + b)*(a + b)*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((-2*b^5*Sin[c + d*x])/(a^4*(a^2 - b^2)*(a + b*Cos[c + d*x])) + (2*Sec[c + d*x]*(3*a^2*Sin[c + d*x] + 11*b^2*Sin[c + d*x]))/(5*a^4) - (6*b*Sec[c + d*x]*Tan[c + d*x])/(5*a^3) + (2*Sec[c + d*x]^2*Tan[c + d*x])/(5*a^2)))/d","C",1
638,1,1282,497,6.3048103,"\int \frac{\cos ^{\frac{5}{2}}(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])^(5/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \sin (c+d x) a^2}{3 b \left(b^2-a^2\right) (a+b \cos (c+d x))^2}+\frac{2 \left(3 a^3 \sin (c+d x)-7 a b^2 \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(a^3-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^3-a^2 b\right) \left(\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)+2 \left(3 a^3-7 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)}{3 (a-b)^2 b (a+b)^2 d}","-\frac{2 a^2 \left(3 a^2-7 b^2\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 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(3 a^2+a b-6 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 b^2 d (a-b) (a+b)^{3/2}}+\frac{2 \left(3 a^2-7 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 b^2 d (a-b) (a+b)^{3/2}}-\frac{2 \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^2*Sin[c + d*x])/(3*b*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (2*(3*a^3*Sin[c + d*x] - 7*a*b^2*Sin[c + d*x]))/(3*b*(-a^2 + b^2)^2*(a + b*Cos[c + d*x]))))/d - ((-4*a*(a^3 - 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^2*b) - 3*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]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[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^3 - 7*a*b^2)*((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
639,1,277,342,6.3791574,"\int \frac{\cos ^{\frac{3}{2}}(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])^(5/2),x]","\frac{2 \left(\sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3+3 a b^2+4 b^3 \cos (c+d x)\right)-\sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right)} (a+b \cos (c+d x)) \left(-\left(a^2+4 a b+3 b^2\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)+4 b \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} (a+b \cos (c+d x))+4 b (a+b) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)\right)}{3 d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^{3/2}}","-\frac{8 a b \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 \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 (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 (a-b) (a+b)^{3/2}}+\frac{8 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 (a-b) (a+b)^{3/2}}",1,"(2*(Sqrt[Cos[c + d*x]]*(a^3 + 3*a*b^2 + 4*b^3*Cos[c + d*x])*Sin[c + d*x] - Sqrt[Cos[(c + d*x)/2]^2]*(a + b*Cos[c + d*x])*(4*b*(a + b)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (a^2 + 4*a*b + 3*b^2)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 4*b*(a + b*Cos[c + d*x])*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Tan[(c + d*x)/2])))/(3*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^(3/2))","A",0
640,1,1273,359,6.253332,"\int \frac{\sqrt{\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])^(5/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(-\frac{2 b \sin (c+d x)}{3 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{2 \left(\sin (c+d x) b^3+3 a^2 \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^3-a^2 b\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(3 a^3+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(b^3+3 a^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)}{3 a (a-b)^2 (a+b)^2 d}","\frac{2 \left(3 a^2+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 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+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-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*b*Sin[c + d*x])/(3*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) - (2*(3*a^2*b*Sin[c + d*x] + b^3*Sin[c + d*x]))/(3*a*(a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/d + ((-4*a*(-(a^2*b) + 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*(3*a^3 + 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]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(3*a^2*b + b^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]])))/(3*a*(a - b)^2*(a + b)^2*d)","C",1
641,1,1296,381,6.2791364,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2}} \, dx","Integrate[1/(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 \sin (c+d x) b^2}{3 a \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{4 \left(3 a^2 b^2 \sin (c+d x)-b^4 \sin (c+d x)\right)}{3 a^2 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}\right)}{d}+\frac{-\frac{4 a \left(3 a^4-5 b^2 a^2+2 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(2 a b^3-6 a^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(2 b^4-6 a^2 b^2\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{3 a^2 (a-b)^2 (a+b)^2 d}","\frac{2 b^2 \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{4 b \left(3 a^2-b^2\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^2-3 a b-2 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 (a-b) (a+b)^{3/2}}+\frac{4 b \left(3 a^2-b^2\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d (a-b) (a+b)^{3/2}}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((2*b^2*Sin[c + d*x])/(3*a*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) + (4*(3*a^2*b^2*Sin[c + d*x] - b^4*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/d + ((-4*a*(3*a^4 - 5*a^2*b^2 + 2*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]]) - 4*a*(-6*a^3*b + 2*a*b^3)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-6*a^2*b^2 + 2*b^4)*((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",1
642,1,1321,398,6.3978552,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}} \, dx","Integrate[1/(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 \sin (c+d x) b^3}{3 a^2 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{2 \left(9 a^2 b^3 \sin (c+d x)-5 b^5 \sin (c+d x)\right)}{3 a^3 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{2 \tan (c+d x)}{a^3}\right)}{d}-\frac{-\frac{4 a \left(8 b^5-17 a^2 b^3+9 a^4 b\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(3 a^5-15 b^2 a^3+8 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 b^5-15 a^2 b^3+3 a^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)}{3 a^3 (a-b)^2 (a+b)^2 d}","\frac{8 b^2 \left(2 a^2-b^2\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 b^2 \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 \left(3 a^4-15 a^2 b^2+8 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}}-\frac{2 \left(3 a^3+9 a^2 b-6 a b^2-8 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 (a-b) (a+b)^{3/2}}",1,"-1/3*((-4*a*(9*a^4*b - 17*a^2*b^3 + 8*b^5)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-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 - 15*a^3*b^2 + 8*a*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*(3*a^4*b - 15*a^2*b^3 + 8*b^5)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]]*EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b)]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (a*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])))/b + (Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*Sqrt[Cos[c + d*x]])))/(a^3*(a - b)^2*(a + b)^2*d) + (Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]*((-2*b^3*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) - (2*(9*a^2*b^3*Sin[c + d*x] - 5*b^5*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) + (2*Tan[c + d*x])/a^3))/d","C",1
643,1,1351,473,6.4896038,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^{5/2}} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*(a + b*Cos[c + d*x])^(5/2)),x]","\frac{-\frac{4 a \left(a^6+15 b^2 a^4-32 b^4 a^2+16 b^6\right) \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-4 a \left(8 b a^5-28 b^3 a^3+16 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 b^6-28 a^2 b^4+8 a^4 b^2\right) \left(\frac{i \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \cos (c+d x)} E\left(i \sinh ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)|-\frac{2 a}{-a-b}\right) \sec (c+d x)}{b \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{\frac{(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac{2 a \left(\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{(a+b) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{a \sqrt{\frac{(a+b) \cot ^2\left(\frac{1}{2} (c+d x)\right)}{b-a}} \sqrt{-\frac{(a+b) \cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}} \csc (c+d x) \Pi \left(-\frac{a}{b};\sin ^{-1}\left(\frac{\sqrt{\frac{(a+b \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)}{a}}}{\sqrt{2}}\right)|-\frac{2 a}{b-a}\right) \sin ^4\left(\frac{1}{2} (c+d x)\right)}{b \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\right)}{b}+\frac{\sqrt{a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt{\cos (c+d x)}}\right)}{3 a^4 (a-b)^2 (a+b)^2 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \sin (c+d x) b^4}{3 a^3 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{16 \tan (c+d x) b}{3 a^4}+\frac{8 \left(3 a^2 b^4 \sin (c+d x)-2 b^6 \sin (c+d x)\right)}{3 a^4 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{2 \sec (c+d x) \tan (c+d x)}{3 a^3}\right)}{d}","\frac{4 b^2 \left(5 a^2-3 b^2\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 b^2 \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{8 b \left(2 a^4-7 a^2 b^2+4 b^4\right) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^5 d (a-b) (a+b)^{3/2}}+\frac{2 \left(a^4-13 a^2 b^2+8 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(a^4+9 a^3 b+16 a^2 b^2-12 a b^3-16 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 (a-b) (a+b)^{3/2}}",1,"((-4*a*(a^6 + 15*a^4*b^2 - 32*a^2*b^4 + 16*b^6)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-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*b - 28*a^3*b^3 + 16*a*b^5)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-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*b^2 - 28*a^2*b^4 + 16*b^6)*((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*b^4*Sin[c + d*x])/(3*a^3*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) + (8*(3*a^2*b^4*Sin[c + d*x] - 2*b^6*Sin[c + d*x]))/(3*a^4*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) - (16*b*Tan[c + d*x])/(3*a^4) + (2*Sec[c + d*x]*Tan[c + d*x])/(3*a^3)))/d","C",0
644,1,131,32,3.6602133,"\int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{2+3 \cos (c+d x)}} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*Sqrt[2 + 3*Cos[c + d*x]]),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{3 \cos (c+d x)+2} \sqrt{\cot ^2\left(\frac{1}{2} (c+d x)\right)} \csc (c+d x) F\left(\left.\sin ^{-1}\left(\frac{1}{2} \sqrt{(3 \cos (c+d x)+2) \csc ^2\left(\frac{1}{2} (c+d x)\right)}\right)\right|-4\right)}{d \sqrt{\frac{-3 \cos (c+d x)-2}{\cos (c+d x)-1}} \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)-1}}}","\frac{2 F\left(\sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)|\frac{1}{5}\right)}{\sqrt{5} d}",1,"(2*Sqrt[Cos[c + d*x]]*Sqrt[2 + 3*Cos[c + d*x]]*Sqrt[Cot[(c + d*x)/2]^2]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[(2 + 3*Cos[c + d*x])*Csc[(c + d*x)/2]^2]/2], -4])/(d*Sqrt[(-2 - 3*Cos[c + d*x])/(-1 + Cos[c + d*x])]*Sqrt[Cos[c + d*x]/(-1 + Cos[c + d*x])])","B",1
645,1,156,25,0.9353275,"\int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{-2+3 \cos (c+d x)}} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*Sqrt[-2 + 3*Cos[c + d*x]]),x]","\frac{4 \sin ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\cot ^2\left(\frac{1}{2} (c+d x)\right)} \csc (c+d x) \sqrt{\cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{-\left((3 \cos (c+d x)-2) \csc ^2\left(\frac{1}{2} (c+d x)\right)\right)} F\left(\sin ^{-1}\left(\frac{1}{2} \sqrt{-\left((3 \cos (c+d x)-2) \csc ^2\left(\frac{1}{2} (c+d x)\right)\right)}\right)|\frac{4}{5}\right)}{\sqrt{5} d \sqrt{\cos (c+d x)} \sqrt{3 \cos (c+d x)-2}}","\frac{2 F\left(\left.\sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)\right|5\right)}{d}",1,"(4*Sqrt[Cot[(c + d*x)/2]^2]*Sqrt[Cos[c + d*x]*Csc[(c + d*x)/2]^2]*Sqrt[-((-2 + 3*Cos[c + d*x])*Csc[(c + d*x)/2]^2)]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[-((-2 + 3*Cos[c + d*x])*Csc[(c + d*x)/2]^2)]/2], 4/5]*Sin[(c + d*x)/2]^4)/(Sqrt[5]*d*Sqrt[Cos[c + d*x]]*Sqrt[-2 + 3*Cos[c + d*x]])","B",1
646,1,143,56,1.1019072,"\int \frac{1}{\sqrt{2-3 \cos (c+d x)} \sqrt{\cos (c+d x)}} \, dx","Integrate[1/(Sqrt[2 - 3*Cos[c + d*x]]*Sqrt[Cos[c + d*x]]),x]","-\frac{4 \sin ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\cot ^2\left(\frac{1}{2} (c+d x)\right)} \csc (c+d x) \sqrt{(2-3 \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)} F\left(\left.\sin ^{-1}\left(\frac{1}{2} \sqrt{\cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}\right)\right|-4\right)}{d \sqrt{2-3 \cos (c+d x)} \sqrt{\cos (c+d x)}}","-\frac{2 \sqrt{-\cos (c+d x)} F\left(\sin ^{-1}\left(\frac{\sin (c+d x)}{1-\cos (c+d x)}\right)|\frac{1}{5}\right)}{\sqrt{5} d \sqrt{\cos (c+d x)}}",1,"(-4*Sqrt[Cot[(c + d*x)/2]^2]*Sqrt[(2 - 3*Cos[c + d*x])*Csc[(c + d*x)/2]^2]*Sqrt[Cos[c + d*x]*Csc[(c + d*x)/2]^2]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[Cos[c + d*x]*Csc[(c + d*x)/2]^2]/2], -4]*Sin[(c + d*x)/2]^4)/(d*Sqrt[2 - 3*Cos[c + d*x]]*Sqrt[Cos[c + d*x]])","B",1
647,1,153,49,1.5214095,"\int \frac{1}{\sqrt{-2-3 \cos (c+d x)} \sqrt{\cos (c+d x)}} \, dx","Integrate[1/(Sqrt[-2 - 3*Cos[c + d*x]]*Sqrt[Cos[c + d*x]]),x]","\frac{4 \sin ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\cot ^2\left(\frac{1}{2} (c+d x)\right)} \csc (c+d x) \sqrt{-\cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{(3 \cos (c+d x)+2) \csc ^2\left(\frac{1}{2} (c+d x)\right)} F\left(\sin ^{-1}\left(\sqrt{\frac{5}{2}} \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)-1}}\right)|\frac{4}{5}\right)}{\sqrt{5} d \sqrt{-3 \cos (c+d x)-2} \sqrt{\cos (c+d x)}}","-\frac{2 \sqrt{-\cos (c+d x)} F\left(\left.\sin ^{-1}\left(\frac{\sin (c+d x)}{1-\cos (c+d x)}\right)\right|5\right)}{d \sqrt{\cos (c+d x)}}",1,"(4*Sqrt[Cot[(c + d*x)/2]^2]*Sqrt[-(Cos[c + d*x]*Csc[(c + d*x)/2]^2)]*Sqrt[(2 + 3*Cos[c + d*x])*Csc[(c + d*x)/2]^2]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[5/2]*Sqrt[Cos[c + d*x]/(-1 + Cos[c + d*x])]], 4/5]*Sin[(c + d*x)/2]^4)/(Sqrt[5]*d*Sqrt[-2 - 3*Cos[c + d*x]]*Sqrt[Cos[c + d*x]])","B",1
648,1,140,58,1.0514361,"\int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{3+2 \cos (c+d x)}} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*Sqrt[3 + 2*Cos[c + d*x]]),x]","\frac{4 \sqrt{\cos (c+d x)} \sqrt{2 \cos (c+d x)+3} \sqrt{-\cot ^2\left(\frac{1}{2} (c+d x)\right)} \csc (c+d x) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{(2 \cos (c+d x)+3) \csc ^2\left(\frac{1}{2} (c+d x)\right)}}{\sqrt{6}}\right)\right|6\right)}{d \sqrt{-\cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{(2 \cos (c+d x)+3) \csc ^2\left(\frac{1}{2} (c+d x)\right)}}","\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)+3}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right)\right|-5\right)}{d}",1,"(4*Sqrt[Cos[c + d*x]]*Sqrt[3 + 2*Cos[c + d*x]]*Sqrt[-Cot[(c + d*x)/2]^2]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[(3 + 2*Cos[c + d*x])*Csc[(c + d*x)/2]^2]/Sqrt[6]], 6])/(d*Sqrt[-(Cos[c + d*x]*Csc[(c + d*x)/2]^2)]*Sqrt[(3 + 2*Cos[c + d*x])*Csc[(c + d*x)/2]^2])","B",1
649,1,144,60,1.0427436,"\int \frac{1}{\sqrt{3-2 \cos (c+d x)} \sqrt{\cos (c+d x)}} \, dx","Integrate[1/(Sqrt[3 - 2*Cos[c + d*x]]*Sqrt[Cos[c + d*x]]),x]","\frac{4 \sin ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\cot ^2\left(\frac{1}{2} (c+d x)\right)} \csc (c+d x) \sqrt{(3-2 \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{-\cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{\cos (c+d x)}{\cos (c+d x)-1}}}{\sqrt{3}}\right)\right|6\right)}{d \sqrt{3-2 \cos (c+d x)} \sqrt{\cos (c+d x)}}","\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{3-2 \cos (c+d x)}}{\sqrt{\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{\sqrt{5} d}",1,"(4*Sqrt[Cot[(c + d*x)/2]^2]*Sqrt[(3 - 2*Cos[c + d*x])*Csc[(c + d*x)/2]^2]*Sqrt[-(Cos[c + d*x]*Csc[(c + d*x)/2]^2)]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[Cos[c + d*x]/(-1 + Cos[c + d*x])]/Sqrt[3]], 6]*Sin[(c + d*x)/2]^4)/(d*Sqrt[3 - 2*Cos[c + d*x]]*Sqrt[Cos[c + d*x]])","B",1
650,1,144,84,1.2561713,"\int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{-3+2 \cos (c+d x)}} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*Sqrt[-3 + 2*Cos[c + d*x]]),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\frac{2 \cos (c+d x)-3}{\cos (c+d x)-1}} \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{-\cot ^2\left(\frac{1}{2} (c+d x)\right)} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{2 \cos (c+d x)-3}{\cos (c+d x)-1}}}{\sqrt{3}}\right)|\frac{6}{5}\right)}{\sqrt{5} d \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)-1}} \sqrt{2 \cos (c+d x)-3}}","-\frac{2 \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \sqrt{-\tan ^2(c+d x)} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)-3}}{\sqrt{-\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{\sqrt{5} d}",1,"(2*Sqrt[Cos[c + d*x]]*Sqrt[(-3 + 2*Cos[c + d*x])/(-1 + Cos[c + d*x])]*Sqrt[-Cot[(c + d*x)/2]^2]*EllipticF[ArcSin[Sqrt[(-3 + 2*Cos[c + d*x])/(-1 + Cos[c + d*x])]/Sqrt[3]], 6/5]*Tan[(c + d*x)/2])/(Sqrt[5]*d*Sqrt[Cos[c + d*x]/(-1 + Cos[c + d*x])]*Sqrt[-3 + 2*Cos[c + d*x]])","A",1
651,1,153,82,1.0579402,"\int \frac{1}{\sqrt{-3-2 \cos (c+d x)} \sqrt{\cos (c+d x)}} \, dx","Integrate[1/(Sqrt[-3 - 2*Cos[c + d*x]]*Sqrt[Cos[c + d*x]]),x]","\frac{4 \sin ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\cot ^2\left(\frac{1}{2} (c+d x)\right)} \csc (c+d x) \sqrt{-\cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{(2 \cos (c+d x)+3) \csc ^2\left(\frac{1}{2} (c+d x)\right)} F\left(\sin ^{-1}\left(\sqrt{\frac{5}{3}} \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)-1}}\right)|\frac{6}{5}\right)}{\sqrt{5} d \sqrt{-2 \cos (c+d x)-3} \sqrt{\cos (c+d x)}}","-\frac{2 \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \sqrt{-\tan ^2(c+d x)} \csc (c+d x) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{-2 \cos (c+d x)-3}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right)\right|-5\right)}{d}",1,"(4*Sqrt[Cot[(c + d*x)/2]^2]*Sqrt[-(Cos[c + d*x]*Csc[(c + d*x)/2]^2)]*Sqrt[(3 + 2*Cos[c + d*x])*Csc[(c + d*x)/2]^2]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[5/3]*Sqrt[Cos[c + d*x]/(-1 + Cos[c + d*x])]], 6/5]*Sin[(c + d*x)/2]^4)/(Sqrt[5]*d*Sqrt[-3 - 2*Cos[c + d*x]]*Sqrt[Cos[c + d*x]])","A",1
652,1,150,54,0.6139355,"\int \frac{1}{\sqrt{-\cos (c+d x)} \sqrt{2+3 \cos (c+d x)}} \, dx","Integrate[1/(Sqrt[-Cos[c + d*x]]*Sqrt[2 + 3*Cos[c + d*x]]),x]","-\frac{4 \sin ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\cot ^2\left(\frac{1}{2} (c+d x)\right)} \csc (c+d x) \sqrt{-\cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{(3 \cos (c+d x)+2) \csc ^2\left(\frac{1}{2} (c+d x)\right)} F\left(\left.\sin ^{-1}\left(\frac{1}{2} \sqrt{(3 \cos (c+d x)+2) \csc ^2\left(\frac{1}{2} (c+d x)\right)}\right)\right|-4\right)}{d \sqrt{-\cos (c+d x)} \sqrt{3 \cos (c+d x)+2}}","\frac{2 \sqrt{\cos (c+d x)} F\left(\sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)|\frac{1}{5}\right)}{\sqrt{5} d \sqrt{-\cos (c+d x)}}",1,"(-4*Sqrt[Cot[(c + d*x)/2]^2]*Sqrt[-(Cos[c + d*x]*Csc[(c + d*x)/2]^2)]*Sqrt[(2 + 3*Cos[c + d*x])*Csc[(c + d*x)/2]^2]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[(2 + 3*Cos[c + d*x])*Csc[(c + d*x)/2]^2]/2], -4]*Sin[(c + d*x)/2]^4)/(d*Sqrt[-Cos[c + d*x]]*Sqrt[2 + 3*Cos[c + d*x]])","B",1
653,1,158,47,0.4006611,"\int \frac{1}{\sqrt{-\cos (c+d x)} \sqrt{-2+3 \cos (c+d x)}} \, dx","Integrate[1/(Sqrt[-Cos[c + d*x]]*Sqrt[-2 + 3*Cos[c + d*x]]),x]","\frac{4 \sin ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\cot ^2\left(\frac{1}{2} (c+d x)\right)} \csc (c+d x) \sqrt{\cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{-\left((3 \cos (c+d x)-2) \csc ^2\left(\frac{1}{2} (c+d x)\right)\right)} F\left(\sin ^{-1}\left(\frac{1}{2} \sqrt{-\left((3 \cos (c+d x)-2) \csc ^2\left(\frac{1}{2} (c+d x)\right)\right)}\right)|\frac{4}{5}\right)}{\sqrt{5} d \sqrt{-\cos (c+d x)} \sqrt{3 \cos (c+d x)-2}}","\frac{2 \sqrt{\cos (c+d x)} F\left(\left.\sin ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+1}\right)\right|5\right)}{d \sqrt{-\cos (c+d x)}}",1,"(4*Sqrt[Cot[(c + d*x)/2]^2]*Sqrt[Cos[c + d*x]*Csc[(c + d*x)/2]^2]*Sqrt[-((-2 + 3*Cos[c + d*x])*Csc[(c + d*x)/2]^2)]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[-((-2 + 3*Cos[c + d*x])*Csc[(c + d*x)/2]^2)]/2], 4/5]*Sin[(c + d*x)/2]^4)/(Sqrt[5]*d*Sqrt[-Cos[c + d*x]]*Sqrt[-2 + 3*Cos[c + d*x]])","B",1
654,1,145,34,0.5527369,"\int \frac{1}{\sqrt{2-3 \cos (c+d x)} \sqrt{-\cos (c+d x)}} \, dx","Integrate[1/(Sqrt[2 - 3*Cos[c + d*x]]*Sqrt[-Cos[c + d*x]]),x]","-\frac{4 \sin ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\cot ^2\left(\frac{1}{2} (c+d x)\right)} \csc (c+d x) \sqrt{(2-3 \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)} F\left(\left.\sin ^{-1}\left(\frac{1}{2} \sqrt{\cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)}\right)\right|-4\right)}{d \sqrt{2-3 \cos (c+d x)} \sqrt{-\cos (c+d x)}}","-\frac{2 F\left(\sin ^{-1}\left(\frac{\sin (c+d x)}{1-\cos (c+d x)}\right)|\frac{1}{5}\right)}{\sqrt{5} d}",1,"(-4*Sqrt[Cot[(c + d*x)/2]^2]*Sqrt[(2 - 3*Cos[c + d*x])*Csc[(c + d*x)/2]^2]*Sqrt[Cos[c + d*x]*Csc[(c + d*x)/2]^2]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[Cos[c + d*x]*Csc[(c + d*x)/2]^2]/2], -4]*Sin[(c + d*x)/2]^4)/(d*Sqrt[2 - 3*Cos[c + d*x]]*Sqrt[-Cos[c + d*x]])","B",1
655,1,155,27,0.4938381,"\int \frac{1}{\sqrt{-2-3 \cos (c+d x)} \sqrt{-\cos (c+d x)}} \, dx","Integrate[1/(Sqrt[-2 - 3*Cos[c + d*x]]*Sqrt[-Cos[c + d*x]]),x]","\frac{4 \sin ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\cot ^2\left(\frac{1}{2} (c+d x)\right)} \csc (c+d x) \sqrt{-\cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{(3 \cos (c+d x)+2) \csc ^2\left(\frac{1}{2} (c+d x)\right)} F\left(\sin ^{-1}\left(\sqrt{\frac{5}{2}} \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)-1}}\right)|\frac{4}{5}\right)}{\sqrt{5} d \sqrt{-3 \cos (c+d x)-2} \sqrt{-\cos (c+d x)}}","-\frac{2 F\left(\left.\sin ^{-1}\left(\frac{\sin (c+d x)}{1-\cos (c+d x)}\right)\right|5\right)}{d}",1,"(4*Sqrt[Cot[(c + d*x)/2]^2]*Sqrt[-(Cos[c + d*x]*Csc[(c + d*x)/2]^2)]*Sqrt[(2 + 3*Cos[c + d*x])*Csc[(c + d*x)/2]^2]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[5/2]*Sqrt[Cos[c + d*x]/(-1 + Cos[c + d*x])]], 4/5]*Sin[(c + d*x)/2]^4)/(Sqrt[5]*d*Sqrt[-2 - 3*Cos[c + d*x]]*Sqrt[-Cos[c + d*x]])","B",1
656,1,154,80,0.6149885,"\int \frac{1}{\sqrt{-\cos (c+d x)} \sqrt{3+2 \cos (c+d x)}} \, dx","Integrate[1/(Sqrt[-Cos[c + d*x]]*Sqrt[3 + 2*Cos[c + d*x]]),x]","-\frac{4 \sin ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{-\cot ^2\left(\frac{1}{2} (c+d x)\right)} \csc (c+d x) \sqrt{-\cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{(2 \cos (c+d x)+3) \csc ^2\left(\frac{1}{2} (c+d x)\right)} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{(2 \cos (c+d x)+3) \csc ^2\left(\frac{1}{2} (c+d x)\right)}}{\sqrt{6}}\right)\right|6\right)}{d \sqrt{-\cos (c+d x)} \sqrt{2 \cos (c+d x)+3}}","\frac{2 \cos ^{\frac{3}{2}}(c+d x) \sqrt{-\tan ^2(c+d x)} \csc (c+d x) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)+3}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right)\right|-5\right)}{d \sqrt{-\cos (c+d x)}}",1,"(-4*Sqrt[-Cot[(c + d*x)/2]^2]*Sqrt[-(Cos[c + d*x]*Csc[(c + d*x)/2]^2)]*Sqrt[(3 + 2*Cos[c + d*x])*Csc[(c + d*x)/2]^2]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[(3 + 2*Cos[c + d*x])*Csc[(c + d*x)/2]^2]/Sqrt[6]], 6]*Sin[(c + d*x)/2]^4)/(d*Sqrt[-Cos[c + d*x]]*Sqrt[3 + 2*Cos[c + d*x]])","A",1
657,1,146,82,0.4833328,"\int \frac{1}{\sqrt{3-2 \cos (c+d x)} \sqrt{-\cos (c+d x)}} \, dx","Integrate[1/(Sqrt[3 - 2*Cos[c + d*x]]*Sqrt[-Cos[c + d*x]]),x]","\frac{4 \sin ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\cot ^2\left(\frac{1}{2} (c+d x)\right)} \csc (c+d x) \sqrt{(3-2 \cos (c+d x)) \csc ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{-\cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{\frac{\cos (c+d x)}{\cos (c+d x)-1}}}{\sqrt{3}}\right)\right|6\right)}{d \sqrt{3-2 \cos (c+d x)} \sqrt{-\cos (c+d x)}}","\frac{2 \cos ^{\frac{3}{2}}(c+d x) \sqrt{-\tan ^2(c+d x)} \csc (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{3-2 \cos (c+d x)}}{\sqrt{\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{\sqrt{5} d \sqrt{-\cos (c+d x)}}",1,"(4*Sqrt[Cot[(c + d*x)/2]^2]*Sqrt[(3 - 2*Cos[c + d*x])*Csc[(c + d*x)/2]^2]*Sqrt[-(Cos[c + d*x]*Csc[(c + d*x)/2]^2)]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[Cos[c + d*x]/(-1 + Cos[c + d*x])]/Sqrt[3]], 6]*Sin[(c + d*x)/2]^4)/(d*Sqrt[3 - 2*Cos[c + d*x]]*Sqrt[-Cos[c + d*x]])","A",1
658,1,160,62,0.6892174,"\int \frac{1}{\sqrt{-\cos (c+d x)} \sqrt{-3+2 \cos (c+d x)}} \, dx","Integrate[1/(Sqrt[-Cos[c + d*x]]*Sqrt[-3 + 2*Cos[c + d*x]]),x]","\frac{4 \sin ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{-\cot ^2\left(\frac{1}{2} (c+d x)\right)} \cot (c+d x) \sqrt{-\cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{-\left((2 \cos (c+d x)-3) \csc ^2\left(\frac{1}{2} (c+d x)\right)\right)} F\left(\sin ^{-1}\left(\frac{\sqrt{\frac{2 \cos (c+d x)-3}{\cos (c+d x)-1}}}{\sqrt{3}}\right)|\frac{6}{5}\right)}{\sqrt{5} d (-\cos (c+d x))^{3/2} \sqrt{2 \cos (c+d x)-3}}","-\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) F\left(\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)-3}}{\sqrt{-\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{\sqrt{5} d}",1,"(4*Sqrt[-Cot[(c + d*x)/2]^2]*Cot[c + d*x]*Sqrt[-(Cos[c + d*x]*Csc[(c + d*x)/2]^2)]*Sqrt[-((-3 + 2*Cos[c + d*x])*Csc[(c + d*x)/2]^2)]*EllipticF[ArcSin[Sqrt[(-3 + 2*Cos[c + d*x])/(-1 + Cos[c + d*x])]/Sqrt[3]], 6/5]*Sin[(c + d*x)/2]^4)/(Sqrt[5]*d*(-Cos[c + d*x])^(3/2)*Sqrt[-3 + 2*Cos[c + d*x]])","B",1
659,1,155,60,0.4614839,"\int \frac{1}{\sqrt{-3-2 \cos (c+d x)} \sqrt{-\cos (c+d x)}} \, dx","Integrate[1/(Sqrt[-3 - 2*Cos[c + d*x]]*Sqrt[-Cos[c + d*x]]),x]","\frac{4 \sin ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\cot ^2\left(\frac{1}{2} (c+d x)\right)} \csc (c+d x) \sqrt{-\cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{(2 \cos (c+d x)+3) \csc ^2\left(\frac{1}{2} (c+d x)\right)} F\left(\sin ^{-1}\left(\sqrt{\frac{5}{3}} \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)-1}}\right)|\frac{6}{5}\right)}{\sqrt{5} d \sqrt{-2 \cos (c+d x)-3} \sqrt{-\cos (c+d x)}}","-\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) F\left(\left.\sin ^{-1}\left(\frac{\sqrt{-2 \cos (c+d x)-3}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right)\right|-5\right)}{d}",1,"(4*Sqrt[Cot[(c + d*x)/2]^2]*Sqrt[-(Cos[c + d*x]*Csc[(c + d*x)/2]^2)]*Sqrt[(3 + 2*Cos[c + d*x])*Csc[(c + d*x)/2]^2]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[5/3]*Sqrt[Cos[c + d*x]/(-1 + Cos[c + d*x])]], 6/5]*Sin[(c + d*x)/2]^4)/(Sqrt[5]*d*Sqrt[-3 - 2*Cos[c + d*x]]*Sqrt[-Cos[c + d*x]])","B",1
660,1,175,77,2.8405147,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{2+3 \cos (c+d x)}} \, dx","Integrate[Sqrt[Cos[c + d*x]]/Sqrt[2 + 3*Cos[c + d*x]],x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{3 \cos (c+d x)+2} \sqrt{\cot ^2\left(\frac{1}{2} (c+d x)\right)} \csc (c+d x) \left(3 F\left(\left.\sin ^{-1}\left(\frac{1}{2} \sqrt{(3 \cos (c+d x)+2) \csc ^2\left(\frac{1}{2} (c+d x)\right)}\right)\right|-4\right)-5 \Pi \left(-\frac{2}{3};\left.\sin ^{-1}\left(\frac{1}{2} \sqrt{(3 \cos (c+d x)+2) \csc ^2\left(\frac{1}{2} (c+d x)\right)}\right)\right|-4\right)\right)}{3 d \sqrt{\frac{-3 \cos (c+d x)-2}{\cos (c+d x)-1}} \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)-1}}}","-\frac{4 \cot (c+d x) \sqrt{-\sec (c+d x)-1} \sqrt{1-\sec (c+d x)} \Pi \left(\frac{5}{3};\left.\sin ^{-1}\left(\frac{\sqrt{3 \cos (c+d x)+2}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right)\right|5\right)}{3 d}",1,"(2*Sqrt[Cos[c + d*x]]*Sqrt[2 + 3*Cos[c + d*x]]*Sqrt[Cot[(c + d*x)/2]^2]*Csc[c + d*x]*(3*EllipticF[ArcSin[Sqrt[(2 + 3*Cos[c + d*x])*Csc[(c + d*x)/2]^2]/2], -4] - 5*EllipticPi[-2/3, ArcSin[Sqrt[(2 + 3*Cos[c + d*x])*Csc[(c + d*x)/2]^2]/2], -4]))/(3*d*Sqrt[(-2 - 3*Cos[c + d*x])/(-1 + Cos[c + d*x])]*Sqrt[Cos[c + d*x]/(-1 + Cos[c + d*x])])","B",1
661,1,140,75,0.691949,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{-2+3 \cos (c+d x)}} \, dx","Integrate[Sqrt[Cos[c + d*x]]/Sqrt[-2 + 3*Cos[c + d*x]],x]","-\frac{4 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{3 \cos (c+d x)-2}{\cos (c+d x)+1}} \left(F\left(\sin ^{-1}\left(\sqrt{5} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{1}{5}\right)-2 \Pi \left(-\frac{1}{5};\sin ^{-1}\left(\sqrt{5} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{1}{5}\right)\right)}{\sqrt{5} d \sqrt{\cos (c+d x)} \sqrt{3 \cos (c+d x)-2}}","-\frac{4 \cot (c+d x) \sqrt{\sec (c+d x)-1} \sqrt{\sec (c+d x)+1} \Pi \left(\frac{1}{3};\sin ^{-1}\left(\frac{\sqrt{3 \cos (c+d x)-2}}{\sqrt{\cos (c+d x)}}\right)|\frac{1}{5}\right)}{3 \sqrt{5} d}",1,"(-4*Cos[(c + d*x)/2]^2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(-2 + 3*Cos[c + d*x])/(1 + Cos[c + d*x])]*(EllipticF[ArcSin[Sqrt[5]*Tan[(c + d*x)/2]], 1/5] - 2*EllipticPi[-1/5, ArcSin[Sqrt[5]*Tan[(c + d*x)/2]], 1/5]))/(Sqrt[5]*d*Sqrt[Cos[c + d*x]]*Sqrt[-2 + 3*Cos[c + d*x]])","A",1
662,1,145,99,1.7673521,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{2-3 \cos (c+d x)}} \, dx","Integrate[Sqrt[Cos[c + d*x]]/Sqrt[2 - 3*Cos[c + d*x]],x]","-\frac{4 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{-\frac{(2-3 \cos (c+d x))^2}{(\cos (c+d x)+1)^2}} \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \left(F\left(\left.\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)\right|5\right)-2 \Pi \left(-1;\left.\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)\right|5\right)\right)}{d \sqrt{2-3 \cos (c+d x)} \sqrt{\cos (c+d x)} \sqrt{\frac{2-3 \cos (c+d x)}{\cos (c+d x)+1}}}","-\frac{4 \cos ^{\frac{3}{2}}(c+d x) \csc (c+d x) \sqrt{\sec (c+d x)-1} \sqrt{\sec (c+d x)+1} \Pi \left(\frac{1}{3};\sin ^{-1}\left(\frac{\sqrt{2-3 \cos (c+d x)}}{\sqrt{-\cos (c+d x)}}\right)|\frac{1}{5}\right)}{3 \sqrt{5} d \sqrt{-\cos (c+d x)}}",1,"(-4*Cos[(c + d*x)/2]^2*Sqrt[-((2 - 3*Cos[c + d*x])^2/(1 + Cos[c + d*x])^2)]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*(EllipticF[ArcSin[Tan[(c + d*x)/2]], 5] - 2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], 5]))/(d*Sqrt[2 - 3*Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Sqrt[(2 - 3*Cos[c + d*x])/(1 + Cos[c + d*x])])","A",1
663,1,155,101,2.0508757,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{-2-3 \cos (c+d x)}} \, dx","Integrate[Sqrt[Cos[c + d*x]]/Sqrt[-2 - 3*Cos[c + d*x]],x]","-\frac{4 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{-\frac{(3 \cos (c+d x)+2)^2}{(\cos (c+d x)+1)^2}} \left(F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{1}{5}\right)-2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{1}{5}\right)\right)}{\sqrt{5} d \sqrt{-3 \cos (c+d x)-2} \sqrt{\cos (c+d x)} \sqrt{-\frac{3 \cos (c+d x)+2}{\cos (c+d x)+1}}}","-\frac{4 \cos ^{\frac{3}{2}}(c+d x) \csc (c+d x) \sqrt{-\sec (c+d x)-1} \sqrt{1-\sec (c+d x)} \Pi \left(\frac{5}{3};\left.\sin ^{-1}\left(\frac{\sqrt{-3 \cos (c+d x)-2}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right)\right|5\right)}{3 d \sqrt{-\cos (c+d x)}}",1,"(-4*Cos[(c + d*x)/2]^2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[-((2 + 3*Cos[c + d*x])^2/(1 + Cos[c + d*x])^2)]*(EllipticF[ArcSin[Tan[(c + d*x)/2]], 1/5] - 2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], 1/5]))/(Sqrt[5]*d*Sqrt[-2 - 3*Cos[c + d*x]]*Sqrt[Cos[c + d*x]]*Sqrt[-((2 + 3*Cos[c + d*x])/(1 + Cos[c + d*x]))])","A",1
664,1,115,73,1.4391955,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{3+2 \cos (c+d x)}} \, dx","Integrate[Sqrt[Cos[c + d*x]]/Sqrt[3 + 2*Cos[c + d*x]],x]","-\frac{2 \sqrt{\cos (c+d x)} \sqrt{2 \cos (c+d x)+3} \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{1}{5}\right)-2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{1}{5}\right)\right)}{\sqrt{5} d \sqrt{(3 \cos (c+d x)+\cos (2 (c+d x))+1) \sec ^4\left(\frac{1}{2} (c+d x)\right)}}","-\frac{3 \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left(\frac{5}{2};\left.\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)+3}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right)\right|-5\right)}{d}",1,"(-2*Sqrt[Cos[c + d*x]]*Sqrt[3 + 2*Cos[c + d*x]]*(EllipticF[ArcSin[Tan[(c + d*x)/2]], -1/5] - 2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], -1/5])*Sec[(c + d*x)/2]^2)/(Sqrt[5]*d*Sqrt[(1 + 3*Cos[c + d*x] + Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^4])","A",1
665,1,117,75,0.8411041,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{3-2 \cos (c+d x)}} \, dx","Integrate[Sqrt[Cos[c + d*x]]/Sqrt[3 - 2*Cos[c + d*x]],x]","-\frac{4 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{3-2 \cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \left(F\left(\left.\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)\right|-5\right)-2 \Pi \left(-1;\left.\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)\right|-5\right)\right)}{d \sqrt{3-2 \cos (c+d x)} \sqrt{\cos (c+d x)}}","\frac{3 \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left(-\frac{1}{2};\sin ^{-1}\left(\frac{\sqrt{3-2 \cos (c+d x)}}{\sqrt{\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{\sqrt{5} d}",1,"(-4*Cos[(c + d*x)/2]^2*Sqrt[(3 - 2*Cos[c + d*x])/(1 + Cos[c + d*x])]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*(EllipticF[ArcSin[Tan[(c + d*x)/2]], -5] - 2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], -5]))/(d*Sqrt[3 - 2*Cos[c + d*x]]*Sqrt[Cos[c + d*x]])","A",1
666,1,135,99,0.9846789,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{-3+2 \cos (c+d x)}} \, dx","Integrate[Sqrt[Cos[c + d*x]]/Sqrt[-3 + 2*Cos[c + d*x]],x]","-\frac{2 i \sqrt{2 \cos (c+d x)-3} \sqrt{\frac{\cos (c+d x)}{5 \cos (c+d x)+5}} \left(F\left(i \sinh ^{-1}\left(\sqrt{5} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{1}{5}\right)-2 \Pi \left(\frac{1}{5};i \sinh ^{-1}\left(\sqrt{5} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{1}{5}\right)\right)}{d \sqrt{\cos (c+d x)} \sqrt{\frac{3-2 \cos (c+d x)}{\cos (c+d x)+1}}}","\frac{3 \cos ^{\frac{3}{2}}(c+d x) \csc (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left(-\frac{1}{2};\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)-3}}{\sqrt{-\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{\sqrt{5} d \sqrt{-\cos (c+d x)}}",1,"((-2*I)*Sqrt[-3 + 2*Cos[c + d*x]]*Sqrt[Cos[c + d*x]/(5 + 5*Cos[c + d*x])]*(EllipticF[I*ArcSinh[Sqrt[5]*Tan[(c + d*x)/2]], -1/5] - 2*EllipticPi[1/5, I*ArcSinh[Sqrt[5]*Tan[(c + d*x)/2]], -1/5]))/(d*Sqrt[Cos[c + d*x]]*Sqrt[(3 - 2*Cos[c + d*x])/(1 + Cos[c + d*x])])","C",1
667,1,113,97,0.5999959,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{-3-2 \cos (c+d x)}} \, dx","Integrate[Sqrt[Cos[c + d*x]]/Sqrt[-3 - 2*Cos[c + d*x]],x]","-\frac{2 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x) (2 \cos (c+d x)+3) \sec ^4\left(\frac{1}{2} (c+d x)\right)} \left(F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{1}{5}\right)-2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{1}{5}\right)\right)}{\sqrt{5} d \sqrt{-2 \cos (c+d x)-3} \sqrt{\cos (c+d x)}}","-\frac{3 \cos ^{\frac{3}{2}}(c+d x) \csc (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left(\frac{5}{2};\left.\sin ^{-1}\left(\frac{\sqrt{-2 \cos (c+d x)-3}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right)\right|-5\right)}{d \sqrt{-\cos (c+d x)}}",1,"(-2*Cos[(c + d*x)/2]^2*(EllipticF[ArcSin[Tan[(c + d*x)/2]], -1/5] - 2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], -1/5])*Sqrt[Cos[c + d*x]*(3 + 2*Cos[c + d*x])*Sec[(c + d*x)/2]^4])/(Sqrt[5]*d*Sqrt[-3 - 2*Cos[c + d*x]]*Sqrt[Cos[c + d*x]])","A",1
668,1,194,99,0.7657597,"\int \frac{\sqrt{-\cos (c+d x)}}{\sqrt{2+3 \cos (c+d x)}} \, dx","Integrate[Sqrt[-Cos[c + d*x]]/Sqrt[2 + 3*Cos[c + d*x]],x]","\frac{4 \sin ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\cot ^2\left(\frac{1}{2} (c+d x)\right)} \csc (c+d x) \sqrt{-\cos (c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{(3 \cos (c+d x)+2) \csc ^2\left(\frac{1}{2} (c+d x)\right)} \left(3 F\left(\left.\sin ^{-1}\left(\frac{1}{2} \sqrt{(3 \cos (c+d x)+2) \csc ^2\left(\frac{1}{2} (c+d x)\right)}\right)\right|-4\right)-5 \Pi \left(-\frac{2}{3};\left.\sin ^{-1}\left(\frac{1}{2} \sqrt{(3 \cos (c+d x)+2) \csc ^2\left(\frac{1}{2} (c+d x)\right)}\right)\right|-4\right)\right)}{3 d \sqrt{-\cos (c+d x)} \sqrt{3 \cos (c+d x)+2}}","-\frac{4 \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)} \Pi \left(\frac{5}{3};\left.\sin ^{-1}\left(\frac{\sqrt{3 \cos (c+d x)+2}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right)\right|5\right)}{3 d}",1,"(4*Sqrt[Cot[(c + d*x)/2]^2]*Sqrt[-(Cos[c + d*x]*Csc[(c + d*x)/2]^2)]*Sqrt[(2 + 3*Cos[c + d*x])*Csc[(c + d*x)/2]^2]*Csc[c + d*x]*(3*EllipticF[ArcSin[Sqrt[(2 + 3*Cos[c + d*x])*Csc[(c + d*x)/2]^2]/2], -4] - 5*EllipticPi[-2/3, ArcSin[Sqrt[(2 + 3*Cos[c + d*x])*Csc[(c + d*x)/2]^2]/2], -4])*Sin[(c + d*x)/2]^4)/(3*d*Sqrt[-Cos[c + d*x]]*Sqrt[2 + 3*Cos[c + d*x]])","A",1
669,1,142,97,0.2235069,"\int \frac{\sqrt{-\cos (c+d x)}}{\sqrt{-2+3 \cos (c+d x)}} \, dx","Integrate[Sqrt[-Cos[c + d*x]]/Sqrt[-2 + 3*Cos[c + d*x]],x]","\frac{4 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{3 \cos (c+d x)-2}{\cos (c+d x)+1}} \left(F\left(\sin ^{-1}\left(\sqrt{5} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{1}{5}\right)-2 \Pi \left(-\frac{1}{5};\sin ^{-1}\left(\sqrt{5} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{1}{5}\right)\right)}{\sqrt{5} d \sqrt{-\cos (c+d x)} \sqrt{3 \cos (c+d x)-2}}","-\frac{4 \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} \Pi \left(\frac{1}{3};\sin ^{-1}\left(\frac{\sqrt{3 \cos (c+d x)-2}}{\sqrt{\cos (c+d x)}}\right)|\frac{1}{5}\right)}{3 \sqrt{5} d}",1,"(4*Cos[(c + d*x)/2]^2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(-2 + 3*Cos[c + d*x])/(1 + Cos[c + d*x])]*(EllipticF[ArcSin[Sqrt[5]*Tan[(c + d*x)/2]], 1/5] - 2*EllipticPi[-1/5, ArcSin[Sqrt[5]*Tan[(c + d*x)/2]], 1/5]))/(Sqrt[5]*d*Sqrt[-Cos[c + d*x]]*Sqrt[-2 + 3*Cos[c + d*x]])","A",1
670,1,147,77,0.5093711,"\int \frac{\sqrt{-\cos (c+d x)}}{\sqrt{2-3 \cos (c+d x)}} \, dx","Integrate[Sqrt[-Cos[c + d*x]]/Sqrt[2 - 3*Cos[c + d*x]],x]","\frac{4 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{-\frac{(2-3 \cos (c+d x))^2}{(\cos (c+d x)+1)^2}} \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \left(F\left(\left.\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)\right|5\right)-2 \Pi \left(-1;\left.\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)\right|5\right)\right)}{d \sqrt{2-3 \cos (c+d x)} \sqrt{-\cos (c+d x)} \sqrt{\frac{2-3 \cos (c+d x)}{\cos (c+d x)+1}}}","-\frac{4 \cot (c+d x) \sqrt{\sec (c+d x)-1} \sqrt{\sec (c+d x)+1} \Pi \left(\frac{1}{3};\sin ^{-1}\left(\frac{\sqrt{2-3 \cos (c+d x)}}{\sqrt{-\cos (c+d x)}}\right)|\frac{1}{5}\right)}{3 \sqrt{5} d}",1,"(4*Cos[(c + d*x)/2]^2*Sqrt[-((2 - 3*Cos[c + d*x])^2/(1 + Cos[c + d*x])^2)]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*(EllipticF[ArcSin[Tan[(c + d*x)/2]], 5] - 2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], 5]))/(d*Sqrt[2 - 3*Cos[c + d*x]]*Sqrt[-Cos[c + d*x]]*Sqrt[(2 - 3*Cos[c + d*x])/(1 + Cos[c + d*x])])","A",1
671,1,156,79,0.508657,"\int \frac{\sqrt{-\cos (c+d x)}}{\sqrt{-2-3 \cos (c+d x)}} \, dx","Integrate[Sqrt[-Cos[c + d*x]]/Sqrt[-2 - 3*Cos[c + d*x]],x]","\frac{4 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{-\frac{(3 \cos (c+d x)+2)^2}{(\cos (c+d x)+1)^2}} \left(F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{1}{5}\right)-2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{1}{5}\right)\right)}{\sqrt{5} d \sqrt{-3 \cos (c+d x)-2} \sqrt{-\cos (c+d x)} \sqrt{\frac{-3 \cos (c+d x)-2}{\cos (c+d x)+1}}}","-\frac{4 \cot (c+d x) \sqrt{-\sec (c+d x)-1} \sqrt{1-\sec (c+d x)} \Pi \left(\frac{5}{3};\left.\sin ^{-1}\left(\frac{\sqrt{-3 \cos (c+d x)-2}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right)\right|5\right)}{3 d}",1,"(4*Cos[(c + d*x)/2]^2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[-((2 + 3*Cos[c + d*x])^2/(1 + Cos[c + d*x])^2)]*(EllipticF[ArcSin[Tan[(c + d*x)/2]], 1/5] - 2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], 1/5]))/(Sqrt[5]*d*Sqrt[-2 - 3*Cos[c + d*x]]*Sqrt[-Cos[c + d*x]]*Sqrt[(-2 - 3*Cos[c + d*x])/(1 + Cos[c + d*x])])","A",1
672,1,117,95,0.5027323,"\int \frac{\sqrt{-\cos (c+d x)}}{\sqrt{3+2 \cos (c+d x)}} \, dx","Integrate[Sqrt[-Cos[c + d*x]]/Sqrt[3 + 2*Cos[c + d*x]],x]","-\frac{2 \sqrt{-\cos (c+d x)} \sqrt{2 \cos (c+d x)+3} \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{1}{5}\right)-2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{1}{5}\right)\right)}{\sqrt{5} d \sqrt{(3 \cos (c+d x)+\cos (2 (c+d x))+1) \sec ^4\left(\frac{1}{2} (c+d x)\right)}}","-\frac{3 \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} \Pi \left(\frac{5}{2};\left.\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)+3}}{\sqrt{5} \sqrt{\cos (c+d x)}}\right)\right|-5\right)}{d}",1,"(-2*Sqrt[-Cos[c + d*x]]*Sqrt[3 + 2*Cos[c + d*x]]*(EllipticF[ArcSin[Tan[(c + d*x)/2]], -1/5] - 2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], -1/5])*Sec[(c + d*x)/2]^2)/(Sqrt[5]*d*Sqrt[(1 + 3*Cos[c + d*x] + Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^4])","A",1
673,1,119,97,0.2233745,"\int \frac{\sqrt{-\cos (c+d x)}}{\sqrt{3-2 \cos (c+d x)}} \, dx","Integrate[Sqrt[-Cos[c + d*x]]/Sqrt[3 - 2*Cos[c + d*x]],x]","\frac{4 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{3-2 \cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \left(F\left(\left.\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)\right|-5\right)-2 \Pi \left(-1;\left.\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)\right|-5\right)\right)}{d \sqrt{3-2 \cos (c+d x)} \sqrt{-\cos (c+d x)}}","\frac{3 \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} \Pi \left(-\frac{1}{2};\sin ^{-1}\left(\frac{\sqrt{3-2 \cos (c+d x)}}{\sqrt{\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{\sqrt{5} d}",1,"(4*Cos[(c + d*x)/2]^2*Sqrt[(3 - 2*Cos[c + d*x])/(1 + Cos[c + d*x])]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*(EllipticF[ArcSin[Tan[(c + d*x)/2]], -5] - 2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], -5]))/(d*Sqrt[3 - 2*Cos[c + d*x]]*Sqrt[-Cos[c + d*x]])","A",1
674,1,140,77,0.1586692,"\int \frac{\sqrt{-\cos (c+d x)}}{\sqrt{-3+2 \cos (c+d x)}} \, dx","Integrate[Sqrt[-Cos[c + d*x]]/Sqrt[-3 + 2*Cos[c + d*x]],x]","\frac{2 i \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{2 \cos (c+d x)-3} \left(F\left(i \sinh ^{-1}\left(\sqrt{5} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{1}{5}\right)-2 \Pi \left(\frac{1}{5};i \sinh ^{-1}\left(\sqrt{5} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{1}{5}\right)\right)}{\sqrt{5} d \sqrt{-\cos (c+d x)} \sqrt{\frac{3-2 \cos (c+d x)}{\cos (c+d x)+1}}}","\frac{3 \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left(-\frac{1}{2};\sin ^{-1}\left(\frac{\sqrt{2 \cos (c+d x)-3}}{\sqrt{-\cos (c+d x)}}\right)|-\frac{1}{5}\right)}{\sqrt{5} d}",1,"((2*I)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[-3 + 2*Cos[c + d*x]]*(EllipticF[I*ArcSinh[Sqrt[5]*Tan[(c + d*x)/2]], -1/5] - 2*EllipticPi[1/5, I*ArcSinh[Sqrt[5]*Tan[(c + d*x)/2]], -1/5]))/(Sqrt[5]*d*Sqrt[-Cos[c + d*x]]*Sqrt[(3 - 2*Cos[c + d*x])/(1 + Cos[c + d*x])])","C",1
675,1,115,75,0.3051899,"\int \frac{\sqrt{-\cos (c+d x)}}{\sqrt{-3-2 \cos (c+d x)}} \, dx","Integrate[Sqrt[-Cos[c + d*x]]/Sqrt[-3 - 2*Cos[c + d*x]],x]","\frac{2 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x) (2 \cos (c+d x)+3) \sec ^4\left(\frac{1}{2} (c+d x)\right)} \left(F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{1}{5}\right)-2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{1}{5}\right)\right)}{\sqrt{5} d \sqrt{-2 \cos (c+d x)-3} \sqrt{-\cos (c+d x)}}","-\frac{3 \cot (c+d x) \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1} \Pi \left(\frac{5}{2};\left.\sin ^{-1}\left(\frac{\sqrt{-2 \cos (c+d x)-3}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right)\right|-5\right)}{d}",1,"(2*Cos[(c + d*x)/2]^2*(EllipticF[ArcSin[Tan[(c + d*x)/2]], -1/5] - 2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], -1/5])*Sqrt[Cos[c + d*x]*(3 + 2*Cos[c + d*x])*Sec[(c + d*x)/2]^4])/(Sqrt[5]*d*Sqrt[-3 - 2*Cos[c + d*x]]*Sqrt[-Cos[c + d*x]])","A",1
676,1,4614,176,21.7014924,"\int \frac{\cos ^{\frac{2}{3}}(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^(2/3)/(a + b*Cos[c + d*x]),x]","\text{Result too large to show}","\frac{a \sin (c+d x) \sqrt[6]{\cos ^2(c+d x)} F_1\left(\frac{1}{2};\frac{1}{6},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \sqrt[3]{\cos (c+d x)}}-\frac{b \sin (c+d x) \cos ^{\frac{2}{3}}(c+d x) F_1\left(\frac{1}{2};-\frac{1}{3},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \sqrt[3]{\cos ^2(c+d x)}}",1,"(9*(a^2 - b^2)*Sin[c + d*x]*((a*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^2)*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] - 2*(3*a^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (b*AppellF1[1/2, 5/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])/(-9*(a^2 - b^2)*AppellF1[1/2, 5/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (6*a^2*AppellF1[3/2, 5/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 5*(a^2 - b^2)*AppellF1[3/2, 11/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2)))/(d*Cos[c + d*x]^(1/3)*(a + b*Cos[c + d*x])*(Sec[c + d*x]^2)^(5/6)*(-b^2 + a^2*Sec[c + d*x]^2)*((9*(a^2 - b^2)*(Sec[c + d*x]^2)^(1/6)*((a*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^2)*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] - 2*(3*a^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (b*AppellF1[1/2, 5/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])/(-9*(a^2 - b^2)*AppellF1[1/2, 5/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (6*a^2*AppellF1[3/2, 5/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 5*(a^2 - b^2)*AppellF1[3/2, 11/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2)))/(-b^2 + a^2*Sec[c + d*x]^2) - (18*a^2*(a^2 - b^2)*(Sec[c + d*x]^2)^(1/6)*Tan[c + d*x]^2*((a*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^2)*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] - 2*(3*a^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (b*AppellF1[1/2, 5/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])/(-9*(a^2 - b^2)*AppellF1[1/2, 5/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (6*a^2*AppellF1[3/2, 5/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 5*(a^2 - b^2)*AppellF1[3/2, 11/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2)))/(-b^2 + a^2*Sec[c + d*x]^2)^2 - (15*(a^2 - b^2)*Tan[c + d*x]^2*((a*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^2)*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] - 2*(3*a^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (b*AppellF1[1/2, 5/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])/(-9*(a^2 - b^2)*AppellF1[1/2, 5/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (6*a^2*AppellF1[3/2, 5/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 5*(a^2 - b^2)*AppellF1[3/2, 11/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2)))/((Sec[c + d*x]^2)^(5/6)*(-b^2 + a^2*Sec[c + d*x]^2)) + (9*(a^2 - b^2)*Tan[c + d*x]*((a*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2]*Tan[c + d*x])/(9*(a^2 - b^2)*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] - 2*(3*a^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (a*Sqrt[Sec[c + d*x]^2]*((-2*a^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(3*(a^2 - b^2)) - (2*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/9))/(9*(a^2 - b^2)*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] - 2*(3*a^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (b*((-2*a^2*AppellF1[3/2, 5/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(3*(a^2 - b^2)) - (5*AppellF1[3/2, 11/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/9))/(-9*(a^2 - b^2)*AppellF1[1/2, 5/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (6*a^2*AppellF1[3/2, 5/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 5*(a^2 - b^2)*AppellF1[3/2, 11/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) - (a*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2]*(-4*(3*a^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Sec[c + d*x]^2*Tan[c + d*x] + 9*(a^2 - b^2)*((-2*a^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(3*(a^2 - b^2)) - (2*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/9) - 2*Tan[c + d*x]^2*(3*a^2*((-12*a^2*AppellF1[5/2, 1/3, 3, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(5*(a^2 - b^2)) - (2*AppellF1[5/2, 4/3, 2, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/5) + (a^2 - b^2)*((-6*a^2*AppellF1[5/2, 4/3, 2, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(5*(a^2 - b^2)) - (8*AppellF1[5/2, 7/3, 1, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/5))))/(9*(a^2 - b^2)*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] - 2*(3*a^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2)^2 - (b*AppellF1[1/2, 5/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*(2*(6*a^2*AppellF1[3/2, 5/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 5*(a^2 - b^2)*AppellF1[3/2, 11/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Sec[c + d*x]^2*Tan[c + d*x] - 9*(a^2 - b^2)*((-2*a^2*AppellF1[3/2, 5/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(3*(a^2 - b^2)) - (5*AppellF1[3/2, 11/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/9) + Tan[c + d*x]^2*(6*a^2*((-12*a^2*AppellF1[5/2, 5/6, 3, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(5*(a^2 - b^2)) - AppellF1[5/2, 11/6, 2, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x]) + 5*(a^2 - b^2)*((-6*a^2*AppellF1[5/2, 11/6, 2, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(5*(a^2 - b^2)) - (11*AppellF1[5/2, 17/6, 1, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/5))))/(-9*(a^2 - b^2)*AppellF1[1/2, 5/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (6*a^2*AppellF1[3/2, 5/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 5*(a^2 - b^2)*AppellF1[3/2, 11/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2)^2))/((Sec[c + d*x]^2)^(5/6)*(-b^2 + a^2*Sec[c + d*x]^2))))","B",0
677,1,4613,176,21.3280723,"\int \frac{\sqrt[3]{\cos (c+d x)}}{a+b \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^(1/3)/(a + b*Cos[c + d*x]),x]","\text{Result too large to show}","\frac{a \sin (c+d x) \sqrt[3]{\cos ^2(c+d x)} F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \cos ^{\frac{2}{3}}(c+d x)}-\frac{b \sin (c+d x) \sqrt[3]{\cos (c+d x)} F_1\left(\frac{1}{2};-\frac{1}{6},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \sqrt[6]{\cos ^2(c+d x)}}",1,"(9*(a^2 - b^2)*Sin[c + d*x]*((a*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^2)*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-6*a^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-a^2 + b^2)*AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (b*AppellF1[1/2, 2/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])/(-9*(a^2 - b^2)*AppellF1[1/2, 2/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 2*(3*a^2*AppellF1[3/2, 2/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 2*(a^2 - b^2)*AppellF1[3/2, 5/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2)))/(d*Cos[c + d*x]^(2/3)*(a + b*Cos[c + d*x])*(Sec[c + d*x]^2)^(2/3)*(-b^2 + a^2*Sec[c + d*x]^2)*((9*(a^2 - b^2)*(Sec[c + d*x]^2)^(1/3)*((a*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^2)*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-6*a^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-a^2 + b^2)*AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (b*AppellF1[1/2, 2/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])/(-9*(a^2 - b^2)*AppellF1[1/2, 2/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 2*(3*a^2*AppellF1[3/2, 2/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 2*(a^2 - b^2)*AppellF1[3/2, 5/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2)))/(-b^2 + a^2*Sec[c + d*x]^2) - (18*a^2*(a^2 - b^2)*(Sec[c + d*x]^2)^(1/3)*Tan[c + d*x]^2*((a*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^2)*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-6*a^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-a^2 + b^2)*AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (b*AppellF1[1/2, 2/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])/(-9*(a^2 - b^2)*AppellF1[1/2, 2/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 2*(3*a^2*AppellF1[3/2, 2/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 2*(a^2 - b^2)*AppellF1[3/2, 5/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2)))/(-b^2 + a^2*Sec[c + d*x]^2)^2 - (12*(a^2 - b^2)*Tan[c + d*x]^2*((a*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^2)*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-6*a^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-a^2 + b^2)*AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (b*AppellF1[1/2, 2/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])/(-9*(a^2 - b^2)*AppellF1[1/2, 2/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 2*(3*a^2*AppellF1[3/2, 2/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 2*(a^2 - b^2)*AppellF1[3/2, 5/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2)))/((Sec[c + d*x]^2)^(2/3)*(-b^2 + a^2*Sec[c + d*x]^2)) + (9*(a^2 - b^2)*Tan[c + d*x]*((a*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2]*Tan[c + d*x])/(9*(a^2 - b^2)*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-6*a^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-a^2 + b^2)*AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (a*Sqrt[Sec[c + d*x]^2]*((-2*a^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(3*(a^2 - b^2)) - (AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/9))/(9*(a^2 - b^2)*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-6*a^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-a^2 + b^2)*AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (b*((-2*a^2*AppellF1[3/2, 2/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(3*(a^2 - b^2)) - (4*AppellF1[3/2, 5/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/9))/(-9*(a^2 - b^2)*AppellF1[1/2, 2/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 2*(3*a^2*AppellF1[3/2, 2/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 2*(a^2 - b^2)*AppellF1[3/2, 5/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) - (a*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2]*(2*(-6*a^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-a^2 + b^2)*AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Sec[c + d*x]^2*Tan[c + d*x] + 9*(a^2 - b^2)*((-2*a^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(3*(a^2 - b^2)) - (AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/9) + Tan[c + d*x]^2*(-6*a^2*((-12*a^2*AppellF1[5/2, 1/6, 3, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(5*(a^2 - b^2)) - (AppellF1[5/2, 7/6, 2, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/5) + (-a^2 + b^2)*((-6*a^2*AppellF1[5/2, 7/6, 2, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(5*(a^2 - b^2)) - (7*AppellF1[5/2, 13/6, 1, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/5))))/(9*(a^2 - b^2)*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-6*a^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-a^2 + b^2)*AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2)^2 - (b*AppellF1[1/2, 2/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*(4*(3*a^2*AppellF1[3/2, 2/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 2*(a^2 - b^2)*AppellF1[3/2, 5/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Sec[c + d*x]^2*Tan[c + d*x] - 9*(a^2 - b^2)*((-2*a^2*AppellF1[3/2, 2/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(3*(a^2 - b^2)) - (4*AppellF1[3/2, 5/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/9) + 2*Tan[c + d*x]^2*(3*a^2*((-12*a^2*AppellF1[5/2, 2/3, 3, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(5*(a^2 - b^2)) - (4*AppellF1[5/2, 5/3, 2, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/5) + 2*(a^2 - b^2)*((-6*a^2*AppellF1[5/2, 5/3, 2, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(5*(a^2 - b^2)) - 2*AppellF1[5/2, 8/3, 1, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x]))))/(-9*(a^2 - b^2)*AppellF1[1/2, 2/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 2*(3*a^2*AppellF1[3/2, 2/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 2*(a^2 - b^2)*AppellF1[3/2, 5/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2)^2))/((Sec[c + d*x]^2)^(2/3)*(-b^2 + a^2*Sec[c + d*x]^2))))","B",0
678,1,4605,176,21.3372151,"\int \frac{1}{\sqrt[3]{\cos (c+d x)} (a+b \cos (c+d x))} \, dx","Integrate[1/(Cos[c + d*x]^(1/3)*(a + b*Cos[c + d*x])),x]","\text{Result too large to show}","\frac{a \sin (c+d x) \cos ^2(c+d x)^{2/3} F_1\left(\frac{1}{2};\frac{2}{3},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \cos ^{\frac{4}{3}}(c+d x)}-\frac{b \sin (c+d x) \sqrt[6]{\cos ^2(c+d x)} F_1\left(\frac{1}{2};\frac{1}{6},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \sqrt[3]{\cos (c+d x)}}",1,"(9*(a^2 - b^2)*Sin[c + d*x]*((a*AppellF1[1/2, -1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^2)*AppellF1[1/2, -1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-6*a^2*AppellF1[3/2, -1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 5/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (b*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])/(-9*(a^2 - b^2)*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 2*(3*a^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2)))/(d*Cos[c + d*x]^(4/3)*(a + b*Cos[c + d*x])*(Sec[c + d*x]^2)^(1/3)*(-b^2 + a^2*Sec[c + d*x]^2)*((9*(a^2 - b^2)*(Sec[c + d*x]^2)^(2/3)*((a*AppellF1[1/2, -1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^2)*AppellF1[1/2, -1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-6*a^2*AppellF1[3/2, -1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 5/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (b*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])/(-9*(a^2 - b^2)*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 2*(3*a^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2)))/(-b^2 + a^2*Sec[c + d*x]^2) - (18*a^2*(a^2 - b^2)*(Sec[c + d*x]^2)^(2/3)*Tan[c + d*x]^2*((a*AppellF1[1/2, -1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^2)*AppellF1[1/2, -1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-6*a^2*AppellF1[3/2, -1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 5/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (b*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])/(-9*(a^2 - b^2)*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 2*(3*a^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2)))/(-b^2 + a^2*Sec[c + d*x]^2)^2 - (6*(a^2 - b^2)*Tan[c + d*x]^2*((a*AppellF1[1/2, -1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^2)*AppellF1[1/2, -1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-6*a^2*AppellF1[3/2, -1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 5/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (b*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])/(-9*(a^2 - b^2)*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 2*(3*a^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2)))/((Sec[c + d*x]^2)^(1/3)*(-b^2 + a^2*Sec[c + d*x]^2)) + (9*(a^2 - b^2)*Tan[c + d*x]*((a*AppellF1[1/2, -1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2]*Tan[c + d*x])/(9*(a^2 - b^2)*AppellF1[1/2, -1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-6*a^2*AppellF1[3/2, -1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 5/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (a*Sqrt[Sec[c + d*x]^2]*((-2*a^2*AppellF1[3/2, -1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(3*(a^2 - b^2)) + (AppellF1[3/2, 5/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/9))/(9*(a^2 - b^2)*AppellF1[1/2, -1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-6*a^2*AppellF1[3/2, -1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 5/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (b*((-2*a^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(3*(a^2 - b^2)) - (2*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/9))/(-9*(a^2 - b^2)*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 2*(3*a^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) - (a*AppellF1[1/2, -1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2]*(2*(-6*a^2*AppellF1[3/2, -1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 5/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Sec[c + d*x]^2*Tan[c + d*x] + 9*(a^2 - b^2)*((-2*a^2*AppellF1[3/2, -1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(3*(a^2 - b^2)) + (AppellF1[3/2, 5/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/9) + Tan[c + d*x]^2*(-6*a^2*((-12*a^2*AppellF1[5/2, -1/6, 3, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(5*(a^2 - b^2)) + (AppellF1[5/2, 5/6, 2, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/5) + (a^2 - b^2)*((-6*a^2*AppellF1[5/2, 5/6, 2, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(5*(a^2 - b^2)) - AppellF1[5/2, 11/6, 1, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x]))))/(9*(a^2 - b^2)*AppellF1[1/2, -1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-6*a^2*AppellF1[3/2, -1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 5/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2)^2 - (b*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*(4*(3*a^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Sec[c + d*x]^2*Tan[c + d*x] - 9*(a^2 - b^2)*((-2*a^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(3*(a^2 - b^2)) - (2*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/9) + 2*Tan[c + d*x]^2*(3*a^2*((-12*a^2*AppellF1[5/2, 1/3, 3, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(5*(a^2 - b^2)) - (2*AppellF1[5/2, 4/3, 2, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/5) + (a^2 - b^2)*((-6*a^2*AppellF1[5/2, 4/3, 2, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(5*(a^2 - b^2)) - (8*AppellF1[5/2, 7/3, 1, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/5))))/(-9*(a^2 - b^2)*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + 2*(3*a^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2)^2))/((Sec[c + d*x]^2)^(1/3)*(-b^2 + a^2*Sec[c + d*x]^2))))","B",0
679,1,4608,176,21.2034849,"\int \frac{1}{\cos ^{\frac{2}{3}}(c+d x) (a+b \cos (c+d x))} \, dx","Integrate[1/(Cos[c + d*x]^(2/3)*(a + b*Cos[c + d*x])),x]","\text{Result too large to show}","\frac{a \sin (c+d x) \cos ^2(c+d x)^{5/6} F_1\left(\frac{1}{2};\frac{5}{6},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \cos ^{\frac{5}{3}}(c+d x)}-\frac{b \sin (c+d x) \sqrt[3]{\cos ^2(c+d x)} F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \cos ^{\frac{2}{3}}(c+d x)}",1,"(9*(a^2 - b^2)*Sin[c + d*x]*((a*AppellF1[1/2, -1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^2)*AppellF1[1/2, -1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] - 2*(3*a^2*AppellF1[3/2, -1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-a^2 + b^2)*AppellF1[3/2, 2/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (b*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])/(-9*(a^2 - b^2)*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (6*a^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2)))/(d*Cos[c + d*x]^(5/3)*(a + b*Cos[c + d*x])*(Sec[c + d*x]^2)^(1/6)*(-b^2 + a^2*Sec[c + d*x]^2)*((9*(a^2 - b^2)*(Sec[c + d*x]^2)^(5/6)*((a*AppellF1[1/2, -1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^2)*AppellF1[1/2, -1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] - 2*(3*a^2*AppellF1[3/2, -1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-a^2 + b^2)*AppellF1[3/2, 2/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (b*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])/(-9*(a^2 - b^2)*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (6*a^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2)))/(-b^2 + a^2*Sec[c + d*x]^2) - (18*a^2*(a^2 - b^2)*(Sec[c + d*x]^2)^(5/6)*Tan[c + d*x]^2*((a*AppellF1[1/2, -1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^2)*AppellF1[1/2, -1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] - 2*(3*a^2*AppellF1[3/2, -1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-a^2 + b^2)*AppellF1[3/2, 2/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (b*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])/(-9*(a^2 - b^2)*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (6*a^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2)))/(-b^2 + a^2*Sec[c + d*x]^2)^2 - (3*(a^2 - b^2)*Tan[c + d*x]^2*((a*AppellF1[1/2, -1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^2)*AppellF1[1/2, -1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] - 2*(3*a^2*AppellF1[3/2, -1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-a^2 + b^2)*AppellF1[3/2, 2/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (b*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])/(-9*(a^2 - b^2)*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (6*a^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2)))/((Sec[c + d*x]^2)^(1/6)*(-b^2 + a^2*Sec[c + d*x]^2)) + (9*(a^2 - b^2)*Tan[c + d*x]*((a*AppellF1[1/2, -1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2]*Tan[c + d*x])/(9*(a^2 - b^2)*AppellF1[1/2, -1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] - 2*(3*a^2*AppellF1[3/2, -1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-a^2 + b^2)*AppellF1[3/2, 2/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (a*Sqrt[Sec[c + d*x]^2]*((-2*a^2*AppellF1[3/2, -1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(3*(a^2 - b^2)) + (2*AppellF1[3/2, 2/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/9))/(9*(a^2 - b^2)*AppellF1[1/2, -1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] - 2*(3*a^2*AppellF1[3/2, -1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-a^2 + b^2)*AppellF1[3/2, 2/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) + (b*((-2*a^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(3*(a^2 - b^2)) - (AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/9))/(-9*(a^2 - b^2)*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (6*a^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2) - (a*AppellF1[1/2, -1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sqrt[Sec[c + d*x]^2]*(-4*(3*a^2*AppellF1[3/2, -1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-a^2 + b^2)*AppellF1[3/2, 2/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Sec[c + d*x]^2*Tan[c + d*x] + 9*(a^2 - b^2)*((-2*a^2*AppellF1[3/2, -1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(3*(a^2 - b^2)) + (2*AppellF1[3/2, 2/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/9) - 2*Tan[c + d*x]^2*(3*a^2*((-12*a^2*AppellF1[5/2, -1/3, 3, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(5*(a^2 - b^2)) + (2*AppellF1[5/2, 2/3, 2, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/5) + (-a^2 + b^2)*((-6*a^2*AppellF1[5/2, 2/3, 2, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(5*(a^2 - b^2)) - (4*AppellF1[5/2, 5/3, 1, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/5))))/(9*(a^2 - b^2)*AppellF1[1/2, -1/3, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] - 2*(3*a^2*AppellF1[3/2, -1/3, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (-a^2 + b^2)*AppellF1[3/2, 2/3, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2)^2 - (b*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*(2*(6*a^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Sec[c + d*x]^2*Tan[c + d*x] - 9*(a^2 - b^2)*((-2*a^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(3*(a^2 - b^2)) - (AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/9) + Tan[c + d*x]^2*(6*a^2*((-12*a^2*AppellF1[5/2, 1/6, 3, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(5*(a^2 - b^2)) - (AppellF1[5/2, 7/6, 2, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/5) + (a^2 - b^2)*((-6*a^2*AppellF1[5/2, 7/6, 2, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/(5*(a^2 - b^2)) - (7*AppellF1[5/2, 13/6, 1, 7/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Sec[c + d*x]^2*Tan[c + d*x])/5))))/(-9*(a^2 - b^2)*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (6*a^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))] + (a^2 - b^2)*AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))])*Tan[c + d*x]^2)^2))/((Sec[c + d*x]^2)^(1/6)*(-b^2 + a^2*Sec[c + d*x]^2))))","B",0
680,0,0,28,33.5612182,"\int \frac{\cos ^{\frac{7}{3}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]^(7/3)/Sqrt[a + b*Cos[c + d*x]],x]","\int \frac{\cos ^{\frac{7}{3}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","\text{Int}\left(\frac{\cos ^{\frac{7}{3}}(c+d x)}{\sqrt{a+b \cos (c+d x)}},x\right)",0,"Integrate[Cos[c + d*x]^(7/3)/Sqrt[a + b*Cos[c + d*x]], x]","A",-1
681,0,0,28,80.1397885,"\int \frac{\cos ^{\frac{5}{3}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]^(5/3)/Sqrt[a + b*Cos[c + d*x]],x]","\int \frac{\cos ^{\frac{5}{3}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","\text{Int}\left(\frac{\cos ^{\frac{5}{3}}(c+d x)}{\sqrt{a+b \cos (c+d x)}},x\right)",0,"Integrate[Cos[c + d*x]^(5/3)/Sqrt[a + b*Cos[c + d*x]], x]","A",-1
682,0,0,28,19.4130558,"\int \frac{\cos ^{\frac{4}{3}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]^(4/3)/Sqrt[a + b*Cos[c + d*x]],x]","\int \frac{\cos ^{\frac{4}{3}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","\text{Int}\left(\frac{\cos ^{\frac{4}{3}}(c+d x)}{\sqrt{a+b \cos (c+d x)}},x\right)",0,"Integrate[Cos[c + d*x]^(4/3)/Sqrt[a + b*Cos[c + d*x]], x]","A",-1
683,0,0,28,9.177468,"\int \frac{\cos ^{\frac{2}{3}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]^(2/3)/Sqrt[a + b*Cos[c + d*x]],x]","\int \frac{\cos ^{\frac{2}{3}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","\text{Int}\left(\frac{\cos ^{\frac{2}{3}}(c+d x)}{\sqrt{a+b \cos (c+d x)}},x\right)",0,"Integrate[Cos[c + d*x]^(2/3)/Sqrt[a + b*Cos[c + d*x]], x]","A",-1
684,0,0,28,2.7428214,"\int \frac{\sqrt[3]{\cos (c+d x)}}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[Cos[c + d*x]^(1/3)/Sqrt[a + b*Cos[c + d*x]],x]","\int \frac{\sqrt[3]{\cos (c+d x)}}{\sqrt{a+b \cos (c+d x)}} \, dx","\text{Int}\left(\frac{\sqrt[3]{\cos (c+d x)}}{\sqrt{a+b \cos (c+d x)}},x\right)",0,"Integrate[Cos[c + d*x]^(1/3)/Sqrt[a + b*Cos[c + d*x]], x]","A",-1
685,0,0,28,1.9309572,"\int \frac{1}{\sqrt[3]{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx","Integrate[1/(Cos[c + d*x]^(1/3)*Sqrt[a + b*Cos[c + d*x]]),x]","\int \frac{1}{\sqrt[3]{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx","\text{Int}\left(\frac{1}{\sqrt[3]{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}},x\right)",0,"Integrate[1/(Cos[c + d*x]^(1/3)*Sqrt[a + b*Cos[c + d*x]]), x]","A",-1
686,0,0,28,0.534362,"\int \frac{1}{\cos ^{\frac{2}{3}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Integrate[1/(Cos[c + d*x]^(2/3)*Sqrt[a + b*Cos[c + d*x]]),x]","\int \frac{1}{\cos ^{\frac{2}{3}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","\text{Int}\left(\frac{1}{\cos ^{\frac{2}{3}}(c+d x) \sqrt{a+b \cos (c+d x)}},x\right)",0,"Integrate[1/(Cos[c + d*x]^(2/3)*Sqrt[a + b*Cos[c + d*x]]), x]","A",-1
687,0,0,28,82.3166769,"\int \frac{1}{\cos ^{\frac{4}{3}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Integrate[1/(Cos[c + d*x]^(4/3)*Sqrt[a + b*Cos[c + d*x]]),x]","\int \frac{1}{\cos ^{\frac{4}{3}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","\text{Int}\left(\frac{1}{\cos ^{\frac{4}{3}}(c+d x) \sqrt{a+b \cos (c+d x)}},x\right)",0,"Integrate[1/(Cos[c + d*x]^(4/3)*Sqrt[a + b*Cos[c + d*x]]), x]","A",-1
688,0,0,28,29.3010616,"\int \frac{1}{\cos ^{\frac{5}{3}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Integrate[1/(Cos[c + d*x]^(5/3)*Sqrt[a + b*Cos[c + d*x]]),x]","\int \frac{1}{\cos ^{\frac{5}{3}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","\text{Int}\left(\frac{1}{\cos ^{\frac{5}{3}}(c+d x) \sqrt{a+b \cos (c+d x)}},x\right)",0,"Integrate[1/(Cos[c + d*x]^(5/3)*Sqrt[a + b*Cos[c + d*x]]), x]","A",-1
689,0,0,28,86.2287762,"\int \frac{1}{\cos ^{\frac{7}{3}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","Integrate[1/(Cos[c + d*x]^(7/3)*Sqrt[a + b*Cos[c + d*x]]),x]","\int \frac{1}{\cos ^{\frac{7}{3}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx","\text{Int}\left(\frac{1}{\cos ^{\frac{7}{3}}(c+d x) \sqrt{a+b \cos (c+d x)}},x\right)",0,"Integrate[1/(Cos[c + d*x]^(7/3)*Sqrt[a + b*Cos[c + d*x]]), x]","A",-1
690,1,97,151,0.3470675,"\int (A+B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(A + B*Cos[c + d*x])*Sec[c + d*x]^(7/2),x]","\frac{\sec ^{\frac{5}{2}}(c+d x) \left(21 A \sin (c+d x)+9 A \sin (3 (c+d x))-36 A \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+10 B \sin (2 (c+d x))+20 B \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{30 d}","\frac{2 A \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{6 A \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{6 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}",1,"(Sec[c + d*x]^(5/2)*(-36*A*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 20*B*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 21*A*Sin[c + d*x] + 10*B*Sin[2*(c + d*x)] + 9*A*Sin[3*(c + d*x)]))/(30*d)","A",1
691,1,85,123,0.2392448,"\int (A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(A + B*Cos[c + d*x])*Sec[c + d*x]^(5/2),x]","\frac{\sec ^{\frac{3}{2}}(c+d x) \left(2 \sin (c+d x) (A+3 B \cos (c+d x))+2 A \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 B \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d}","\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 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 d}+\frac{2 B \sin (c+d x) \sqrt{\sec (c+d x)}}{d}-\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(Sec[c + d*x]^(3/2)*(-6*B*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 2*A*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 2*(A + 3*B*Cos[c + d*x])*Sin[c + d*x]))/(3*d)","A",1
692,1,71,97,0.1105243,"\int (A+B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2),x]","\frac{2 \sqrt{\sec (c+d x)} \left(A \sin (c+d x)-A \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d}","\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{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)}{d}+\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*Sqrt[Sec[c + d*x]]*(-(A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]) + B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + A*Sin[c + d*x]))/d","A",1
693,1,52,75,0.0741607,"\int (A+B \cos (c+d x)) \sqrt{\sec (c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x])*Sqrt[Sec[c + d*x]],x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{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)}{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*Sqrt[Cos[c + d*x]]*(B*EllipticE[(c + d*x)/2, 2] + A*EllipticF[(c + d*x)/2, 2])*Sqrt[Sec[c + d*x]])/d","A",1
694,1,76,101,0.1390073,"\int \frac{A+B \cos (c+d x)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(6 A \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+B \left(\sin (2 (c+d x))+2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)\right)}{3 d}","\frac{2 A \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 \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,"(Sqrt[Sec[c + d*x]]*(6*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + B*(2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + Sin[2*(c + d*x)])))/(3*d)","A",1
695,1,88,127,0.3232704,"\int \frac{A+B \cos (c+d x)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(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)) (5 A+3 B \cos (c+d x))+10 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+18 B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d}","\frac{2 A \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\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 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,"(Sqrt[Sec[c + d*x]]*(18*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 10*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (5*A + 3*B*Cos[c + d*x])*Sin[2*(c + d*x)]))/(15*d)","A",1
696,1,99,151,0.5645042,"\int \frac{A+B \cos (c+d x)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(A + B*Cos[c + d*x])/Sec[c + d*x]^(5/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) (42 A \cos (c+d x)+15 B \cos (2 (c+d x))+65 B)+252 A \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+100 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 A \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{6 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{10 B \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{10 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}",1,"(Sqrt[Sec[c + d*x]]*(252*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 100*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (65*B + 42*A*Cos[c + d*x] + 15*B*Cos[2*(c + d*x)])*Sin[2*(c + d*x)]))/(210*d)","A",1
697,1,139,200,0.8716063,"\int (a+b \cos (c+d x))^2 \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(9/2),x]","\frac{\sec ^{\frac{7}{2}}(c+d x) \left(20 \left(5 a^2+7 b^2\right) \cos ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) \left(5 \left(5 a^2+7 b^2\right) \cos (2 (c+d x))+55 a^2+273 a b \cos (c+d x)+63 a b \cos (3 (c+d x))+35 b^2\right)-504 a b \cos ^{\frac{7}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 \left(5 a^2+7 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 \left(5 a^2+7 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)}{21 d}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{4 a b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{12 a b \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{12 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}",1,"(Sec[c + d*x]^(7/2)*(-504*a*b*Cos[c + d*x]^(7/2)*EllipticE[(c + d*x)/2, 2] + 20*(5*a^2 + 7*b^2)*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] + 2*(55*a^2 + 35*b^2 + 273*a*b*Cos[c + d*x] + 5*(5*a^2 + 7*b^2)*Cos[2*(c + d*x)] + 63*a*b*Cos[3*(c + d*x)])*Sin[c + d*x]))/(210*d)","A",1
698,1,126,175,1.3129273,"\int (a+b \cos (c+d x))^2 \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(7/2),x]","\frac{\sec ^{\frac{5}{2}}(c+d x) \left(-12 \left(3 a^2+5 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+5 b^2\right) \cos (2 (c+d x))+15 \left(a^2+b^2\right)+20 a b \cos (c+d x)\right)+40 a b \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{30 d}","\frac{2 \left(3 a^2+5 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{2 \left(3 a^2+5 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 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{4 a b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{4 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}",1,"(Sec[c + d*x]^(5/2)*(-12*(3*a^2 + 5*b^2)*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 40*a*b*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 2*(15*(a^2 + b^2) + 20*a*b*Cos[c + d*x] + 3*(3*a^2 + 5*b^2)*Cos[2*(c + d*x)])*Sin[c + d*x]))/(30*d)","A",1
699,1,93,135,0.3357951,"\int (a+b \cos (c+d x))^2 \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2),x]","\frac{2 \sec ^{\frac{3}{2}}(c+d x) \left(\left(a^2+3 b^2\right) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 a b \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+a \sin (c+d x) (a+6 b \cos (c+d x))\right)}{3 d}","\frac{2 \left(a^2+3 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 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{4 a b \sin (c+d x) \sqrt{\sec (c+d x)}}{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)}{d}",1,"(2*Sec[c + d*x]^(3/2)*(-6*a*b*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + (a^2 + 3*b^2)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + a*(a + 6*b*Cos[c + d*x])*Sin[c + d*x]))/(3*d)","A",1
700,1,83,108,0.1978043,"\int (a+b \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2),x]","\frac{2 \sqrt{\sec (c+d x)} \left(a \left(a \sin (c+d x)+2 b \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)-\left(a^2-b^2\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d}","-\frac{2 \left(a^2-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)}{d}+\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{4 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}",1,"(2*Sqrt[Sec[c + d*x]]*(-((a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]) + a*(2*b*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + a*Sin[c + d*x])))/d","A",1
701,1,87,112,0.1867926,"\int (a+b \cos (c+d x))^2 \sqrt{\sec (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(2 \left(3 a^2+b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+12 a b \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b^2 \sin (2 (c+d x))\right)}{3 d}","\frac{2 \left(3 a^2+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{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)}{d}+\frac{2 b^2 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Sec[c + d*x]]*(12*a*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*(3*a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + b^2*Sin[2*(c + d*x)]))/(3*d)","A",1
702,1,100,141,0.4554707,"\int \frac{(a+b \cos (c+d x))^2}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(a + b*Cos[c + d*x])^2/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(6 \left(5 a^2+3 b^2\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+3 b \cos (c+d x))+20 a b \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d}","\frac{2 \left(5 a^2+3 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{4 a b \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{4 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 b^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(6*(5*a^2 + 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 20*a*b*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + b*(10*a + 3*b*Cos[c + d*x])*Sin[2*(c + d*x)]))/(15*d)","A",1
703,1,120,175,0.681858,"\int \frac{(a+b \cos (c+d x))^2}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^2/Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) \left(70 a^2+84 a b \cos (c+d x)+15 b^2 \cos (2 (c+d x))+65 b^2\right)+20 \left(7 a^2+5 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+504 a b \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+5 b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left(7 a^2+5 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)}{21 d}+\frac{4 a b \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{12 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^2 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(504*a*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 20*(7*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (70*a^2 + 65*b^2 + 84*a*b*Cos[c + d*x] + 15*b^2*Cos[2*(c + d*x)])*Sin[2*(c + d*x)]))/(210*d)","A",1
704,1,135,200,1.1089578,"\int \frac{(a+b \cos (c+d x))^2}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^2/Sec[c + d*x]^(5/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) \left(7 \left(36 a^2+43 b^2\right) \cos (c+d x)+5 b (36 a \cos (2 (c+d x))+156 a+7 b \cos (3 (c+d x)))\right)+168 \left(9 a^2+7 b^2\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+1200 a b \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{1260 d}","\frac{2 \left(9 a^2+7 b^2\right) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(9 a^2+7 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)}{15 d}+\frac{4 a b \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{20 a b \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{20 a 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{2 b^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(168*(9*a^2 + 7*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 1200*a*b*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (7*(36*a^2 + 43*b^2)*Cos[c + d*x] + 5*b*(156*a + 36*a*Cos[2*(c + d*x)] + 7*b*Cos[3*(c + d*x)]))*Sin[2*(c + d*x)]))/(1260*d)","A",1
705,1,191,234,1.1977484,"\int (a+b \cos (c+d x))^3 \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(9/2),x]","\frac{\sec ^{\frac{7}{2}}(c+d x) \left(30 a^3 \sin (c+d x)+50 a^3 \sin (c+d x) \cos ^2(c+d x)+10 a \left(5 a^2+21 b^2\right) \cos ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-42 b \left(9 a^2+5 b^2\right) \cos ^{\frac{7}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+63 a^2 b \sin (2 (c+d x))+378 a^2 b \sin (c+d x) \cos ^3(c+d x)+210 a b^2 \sin (c+d x) \cos ^2(c+d x)+210 b^3 \sin (c+d x) \cos ^3(c+d x)\right)}{105 d}","\frac{2 a \left(5 a^2+21 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 b \left(9 a^2+5 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a \left(5 a^2+21 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)}{21 d}-\frac{2 b \left(9 a^2+5 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 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+b)}{7 d}+\frac{32 a^2 b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}",1,"(Sec[c + d*x]^(7/2)*(-42*b*(9*a^2 + 5*b^2)*Cos[c + d*x]^(7/2)*EllipticE[(c + d*x)/2, 2] + 10*a*(5*a^2 + 21*b^2)*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] + 30*a^3*Sin[c + d*x] + 50*a^3*Cos[c + d*x]^2*Sin[c + d*x] + 210*a*b^2*Cos[c + d*x]^2*Sin[c + d*x] + 378*a^2*b*Cos[c + d*x]^3*Sin[c + d*x] + 210*b^3*Cos[c + d*x]^3*Sin[c + d*x] + 63*a^2*b*Sin[2*(c + d*x)]))/(105*d)","A",1
706,1,134,189,1.6729679,"\int (a+b \cos (c+d x))^3 \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(7/2),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(5 b \left(a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-3 a \left(a^2+5 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{a \sin (c+d x) \left(3 \left(a^2+5 b^2\right) \cos (2 (c+d x))+5 \left(a^2+3 b^2\right)+10 a b \cos (c+d x)\right)}{2 \cos ^{\frac{5}{2}}(c+d x)}\right)}{5 d}","\frac{6 a \left(a^2+5 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 b \left(a^2+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)}{d}-\frac{6 a \left(a^2+5 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{8 a^2 b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+b)}{5 d}",1,"(2*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-3*a*(a^2 + 5*b^2)*EllipticE[(c + d*x)/2, 2] + 5*b*(a^2 + b^2)*EllipticF[(c + d*x)/2, 2] + (a*(5*(a^2 + 3*b^2) + 10*a*b*Cos[c + d*x] + 3*(a^2 + 5*b^2)*Cos[2*(c + d*x)])*Sin[c + d*x])/(2*Cos[c + d*x]^(5/2))))/(5*d)","A",1
707,1,106,160,0.5057303,"\int (a+b \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(5/2),x]","\frac{\sec ^{\frac{3}{2}}(c+d x) \left(6 b \left(b^2-3 a^2\right) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+a \left(2 \left(a^2+9 b^2\right) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 a \sin (c+d x) (a+9 b \cos (c+d x))\right)\right)}{3 d}","\frac{2 a \left(a^2+9 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 b \left(3 a^2-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)}{d}+\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+b)}{3 d}+\frac{16 a^2 b \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}",1,"(Sec[c + d*x]^(3/2)*(6*b*(-3*a^2 + b^2)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + a*(2*(a^2 + 9*b^2)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 2*a*(a + 9*b*Cos[c + d*x])*Sin[c + d*x])))/(3*d)","A",1
708,1,108,166,0.6309246,"\int (a+b \cos (c+d x))^3 \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \left(2 \sin (c+d x) \left(3 a^3+b^3 \cos (c+d x)\right)+2 b \left(9 a^2+b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 a \left(a^2-3 b^2\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d}","\frac{2 a \left(3 a^2-b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 b \left(9 a^2+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 a \left(a^2-3 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)}{d}+\frac{2 b^2 \sin (c+d x) (a \sec (c+d x)+b)}{3 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Sec[c + d*x]]*(-6*a*(a^2 - 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*b*(9*a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(3*a^3 + b^3*Cos[c + d*x])*Sin[c + d*x]))/(3*d)","A",1
709,1,106,156,0.4670335,"\int (a+b \cos (c+d x))^3 \sqrt{\sec (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(10 a \left(a^2+b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 b \left(5 a^2+b^2\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b^2 \sin (2 (c+d x)) (5 a+b \cos (c+d x))\right)}{5 d}","\frac{2 a \left(a^2+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)}{d}+\frac{6 b \left(5 a^2+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 \sin (c+d x) (a \sec (c+d x)+b)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 a b^2 \sin (c+d x)}{5 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Sec[c + d*x]]*(6*b*(5*a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 10*a*(a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + b^2*(5*a + b*Cos[c + d*x])*Sin[2*(c + d*x)]))/(5*d)","A",1
710,1,132,199,0.9559882,"\int \frac{(a+b \cos (c+d x))^3}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(a + b*Cos[c + d*x])^3/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(b \sin (2 (c+d x)) \left(210 a^2+126 a b \cos (c+d x)+15 b^2 \cos (2 (c+d x))+65 b^2\right)+20 b \left(21 a^2+5 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+84 a \left(5 a^2+9 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 b \left(21 a^2+5 b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 b \left(21 a^2+5 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)}{21 d}+\frac{2 a \left(5 a^2+9 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 \sin (c+d x) (a \sec (c+d x)+b)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{32 a b^2 \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(84*a*(5*a^2 + 9*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 20*b*(21*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + b*(210*a^2 + 65*b^2 + 126*a*b*Cos[c + d*x] + 15*b^2*Cos[2*(c + d*x)])*Sin[2*(c + d*x)]))/(210*d)","A",1
711,1,159,234,1.2380892,"\int \frac{(a+b \cos (c+d x))^3}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^3/Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \left(120 a \left(7 a^2+15 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+168 b \left(27 a^2+7 b^2\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (2 (c+d x)) \left(5 \left(84 a^3+54 a b^2 \cos (2 (c+d x))+234 a b^2+7 b^3 \cos (3 (c+d x))\right)+7 b \left(108 a^2+43 b^2\right) \cos (c+d x)\right)\right)}{1260 d}","\frac{2 b \left(27 a^2+7 b^2\right) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left(7 a^2+15 b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a \left(7 a^2+15 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)}{21 d}+\frac{2 b \left(27 a^2+7 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)}{15 d}+\frac{2 b^2 \sin (c+d x) (a \sec (c+d x)+b)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{40 a b^2 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(168*b*(27*a^2 + 7*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 120*a*(7*a^2 + 15*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (7*b*(108*a^2 + 43*b^2)*Cos[c + d*x] + 5*(84*a^3 + 234*a*b^2 + 54*a*b^2*Cos[2*(c + d*x)] + 7*b^3*Cos[3*(c + d*x)]))*Sin[2*(c + d*x)]))/(1260*d)","A",1
712,1,165,188,3.0197964,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[Sec[c + d*x]^(5/2)/(a + b*Cos[c + d*x]),x]","-\frac{\cot (c+d x) \left(-2 \left(a^2+3 a b+3 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 \sec ^{\frac{5}{2}}(c+d x)+a^2 \cos (2 (c+d x)) \sec ^{\frac{5}{2}}(c+d x)+6 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 b \sqrt{-\tan ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{3 a^3 d}","\frac{2 b^2 \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 b \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 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)}{a^2 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}",1,"-1/3*(Cot[c + d*x]*(-(a^2*Sec[c + d*x]^(5/2)) + a^2*Cos[2*(c + d*x)]*Sec[c + d*x]^(5/2) + 6*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] - 2*(a^2 + 3*a*b + 3*b^2)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] + 6*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2]))/(a^3*d)","A",1
713,1,83,117,5.1603129,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[Sec[c + d*x]^(3/2)/(a + b*Cos[c + d*x]),x]","\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) \left(-(a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+b \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+a E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a^2 d}","-\frac{2 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 \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(2*Cot[c + d*x]*(a*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1] - (a + b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] + b*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*Sqrt[-Tan[c + d*x]^2])/(a^2*d)","A",1
714,1,63,49,0.3360946,"\int \frac{\sqrt{\sec (c+d x)}}{a+b \cos (c+d x)} \, dx","Integrate[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(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 d}","\frac{2 \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)}{d (a+b)}",1,"(2*Cot[c + d*x]*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*Sqrt[-Tan[c + d*x]^2])/(a*d)","A",1
715,1,47,93,0.2438839,"\int \frac{1}{(a+b \cos (c+d x)) \sqrt{\sec (c+d x)}} \, dx","Integrate[1/((a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]]),x]","\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{b d}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}-\frac{2 a \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)}",1,"(2*Cot[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2])/(b*d)","A",1
716,1,176,135,6.5186876,"\int \frac{1}{(a+b \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[1/((a + b*Cos[c + d*x])*Sec[c + d*x]^(3/2)),x]","\frac{\cot (c+d x) \left(-2 a \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 \sec ^{\frac{7}{2}}(c+d x)-b \sec ^{\frac{3}{2}}(c+d x)+b \cos (2 (c+d x)) \sec ^{\frac{7}{2}}(c+d x)-b \cos (2 (c+d x)) \sec ^{\frac{3}{2}}(c+d x)+2 b \sqrt{-\tan ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-2 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^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}-\frac{2 a \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 \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*Sec[c + d*x]^(3/2)) - b*Cos[2*(c + d*x)]*Sec[c + d*x]^(3/2) + b*Sec[c + d*x]^(7/2) + b*Cos[2*(c + d*x)]*Sec[c + d*x]^(7/2) - 2*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] + 2*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] - 2*a*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2]))/(b^2*d)","A",1
717,1,196,172,6.7868326,"\int \frac{1}{(a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[1/((a + b*Cos[c + d*x])*Sec[c + d*x]^(5/2)),x]","-\frac{\cot (c+d x) \left(-12 a^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 b \sec ^{\frac{3}{2}}(c+d x)-6 a b \cos (2 (c+d x)) \sec ^{\frac{3}{2}}(c+d x)+4 b (3 a-b) \sqrt{-\tan ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-12 a b \sqrt{-\tan ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-b^2 \sqrt{\sec (c+d x)}+b^2 \cos (3 (c+d x)) \sec ^{\frac{3}{2}}(c+d x)\right)}{6 b^3 d}","-\frac{2 a^3 \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^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^3 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)}{b^2 d}+\frac{2 \sin (c+d x)}{3 b d \sqrt{\sec (c+d x)}}",1,"-1/6*(Cot[c + d*x]*(-(b^2*Sqrt[Sec[c + d*x]]) + 6*a*b*Sec[c + d*x]^(3/2) - 6*a*b*Cos[2*(c + d*x)]*Sec[c + d*x]^(3/2) + b^2*Cos[3*(c + d*x)]*Sec[c + d*x]^(3/2) - 12*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] + 4*(3*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] - 12*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2]))/(b^3*d)","A",1
718,1,655,341,6.7981445,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^2,x]","\frac{\sqrt{\sec (c+d x)} \left(-\frac{b^3 \sin (c+d x)}{a^2 \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{2 \tan (c+d x)}{3 a^2}-\frac{b \left(4 a^2-5 b^2\right) \sin (c+d x)}{a^3 \left(a^2-b^2\right)}\right)}{d}+\frac{\frac{2 \left(40 a b^3-28 a^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{\left(15 b^4-12 a^2 b^2\right) \sin (c+d x) \cos (2 (c+d x)) (a \sec (c+d x)+b) \left(-4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a b^2 \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{2 \left(-4 a^4-44 a^2 b^2+45 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^2 \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^2-5 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^2-5 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{b \left(4 a^2-5 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^3 d \left(a^2-b^2\right)}+\frac{b \left(4 a^2-5 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^3 d \left(a^2-b^2\right)}+\frac{b^2 \left(7 a^2-5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}",1,"((2*(-4*a^4 - 44*a^2*b^2 + 45*b^4)*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*b + 40*a*b^3)*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*b^2 + 15*b^4)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(12*a^3*(-a + b)*(a + b)*d) + (Sqrt[Sec[c + d*x]]*(-((b*(4*a^2 - 5*b^2)*Sin[c + d*x])/(a^3*(a^2 - b^2))) - (b^3*Sin[c + d*x])/(a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])) + (2*Tan[c + d*x])/(3*a^2)))/d","A",0
719,1,351,277,4.6255848,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^2,x]","\frac{\frac{2 a \sin (c+d x) \left(2 a \left(a^2-b^2\right) \sec (c+d x)+2 a^2 b-3 b^3\right)}{\left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}+\frac{\cot (c+d x) \left(-2 a^3 \sec ^{\frac{3}{2}}(c+d x)+2 a^3 \cos (2 (c+d x)) \sec ^{\frac{3}{2}}(c+d x)+2 a \left(2 a^2-3 b^2\right) \sqrt{-\tan ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+10 a^2 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 \left(2 a^3+4 a^2 b-3 a b^2-3 b^3\right) \sqrt{-\tan ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-6 b^3 \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+3 a b^2 \sec ^{\frac{3}{2}}(c+d x)-3 a b^2 \cos (2 (c+d x)) \sec ^{\frac{3}{2}}(c+d x)\right)}{(a-b) (a+b)}}{2 a^3 d}","\frac{b^2 \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^2-3 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d \left(a^2-b^2\right)}+\frac{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^2-3 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{b \left(5 a^2-3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a-b) (a+b)^2}",1,"((2*a*(2*a^2*b - 3*b^3 + 2*a*(a^2 - b^2)*Sec[c + d*x])*Sin[c + d*x])/((a^2 - b^2)*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]]) + (Cot[c + d*x]*(-2*a^3*Sec[c + d*x]^(3/2) + 3*a*b^2*Sec[c + d*x]^(3/2) + 2*a^3*Cos[2*(c + d*x)]*Sec[c + d*x]^(3/2) - 3*a*b^2*Cos[2*(c + d*x)]*Sec[c + d*x]^(3/2) + 2*a*(2*a^2 - 3*b^2)*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] - 2*(2*a^3 + 4*a^2*b - 3*a*b^2 - 3*b^3)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] + 10*a^2*b*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] - 6*b^3*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2]))/((a - b)*(a + b)))/(2*a^3*d)","A",1
720,1,584,217,6.6698124,"\int \frac{\sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^2} \, dx","Integrate[Sqrt[Sec[c + d*x]]/(a + b*Cos[c + d*x])^2,x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{b \sin (c+d x)}{a \left(a^2-b^2\right)}+\frac{b \sin (c+d x)}{\left(b^2-a^2\right) (a+b \cos (c+d x))}\right)}{d}+\frac{\frac{2 \left(3 b^2-4 a^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{\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 \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{8 a \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{\left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}}{4 a d (b-a) (a+b)}","\frac{b^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{a d \left(a^2-b^2\right) (a \sec (c+d x)+b)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \left(a^2-b^2\right)}-\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 d \left(a^2-b^2\right)}+\frac{\left(3 a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a d (a-b) (a+b)^2}",1,"(Sqrt[Sec[c + d*x]]*((b*Sin[c + d*x])/(a*(a^2 - b^2)) + (b*Sin[c + d*x])/((-a^2 + b^2)*(a + b*Cos[c + d*x]))))/d + ((2*(-4*a^2 + 3*b^2)*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)) + (8*a*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/((a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (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*(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)","B",0
721,1,574,208,6.6730976,"\int \frac{1}{(a+b \cos (c+d x))^2 \sqrt{\sec (c+d x)}} \, dx","Integrate[1/((a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{a \sin (c+d x)}{\left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\sin (c+d x)}{a^2-b^2}\right)}{d}+\frac{\frac{\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 \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{8 a \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 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{b \sin (c+d x) \sqrt{\sec (c+d x)}}{d \left(a^2-b^2\right) (a \sec (c+d x)+b)}+\frac{a \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{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \left(a^2-b^2\right)}-\frac{\left(a^2+b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a-b) (a+b)^2}",1,"(Sqrt[Sec[c + d*x]]*(-(Sin[c + d*x]/(a^2 - b^2)) + (a*Sin[c + d*x])/((a^2 - b^2)*(a + b*Cos[c + d*x]))))/d + ((-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)) + (8*a*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)) + (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*(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)","B",0
722,1,250,223,5.6442624,"\int \frac{1}{(a+b \cos (c+d x))^2 \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[1/((a + b*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)),x]","\frac{\cos (2 (c+d x)) \csc (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(-\left(a^2-3 b^2\right) \sqrt{-\tan ^2(c+d x)} \sqrt{\sec (c+d x)} (a+b \cos (c+d x)) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-b (b-a) \sqrt{-\tan ^2(c+d x)} \sqrt{\sec (c+d x)} (a+b \cos (c+d x)) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+a b \left(a \tan ^2(c+d x)-\sqrt{-\tan ^2(c+d x)} \sqrt{\sec (c+d x)} (a+b \cos (c+d x)) E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)\right)}{b^2 d (a-b) (a+b) \left(\sec ^2(c+d x)-2\right) (a \sec (c+d x)+b)}","\frac{a \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^2-2 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)}{b^2 d \left(a^2-b^2\right)}-\frac{a \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{a \left(a^2-3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a-b) (a+b)^2}",1,"(Cos[2*(c + d*x)]*Csc[c + d*x]*Sec[c + d*x]^(3/2)*(-(b*(-a + b)*(a + b*Cos[c + d*x])*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[-Tan[c + d*x]^2]) - (a^2 - 3*b^2)*(a + b*Cos[c + d*x])*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[-Tan[c + d*x]^2] + a*b*(a*Tan[c + d*x]^2 - (a + b*Cos[c + d*x])*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[-Tan[c + d*x]^2])))/((a - b)*b^2*(a + b)*d*(b + a*Sec[c + d*x])*(-2 + Sec[c + d*x]^2))","A",1
723,1,319,245,6.4291934,"\int \frac{1}{(a+b \cos (c+d x))^2 \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[1/((a + b*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)),x]","\frac{\frac{4 a^2 \sin (c+d x)}{b \left(b^2-a^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))}-\frac{2 \cot (c+d x) \left(6 a^3 \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b \left(-3 a^2+a b+2 b^2\right) \sqrt{-\tan ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b \left(3 a^2-2 b^2\right) \sqrt{-\tan ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-3 a^2 b \sec ^{\frac{3}{2}}(c+d x)+3 a^2 b \cos (2 (c+d x)) \sec ^{\frac{3}{2}}(c+d x)-10 a b^2 \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b^3 \sec ^{\frac{3}{2}}(c+d x)-2 b^3 \cos (2 (c+d x)) \sec ^{\frac{3}{2}}(c+d x)\right)}{b^3 (a-b) (a+b)}}{4 d}","-\frac{a^2 \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^2-2 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)}{b^2 d \left(a^2-b^2\right)}-\frac{a \left(3 a^2-4 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)}{b^3 d \left(a^2-b^2\right)}+\frac{a^2 \left(3 a^2-5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}",1,"((4*a^2*Sin[c + d*x])/(b*(-a^2 + b^2)*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]]) - (2*Cot[c + d*x]*(-3*a^2*b*Sec[c + d*x]^(3/2) + 2*b^3*Sec[c + d*x]^(3/2) + 3*a^2*b*Cos[2*(c + d*x)]*Sec[c + d*x]^(3/2) - 2*b^3*Cos[2*(c + d*x)]*Sec[c + d*x]^(3/2) + 2*b*(3*a^2 - 2*b^2)*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] + 2*b*(-3*a^2 + a*b + 2*b^2)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] + 6*a^3*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] - 10*a*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2]))/((a - b)*b^3*(a + b)))/(4*d)","A",1
724,1,747,455,6.8343525,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^3,x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{2 \tan (c+d x)}{3 a^3}-\frac{b^3 \sin (c+d x)}{2 a^2 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{b \left(24 a^4-65 a^2 b^2+35 b^4\right) \sin (c+d x)}{4 a^4 \left(a^2-b^2\right)^2}-\frac{3 \left(5 a^2 b^3 \sin (c+d x)-3 b^5 \sin (c+d x)\right)}{4 a^3 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}\right)}{d}+\frac{\frac{2 \left(160 a^5 b-512 a^3 b^3+280 a b^5\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{b \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{\left(72 a^4 b^2-195 a^2 b^4+105 b^6\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(16 a^6+328 a^4 b^2-641 a^2 b^4+315 b^6\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))}}{48 a^4 d (a-b)^2 (a+b)^2}","\frac{b^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{2 a d \left(a^2-b^2\right) (a \sec (c+d x)+b)^2}+\frac{b^2 \left(13 a^2-7 b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 (a \sec (c+d x)+b)}-\frac{b \left(24 a^4-65 a^2 b^2+35 b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{b \left(24 a^4-65 a^2 b^2+35 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{b^2 \left(63 a^4-86 a^2 b^2+35 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}+\frac{\left(8 a^4-61 a^2 b^2+35 b^4\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{12 a^3 d \left(a^2-b^2\right)^2}+\frac{\left(8 a^4-61 a^2 b^2+35 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 a^3 d \left(a^2-b^2\right)^2}",1,"((2*(16*a^6 + 328*a^4*b^2 - 641*a^2*b^4 + 315*b^6)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(160*a^5*b - 512*a^3*b^3 + 280*a*b^5)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((72*a^4*b^2 - 195*a^2*b^4 + 105*b^6)*Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(48*a^4*(a - b)^2*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(-1/4*(b*(24*a^4 - 65*a^2*b^2 + 35*b^4)*Sin[c + d*x])/(a^4*(a^2 - b^2)^2) - (b^3*Sin[c + d*x])/(2*a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) - (3*(5*a^2*b^3*Sin[c + d*x] - 3*b^5*Sin[c + d*x]))/(4*a^3*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) + (2*Tan[c + d*x])/(3*a^3)))/d","A",0
725,1,723,388,6.8971329,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^3,x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{b^2 \sin (c+d x)}{2 a \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{11 a^2 b^2 \sin (c+d x)-5 b^4 \sin (c+d x)}{4 a^2 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{\left(8 a^4-29 a^2 b^2+15 b^4\right) \sin (c+d x)}{4 a^3 \left(a^2-b^2\right)^2}\right)}{d}-\frac{\frac{2 \left(16 a^5-80 a^3 b^2+40 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) \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(56 a^4 b-95 a^2 b^3+45 b^5\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(8 a^4 b-29 a^2 b^3+15 b^5\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^3 d (a-b)^2 (a+b)^2}","\frac{b^2 \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^2 \left(11 a^2-5 b^2\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{b \left(11 a^2-5 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)}{4 a^2 d \left(a^2-b^2\right)^2}+\frac{\left(8 a^4-29 a^2 b^2+15 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^4-29 a^2 b^2+15 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{b \left(35 a^4-38 a^2 b^2+15 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d (a-b)^2 (a+b)^3}",1,"-1/16*((2*(56*a^4*b - 95*a^2*b^3 + 45*b^5)*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 - 80*a^3*b^2 + 40*a*b^4)*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*b - 29*a^2*b^3 + 15*b^5)*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 - 29*a^2*b^2 + 15*b^4)*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2) + (b^2*Sin[c + d*x])/(2*a*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) + (11*a^2*b^2*Sin[c + d*x] - 5*b^4*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/d","A",0
726,1,694,321,6.7680959,"\int \frac{\sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^3} \, dx","Integrate[Sqrt[Sec[c + d*x]]/(a + b*Cos[c + d*x])^3,x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{3 b \left(3 a^2-b^2\right) \sin (c+d x)}{4 a^2 \left(a^2-b^2\right)^2}-\frac{b \sin (c+d x)}{2 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{b^3 \sin (c+d x)-7 a^2 b \sin (c+d x)}{4 a \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}\right)}{d}+\frac{\frac{2 \left(8 a b^3-32 a^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{\left(3 b^4-9 a^2 b^2\right) \sin (c+d x) \cos (2 (c+d x)) (a \sec (c+d x)+b) \left(-4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a b^2 \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{2 \left(16 a^4-19 a^2 b^2+9 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))}}{16 a^2 d (a-b)^2 (a+b)^2}","\frac{b^2 \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{3 b^2 \left(3 a^2-b^2\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(7 a^2-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)}{4 a d \left(a^2-b^2\right)^2}-\frac{3 b \left(3 a^2-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)}{4 a^2 d \left(a^2-b^2\right)^2}+\frac{3 \left(5 a^4-2 a^2 b^2+b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d (a-b)^2 (a+b)^3}",1,"((2*(16*a^4 - 19*a^2*b^2 + 9*b^4)*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*b + 8*a*b^3)*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*b^2 + 3*b^4)*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]]*((3*b*(3*a^2 - b^2)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2) - (b*Sin[c + d*x])/(2*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) + (-7*a^2*b*Sin[c + d*x] + b^3*Sin[c + d*x])/(4*a*(a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/d","B",0
727,1,395,317,5.9833486,"\int \frac{1}{(a+b \cos (c+d x))^3 \sqrt{\sec (c+d x)}} \, dx","Integrate[1/((a + b*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]),x]","\frac{\frac{2 \cot (c+d x) \left(6 a^4 \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+5 a^3 b \sec ^{\frac{3}{2}}(c+d x)-5 a^3 b \cos (2 (c+d x)) \sec ^{\frac{3}{2}}(c+d x)-2 a b \left(5 a^2+b^2\right) \sqrt{-\tan ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+20 a^2 b^2 \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b \left(5 a^3-7 a^2 b+a b^2+b^3\right) \sqrt{-\tan ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-2 b^4 \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+a b^3 \sec ^{\frac{3}{2}}(c+d x)-a b^3 \cos (2 (c+d x)) \sec ^{\frac{3}{2}}(c+d x)\right)}{a^2 b (a-b)^2 (a+b)^2}-\frac{4 b \sin (c+d x) \left(7 a^3+b \left(5 a^2+b^2\right) \cos (c+d x)-a b^2\right)}{a \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}}{16 d}","\frac{b^2 \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{b \left(7 a^2-b^2\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{3 \left(a^2+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)}{4 b d \left(a^2-b^2\right)^2}+\frac{\left(5 a^2+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)}{4 a d \left(a^2-b^2\right)^2}-\frac{\left(3 a^4+10 a^2 b^2-b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b d (a-b)^2 (a+b)^3}",1,"((-4*b*(7*a^3 - a*b^2 + b*(5*a^2 + b^2)*Cos[c + d*x])*Sin[c + d*x])/(a*(a^2 - b^2)^2*(a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) + (2*Cot[c + d*x]*(5*a^3*b*Sec[c + d*x]^(3/2) + a*b^3*Sec[c + d*x]^(3/2) - 5*a^3*b*Cos[2*(c + d*x)]*Sec[c + d*x]^(3/2) - a*b^3*Cos[2*(c + d*x)]*Sec[c + d*x]^(3/2) - 2*a*b*(5*a^2 + b^2)*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] + 2*b*(5*a^3 - 7*a^2*b + a*b^2 + b^3)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] + 6*a^4*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] + 20*a^2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] - 2*b^4*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2]))/(a^2*(a - b)^2*b*(a + b)^2))/(16*d)","A",1
728,1,665,302,6.7497632,"\int \frac{1}{(a+b \cos (c+d x))^3 \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[1/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{\left(a^2+5 b^2\right) \sin (c+d x)}{4 b \left(b^2-a^2\right)^2}+\frac{a^2 \sin (c+d x)}{2 b \left(b^2-a^2\right) (a+b \cos (c+d x))^2}+\frac{a^3 \sin (c+d x)-7 a b^2 \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(-5 a^2-b^2\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a \left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{\left(a^2+5 b^2\right) \sin (c+d x) \cos (2 (c+d x)) (a \sec (c+d x)+b) \left(-4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a b^2 \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{48 a \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{\left(1-\cos ^2(c+d x)\right) (a+b \cos (c+d x))}}{16 d (a-b)^2 (a+b)^2}","\frac{3 \left(a^2+b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 d \left(a^2-b^2\right)^2 (a \sec (c+d x)+b)}-\frac{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{a \left(a^2-7 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)}{4 b^2 d \left(a^2-b^2\right)^2}-\frac{\left(a^2+5 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)}{4 b d \left(a^2-b^2\right)^2}-\frac{\left(a^4-10 a^2 b^2-3 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d (a-b)^2 (a+b)^3}",1,"-1/16*((2*(-5*a^2 - b^2)*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)) + (48*a*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/((a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((a^2 + 5*b^2)*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*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(((a^2 + 5*b^2)*Sin[c + d*x])/(4*b*(-a^2 + b^2)^2) + (a^2*Sin[c + d*x])/(2*b*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + (a^3*Sin[c + d*x] - 7*a*b^2*Sin[c + d*x])/(4*b*(-a^2 + b^2)^2*(a + b*Cos[c + d*x]))))/d","B",0
729,1,280,319,5.6571502,"\int \frac{1}{(a+b \cos (c+d x))^3 \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[1/((a + b*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)),x]","\frac{\frac{2 a b^2 \sin (c+d x) \left(a^3+3 b \left(a^2-3 b^2\right) \cos (c+d x)-7 a b^2\right)}{\left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^2}+\frac{\cot (c+d x) \left(-6 a b \left(a^2-3 b^2\right) \sin ^2(c+d x) \sec ^{\frac{3}{2}}(c+d x)+6 a b \left(a^2-3 b^2\right) \sqrt{-\tan ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+6 \left(a^4-2 a^2 b^2+5 b^4\right) \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-2 b \left(3 a^3-a^2 b-9 a b^2+7 b^3\right) \sqrt{-\tan ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{(a-b)^2 (a+b)^2}}{8 b^3 d}","\frac{a \left(a^2-7 b^2\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{a \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{3 a \left(a^2-3 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)}{4 b^2 d \left(a^2-b^2\right)^2}+\frac{\left(3 a^4-5 a^2 b^2+8 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d \left(a^2-b^2\right)^2}-\frac{3 a \left(a^4-2 a^2 b^2+5 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 b}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d (a-b)^2 (a+b)^3}",1,"((2*a*b^2*(a^3 - 7*a*b^2 + 3*b*(a^2 - 3*b^2)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]) + (Cot[c + d*x]*(-6*a*b*(a^2 - 3*b^2)*Sec[c + d*x]^(3/2)*Sin[c + d*x]^2 + 6*a*b*(a^2 - 3*b^2)*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] - 2*b*(3*a^3 - a^2*b - 9*a*b^2 + 7*b^3)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] + 6*(a^4 - 2*a^2*b^2 + 5*b^4)*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(8*b^3*d)","A",1
730,1,353,369,13.0776515,"\int \sqrt{a+b \cos (c+d x)} \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(7/2),x]","\frac{2 \left(\sqrt{\sec (c+d x)} (a+b \cos (c+d x)) \left(\left(9 a^2-2 b^2\right) \sin (c+d x)+a \tan (c+d x) (3 a \sec (c+d x)+b)\right)+\frac{\sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(-\left(\left(9 a^2-2 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 \left(9 a^2+7 a b-2 b^2\right) \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 \left(9 a^3+9 a^2 b-2 a b^2-2 b^3\right) \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)}{15 a^2 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 (a-b) \sqrt{a+b} (9 a+2 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^2-2 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 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}+\frac{2 b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{15 a d}",1,"(2*((Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(9*a^3 + 9*a^2*b - 2*a*b^2 - 2*b^3)*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*(9*a^2 + 7*a*b - 2*b^2)*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]))] - (9*a^2 - 2*b^2)*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 + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*((9*a^2 - 2*b^2)*Sin[c + d*x] + a*(b + 3*a*Sec[c + d*x])*Tan[c + d*x])))/(15*a^2*d*Sqrt[a + b*Cos[c + d*x]])","A",0
731,1,301,311,10.99304,"\int \sqrt{a+b \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[Sqrt[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 b \sin (c+d x)}{3 a}+\frac{2}{3} \tan (c+d x)\right)}{d}-\frac{2 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(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+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 b (a+b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{3 a d \sqrt{a+b \cos (c+d x)}}","\frac{2 b (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{2 (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}} F\left(\sin ^{-1}\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)}}",1,"(-2*Cos[(c + d*x)/2]^2*Sqrt[Sec[c + d*x]]*(2*b*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 2*a*(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)] + 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[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*b*Sin[c + d*x])/(3*a) + (2*Tan[c + d*x])/3))/d","A",1
732,1,215,269,6.0491712,"\int \sqrt{a+b \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2),x]","\frac{2 \left(\sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))+\frac{-\tan \left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))-\frac{(a+b) \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} \left(E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{\sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}}}}{\sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)}}\right)}{d \sqrt{a+b \cos (c+d x)}}","\frac{2 (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 (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}} F\left(\sin ^{-1}\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 + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*Sin[c + d*x] + (-(((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)] - EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]) - (a + b*Cos[c + d*x])*Tan[(c + d*x)/2])/(Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))/(d*Sqrt[a + b*Cos[c + d*x]])","A",0
733,1,146,155,1.3635372,"\int \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]],x]","\frac{2 \sqrt{\sec (c+d x)} \sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \cos (c+d x)} \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 (a+b) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))}{a+b}}}","-\frac{2 \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\sec (c+d x)} \sqrt{\frac{a (1-\cos (c+d x))}{a+b \cos (c+d x)}} \sqrt{\frac{a (\cos (c+d x)+1)}{a+b \cos (c+d x)}} (a+b \cos (c+d x)) \Pi \left(\frac{b}{a+b};\sin ^{-1}\left(\frac{\sqrt{a+b} \sqrt{\cos (c+d x)}}{\sqrt{a+b \cos (c+d x)}}\right)|-\frac{a-b}{a+b}\right)}{d \sqrt{a+b}}",1,"(2*Sqrt[a + b*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[Sec[c + d*x]])/((a + b)*d*Sqrt[((a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)])","A",1
734,1,565,431,13.8522901,"\int \frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]/Sqrt[Sec[c + d*x]],x]","-\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(2 a \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-(a+b) \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 \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 \sqrt{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 \tan ^5\left(\frac{1}{2} (c+d x)\right)-a \tan \left(\frac{1}{2} (c+d x)\right)-b \tan ^5\left(\frac{1}{2} (c+d x)\right)+2 b \tan ^3\left(\frac{1}{2} (c+d x)\right)-b \tan \left(\frac{1}{2} (c+d x)\right)\right)}{d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}+\frac{\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)}{d \sqrt{\sec (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}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a d \sqrt{\sec (c+d x)}}-\frac{a \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,"-((Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-(a*Tan[(c + d*x)/2]) - b*Tan[(c + d*x)/2] + 2*b*Tan[(c + d*x)/2]^3 + a*Tan[(c + d*x)/2]^5 - b*Tan[(c + d*x)/2]^5 - 2*a*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*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)*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*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
735,1,1113,498,18.15685,"\int \frac{\sqrt{a+b \cos (c+d x)}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]]/Sec[c + d*x]^(3/2),x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (2 (c+d x))}{4 d}+\frac{a^2 \sqrt{\frac{a-b}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-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}} \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 i a^2 \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+8 i b^2 \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-a^2 \sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-a b \sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-i a (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}}+2 i \left(a^2+b a-2 b^2\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}}-2 i a^2 \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+8 i b^2 \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}}{4 b \sqrt{\frac{a-b}{a+b}} d \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\sqrt{a+b} \left(a^2-4 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d \sqrt{\sec (c+d x)}}+\frac{a \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b d}+\frac{\sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} (a+2 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} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[2*(c + d*x)])/(4*d) + (-(a^2*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]) - a*b*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2] + 2*a*b*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^3 + a^2*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^5 - a*b*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^5 - (2*I)*a^2*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*I)*b^2*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*a^2*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*I)*b^2*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - I*a*(a - b)*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^2 + a*b - 2*b^2)*EllipticF[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)])/(4*b*Sqrt[(a - b)/(a + b)]*d*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","C",0
736,1,441,427,13.8810668,"\int (a+b \cos (c+d x))^{3/2} \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(9/2),x]","\frac{\sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)} \left(-\frac{4 b \left(3 b^2-41 a^2\right) \sin (c+d x)}{105 a^2}+\frac{2 \sec (c+d x) \left(25 a^2 \sin (c+d x)+3 b^2 \sin (c+d x)\right)}{105 a}+\frac{2}{7} a \tan (c+d x) \sec ^2(c+d x)+\frac{16}{35} b \tan (c+d x) \sec (c+d x)\right)}{d}+\frac{4 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(b \left(3 b^2-41 a^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))+a \left(25 a^3+82 a^2 b+51 a b^2-6 b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+2 b \left(-41 a^3-41 a^2 b+3 a b^2+3 b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{105 a^2 d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(25 a^2+3 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{105 a d}+\frac{2 (a-b) \sqrt{a+b} \left(25 a^2-57 a b-6 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^2 d \sqrt{\sec (c+d x)}}+\frac{4 b (a-b) \sqrt{a+b} \left(41 a^2-3 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{105 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 a \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{7 d}+\frac{16 b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{35 d}",1,"(4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*b*(-41*a^3 - 41*a^2*b + 3*a*b^2 + 3*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*(25*a^3 + 82*a^2*b + 51*a*b^2 - 6*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + b*(-41*a^2 + 3*b^2)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*a^2*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((-4*b*(-41*a^2 + 3*b^2)*Sin[c + d*x])/(105*a^2) + (2*Sec[c + d*x]*(25*a^2*Sin[c + d*x] + 3*b^2*Sin[c + d*x]))/(105*a) + (16*b*Sec[c + d*x]*Tan[c + d*x])/35 + (2*a*Sec[c + d*x]^2*Tan[c + d*x])/7))/d","A",0
737,1,345,365,11.8698265,"\int (a+b \cos (c+d x))^{3/2} \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(7/2),x]","\frac{2 \left(\sqrt{\sec (c+d x)} (a+b \cos (c+d x)) \left(\left(3 a^2+b^2\right) \sin (c+d x)+a \tan (c+d x) (a \sec (c+d x)+2 b)\right)+\frac{\sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(-\left(\left(3 a^2+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 \left(3 a^2+4 a b+b^2\right) \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 \left(3 a^3+3 a^2 b+a b^2+b^3\right) \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)}{5 a d \sqrt{a+b \cos (c+d x)}}","\frac{2 (a-b) \sqrt{a+b} \left(3 a^2+b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 a \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}+\frac{4 b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}-\frac{2 (a-b) (3 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}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{5 a d \sqrt{\sec (c+d x)}}",1,"(2*((Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(3*a^3 + 3*a^2*b + a*b^2 + b^3)*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*(3*a^2 + 4*a*b + b^2)*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]))] - (3*a^2 + b^2)*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 + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*((3*a^2 + b^2)*Sin[c + d*x] + a*(2*b + a*Sec[c + d*x])*Tan[c + d*x])))/(5*a*d*Sqrt[a + b*Cos[c + d*x]])","A",0
738,1,291,317,7.2159076,"\int (a+b \cos (c+d x))^{3/2} \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2),x]","\frac{\sqrt{\sec (c+d x)} \left(4 \cos ^2\left(\frac{1}{2} (c+d x)\right) \left(\left(a^2+4 a b+3 b^2\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+b \left(\sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))-4 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)+2 \tan (c+d x) (a+b \cos (c+d x)) (a+4 b \cos (c+d x))\right)}{3 d \sqrt{a+b \cos (c+d x)}}","\frac{2 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{2 (a-3 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}} F\left(\sin ^{-1}\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{8 b (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Sec[c + d*x]]*(4*Cos[(c + d*x)/2]^2*(-4*b*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (a^2 + 4*a*b + 3*b^2)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + b*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^3*(Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2])) + 2*(a + b*Cos[c + d*x])*(a + 4*b*Cos[c + d*x])*Tan[c + d*x]))/(3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
739,1,639,397,17.487741,"\int (a+b \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2),x]","\frac{2 \left(-\left(a^2+2 a b-b^2\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)+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^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+a^2 \tan \left(\frac{1}{2} (c+d x)\right)-2 b^2 \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^2 \sqrt{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)+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)\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 (a-2 b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{d \sqrt{\sec (c+d x)}}+\frac{2 (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)}{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^2*Tan[(c + d*x)/2] + a*b*Tan[(c + d*x)/2] - 2*a*b*Tan[(c + d*x)/2]^3 - a^2*Tan[(c + d*x)/2]^5 + a*b*Tan[(c + d*x)/2]^5 - 2*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)] - 2*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)] + a*(a + b)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (a^2 + 2*a*b - b^2)*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",1
740,1,322,435,11.6692225,"\int (a+b \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]],x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(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))+4 a (a-2 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 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)+12 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)}} \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{a+b \cos (c+d x)}}","\frac{b \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}+\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{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)}}-\frac{3 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}} \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,"(Cos[(c + d*x)/2]^2*Sqrt[Sec[c + d*x]]*(2*b*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 4*a*(a - 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)] + 12*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]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + b*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(d*Sqrt[a + b*Cos[c + d*x]])","A",1
741,1,845,493,17.8038214,"\int \frac{(a+b \cos (c+d x))^{3/2}}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2)/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{-5 a^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+5 a b \tan ^5\left(\frac{1}{2} (c+d x)\right)-10 a b \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 a^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^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)+5 a^2 \tan \left(\frac{1}{2} (c+d x)\right)+5 a b \tan \left(\frac{1}{2} (c+d x)\right)+5 a (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^2-b a+2 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}}+6 a^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^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}}}{4 d \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","-\frac{\sqrt{a+b} \left(3 a^2+4 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b d \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}{2 d}+\frac{3 a \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}+\frac{\sqrt{a+b} (5 a+2 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{5 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 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) - (5*a^2*Tan[(c + d*x)/2] + 5*a*b*Tan[(c + d*x)/2] - 10*a*b*Tan[(c + d*x)/2]^3 - 5*a^2*Tan[(c + d*x)/2]^5 + 5*a*b*Tan[(c + d*x)/2]^5 + 6*a^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 8*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*a^2*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*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)] + 5*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)] - 2*(4*a^2 - a*b + 2*b^2)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)])/(4*d*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","A",1
742,1,961,568,17.6230428,"\int \frac{(a+b \cos (c+d x))^{3/2}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(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{1}{12} b \sin (c+d x)+\frac{7}{24} a \sin (2 (c+d x))+\frac{1}{12} b \sin (3 (c+d x))\right)}{d}+\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(-3 a^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)+16 b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 a b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 a^2 b \tan ^5\left(\frac{1}{2} (c+d x)\right)-32 b^3 \tan ^3\left(\frac{1}{2} (c+d x)\right)-6 a^2 b \tan ^3\left(\frac{1}{2} (c+d x)\right)-6 a^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)+72 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)+3 a^3 \tan \left(\frac{1}{2} (c+d x)\right)+16 b^3 \tan \left(\frac{1}{2} (c+d x)\right)+16 a b^2 \tan \left(\frac{1}{2} (c+d x)\right)+3 a^2 b \tan \left(\frac{1}{2} (c+d x)\right)+\left(3 a^3+3 b a^2+16 b^2 a+16 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 a (7 a-26 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}}-6 a^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}}+72 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}}\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+16 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{24 b d}-\frac{(a-b) \sqrt{a+b} \left(3 a^2+16 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)}{24 a b d \sqrt{\sec (c+d x)}}+\frac{a \sqrt{a+b} \left(a^2-12 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) (a+b \cos (c+d x))^{3/2}}{3 d \sqrt{\sec (c+d x)}}+\frac{a \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} (a+2 b) (3 a+8 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 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 + (7*a*Sin[2*(c + d*x)])/24 + (b*Sin[3*(c + d*x)])/12))/d + (Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(3*a^3*Tan[(c + d*x)/2] + 3*a^2*b*Tan[(c + d*x)/2] + 16*a*b^2*Tan[(c + d*x)/2] + 16*b^3*Tan[(c + d*x)/2] - 6*a^2*b*Tan[(c + d*x)/2]^3 - 32*b^3*Tan[(c + d*x)/2]^3 - 3*a^3*Tan[(c + d*x)/2]^5 + 3*a^2*b*Tan[(c + d*x)/2]^5 - 16*a*b^2*Tan[(c + d*x)/2]^5 + 16*b^3*Tan[(c + d*x)/2]^5 - 6*a^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)] + 72*a*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 6*a^3*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*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)] + (3*a^3 + 3*a^2*b + 16*a*b^2 + 16*b^3)*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*(7*a - 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)])","A",1
743,1,521,494,16.2915199,"\int (a+b \cos (c+d x))^{5/2} \sec ^{\frac{11}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(11/2),x]","\frac{\sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \sec (c+d x) \left(163 a^2 b \sin (c+d x)+5 b^3 \sin (c+d x)\right)}{315 a}+\frac{2}{315} \sec ^2(c+d x) \left(49 a^2 \sin (c+d x)+75 b^2 \sin (c+d x)\right)+\frac{2}{9} a^2 \tan (c+d x) \sec ^3(c+d x)+\frac{2 \left(147 a^4+279 a^2 b^2-10 b^4\right) \sin (c+d x)}{315 a^2}+\frac{38}{63} a b \tan (c+d x) \sec ^2(c+d x)\right)}{d}+\frac{2 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(-\left(\left(147 a^4+279 a^2 b^2-10 b^4\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 \left(147 a^4+261 a^3 b+279 a^2 b^2+155 a b^3-10 b^4\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-2 \left(147 a^5+147 a^4 b+279 a^3 b^2+279 a^2 b^3-10 a b^4-10 b^5\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{315 a^2 d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(49 a^2+75 b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 d}+\frac{2 b \left(163 a^2+5 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{315 a d}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{9 d}-\frac{2 (a-b) \sqrt{a+b} \left(147 a^3-114 a^2 b+165 a b^2+10 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+279 a^2 b^2-10 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{38 a b \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{63 d}",1,"(2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*(147*a^5 + 147*a^4*b + 279*a^3*b^2 + 279*a^2*b^3 - 10*a*b^4 - 10*b^5)*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*(147*a^4 + 261*a^3*b + 279*a^2*b^2 + 155*a*b^3 - 10*b^4)*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 + 279*a^2*b^2 - 10*b^4)*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]) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*(147*a^4 + 279*a^2*b^2 - 10*b^4)*Sin[c + d*x])/(315*a^2) + (2*Sec[c + d*x]^2*(49*a^2*Sin[c + d*x] + 75*b^2*Sin[c + d*x]))/315 + (2*Sec[c + d*x]*(163*a^2*b*Sin[c + d*x] + 5*b^3*Sin[c + d*x]))/(315*a) + (38*a*b*Sec[c + d*x]^2*Tan[c + d*x])/63 + (2*a^2*Sec[c + d*x]^3*Tan[c + d*x])/9))/d","A",0
744,1,443,427,13.8309348,"\int (a+b \cos (c+d x))^{5/2} \sec ^{\frac{9}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(9/2),x]","\frac{\sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 b \left(29 a^2+3 b^2\right) \sin (c+d x)}{21 a}+\frac{2}{21} \sec (c+d x) \left(5 a^2 \sin (c+d x)+9 b^2 \sin (c+d x)\right)+\frac{2}{7} a^2 \tan (c+d x) \sec ^2(c+d x)+\frac{6}{7} a b \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(-b \left(29 a^2+3 b^2\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))+2 a \left(5 a^3+29 a^2 b+27 a b^2+3 b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-2 b \left(29 a^3+29 a^2 b+3 a b^2+3 b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{21 a d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \cos (c+d x)}}","\frac{2 \left(5 a^2+9 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{21 d}+\frac{2 (a-b) \sqrt{a+b} \left(5 a^2-24 a b+3 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)}{21 a d \sqrt{\sec (c+d x)}}+\frac{2 b (a-b) \sqrt{a+b} \left(29 a^2+3 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{21 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{7 d}+\frac{6 a b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{7 d}",1,"(2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-2*b*(29*a^3 + 29*a^2*b + 3*a*b^2 + 3*b^3)*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*(5*a^3 + 29*a^2*b + 27*a*b^2 + 3*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - b*(29*a^2 + 3*b^2)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(21*a*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*b*(29*a^2 + 3*b^2)*Sin[c + d*x])/(21*a) + (2*Sec[c + d*x]*(5*a^2*Sin[c + d*x] + 9*b^2*Sin[c + d*x]))/21 + (6*a*b*Sec[c + d*x]*Tan[c + d*x])/7 + (2*a^2*Sec[c + d*x]^2*Tan[c + d*x])/7))/d","A",0
745,1,376,378,15.1690976,"\int (a+b \cos (c+d x))^{5/2} \sec ^{\frac{7}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(7/2),x]","\frac{2 \sqrt{\sec (c+d x)} (a+b \cos (c+d x)) \left(\left(9 a^2+23 b^2\right) \sin (c+d x)+a \tan (c+d x) (3 a \sec (c+d x)+11 b)\right)+\frac{2 \left(\left(9 a^2+23 b^2\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))-2 \left(9 a^3+17 a^2 b+23 a b^2+15 b^3\right) \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 \left(9 a^3+9 a^2 b+23 a b^2+23 b^3\right) \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)}{\left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)}}}{15 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 (a-b) \sqrt{a+b} \left(9 a^2-8 a b+15 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 d \sqrt{\sec (c+d x)}}+\frac{2 (a-b) \sqrt{a+b} \left(9 a^2+23 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 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{5 d}+\frac{22 a b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}",1,"((2*(2*(9*a^3 + 9*a^2*b + 23*a*b^2 + 23*b^3)*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*(9*a^3 + 17*a^2*b + 23*a*b^2 + 15*b^3)*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]))] + (9*a^2 + 23*b^2)*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]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)) + 2*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*((9*a^2 + 23*b^2)*Sin[c + d*x] + a*(11*b + 3*a*Sec[c + d*x])*Tan[c + d*x]))/(15*d*Sqrt[a + b*Cos[c + d*x]])","A",0
746,1,399,452,11.9953829,"\int (a+b \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*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^2 \tan (c+d x)+\frac{14}{3} a b \sin (c+d x)\right)}{d}-\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(-4 \left(a^3+7 a^2 b+9 a b^2-3 b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-24 b^3 \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+14 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))+28 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 d \sqrt{a+b \cos (c+d x)}}","\frac{2 \sqrt{a+b} \left(a^2-7 a b+9 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 d \sqrt{\sec (c+d x)}}+\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}-\frac{2 b^2 \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)}}+\frac{14 b (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 d \sqrt{\sec (c+d x)}}",1,"-1/3*(Cos[(c + d*x)/2]^2*Sqrt[Sec[c + d*x]]*(28*a*b*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 4*(a^3 + 7*a^2*b + 9*a*b^2 - 3*b^3)*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)] - 24*b^3*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 14*a*b*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(d*Sqrt[a + b*Cos[c + d*x]]) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((14*a*b*Sin[c + d*x])/3 + (2*a^2*Tan[c + d*x])/3))/d","A",1
747,1,736,505,16.6791385,"\int (a+b \cos (c+d x))^{5/2} \sec ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2),x]","\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}+\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(2 a^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a^3 \tan \left(\frac{1}{2} (c+d x)\right)+2 a \left(a^2+3 a b-3 b^2\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)+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)-2 a^2 b \tan ^5\left(\frac{1}{2} (c+d x)\right)+4 a^2 b \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 a^2 b \tan \left(\frac{1}{2} (c+d x)\right)-\left(2 a^3+2 a^2 b-a b^2-b^3\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)+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 b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+a b^2 \tan \left(\frac{1}{2} (c+d x)\right)+10 a b^2 \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)+10 a b^2 \sqrt{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^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 b^3 \tan ^3\left(\frac{1}{2} (c+d x)\right)+b^3 \tan \left(\frac{1}{2} (c+d x)\right)\right)}{d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","-\frac{\left(2 a^2-b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{\sqrt{a+b} \left(2 a^2-6 a b-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)}{d \sqrt{\sec (c+d x)}}+\frac{(a-b) \sqrt{a+b} \left(2 a^2-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 d \sqrt{\sec (c+d x)}}+\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{d}-\frac{5 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)}{d \sqrt{\sec (c+d x)}}",1,"(2*a^2*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^3*Tan[(c + d*x)/2] - 2*a^2*b*Tan[(c + d*x)/2] + a*b^2*Tan[(c + d*x)/2] + b^3*Tan[(c + d*x)/2] + 4*a^2*b*Tan[(c + d*x)/2]^3 - 2*b^3*Tan[(c + d*x)/2]^3 + 2*a^3*Tan[(c + d*x)/2]^5 - 2*a^2*b*Tan[(c + d*x)/2]^5 - a*b^2*Tan[(c + d*x)/2]^5 + b^3*Tan[(c + d*x)/2]^5 + 10*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)] + 10*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)] - (2*a^3 + 2*a^2*b - a*b^2 - b^3)*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*(a^2 + 3*a*b - 3*b^2)*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
748,1,3679,503,22.3196105,"\int (a+b \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]],x]","\text{Result too large to show}","\frac{\sqrt{a+b} \left(8 a^2+9 a b+2 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d \sqrt{\sec (c+d x)}}-\frac{\sqrt{a+b} \left(15 a^2+4 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 d \sqrt{\sec (c+d x)}}+\frac{b^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 d \sqrt{\sec (c+d x)}}+\frac{9 a b \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{9 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)}{4 d \sqrt{\sec (c+d x)}}",1,"(b^2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[2*(c + d*x)])/(4*d) + (((3*a^2*b)/(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + b^3/(2*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^3*Sqrt[Sec[c + d*x]])/Sqrt[a + b*Cos[c + d*x]] + (11*a*b^2*Sqrt[Sec[c + d*x]])/(8*Sqrt[a + b*Cos[c + d*x]]) + (9*a*b^2*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(8*Sqrt[a + b*Cos[c + d*x]]))*(-18*a*b*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 4*(4*a^3 - 12*a^2*b + a*b^2 - 2*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 4*b*(15*a^2 + 4*b^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]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 9*a*b*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(4*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)*(-1/4*(Sqrt[Sec[(c + d*x)/2]^2]*Tan[(c + d*x)/2]*(-18*a*b*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 4*(4*a^3 - 12*a^2*b + a*b^2 - 2*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 4*b*(15*a^2 + 4*b^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]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 9*a*b*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)^2) + (b*Sin[c + d*x]*(-18*a*b*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 4*(4*a^3 - 12*a^2*b + a*b^2 - 2*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 4*b*(15*a^2 + 4*b^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]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 9*a*b*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(8*(a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)) - (Tan[(c + d*x)/2]*(-18*a*b*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 4*(4*a^3 - 12*a^2*b + a*b^2 - 2*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 4*b*(15*a^2 + 4*b^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]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 9*a*b*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(8*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)) + ((-9*a*b*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 - (9*a*b*(a + b)*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (2*(4*a^3 - 12*a^2*b + a*b^2 - 2*b^3)*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])] - (2*b*(15*a^2 + 4*b^2)*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*((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])] - (9*a*b*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (2*(4*a^3 - 12*a^2*b + a*b^2 - 2*b^3)*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]))] - (2*b*(15*a^2 + 4*b^2)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*(-((b*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((a + b*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + 9*a*b^2*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + 9*a*b*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - 9*a*b*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 - (2*(4*a^3 - 12*a^2*b + a*b^2 - 2*b^3)*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)]) - (2*b*(15*a^2 + 4*b^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]))]*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)]) - (9*a*b*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*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])/(4*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)) - ((-18*a*b*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 4*(4*a^3 - 12*a^2*b + a*b^2 - 2*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 4*b*(15*a^2 + 4*b^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]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - 9*a*b*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(8*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(-1 + Tan[(c + d*x)/2]^2))))","B",0
749,1,970,566,17.7046544,"\int \frac{(a+b \cos (c+d x))^{5/2}}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2)/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{1}{12} \sin (c+d x) b^2+\frac{1}{12} \sin (3 (c+d x)) b^2+\frac{13}{24} a \sin (2 (c+d x)) b\right)}{d}+\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(-33 a^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)+16 b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 a b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+33 a^2 b \tan ^5\left(\frac{1}{2} (c+d x)\right)-32 b^3 \tan ^3\left(\frac{1}{2} (c+d x)\right)-66 a^2 b \tan ^3\left(\frac{1}{2} (c+d x)\right)+30 a^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+120 a b^2 \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)+33 a^3 \tan \left(\frac{1}{2} (c+d x)\right)+16 b^3 \tan \left(\frac{1}{2} (c+d x)\right)+16 a b^2 \tan \left(\frac{1}{2} (c+d x)\right)+33 a^2 b \tan \left(\frac{1}{2} (c+d x)\right)+\left(33 a^3+33 b a^2+16 b^2 a+16 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 a \left(24 a^2-13 b a+38 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}}+30 a^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+120 a b^2 \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+16 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{24 d}+\frac{\sqrt{a+b} \left(33 a^2+26 a b+16 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)}{24 d \sqrt{\sec (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left(33 a^2+16 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)}{24 a d \sqrt{\sec (c+d x)}}-\frac{5 a \sqrt{a+b} \left(a^2+4 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{8 b d \sqrt{\sec (c+d x)}}+\frac{b^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{13 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{12 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((b^2*Sin[c + d*x])/12 + (13*a*b*Sin[2*(c + d*x)])/24 + (b^2*Sin[3*(c + d*x)])/12))/d + (Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(33*a^3*Tan[(c + d*x)/2] + 33*a^2*b*Tan[(c + d*x)/2] + 16*a*b^2*Tan[(c + d*x)/2] + 16*b^3*Tan[(c + d*x)/2] - 66*a^2*b*Tan[(c + d*x)/2]^3 - 32*b^3*Tan[(c + d*x)/2]^3 - 33*a^3*Tan[(c + d*x)/2]^5 + 33*a^2*b*Tan[(c + d*x)/2]^5 - 16*a*b^2*Tan[(c + d*x)/2]^5 + 16*b^3*Tan[(c + d*x)/2]^5 + 30*a^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 120*a*b^2*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*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*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)] + (33*a^3 + 33*a^2*b + 16*a*b^2 + 16*b^3)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*(24*a^2 - 13*a*b + 38*b^2)*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)])","A",0
750,1,1642,638,16.6081114,"\int \frac{(a+b \cos (c+d x))^{5/2}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(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{1}{32} \sin (4 (c+d x)) b^2+\frac{17}{96} a \sin (c+d x) b+\frac{17}{96} a \sin (3 (c+d x)) b+\frac{1}{192} \left(59 a^2+48 b^2\right) \sin (2 (c+d x))\right)}{d}+\frac{15 a^4 \sqrt{\frac{a-b}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-284 a b^3 \sqrt{\frac{a-b}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+284 a^2 b^2 \sqrt{\frac{a-b}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-15 a^3 b \sqrt{\frac{a-b}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+568 a b^3 \sqrt{\frac{a-b}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)+30 a^3 b \sqrt{\frac{a-b}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)-30 i a^4 \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)+288 i b^4 \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)+720 i a^2 b^2 \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-15 a^4 \sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-284 a b^3 \sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-284 a^2 b^2 \sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-15 a^3 b \sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-i a \left(15 a^3-15 b a^2+284 b^2 a-284 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}}+2 i \left(15 a^4+59 b a^3-38 b^2 a^2+36 b^3 a-72 b^4\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}}-30 i a^4 \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}}+288 i b^4 \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}}+720 i a^2 b^2 \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}}{192 b \sqrt{\frac{a-b}{a+b}} d \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\left(59 a^2+36 b^2\right) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{96 d \sqrt{\sec (c+d x)}}+\frac{a \left(15 a^2+284 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{192 b d}-\frac{(a-b) \sqrt{a+b} \left(15 a^2+284 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)}{192 b d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left(5 a^4-120 a^2 b^2-48 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}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{64 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sqrt{a+b} \left(15 a^3+118 a^2 b+284 a b^2+72 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)}{192 b d \sqrt{\sec (c+d x)}}+\frac{b^2 \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{4 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{17 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{24 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((17*a*b*Sin[c + d*x])/96 + ((59*a^2 + 48*b^2)*Sin[2*(c + d*x)])/192 + (17*a*b*Sin[3*(c + d*x)])/96 + (b^2*Sin[4*(c + d*x)])/32))/d + (-15*a^4*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2] - 15*a^3*b*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2] - 284*a^2*b^2*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2] - 284*a*b^3*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2] + 30*a^3*b*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^3 + 568*a*b^3*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^3 + 15*a^4*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^5 - 15*a^3*b*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^5 + 284*a^2*b^2*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^5 - 284*a*b^3*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^5 - (30*I)*a^4*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (720*I)*a^2*b^2*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (288*I)*b^4*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (30*I)*a^4*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)] + (720*I)*a^2*b^2*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (288*I)*b^4*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - I*a*(15*a^3 - 15*a^2*b + 284*a*b^2 - 284*b^3)*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)*(15*a^4 + 59*a^3*b - 38*a^2*b^2 + 36*a*b^3 - 72*b^4)*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)])/(192*b*Sqrt[(a - b)/(a + b)]*d*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","C",0
751,1,322,314,14.0441918,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[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 \tan (c+d x)}{3 a}-\frac{4 b \sin (c+d x)}{3 a^2}\right)}{d}+\frac{4 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(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))+a (a-2 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 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^2 d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \cos (c+d x)}}","-\frac{4 b (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^3 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{a+b} (a+2 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 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}}{3 a d}",1,"(4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*b*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*(a - 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)] + 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]]*((-4*b*Sin[c + d*x])/(3*a^2) + (2*Tan[c + d*x])/(3*a)))/d","A",0
752,1,296,264,12.5945865,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[Sec[c + d*x]^(3/2)/Sqrt[a + b*Cos[c + d*x]],x]","\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)}}","\frac{2 (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} \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*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
753,1,103,129,0.8733824,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[Sqrt[Sec[c + d*x]]/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \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 \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*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
754,1,146,136,1.7928498,"\int \frac{1}{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Integrate[1/(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","-\frac{2 \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 \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[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
755,1,507,474,12.1011051,"\int \frac{1}{\sqrt{a+b \cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[1/(Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(3/2)),x]","\frac{\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 \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 \sin (c+d x) \sqrt{\sec (c+d x)}}{b d \sqrt{a+b \cos (c+d x)}}+\frac{\sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}+\frac{\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{(a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{a b d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Cos[c + d*x]/(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
756,1,1153,505,18.9201465,"\int \frac{1}{\sqrt{a+b \cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[1/(Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^(5/2)),x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (2 (c+d x))}{4 b 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(-3 a^2 \sqrt{\frac{a-b}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 a b \sqrt{\frac{a-b}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-6 a b \sqrt{\frac{a-b}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 i a^2 \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+8 i b^2 \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+3 a^2 \sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+3 a b \sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+3 i a (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}}-2 i \left(3 a^2-b a+2 b^2\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^2 \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+8 i b^2 \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{4 b^2 \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} \left(3 a^2+4 b^2\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^3 d \sqrt{\sec (c+d x)}}-\frac{3 a \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{4 b^2 d}-\frac{(3 a-2 b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d \sqrt{\sec (c+d x)}}+\frac{3 (a-b) \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{4 b^2 d \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) \sqrt{a+b \cos (c+d x)}}{2 b d \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sin[2*(c + d*x)])/(4*b*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)]*(3*a^2*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2] + 3*a*b*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2] - 6*a*b*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^3 - 3*a^2*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^5 + 3*a*b*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^5 + (6*I)*a^2*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*I)*b^2*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (6*I)*a^2*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*I)*b^2*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (3*I)*a*(a - b)*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)*(3*a^2 - a*b + 2*b^2)*EllipticF[I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)]))/(4*b^2*Sqrt[(a - b)/(a + b)]*d*(-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
757,1,440,397,14.4429291,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)} \left(-\frac{2 b^3 \sin (c+d x)}{a^2 \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{2 \tan (c+d x)}{3 a^2}-\frac{2 b \left(5 a^2-8 b^2\right) \sin (c+d x)}{3 a^3 \left(a^2-b^2\right)}\right)}{d}-\frac{2 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(b \left(8 b^2-5 a^2\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))-2 a \left(a^3-5 a^2 b+2 a b^2+8 b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+2 b \left(-5 a^3-5 a^2 b+8 a b^2+8 b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{3 a^3 d \left(a^2-b^2\right) \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \cos (c+d x)}}","\frac{2 (a+2 b) (a+4 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 b^2 \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(a^2-4 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 \left(5 a^2-8 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^4 d \sqrt{a+b} \sqrt{\sec (c+d x)}}",1,"(-2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*b*(-5*a^3 - 5*a^2*b + 8*a*b^2 + 8*b^3)*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^3 - 5*a^2*b + 2*a*b^2 + 8*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + b*(-5*a^2 + 8*b^2)*Cos[c + d*x]*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*a^3*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((-2*b*(5*a^2 - 8*b^2)*Sin[c + d*x])/(3*a^3*(a^2 - b^2)) - (2*b^3*Sin[c + d*x])/(a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])) + (2*Tan[c + d*x])/(3*a^2)))/d","A",0
758,1,369,325,12.1305066,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 \left(\sin (c+d x) \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\sec (c+d x)} \left(\left(a^2-2 b^2\right) (a+b \cos (c+d x))+a b^2\right)-\sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(\left(a^2-2 b^2\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))-2 a \left(a^2-a b-2 b^2\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+2 \left(a^3+a^2 b-2 a b^2-2 b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)\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^2 \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+2 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^2-2 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,"(2*((a*b^2 + (a^2 - 2*b^2)*(a + b*Cos[c + d*x]))*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Sec[c + d*x]]*Sin[c + d*x] - Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a^3 + a^2*b - 2*a*b^2 - 2*b^3)*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)*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 - 2*b^2)*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
759,1,237,307,8.4567572,"\int \frac{\sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[Sqrt[Sec[c + d*x]]/(a + b*Cos[c + d*x])^(3/2),x]","\frac{2 \sqrt{\sec (c+d x)} \left(b (b-a) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right)+2 a (a+b) \cos ^2\left(\frac{1}{2} (c+d x)\right) \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)-2 b (a+b) \cos ^2\left(\frac{1}{2} (c+d x)\right) \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)}{a d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}","-\frac{2 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 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 \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*Sqrt[Sec[c + d*x]]*(-2*b*(a + b)*Cos[(c + d*x)/2]^2*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[(1 + Sec[c + d*x])^(-1)] + 2*a*(a + b)*Cos[(c + d*x)/2]^2*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[(1 + Sec[c + d*x])^(-1)] + b*(-a + b)*Cos[c + d*x]*Tan[(c + d*x)/2]))/(a*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])","A",1
760,1,235,306,4.017565,"\int \frac{1}{(a+b \cos (c+d x))^{3/2} \sqrt{\sec (c+d x)}} \, dx","Integrate[1/((a + b*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]),x]","\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) \left((a-b) \sin (c+d x) \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)}}-(a+b \cos (c+d x)) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+(a+b \cos (c+d x)) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{d \left(a^2-b^2\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}}}","\frac{2 a \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 \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)}}-\frac{2 \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{a+b} \sqrt{\sec (c+d x)}}",1,"(Sec[(c + d*x)/2]^2*((a + b*Cos[c + d*x])*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (a + b*Cos[c + d*x])*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + (a - b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sin[c + d*x]))/((a^2 - b^2)*d*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[a + b*Cos[c + d*x]]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sqrt[Sec[c + d*x]])","A",1
761,1,1175,447,18.0540425,"\int \frac{1}{(a+b \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[1/((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 \sin (c+d x) a^2}{b \left(b^2-a^2\right) (a+b \cos (c+d x))}+\frac{2 \sin (c+d x) a}{b \left(a^2-b^2\right)}\right)}{d}-\frac{2 \left(a^2 \sqrt{\frac{a-b}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-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}} \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 i a^2 \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+2 i b^2 \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-a^2 \sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-a b \sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-i a (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 \left(2 a^2-b a-b^2\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}}-2 i a^2 \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 i b^2 \Pi \left(\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|-\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{b \sqrt{\frac{a-b}{a+b}} \left(a^2-b^2\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)-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^2 \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 \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{2 \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{a+b} \sqrt{\sec (c+d x)}}+\frac{2 \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)}{b d \sqrt{a+b} \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*a*Sin[c + d*x])/(b*(a^2 - b^2)) + (2*a^2*Sin[c + d*x])/(b*(-a^2 + b^2)*(a + b*Cos[c + d*x]))))/d - (2*(-(a^2*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]) - a*b*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2] + 2*a*b*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^3 + a^2*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^5 - a*b*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^5 - (2*I)*a^2*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*b^2*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*a^2*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*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - I*a*(a - b)*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*(2*a^2 - a*b - b^2)*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*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","C",0
762,1,1025,525,15.1937801,"\int \frac{1}{(a+b \cos (c+d x))^{3/2} \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[1/((a + b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(5/2)),x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(-\frac{2 \sin (c+d x) a^3}{b^2 \left(b^2-a^2\right) (a+b \cos (c+d x))}-\frac{2 \sin (c+d x) a^2}{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(3 a^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)+b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-a b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 a^2 b \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 b^3 \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 a^2 b \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 a^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)-6 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)-3 a^3 \tan \left(\frac{1}{2} (c+d x)\right)+b^3 \tan \left(\frac{1}{2} (c+d x)\right)+a b^2 \tan \left(\frac{1}{2} (c+d x)\right)-3 a^2 b \tan \left(\frac{1}{2} (c+d x)\right)-\left(3 a^3+3 b a^2-b^2 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 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}}+6 a^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}}-6 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}}\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{2 a^2 \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\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{\left(3 a^2-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 b^2 d \sqrt{a+b} \sqrt{\sec (c+d x)}}+\frac{3 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}} \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{(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^2*Sin[c + d*x])/(b^2*(a^2 - b^2)) - (2*a^3*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)]*(-3*a^3*Tan[(c + d*x)/2] - 3*a^2*b*Tan[(c + d*x)/2] + a*b^2*Tan[(c + d*x)/2] + b^3*Tan[(c + d*x)/2] + 6*a^2*b*Tan[(c + d*x)/2]^3 - 2*b^3*Tan[(c + d*x)/2]^3 + 3*a^3*Tan[(c + d*x)/2]^5 - 3*a^2*b*Tan[(c + d*x)/2]^5 - a*b^2*Tan[(c + d*x)/2]^5 + b^3*Tan[(c + d*x)/2]^5 + 6*a^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*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*a^3*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*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)] - (3*a^3 + 3*a^2*b - a*b^2 - b^3)*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 + 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)))","A",0
763,1,546,513,17.5467375,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^(5/2)/(a + b*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 \tan (c+d x)}{3 a^3}-\frac{2 b^3 \sin (c+d x)}{3 a^2 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{8 b \left(2 a^4-7 a^2 b^2+4 b^4\right) \sin (c+d x)}{3 a^4 \left(a^2-b^2\right)^2}-\frac{2 \left(11 a^2 b^3 \sin (c+d x)-7 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{4 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(2 b \left(2 a^4-7 a^2 b^2+4 b^4\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))+a \left(a^5-8 a^4 b+7 a^3 b^2+28 a^2 b^3-4 a b^4-16 b^5\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+4 b \left(2 a^5+2 a^4 b-7 a^3 b^2-7 a^2 b^3+4 a b^4+4 b^5\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{3 a^4 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{4 b^2 \left(5 a^2-3 b^2\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 b^2 \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{8 b \left(2 a^4-7 a^2 b^2+4 b^4\right) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{3 a^5 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}+\frac{2 \left(a^4-13 a^2 b^2+8 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(a^4+9 a^3 b+16 a^2 b^2-12 a b^3-16 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 (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}",1,"(4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(4*b*(2*a^5 + 2*a^4*b - 7*a^3*b^2 - 7*a^2*b^3 + 4*a*b^4 + 4*b^5)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*(a^5 - 8*a^4*b + 7*a^3*b^2 + 28*a^2*b^3 - 4*a*b^4 - 16*b^5)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*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]) + (Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((-8*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*Sin[c + d*x])/(3*a^4*(a^2 - b^2)^2) - (2*b^3*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) - (2*(11*a^2*b^3*Sin[c + d*x] - 7*b^5*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)^2*(a + b*Cos[c + d*x])) + (2*Tan[c + d*x])/(3*a^3)))/d","A",0
764,1,525,438,17.2659413,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^(3/2)/(a + b*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{2 b^2 \sin (c+d x)}{3 a \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{8 \left(2 a^2 b^2 \sin (c+d x)-b^4 \sin (c+d x)\right)}{3 a^2 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}+\frac{2 \left(3 a^4-15 a^2 b^2+8 b^4\right) \sin (c+d x)}{3 a^3 \left(a^2-b^2\right)^2}\right)}{d}+\frac{2 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(-\left(\left(3 a^4-15 a^2 b^2+8 b^4\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 \left(3 a^4-6 a^3 b-15 a^2 b^2+2 a b^3+8 b^4\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-2 \left(3 a^5+3 a^4 b-15 a^3 b^2-15 a^2 b^3+8 a b^4+8 b^5\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \cos (c+d x)}}","\frac{8 b^2 \left(2 a^2-b^2\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 b^2 \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 \left(3 a^4-15 a^2 b^2+8 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)}}-\frac{2 \left(3 a^3+9 a^2 b-6 a b^2-8 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 (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 - 15*a^2*b^2 + 8*b^4)*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2) + (2*b^2*Sin[c + d*x])/(3*a*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) + (8*(2*a^2*b^2*Sin[c + d*x] - b^4*Sin[c + d*x]))/(3*a^2*(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*(3*a^5 + 3*a^4*b - 15*a^3*b^2 - 15*a^2*b^3 + 8*a*b^4 + 8*b^5)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(3*a^4 - 6*a^3*b - 15*a^2*b^2 + 2*a*b^3 + 8*b^4)*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 - 15*a^2*b^2 + 8*b^4)*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])","A",0
765,1,471,421,14.4537099,"\int \frac{\sqrt{\sec (c+d x)}}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[Sqrt[Sec[c + d*x]]/(a + b*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)} \left(\frac{4 b \left(3 a^2-b^2\right) \sin (c+d x)}{3 a^2 \left(a^2-b^2\right)^2}-\frac{2 b \sin (c+d x)}{3 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{2 \left(5 a^2 b \sin (c+d x)-b^3 \sin (c+d x)\right)}{3 a \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}\right)}{d}+\frac{4 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(b \left(b^2-3 a^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))+a \left(3 a^3+6 a^2 b+a b^2-2 b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+2 b \left(-3 a^3-3 a^2 b+a b^2+b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{3 d \left(a^3-a b^2\right)^2 \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \cos (c+d x)}}","\frac{2 b^2 \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{4 b \left(3 a^2-b^2\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^2-3 a b-2 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 (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}+\frac{4 b \left(3 a^2-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^3 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((4*b*(3*a^2 - b^2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2) - (2*b*Sin[c + d*x])/(3*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) - (2*(5*a^2*b*Sin[c + d*x] - b^3*Sin[c + d*x]))/(3*a*(a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/d + (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*b*(-3*a^3 - 3*a^2*b + a*b^2 + b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + a*(3*a^3 + 6*a^2*b + a*b^2 - 2*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + b*(-3*a^2 + b^2)*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])","A",0
766,1,455,399,13.5051131,"\int \frac{1}{(a+b \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)}} \, dx","Integrate[1/((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+b^2\right) \sin (c+d x)}{3 a \left(a^2-b^2\right)^2}+\frac{2 a \sin (c+d x)}{3 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{4 \left(a^2 \sin (c+d x)+b^2 \sin (c+d x)\right)}{3 \left(a^2-b^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+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 \left(3 a^2+4 a b+b^2\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-2 \left(3 a^3+3 a^2 b+a b^2+b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{3 a 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+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 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+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 (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)}{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 + b^2)*Sin[c + d*x])/(3*a*(a^2 - b^2)^2) + (2*a*Sin[c + d*x])/(3*(a^2 - b^2)*(a + b*Cos[c + d*x])^2) + (4*(a^2*Sin[c + d*x] + b^2*Sin[c + d*x]))/(3*(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*(3*a^3 + 3*a^2*b + a*b^2 + b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + 2*a*(3*a^2 + 4*a*b + b^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)] - (3*a^2 + b^2)*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
767,1,359,382,8.4053023,"\int \frac{1}{(a+b \cos (c+d x))^{5/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[1/((a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)),x]","-\frac{2 \sqrt{\sec (c+d x)} \left(a^2 \left(a^2-b^2\right) \sin (c+d x)-a \left(a^2-5 b^2\right) \sin (c+d x) (a+b \cos (c+d x))+2 b \cos ^2\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x)) \left(-\left(a^2+4 a b+3 b^2\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a+b \cos (c+d x)}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-b \left(\sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) (a+b \cos (c+d x))+4 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)-4 b^2 \sin (c+d x) (a+b \cos (c+d x))^2\right)}{3 b d \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^{3/2}}","-\frac{8 a b \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 \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 (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)}}+\frac{8 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 (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}",1,"(-2*Sqrt[Sec[c + d*x]]*(a^2*(a^2 - b^2)*Sin[c + d*x] - a*(a^2 - 5*b^2)*(a + b*Cos[c + d*x])*Sin[c + d*x] - 4*b^2*(a + b*Cos[c + d*x])^2*Sin[c + d*x] + 2*b*Cos[(c + d*x)/2]^2*(a + b*Cos[c + d*x])*(4*b*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - (a^2 + 4*a*b + 3*b^2)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(a + b*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - b*(a + b*Cos[c + d*x])*Sec[(c + d*x)/2]^3*(Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2]))))/(3*b*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^(3/2))","A",1
768,1,1716,557,14.4536539,"\int \frac{1}{(a+b \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[1/((a + b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)),x]","\frac{\sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \left(-\frac{2 \sin (c+d x) a^3}{3 b^2 \left(b^2-a^2\right) (a+b \cos (c+d x))^2}+\frac{2 \left(3 a^2-7 b^2\right) \sin (c+d x) a}{3 b^2 \left(a^2-b^2\right)^2}-\frac{8 \left(a^4 \sin (c+d x)-2 a^2 b^2 \sin (c+d x)\right)}{3 b^2 \left(b^2-a^2\right)^2 (a+b \cos (c+d x))}\right)}{d}+\frac{2 \left(-3 a^4 \sqrt{\frac{a-b}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-7 a b^3 \sqrt{\frac{a-b}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+7 a^2 b^2 \sqrt{\frac{a-b}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 a^3 b \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}} \tan ^3\left(\frac{1}{2} (c+d x)\right)-6 a^3 b \sqrt{\frac{a-b}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 i a^4 \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 \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 \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}} \tan \left(\frac{1}{2} (c+d x)\right)-7 a b^3 \sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-7 a^2 b^2 \sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+3 a^3 b \sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+i a \left(3 a^3-3 b a^2-7 b^2 a+7 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 \left(6 a^4-2 b a^3-13 b^2 a^2+6 b^3 a+3 b^4\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 \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 \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 \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 \sqrt{\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{\frac{a \tan ^2\left(\frac{1}{2} (c+d x)\right)-b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(\tan ^4\left(\frac{1}{2} (c+d x)\right)-1\right)}","-\frac{2 a^2 \left(3 a^2-7 b^2\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 a^2 \sin (c+d x)}{3 b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \cos (c+d x))^{3/2}}-\frac{2 \left(3 a^2+a b-6 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 b^2 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}+\frac{2 \left(3 a^2-7 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 b^2 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{b};\sin ^{-1}\left(\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right)|-\frac{a+b}{a-b}\right)}{b^3 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*a*(3*a^2 - 7*b^2)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2) - (2*a^3*Sin[c + d*x])/(3*b^2*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) - (8*(a^4*Sin[c + d*x] - 2*a^2*b^2*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)^2*(a + b*Cos[c + d*x]))))/d + (2*(3*a^4*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2] + 3*a^3*b*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2] - 7*a^2*b^2*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2] - 7*a*b^3*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2] - 6*a^3*b*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^3 + 14*a*b^3*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^3 - 3*a^4*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^5 + 3*a^3*b*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^5 + 7*a^2*b^2*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^5 - 7*a*b^3*Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]^5 + (6*I)*a^4*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (12*I)*a^2*b^2*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (6*I)*b^4*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (6*I)*a^4*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] - (12*I)*a^2*b^2*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + (6*I)*b^4*EllipticPi[(a + b)/(a - b), I*ArcSinh[Sqrt[(a - b)/(a + b)]*Tan[(c + d*x)/2]], -((a + b)/(a - b))]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(a + b)] + I*a*(3*a^3 - 3*a^2*b - 7*a*b^2 + 7*b^3)*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*(6*a^4 - 2*a^3*b - 13*a^2*b^2 + 6*a*b^3 + 3*b^4)*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*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*Sqrt[(a + b + a*Tan[(c + d*x)/2]^2 - b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(-1 + Tan[(c + d*x)/2]^4))","C",0
769,1,242,330,1.7785504,"\int \cos ^m(c+d x) (a+b \cos (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^m*(a + b*Cos[c + d*x])^4,x]","\frac{\sqrt{\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+1}(c+d x) \left(b \cos (c+d x) \left(b \cos (c+d x) \left(b \cos (c+d x) \left(-\frac{4 a \, _2F_1\left(\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};\cos ^2(c+d x)\right)}{m+4}-\frac{b \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+5}{2};\frac{m+7}{2};\cos ^2(c+d x)\right)}{m+5}\right)-\frac{6 a^2 \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\cos ^2(c+d x)\right)}{m+3}\right)-\frac{4 a^3 \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{m+2}\right)-\frac{a^4 \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{m+1}\right)}{d}","-\frac{4 a b \left(a^2 (m+3)+b^2 (m+2)\right) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{b^2 \left(a^2 (5 m+22)+b^2 (m+3)\right) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2) (m+4)}-\frac{\left(a^4 \left(m^2+6 m+8\right)+6 a^2 b^2 \left(m^2+5 m+4\right)+b^4 \left(m^2+4 m+3\right)\right) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) (m+4) \sqrt{\sin ^2(c+d x)}}+\frac{2 a b^3 (m+5) \sin (c+d x) \cos ^{m+2}(c+d x)}{d (m+3) (m+4)}+\frac{b^2 \sin (c+d x) \cos ^{m+1}(c+d x) (a+b \cos (c+d x))^2}{d (m+4)}",1,"(Cos[c + d*x]^(1 + m)*Csc[c + d*x]*(-((a^4*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2])/(1 + m)) + b*Cos[c + d*x]*((-4*a^3*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2])/(2 + m) + b*Cos[c + d*x]*((-6*a^2*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, Cos[c + d*x]^2])/(3 + m) + b*Cos[c + d*x]*((-4*a*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, Cos[c + d*x]^2])/(4 + m) - (b*Cos[c + d*x]*Hypergeometric2F1[1/2, (5 + m)/2, (7 + m)/2, Cos[c + d*x]^2])/(5 + m)))))*Sqrt[Sin[c + d*x]^2])/d","A",1
770,1,197,250,0.9053336,"\int \cos ^m(c+d x) (a+b \cos (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^m*(a + b*Cos[c + d*x])^3,x]","\frac{\sqrt{\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+1}(c+d x) \left(b \cos (c+d x) \left(b \cos (c+d x) \left(-\frac{3 a \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\cos ^2(c+d x)\right)}{m+3}-\frac{b \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};\cos ^2(c+d x)\right)}{m+4}\right)-\frac{3 a^2 \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{m+2}\right)-\frac{a^3 \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{m+1}\right)}{d}","-\frac{a \left(a^2 (m+2)+3 b^2 (m+1)\right) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) \sqrt{\sin ^2(c+d x)}}-\frac{b \left(3 a^2 (m+3)+b^2 (m+2)\right) \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{a b^2 (2 m+7) \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2) (m+3)}+\frac{b^2 \sin (c+d x) \cos ^{m+1}(c+d x) (a+b \cos (c+d x))}{d (m+3)}",1,"(Cos[c + d*x]^(1 + m)*Csc[c + d*x]*(-((a^3*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2])/(1 + m)) + b*Cos[c + d*x]*((-3*a^2*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2])/(2 + m) + b*Cos[c + d*x]*((-3*a*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, Cos[c + d*x]^2])/(3 + m) - (b*Cos[c + d*x]*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, Cos[c + d*x]^2])/(4 + m))))*Sqrt[Sin[c + d*x]^2])/d","A",1
771,1,168,179,0.3569845,"\int \cos ^m(c+d x) (a+b \cos (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^m*(a + b*Cos[c + d*x])^2,x]","-\frac{\sqrt{\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+1}(c+d x) \left(a^2 \left(m^2+5 m+6\right) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)+b (m+1) \cos (c+d x) \left(2 a (m+3) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)+b (m+2) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\cos ^2(c+d x)\right)\right)\right)}{d (m+1) (m+2) (m+3)}","-\frac{\left(a^2 (m+2)+b^2 (m+1)\right) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) (m+2) \sqrt{\sin ^2(c+d x)}}-\frac{2 a b \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) \sqrt{\sin ^2(c+d x)}}+\frac{b^2 \sin (c+d x) \cos ^{m+1}(c+d x)}{d (m+2)}",1,"-((Cos[c + d*x]^(1 + m)*Csc[c + d*x]*(a^2*(6 + 5*m + m^2)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2] + b*(1 + m)*Cos[c + d*x]*(2*a*(3 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2] + b*(2 + m)*Cos[c + d*x]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, Cos[c + d*x]^2]))*Sqrt[Sin[c + d*x]^2])/(d*(1 + m)*(2 + m)*(3 + m)))","A",1
772,1,112,131,0.1702311,"\int \cos ^m(c+d x) (a+b \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^m*(a + b*Cos[c + d*x]),x]","-\frac{\sqrt{\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+1}(c+d x) \left(a (m+2) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)+b (m+1) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)\right)}{d (m+1) (m+2)}","-\frac{a \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{d (m+1) \sqrt{\sin ^2(c+d x)}}-\frac{b \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(c+d x)\right)}{d (m+2) \sqrt{\sin ^2(c+d x)}}",1,"-((Cos[c + d*x]^(1 + m)*Csc[c + d*x]*(a*(2 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2] + b*(1 + m)*Cos[c + d*x]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(1 + m)*(2 + m)))","A",1
773,1,6703,190,24.6163336,"\int \frac{\cos ^m(c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[Cos[c + d*x]^m/(a + b*Cos[c + d*x]),x]","\text{Result too large to show}","\frac{a \sin (c+d x) \cos ^{m-1}(c+d x) \cos ^2(c+d x)^{\frac{1-m}{2}} F_1\left(\frac{1}{2};\frac{1-m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)}-\frac{b \sin (c+d x) \cos ^m(c+d x) \cos ^2(c+d x)^{-m/2} F_1\left(\frac{1}{2};-\frac{m}{2},1;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)}",1,"Result too large to show","B",0
774,1,7214,294,26.4603762,"\int \frac{\cos ^m(c+d x)}{(a+b \cos (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^m/(a + b*Cos[c + d*x])^2,x]","\text{Result too large to show}","\frac{b^2 \sin (c+d x) \cos ^{m+1}(c+d x) \cos ^2(c+d x)^{\frac{1}{2} (-m-1)} F_1\left(\frac{1}{2};\frac{1}{2} (-m-1),2;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)^2}+\frac{a^2 \sin (c+d x) \cos ^{m-1}(c+d x) \cos ^2(c+d x)^{\frac{1-m}{2}} F_1\left(\frac{1}{2};\frac{1-m}{2},2;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)^2}-\frac{2 a b \sin (c+d x) \cos ^m(c+d x) \cos ^2(c+d x)^{-m/2} F_1\left(\frac{1}{2};-\frac{m}{2},2;\frac{3}{2};\sin ^2(c+d x),-\frac{b^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)^2}",1,"Result too large to show","B",0
775,1,222,282,0.8818873,"\int (a+b \cos (c+d x))^3 \sec ^m(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^3*Sec[c + d*x]^m,x]","\frac{\sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sec ^{m-4}(c+d x) \left(\frac{1}{2} a (m-3) \sec ^3(c+d x) \left(2 a (m-2) \left(a (m-1) \, _2F_1\left(\frac{1}{2},\frac{m}{2};\frac{m+2}{2};\sec ^2(c+d x)\right)+3 b m \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m-1}{2};\frac{m+1}{2};\sec ^2(c+d x)\right)\right)+6 b^2 (m-1) m \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m-2}{2};\frac{m}{2};\sec ^2(c+d x)\right)\right)+b^3 m \left(m^2-3 m+2\right) \, _2F_1\left(\frac{1}{2},\frac{m-3}{2};\frac{m-1}{2};\sec ^2(c+d x)\right)\right)}{d (m-3) (m-2) (m-1) m}","-\frac{b \left(3 a^2 (3-m)+b^2 (2-m)\right) \sin (c+d x) \sec ^{m-4}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{4-m}{2};\frac{6-m}{2};\cos ^2(c+d x)\right)}{d (2-m) (4-m) \sqrt{\sin ^2(c+d x)}}-\frac{a \left(a^2 (2-m)+3 b^2 (1-m)\right) \sin (c+d x) \sec ^{m-3}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{3-m}{2};\frac{5-m}{2};\cos ^2(c+d x)\right)}{d (1-m) (3-m) \sqrt{\sin ^2(c+d x)}}-\frac{a^2 \sin (c+d x) \sec ^{m-2}(c+d x) (a \sec (c+d x)+b)}{d (1-m)}-\frac{a^2 b (1-2 m) \sin (c+d x) \sec ^{m-2}(c+d x)}{d (1-m) (2-m)}",1,"(Csc[c + d*x]*Sec[c + d*x]^(-4 + m)*(b^3*m*(2 - 3*m + m^2)*Hypergeometric2F1[1/2, (-3 + m)/2, (-1 + m)/2, Sec[c + d*x]^2] + (a*(-3 + m)*(6*b^2*(-1 + m)*m*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (-2 + m)/2, m/2, Sec[c + d*x]^2] + 2*a*(-2 + m)*(3*b*m*Cos[c + d*x]*Hypergeometric2F1[1/2, (-1 + m)/2, (1 + m)/2, Sec[c + d*x]^2] + a*(-1 + m)*Hypergeometric2F1[1/2, m/2, (2 + m)/2, Sec[c + d*x]^2]))*Sec[c + d*x]^3)/2)*Sqrt[-Tan[c + d*x]^2])/(d*(-3 + m)*(-2 + m)*(-1 + m)*m)","A",1
776,1,159,200,0.3433044,"\int (a+b \cos (c+d x))^2 \sec ^m(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])^2*Sec[c + d*x]^m,x]","\frac{\sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sec ^{m-3}(c+d x) \left(a (m-2) \sec ^2(c+d x) \left(a (m-1) \, _2F_1\left(\frac{1}{2},\frac{m}{2};\frac{m+2}{2};\sec ^2(c+d x)\right)+2 b m \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m-1}{2};\frac{m+1}{2};\sec ^2(c+d x)\right)\right)+b^2 (m-1) m \, _2F_1\left(\frac{1}{2},\frac{m-2}{2};\frac{m}{2};\sec ^2(c+d x)\right)\right)}{d (m-2) (m-1) m}","-\frac{\left(a^2 (2-m)+b^2 (1-m)\right) \sin (c+d x) \sec ^{m-3}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{3-m}{2};\frac{5-m}{2};\cos ^2(c+d x)\right)}{d (1-m) (3-m) \sqrt{\sin ^2(c+d x)}}-\frac{a^2 \sin (c+d x) \sec ^{m-1}(c+d x)}{d (1-m)}-\frac{2 a b \sin (c+d x) \sec ^{m-2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{2-m}{2};\frac{4-m}{2};\cos ^2(c+d x)\right)}{d (2-m) \sqrt{\sin ^2(c+d x)}}",1,"(Csc[c + d*x]*Sec[c + d*x]^(-3 + m)*(b^2*(-1 + m)*m*Hypergeometric2F1[1/2, (-2 + m)/2, m/2, Sec[c + d*x]^2] + a*(-2 + m)*(2*b*m*Cos[c + d*x]*Hypergeometric2F1[1/2, (-1 + m)/2, (1 + m)/2, Sec[c + d*x]^2] + a*(-1 + m)*Hypergeometric2F1[1/2, m/2, (2 + m)/2, Sec[c + d*x]^2])*Sec[c + d*x]^2)*Sqrt[-Tan[c + d*x]^2])/(d*(-2 + m)*(-1 + m)*m)","A",1
777,1,107,143,0.1995857,"\int (a+b \cos (c+d x)) \sec ^m(c+d x) \, dx","Integrate[(a + b*Cos[c + d*x])*Sec[c + d*x]^m,x]","\frac{\sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sec ^{m-1}(c+d x) \left(a (m-1) \, _2F_1\left(\frac{1}{2},\frac{m}{2};\frac{m+2}{2};\sec ^2(c+d x)\right)+b m \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m-1}{2};\frac{m+1}{2};\sec ^2(c+d x)\right)\right)}{d (m-1) m}","-\frac{a \sin (c+d x) \sec ^{m-1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1-m}{2};\frac{3-m}{2};\cos ^2(c+d x)\right)}{d (1-m) \sqrt{\sin ^2(c+d x)}}-\frac{b \sin (c+d x) \sec ^{m-2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{2-m}{2};\frac{4-m}{2};\cos ^2(c+d x)\right)}{d (2-m) \sqrt{\sin ^2(c+d x)}}",1,"(Csc[c + d*x]*(b*m*Cos[c + d*x]*Hypergeometric2F1[1/2, (-1 + m)/2, (1 + m)/2, Sec[c + d*x]^2] + a*(-1 + m)*Hypergeometric2F1[1/2, m/2, (2 + m)/2, Sec[c + d*x]^2])*Sec[c + d*x]^(-1 + m)*Sqrt[-Tan[c + d*x]^2])/(d*(-1 + m)*m)","A",1
778,1,47,26,0.079993,"\int \frac{\sqrt{1-\cos (x)}}{\sqrt{a-\cos (x)}} \, dx","Integrate[Sqrt[1 - Cos[x]]/Sqrt[a - Cos[x]],x]","i \sqrt{2-2 \cos (x)} \csc \left(\frac{x}{2}\right) \log \left(\sqrt{a-\cos (x)}+i \sqrt{2} \cos \left(\frac{x}{2}\right)\right)","-2 \tan ^{-1}\left(\frac{\sin (x)}{\sqrt{1-\cos (x)} \sqrt{a-\cos (x)}}\right)",1,"I*Sqrt[2 - 2*Cos[x]]*Csc[x/2]*Log[I*Sqrt[2]*Cos[x/2] + Sqrt[a - Cos[x]]]","C",1
779,1,64,65,0.0724432,"\int \sqrt{\frac{1-\cos (x)}{a-\cos (x)}} \, dx","Integrate[Sqrt[(1 - Cos[x])/(a - Cos[x])],x]","-\sqrt{2} \csc \left(\frac{x}{2}\right) \sqrt{\frac{\cos (x)-1}{\cos (x)-a}} \sqrt{\cos (x)-a} \log \left(\sqrt{\cos (x)-a}+\sqrt{2} \cos \left(\frac{x}{2}\right)\right)","-\frac{2 \sqrt{\frac{1-\cos (x)}{a-\cos (x)}} \sqrt{a-\cos (x)} \tan ^{-1}\left(\frac{\sin (x)}{\sqrt{1-\cos (x)} \sqrt{a-\cos (x)}}\right)}{\sqrt{1-\cos (x)}}",1,"-(Sqrt[2]*Sqrt[(-1 + Cos[x])/(-a + Cos[x])]*Sqrt[-a + Cos[x]]*Csc[x/2]*Log[Sqrt[2]*Cos[x/2] + Sqrt[-a + Cos[x]]])","A",1
780,1,29,37,0.0598316,"\int (a+a \cos (c+d x)) \left(-\frac{B}{2}+B \cos (c+d x)\right) \, dx","Integrate[(a + a*Cos[c + d*x])*(-1/2*B + B*Cos[c + d*x]),x]","\frac{a B (2 \sin (c+d x)+\sin (2 (c+d x))+2 c)}{4 d}","\frac{a B \sin (c+d x)}{2 d}+\frac{a B \sin (c+d x) \cos (c+d x)}{2 d}",1,"(a*B*(2*c + 2*Sin[c + d*x] + Sin[2*(c + d*x)]))/(4*d)","A",1
781,1,31,26,0.2135979,"\int (a+a \cos (c+d x))^4 \left(-\frac{4 B}{5}+B \cos (c+d x)\right) \, dx","Integrate[(a + a*Cos[c + d*x])^4*((-4*B)/5 + B*Cos[c + d*x]),x]","\frac{a^4 B \sin ^9(c+d x) \csc ^8\left(\frac{1}{2} (c+d x)\right)}{80 d}","\frac{B \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d}",1,"(a^4*B*Csc[(c + d*x)/2]^8*Sin[c + d*x]^9)/(80*d)","A",1
782,1,28,28,0.2036182,"\int (a+a \cos (c+d x))^n \left(-\frac{B n}{1+n}+B \cos (c+d x)\right) \, dx","Integrate[(a + a*Cos[c + d*x])^n*(-((B*n)/(1 + n)) + B*Cos[c + d*x]),x]","\frac{B \sin (c+d x) (a (\cos (c+d x)+1))^n}{d (n+1)}","\frac{B \sin (c+d x) (a \cos (c+d x)+a)^n}{d (n+1)}",1,"(B*(a*(1 + Cos[c + d*x]))^n*Sin[c + d*x])/(d*(1 + n))","A",1
783,1,27,26,0.1282122,"\int \frac{-\frac{3 B}{2}+B \cos (c+d x)}{(a+a \cos (c+d x))^3} \, dx","Integrate[((-3*B)/2 + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^3,x]","-\frac{B \sin (c+d x)}{2 a^3 d (\cos (c+d x)+1)^3}","-\frac{B \sin (c+d x)}{2 d (a \cos (c+d x)+a)^3}",1,"-1/2*(B*Sin[c + d*x])/(a^3*d*(1 + Cos[c + d*x])^3)","A",1
784,1,45,28,0.1150443,"\int (a+a \cos (c+d x))^{3/2} \left(-\frac{3 B}{5}+B \cos (c+d x)\right) \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2)*((-3*B)/5 + B*Cos[c + d*x]),x]","\frac{8 a B \sin \left(\frac{1}{2} (c+d x)\right) \cos ^3\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{5 d}","\frac{2 B \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}",1,"(8*a*B*Cos[(c + d*x)/2]^3*Sqrt[a*(1 + Cos[c + d*x])]*Sin[(c + d*x)/2])/(5*d)","A",1
785,1,33,26,0.0477087,"\int \frac{B+B \cos (c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[(B + B*Cos[c + d*x])/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 B \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{a d}","\frac{2 B \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}",1,"(2*B*Sqrt[a*(1 + Cos[c + d*x])]*Tan[(c + d*x)/2])/(a*d)","A",1
786,1,28,28,0.0755775,"\int \frac{-\frac{5 B}{3}+B \cos (c+d x)}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[((-5*B)/3 + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(5/2),x]","-\frac{2 B \sin (c+d x)}{3 d (a (\cos (c+d x)+1))^{5/2}}","-\frac{2 B \sin (c+d x)}{3 d (a \cos (c+d x)+a)^{5/2}}",1,"(-2*B*Sin[c + d*x])/(3*d*(a*(1 + Cos[c + d*x]))^(5/2))","A",1
787,1,164,104,0.6582942,"\int (a+a \cos (c+d x))^{2/3} (A+B \cos (c+d x)) \, dx","Integrate[(a + a*Cos[c + d*x])^(2/3)*(A + B*Cos[c + d*x]),x]","\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a (\cos (c+d x)+1))^{2/3} \left(3\ 2^{5/6} \sin (c+d x) (5 A+2 B \cos (c+d x)+4 B) \sqrt[6]{1-\cos \left(d x-2 \tan ^{-1}\left(\cot \left(\frac{c}{2}\right)\right)\right)}-2 (5 A+2 B) \sin \left(d x-2 \tan ^{-1}\left(\cot \left(\frac{c}{2}\right)\right)\right) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\cos ^2\left(\frac{d x}{2}-\tan ^{-1}\left(\cot \left(\frac{c}{2}\right)\right)\right)\right)\right)}{20\ 2^{5/6} d \sqrt[6]{1-\cos \left(d x-2 \tan ^{-1}\left(\cot \left(\frac{c}{2}\right)\right)\right)}}","\frac{2 \sqrt[6]{2} (5 A+2 B) \sin (c+d x) (a \cos (c+d x)+a)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{5 d (\cos (c+d x)+1)^{7/6}}+\frac{3 B \sin (c+d x) (a \cos (c+d x)+a)^{2/3}}{5 d}",1,"((a*(1 + Cos[c + d*x]))^(2/3)*Sec[(c + d*x)/2]^2*(3*2^(5/6)*(5*A + 4*B + 2*B*Cos[c + d*x])*(1 - Cos[d*x - 2*ArcTan[Cot[c/2]]])^(1/6)*Sin[c + d*x] - 2*(5*A + 2*B)*Hypergeometric2F1[1/2, 5/6, 3/2, Cos[(d*x)/2 - ArcTan[Cot[c/2]]]^2]*Sin[d*x - 2*ArcTan[Cot[c/2]]]))/(20*2^(5/6)*d*(1 - Cos[d*x - 2*ArcTan[Cot[c/2]]])^(1/6))","A",1
788,1,213,102,3.5306544,"\int \sqrt[3]{a+a \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[(a + a*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x]),x]","\frac{3 \sqrt[3]{a (\cos (c+d x)+1)} \left(\frac{2 (4 A+B) \csc \left(\frac{c}{4}\right) \sec \left(\frac{c}{4}\right) \sqrt[3]{i \sin (c) e^{i d x}+\cos (c) e^{i d x}+1} \left(2 \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{2}{3};-e^{i d x} (\cos (c)+i \sin (c))\right)+e^{i d x} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-e^{i d x} (\cos (c)+i \sin (c))\right)\right)}{i \sin \left(\frac{c}{2}\right) \left(-1+e^{i d x}\right)+\cos \left(\frac{c}{2}\right) \left(1+e^{i d x}\right)}-8 (4 A+B) \cot \left(\frac{c}{2}\right)+8 B \sin (c) \cos (d x)+8 B \cos (c) \sin (d x)\right)}{32 d}","\frac{(4 A+B) \sin (c+d x) \sqrt[3]{a \cos (c+d x)+a} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{2 \sqrt[6]{2} d (\cos (c+d x)+1)^{5/6}}+\frac{3 B \sin (c+d x) \sqrt[3]{a \cos (c+d x)+a}}{4 d}",1,"(3*(a*(1 + Cos[c + d*x]))^(1/3)*(-8*(4*A + B)*Cot[c/2] + 8*B*Cos[d*x]*Sin[c] + (2*(4*A + B)*Csc[c/4]*(2*Hypergeometric2F1[-1/3, 1/3, 2/3, -(E^(I*d*x)*(Cos[c] + I*Sin[c]))] + E^(I*d*x)*Hypergeometric2F1[1/3, 2/3, 5/3, -(E^(I*d*x)*(Cos[c] + I*Sin[c]))])*Sec[c/4]*(1 + E^(I*d*x)*Cos[c] + I*E^(I*d*x)*Sin[c])^(1/3))/((1 + E^(I*d*x))*Cos[c/2] + I*(-1 + E^(I*d*x))*Sin[c/2]) + 8*B*Cos[c]*Sin[d*x]))/(32*d)","C",1
789,1,133,101,0.2719159,"\int \frac{A+B \cos (c+d x)}{\sqrt[3]{a+a \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(1/3),x]","\frac{3\ 2^{5/6} B \sin (c+d x) \sqrt[6]{1-\cos \left(d x-2 \tan ^{-1}\left(\cot \left(\frac{c}{2}\right)\right)\right)}-2 (2 A-B) \sin \left(d x-2 \tan ^{-1}\left(\cot \left(\frac{c}{2}\right)\right)\right) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\cos ^2\left(\frac{d x}{2}-\tan ^{-1}\left(\cot \left(\frac{c}{2}\right)\right)\right)\right)}{4 d \sqrt[3]{a (\cos (c+d x)+1)} \sqrt[6]{\sin ^2\left(\frac{d x}{2}-\tan ^{-1}\left(\cot \left(\frac{c}{2}\right)\right)\right)}}","\frac{(2 A-B) \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{2^{5/6} d \sqrt[6]{\cos (c+d x)+1} \sqrt[3]{a \cos (c+d x)+a}}+\frac{3 B \sin (c+d x)}{2 d \sqrt[3]{a \cos (c+d x)+a}}",1,"(3*2^(5/6)*B*(1 - Cos[d*x - 2*ArcTan[Cot[c/2]]])^(1/6)*Sin[c + d*x] - 2*(2*A - B)*Hypergeometric2F1[1/2, 5/6, 3/2, Cos[(d*x)/2 - ArcTan[Cot[c/2]]]^2]*Sin[d*x - 2*ArcTan[Cot[c/2]]])/(4*d*(a*(1 + Cos[c + d*x]))^(1/3)*(Sin[(d*x)/2 - ArcTan[Cot[c/2]]]^2)^(1/6))","A",1
790,1,197,105,1.4200497,"\int \frac{A+B \cos (c+d x)}{(a+a \cos (c+d x))^{2/3}} \, dx","Integrate[(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^(2/3),x]","\frac{3 \cos \left(\frac{1}{2} (c+d x)\right) \left(-4 \csc \left(\frac{c}{2}\right) \left((3 B-2 A) \cos \left(\frac{d x}{2}\right)+B \cos \left(c+\frac{d x}{2}\right)\right)-(A-2 B) \csc \left(\frac{c}{4}\right) \sec \left(\frac{c}{4}\right) e^{-\frac{1}{2} i d x} \sqrt[3]{i \sin (c) e^{i d x}+\cos (c) e^{i d x}+1} \left(2 \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{2}{3};-e^{i d x} (\cos (c)+i \sin (c))\right)+e^{i d x} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-e^{i d x} (\cos (c)+i \sin (c))\right)\right)\right)}{4 d (a (\cos (c+d x)+1))^{2/3}}","\frac{3 (A-B) \sin (c+d x)}{d (a \cos (c+d x)+a)^{2/3}}-\frac{2^{5/6} (A-2 B) \sin (c+d x) \sqrt[3]{a \cos (c+d x)+a} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{a d (\cos (c+d x)+1)^{5/6}}",1,"(3*Cos[(c + d*x)/2]*(-4*((-2*A + 3*B)*Cos[(d*x)/2] + B*Cos[c + (d*x)/2])*Csc[c/2] - ((A - 2*B)*Csc[c/4]*(2*Hypergeometric2F1[-1/3, 1/3, 2/3, -(E^(I*d*x)*(Cos[c] + I*Sin[c]))] + E^(I*d*x)*Hypergeometric2F1[1/3, 2/3, 5/3, -(E^(I*d*x)*(Cos[c] + I*Sin[c]))])*Sec[c/4]*(1 + E^(I*d*x)*Cos[c] + I*E^(I*d*x)*Sin[c])^(1/3))/E^((I/2)*d*x)))/(4*d*(a*(1 + Cos[c + d*x]))^(2/3))","C",1
791,1,64,63,0.1338528,"\int \frac{\frac{b B}{a}+B \cos (c+d x)}{a+b \cos (c+d x)} \, dx","Integrate[((b*B)/a + B*Cos[c + d*x])/(a + b*Cos[c + d*x]),x]","\frac{B \left(2 \sqrt{b^2-a^2} \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)+a (c+d x)\right)}{a b d}","\frac{B x}{b}-\frac{2 B \sqrt{a-b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a b d}",1,"(B*(a*(c + d*x) + 2*Sqrt[-a^2 + b^2]*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]]))/(a*b*d)","A",1
792,1,22,22,0.1053555,"\int \frac{a+b \cos (c+d x)}{(b+a \cos (c+d x))^2} \, dx","Integrate[(a + b*Cos[c + d*x])/(b + a*Cos[c + d*x])^2,x]","\frac{\sin (c+d x)}{d (a \cos (c+d x)+b)}","\frac{\sin (c+d x)}{d (a \cos (c+d x)+b)}",1,"Sin[c + d*x]/(d*(b + a*Cos[c + d*x]))","A",1
793,1,31,47,0.0638364,"\int \frac{3+\cos (c+d x)}{2-\cos (c+d x)} \, dx","Integrate[(3 + Cos[c + d*x])/(2 - Cos[c + d*x]),x]","\frac{10 \tan ^{-1}\left(\sqrt{3} \tan \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{3} d}-x","\frac{10 \tan ^{-1}\left(\frac{\sin (c+d x)}{-\cos (c+d x)+\sqrt{3}+2}\right)}{\sqrt{3} d}+\frac{5 x}{\sqrt{3}}-x",1,"-x + (10*ArcTan[Sqrt[3]*Tan[(c + d*x)/2]])/(Sqrt[3]*d)","A",1
794,1,58,58,0.0754857,"\int \frac{a B+b B \cos (c+d x)}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[(a*B + b*B*Cos[c + d*x])/Sqrt[a + b*Cos[c + d*x]],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}}}","\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*B*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])","A",1
795,1,259,229,2.1099272,"\int (a+b \cos (c+d x))^{2/3} (A+B \cos (c+d x)) \, dx","Integrate[(a + b*Cos[c + d*x])^(2/3)*(A + B*Cos[c + d*x]),x]","\frac{3 (a+b \cos (c+d x))^{2/3} \left(5 B \left(a^2-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}} F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)-(2 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}} (a+b \cos (c+d x)) F_1\left(\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)+5 b^2 B \sin (c+d x)\right)}{25 b^2 d}","\frac{\sqrt{2} (A b-a B) \sin (c+d x) (a+b \cos (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{b d \sqrt{\cos (c+d x)+1} \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3}}+\frac{\sqrt{2} B (a+b) \sin (c+d x) (a+b \cos (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{5}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{b d \sqrt{\cos (c+d x)+1} \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3}}",1,"(3*(a + b*Cos[c + d*x])^(2/3)*(5*(a^2 - b^2)*B*AppellF1[2/3, 1/2, 1/2, 5/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Cos[c + d*x]))/(a - b))]*Csc[c + d*x] - (5*A*b + 2*a*B)*AppellF1[5/3, 1/2, 1/2, 8/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*(a + b*Cos[c + d*x])*Csc[c + d*x] + 5*b^2*B*Sin[c + d*x]))/(25*b^2*d)","A",0
796,1,253,229,1.9159105,"\int \sqrt[3]{a+b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[(a + b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x]),x]","-\frac{3 \csc (c+d x) \sqrt[3]{a+b \cos (c+d x)} \left(4 B \left(b^2-a^2\right) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\cos (c+d x)+1)}{a-b}} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)+(a B+4 A b) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} (a+b \cos (c+d x)) F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)-4 b^2 B \sin ^2(c+d x)\right)}{16 b^2 d}","\frac{\sqrt{2} (A b-a B) \sin (c+d x) \sqrt[3]{a+b \cos (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{b d \sqrt{\cos (c+d x)+1} \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\sqrt{2} B (a+b) \sin (c+d x) \sqrt[3]{a+b \cos (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{b d \sqrt{\cos (c+d x)+1} \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(-3*(a + b*Cos[c + d*x])^(1/3)*Csc[c + d*x]*(4*(-a^2 + b^2)*B*AppellF1[1/3, 1/2, 1/2, 4/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Cos[c + d*x]))/(a - b))] + (4*A*b + a*B)*AppellF1[4/3, 1/2, 1/2, 7/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*(a + b*Cos[c + d*x]) - 4*b^2*B*Sin[c + d*x]^2))/(16*b^2*d)","A",0
797,1,189,226,0.4626976,"\int \frac{A+B \cos (c+d x)}{\sqrt[3]{a+b \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(1/3),x]","-\frac{3 \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} (a+b \cos (c+d x))^{2/3} \left(5 (A b-a B) F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)+2 B (a+b \cos (c+d x)) F_1\left(\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)\right)}{10 b^2 d}","\frac{\sqrt{2} (A b-a B) \sin (c+d x) \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{b d \sqrt{\cos (c+d x)+1} \sqrt[3]{a+b \cos (c+d x)}}+\frac{\sqrt{2} B \sin (c+d x) (a+b \cos (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{b d \sqrt{\cos (c+d x)+1} \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3}}",1,"(-3*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*(a + b*Cos[c + d*x])^(2/3)*(5*(A*b - a*B)*AppellF1[2/3, 1/2, 1/2, 5/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)] + 2*B*AppellF1[5/3, 1/2, 1/2, 8/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*(a + b*Cos[c + d*x]))*Csc[c + d*x])/(10*b^2*d)","A",0
798,1,188,226,0.4761522,"\int \frac{A+B \cos (c+d x)}{(a+b \cos (c+d x))^{2/3}} \, dx","Integrate[(A + B*Cos[c + d*x])/(a + b*Cos[c + d*x])^(2/3),x]","-\frac{3 \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \sqrt[3]{a+b \cos (c+d x)} \left(4 (A b-a B) F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)+B (a+b \cos (c+d x)) F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)\right)}{4 b^2 d}","\frac{\sqrt{2} (A b-a B) \sin (c+d x) \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{b d \sqrt{\cos (c+d x)+1} (a+b \cos (c+d x))^{2/3}}+\frac{\sqrt{2} B \sin (c+d x) \sqrt[3]{a+b \cos (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{b d \sqrt{\cos (c+d x)+1} \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(-3*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*(a + b*Cos[c + d*x])^(1/3)*(4*(A*b - a*B)*AppellF1[1/3, 1/2, 1/2, 4/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)] + B*AppellF1[4/3, 1/2, 1/2, 7/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*(a + b*Cos[c + d*x]))*Csc[c + d*x])/(4*b^2*d)","A",0
799,1,100,168,0.583963,"\int \cos ^2(c+d x) \sqrt{b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{\sqrt{b \cos (c+d x)} \left(2 \sin (c+d x) \sqrt{\cos (c+d x)} (42 A \cos (c+d x)+15 B \cos (2 (c+d x))+65 B)+252 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+100 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d \sqrt{\cos (c+d x)}}","\frac{2 A \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b d}+\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^2 d}+\frac{10 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d}+\frac{10 b B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[b*Cos[c + d*x]]*(252*A*EllipticE[(c + d*x)/2, 2] + 100*B*EllipticF[(c + d*x)/2, 2] + 2*Sqrt[Cos[c + d*x]]*(65*B + 42*A*Cos[c + d*x] + 15*B*Cos[2*(c + d*x)])*Sin[c + d*x]))/(210*d*Sqrt[Cos[c + d*x]])","A",1
800,1,91,139,0.3094731,"\int \cos (c+d x) \sqrt{b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]*Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{2 (b \cos (c+d x))^{3/2} \left(\sin (c+d x) \sqrt{\cos (c+d x)} (5 A+3 B \cos (c+d x))+5 A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+9 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 b d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 A \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}+\frac{2 A b \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b d}+\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}",1,"(2*(b*Cos[c + d*x])^(3/2)*(9*B*EllipticE[(c + d*x)/2, 2] + 5*A*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(5*A + 3*B*Cos[c + d*x])*Sin[c + d*x]))/(15*b*d*Cos[c + d*x]^(3/2))","A",1
801,1,75,108,0.1114035,"\int \sqrt{b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{2 \sqrt{b \cos (c+d x)} \left(3 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+B \left(F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)}\right)\right)}{3 d \sqrt{\cos (c+d x)}}","\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}+\frac{2 b B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}",1,"(2*Sqrt[b*Cos[c + d*x]]*(3*A*EllipticE[(c + d*x)/2, 2] + B*(EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*Sin[c + d*x])))/(3*d*Sqrt[Cos[c + d*x]])","A",1
802,1,55,80,0.0855304,"\int \sqrt{b \cos (c+d x)} (A+B \cos (c+d x)) \sec (c+d x) \, dx","Integrate[Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{2 b \sqrt{\cos (c+d x)} \left(A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d \sqrt{b \cos (c+d x)}}","\frac{2 A b \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(2*b*Sqrt[Cos[c + d*x]]*(B*EllipticE[(c + d*x)/2, 2] + A*EllipticF[(c + d*x)/2, 2]))/(d*Sqrt[b*Cos[c + d*x]])","A",1
803,1,73,105,0.1904688,"\int \sqrt{b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Integrate[Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{2 \sqrt{b \cos (c+d x)} \left(-A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{A \sin (c+d x)}{\sqrt{\cos (c+d x)}}+B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d \sqrt{\cos (c+d x)}}","\frac{2 A b \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 b B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}",1,"(2*Sqrt[b*Cos[c + d*x]]*(-(A*EllipticE[(c + d*x)/2, 2]) + B*EllipticF[(c + d*x)/2, 2] + (A*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(d*Sqrt[Cos[c + d*x]])","A",1
804,1,85,136,0.2648845,"\int \sqrt{b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Integrate[Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{2 b \left(A \tan (c+d x)+A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 B \sin (c+d x)-3 B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d \sqrt{b \cos (c+d x)}}","\frac{2 A b^2 \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 A b \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 b B \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(2*b*(-3*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 3*B*Sin[c + d*x] + A*Tan[c + d*x]))/(3*d*Sqrt[b*Cos[c + d*x]])","A",1
805,1,107,169,0.4899366,"\int \sqrt{b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Integrate[Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{2 \sec ^2(c+d x) \sqrt{b \cos (c+d x)} \left(\frac{9}{2} A \sin (2 (c+d x))+3 A \tan (c+d x)-9 A \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+5 B \sin (c+d x)+5 B \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d}","\frac{2 A b^3 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{6 A b \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b^2 B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 b B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}",1,"(2*Sqrt[b*Cos[c + d*x]]*Sec[c + d*x]^2*(-9*A*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 5*B*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 5*B*Sin[c + d*x] + (9*A*Sin[2*(c + d*x)])/2 + 3*A*Tan[c + d*x]))/(15*d)","A",1
806,1,103,169,0.3023826,"\int \cos (c+d x) (b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]*(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{(b \cos (c+d x))^{5/2} \left(2 \sin (c+d x) \sqrt{\cos (c+d x)} (42 A \cos (c+d x)+15 B \cos (2 (c+d x))+65 B)+252 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+100 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 b d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 A \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 d}+\frac{6 A b E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{10 b^2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b d}+\frac{10 b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d}",1,"((b*Cos[c + d*x])^(5/2)*(252*A*EllipticE[(c + d*x)/2, 2] + 100*B*EllipticF[(c + d*x)/2, 2] + 2*Sqrt[Cos[c + d*x]]*(65*B + 42*A*Cos[c + d*x] + 15*B*Cos[2*(c + d*x)])*Sin[c + d*x]))/(210*b*d*Cos[c + d*x]^(5/2))","A",1
807,1,88,140,0.0527717,"\int (b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Integrate[(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{2 (b \cos (c+d x))^{3/2} \left(\sin (c+d x) \sqrt{\cos (c+d x)} (5 A+3 B \cos (c+d x))+5 A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+9 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 A b^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 d}+\frac{6 b B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}",1,"(2*(b*Cos[c + d*x])^(3/2)*(9*B*EllipticE[(c + d*x)/2, 2] + 5*A*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(5*A + 3*B*Cos[c + d*x])*Sin[c + d*x]))/(15*d*Cos[c + d*x]^(3/2))","A",1
808,1,76,112,0.0430935,"\int (b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec (c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{2 b \sqrt{b \cos (c+d x)} \left(3 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+B \left(F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)}\right)\right)}{3 d \sqrt{\cos (c+d x)}}","\frac{2 A b E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 b^2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}",1,"(2*b*Sqrt[b*Cos[c + d*x]]*(3*A*EllipticE[(c + d*x)/2, 2] + B*(EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*Sin[c + d*x])))/(3*d*Sqrt[Cos[c + d*x]])","A",1
809,1,57,83,0.0667809,"\int (b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{2 b^2 \sqrt{\cos (c+d x)} \left(A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d \sqrt{b \cos (c+d x)}}","\frac{2 A b^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}+\frac{2 b B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(2*b^2*Sqrt[Cos[c + d*x]]*(B*EllipticE[(c + d*x)/2, 2] + A*EllipticF[(c + d*x)/2, 2]))/(d*Sqrt[b*Cos[c + d*x]])","A",1
810,1,73,110,0.150431,"\int (b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{2 (b \cos (c+d x))^{3/2} \left(-A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{A \sin (c+d x)}{\sqrt{\cos (c+d x)}}+B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 A b^2 \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 A b E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 b^2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}",1,"(2*(b*Cos[c + d*x])^(3/2)*(-(A*EllipticE[(c + d*x)/2, 2]) + B*EllipticF[(c + d*x)/2, 2] + (A*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(d*Cos[c + d*x]^(3/2))","A",1
811,1,87,141,0.1596174,"\int (b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{2 b^2 \left(A \tan (c+d x)+A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 B \sin (c+d x)-3 B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d \sqrt{b \cos (c+d x)}}","\frac{2 A b^3 \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 A b^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 b^2 B \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 b B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(2*b^2*(-3*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 3*B*Sin[c + d*x] + A*Tan[c + d*x]))/(3*d*Sqrt[b*Cos[c + d*x]])","A",1
812,1,107,174,0.2853448,"\int (b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^5,x]","\frac{2 \sec ^3(c+d x) (b \cos (c+d x))^{3/2} \left(\frac{9}{2} A \sin (2 (c+d x))+3 A \tan (c+d x)-9 A \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+5 B \sin (c+d x)+5 B \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d}","\frac{2 A b^4 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{6 A b^2 \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{6 A b E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b^3 B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 b^2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}",1,"(2*(b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*(-9*A*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 5*B*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 5*B*Sin[c + d*x] + (9*A*Sin[2*(c + d*x)])/2 + 3*A*Tan[c + d*x]))/(15*d)","A",1
813,1,100,171,0.0644161,"\int (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx","Integrate[(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]),x]","\frac{(b \cos (c+d x))^{5/2} \left(2 \sin (c+d x) \sqrt{\cos (c+d x)} (42 A \cos (c+d x)+15 B \cos (2 (c+d x))+65 B)+252 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+100 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{6 A b^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 A b \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 d}+\frac{10 b^3 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}+\frac{10 b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 d}",1,"((b*Cos[c + d*x])^(5/2)*(252*A*EllipticE[(c + d*x)/2, 2] + 100*B*EllipticF[(c + d*x)/2, 2] + 2*Sqrt[Cos[c + d*x]]*(65*B + 42*A*Cos[c + d*x] + 15*B*Cos[2*(c + d*x)])*Sin[c + d*x]))/(210*d*Cos[c + d*x]^(5/2))","A",1
814,1,89,145,0.0648704,"\int (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec (c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","\frac{2 b (b \cos (c+d x))^{3/2} \left(\sin (c+d x) \sqrt{\cos (c+d x)} (5 A+3 B \cos (c+d x))+5 A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+9 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 A b^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}+\frac{6 b^2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 d}",1,"(2*b*(b*Cos[c + d*x])^(3/2)*(9*B*EllipticE[(c + d*x)/2, 2] + 5*A*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(5*A + 3*B*Cos[c + d*x])*Sin[c + d*x]))/(15*d*Cos[c + d*x]^(3/2))","A",1
815,1,78,116,0.0458688,"\int (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{2 b^2 \sqrt{b \cos (c+d x)} \left(3 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+B \left(F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)}\right)\right)}{3 d \sqrt{\cos (c+d x)}}","\frac{2 A b^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 b^3 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}",1,"(2*b^2*Sqrt[b*Cos[c + d*x]]*(3*A*EllipticE[(c + d*x)/2, 2] + B*(EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*Sin[c + d*x])))/(3*d*Sqrt[Cos[c + d*x]])","A",1
816,1,54,85,0.1037102,"\int (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{2 (b \cos (c+d x))^{5/2} \left(A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 A b^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}+\frac{2 b^2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(2*(b*Cos[c + d*x])^(5/2)*(B*EllipticE[(c + d*x)/2, 2] + A*EllipticF[(c + d*x)/2, 2]))/(d*Cos[c + d*x]^(5/2))","A",1
817,1,73,112,0.2136429,"\int (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","\frac{2 (b \cos (c+d x))^{5/2} \left(-A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{A \sin (c+d x)}{\sqrt{\cos (c+d x)}}+B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 A b^3 \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 A b^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 b^3 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}",1,"(2*(b*Cos[c + d*x])^(5/2)*(-(A*EllipticE[(c + d*x)/2, 2]) + B*EllipticF[(c + d*x)/2, 2] + (A*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(d*Cos[c + d*x]^(5/2))","A",1
818,1,87,143,0.2111033,"\int (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^5,x]","\frac{2 b^3 \left(A \tan (c+d x)+A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 B \sin (c+d x)-3 B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d \sqrt{b \cos (c+d x)}}","\frac{2 A b^4 \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 A b^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 b^3 B \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 b^2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(2*b^3*(-3*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 3*B*Sin[c + d*x] + A*Tan[c + d*x]))/(3*d*Sqrt[b*Cos[c + d*x]])","A",1
819,1,102,176,0.285732,"\int (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^6(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^6,x]","-\frac{2 b^4 \left(-\frac{9}{2} A \sin (2 (c+d x))-3 A \tan (c+d x)+9 A \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)-5 B \sin (c+d x)-5 B \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d (b \cos (c+d x))^{3/2}}","\frac{2 A b^5 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{6 A b^3 \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{6 A b^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b^4 B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 b^3 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}",1,"(-2*b^4*(9*A*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] - 5*B*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] - 5*B*Sin[c + d*x] - (9*A*Sin[2*(c + d*x)])/2 - 3*A*Tan[c + d*x]))/(15*d*(b*Cos[c + d*x])^(3/2))","A",1
820,1,101,173,0.4526104,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/Sqrt[b*Cos[c + d*x]],x]","\frac{\sin (2 (c+d x)) (42 A \cos (c+d x)+15 B \cos (2 (c+d x))+65 B)+252 A \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+100 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{210 d \sqrt{b \cos (c+d x)}}","\frac{2 A \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^2 d}+\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^3 d}+\frac{10 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 b d}+\frac{10 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}",1,"(252*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 100*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (65*B + 42*A*Cos[c + d*x] + 15*B*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(210*d*Sqrt[b*Cos[c + d*x]])","A",1
821,1,88,144,0.2056948,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/Sqrt[b*Cos[c + d*x]],x]","\frac{2 \sqrt{\cos (c+d x)} \left(\sin (c+d x) \sqrt{\cos (c+d x)} (5 A+3 B \cos (c+d x))+5 A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+9 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d \sqrt{b \cos (c+d x)}}","\frac{2 A \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b d}+\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^2 d}+\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b d \sqrt{\cos (c+d x)}}",1,"(2*Sqrt[Cos[c + d*x]]*(9*B*EllipticE[(c + d*x)/2, 2] + 5*A*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(5*A + 3*B*Cos[c + d*x])*Sin[c + d*x]))/(15*d*Sqrt[b*Cos[c + d*x]])","A",1
822,1,78,113,0.0581047,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/Sqrt[b*Cos[c + d*x]],x]","\frac{2 \sqrt{b \cos (c+d x)} \left(3 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+B \left(F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)}\right)\right)}{3 b d \sqrt{\cos (c+d x)}}","\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b d}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}",1,"(2*Sqrt[b*Cos[c + d*x]]*(3*A*EllipticE[(c + d*x)/2, 2] + B*(EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*Sin[c + d*x])))/(3*b*d*Sqrt[Cos[c + d*x]])","A",1
823,1,54,82,0.0537676,"\int \frac{A+B \cos (c+d x)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/Sqrt[b*Cos[c + d*x]],x]","\frac{2 \sqrt{\cos (c+d x)} \left(A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d \sqrt{b \cos (c+d x)}}","\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}",1,"(2*Sqrt[Cos[c + d*x]]*(B*EllipticE[(c + d*x)/2, 2] + A*EllipticF[(c + d*x)/2, 2]))/(d*Sqrt[b*Cos[c + d*x]])","A",1
824,1,73,106,0.1030919,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x])/Sqrt[b*Cos[c + d*x]],x]","\frac{2 \left(A \sin (c+d x)-A \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d \sqrt{b \cos (c+d x)}}","\frac{2 A \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}",1,"(2*(-(A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]) + B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + A*Sin[c + d*x]))/(d*Sqrt[b*Cos[c + d*x]])","A",1
825,1,84,135,0.160475,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/Sqrt[b*Cos[c + d*x]],x]","\frac{2 \left(A \tan (c+d x)+A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 B \sin (c+d x)-3 B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d \sqrt{b \cos (c+d x)}}","\frac{2 A b \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}",1,"(2*(-3*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 3*B*Sin[c + d*x] + A*Tan[c + d*x]))/(3*d*Sqrt[b*Cos[c + d*x]])","A",1
826,1,101,168,0.255908,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/Sqrt[b*Cos[c + d*x]],x]","\frac{2 \left(9 A \sin (c+d x)-9 A \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 A \tan (c+d x) \sec (c+d x)+5 B \tan (c+d x)+5 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d \sqrt{b \cos (c+d x)}}","\frac{2 A b^2 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{6 A \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b d \sqrt{\cos (c+d x)}}+\frac{2 b B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}",1,"(2*(-9*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 5*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 9*A*Sin[c + d*x] + 5*B*Tan[c + d*x] + 3*A*Sec[c + d*x]*Tan[c + d*x]))/(15*d*Sqrt[b*Cos[c + d*x]])","A",1
827,1,104,176,0.1691809,"\int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^4*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(3/2),x]","\frac{\sin (2 (c+d x)) (42 A \cos (c+d x)+15 B \cos (2 (c+d x))+65 B)+252 A \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+100 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{210 b d \sqrt{b \cos (c+d x)}}","\frac{2 A \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^3 d}+\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^2 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^4 d}+\frac{10 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 b^2 d}+\frac{10 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b d \sqrt{b \cos (c+d x)}}",1,"(252*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 100*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (65*B + 42*A*Cos[c + d*x] + 15*B*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(210*b*d*Sqrt[b*Cos[c + d*x]])","A",1
828,1,88,147,0.1459449,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(3/2),x]","\frac{2 \cos ^{\frac{3}{2}}(c+d x) \left(\sin (c+d x) \sqrt{\cos (c+d x)} (5 A+3 B \cos (c+d x))+5 A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+9 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d (b \cos (c+d x))^{3/2}}","\frac{2 A \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b^2 d}+\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^3 d}+\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^2 d \sqrt{\cos (c+d x)}}",1,"(2*Cos[c + d*x]^(3/2)*(9*B*EllipticE[(c + d*x)/2, 2] + 5*A*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(5*A + 3*B*Cos[c + d*x])*Sin[c + d*x]))/(15*d*(b*Cos[c + d*x])^(3/2))","A",1
829,1,75,116,0.0983006,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(3/2),x]","\frac{2 \cos ^{\frac{3}{2}}(c+d x) \left(3 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+B \left(F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)}\right)\right)}{3 d (b \cos (c+d x))^{3/2}}","\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b^2 d}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d \sqrt{b \cos (c+d x)}}",1,"(2*Cos[c + d*x]^(3/2)*(3*A*EllipticE[(c + d*x)/2, 2] + B*(EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*Sin[c + d*x])))/(3*d*(b*Cos[c + d*x])^(3/2))","A",1
830,1,57,85,0.0526217,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(3/2),x]","\frac{2 \sqrt{\cos (c+d x)} \left(A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{b d \sqrt{b \cos (c+d x)}}","\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \sqrt{b \cos (c+d x)}}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}",1,"(2*Sqrt[Cos[c + d*x]]*(B*EllipticE[(c + d*x)/2, 2] + A*EllipticF[(c + d*x)/2, 2]))/(b*d*Sqrt[b*Cos[c + d*x]])","A",1
831,1,76,112,0.0746104,"\int \frac{A+B \cos (c+d x)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(3/2),x]","\frac{2 \left(A \sin (c+d x)-A \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{b d \sqrt{b \cos (c+d x)}}","-\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x)}{b d \sqrt{b \cos (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \sqrt{b \cos (c+d x)}}",1,"(2*(-(A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]) + B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + A*Sin[c + d*x]))/(b*d*Sqrt[b*Cos[c + d*x]])","A",1
832,1,87,140,0.0758484,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x])/(b*Cos[c + d*x])^(3/2),x]","\frac{2 \left(A \tan (c+d x)+A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 B \sin (c+d x)-3 B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 b d \sqrt{b \cos (c+d x)}}","\frac{2 A \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d \sqrt{b \cos (c+d x)}}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x)}{b d \sqrt{b \cos (c+d x)}}",1,"(2*(-3*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 3*B*Sin[c + d*x] + A*Tan[c + d*x]))/(3*b*d*Sqrt[b*Cos[c + d*x]])","A",1
833,1,104,171,0.1119175,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(b*Cos[c + d*x])^(3/2),x]","\frac{2 \left(9 A \sin (c+d x)-9 A \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 A \tan (c+d x) \sec (c+d x)+5 B \tan (c+d x)+5 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 b d \sqrt{b \cos (c+d x)}}","-\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^2 d \sqrt{\cos (c+d x)}}+\frac{6 A \sin (c+d x)}{5 b d \sqrt{b \cos (c+d x)}}+\frac{2 A b \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d \sqrt{b \cos (c+d x)}}",1,"(2*(-9*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 5*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 9*A*Sin[c + d*x] + 5*B*Tan[c + d*x] + 3*A*Sec[c + d*x]*Tan[c + d*x]))/(15*b*d*Sqrt[b*Cos[c + d*x]])","A",1
834,1,104,176,0.0950454,"\int \frac{\cos ^5(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^5*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(5/2),x]","\frac{\sin (2 (c+d x)) (42 A \cos (c+d x)+15 B \cos (2 (c+d x))+65 B)+252 A \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+100 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{210 b^2 d \sqrt{b \cos (c+d x)}}","\frac{2 A \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^4 d}+\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^3 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^5 d}+\frac{10 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 b^3 d}+\frac{10 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b^2 d \sqrt{b \cos (c+d x)}}",1,"(252*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 100*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (65*B + 42*A*Cos[c + d*x] + 15*B*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(210*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
835,1,91,147,0.0891802,"\int \frac{\cos ^4(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^4*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(5/2),x]","\frac{2 \sqrt{\cos (c+d x)} \left(\sin (c+d x) \sqrt{\cos (c+d x)} (5 A+3 B \cos (c+d x))+5 A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+9 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 b^2 d \sqrt{b \cos (c+d x)}}","\frac{2 A \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b^3 d}+\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^4 d}+\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^3 d \sqrt{\cos (c+d x)}}",1,"(2*Sqrt[Cos[c + d*x]]*(9*B*EllipticE[(c + d*x)/2, 2] + 5*A*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(5*A + 3*B*Cos[c + d*x])*Sin[c + d*x]))/(15*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
836,1,78,116,0.0822368,"\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(5/2),x]","\frac{2 \sqrt{\cos (c+d x)} \left(3 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+B \left(F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)}\right)\right)}{3 b^2 d \sqrt{b \cos (c+d x)}}","\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^3 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b^3 d}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^2 d \sqrt{b \cos (c+d x)}}",1,"(2*Sqrt[Cos[c + d*x]]*(3*A*EllipticE[(c + d*x)/2, 2] + B*(EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*Sin[c + d*x])))/(3*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
837,1,57,85,0.0530538,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(5/2),x]","\frac{2 \sqrt{\cos (c+d x)} \left(A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{b^2 d \sqrt{b \cos (c+d x)}}","\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^3 d \sqrt{\cos (c+d x)}}",1,"(2*Sqrt[Cos[c + d*x]]*(B*EllipticE[(c + d*x)/2, 2] + A*EllipticF[(c + d*x)/2, 2]))/(b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
838,1,76,112,0.0683671,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(A \sin (c+d x)-A \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{b^2 d \sqrt{b \cos (c+d x)}}","-\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^3 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x)}{b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \sqrt{b \cos (c+d x)}}",1,"(2*(-(A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]) + B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + A*Sin[c + d*x]))/(b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
839,1,87,143,0.0681825,"\int \frac{A+B \cos (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(A \tan (c+d x)+A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 B \sin (c+d x)-3 B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 b^2 d \sqrt{b \cos (c+d x)}}","\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 A \sin (c+d x)}{3 b d (b \cos (c+d x))^{3/2}}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^3 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x)}{b^2 d \sqrt{b \cos (c+d x)}}",1,"(2*(-3*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 3*B*Sin[c + d*x] + A*Tan[c + d*x]))/(3*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
840,1,104,173,0.078744,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x])/(b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(9 A \sin (c+d x)-9 A \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 A \tan (c+d x) \sec (c+d x)+5 B \tan (c+d x)+5 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 b^2 d \sqrt{b \cos (c+d x)}}","-\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^3 d \sqrt{\cos (c+d x)}}+\frac{6 A \sin (c+d x)}{5 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 A \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x)}{3 b d (b \cos (c+d x))^{3/2}}",1,"(2*(-9*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 5*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 9*A*Sin[c + d*x] + 5*B*Tan[c + d*x] + 3*A*Sec[c + d*x]*Tan[c + d*x]))/(15*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
841,1,104,176,0.0759421,"\int \frac{A+B \cos (c+d x)}{(b \cos (c+d x))^{7/2}} \, dx","Integrate[(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(7/2),x]","\frac{2 \left(9 A \sin (c+d x)-9 A \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 A \tan (c+d x) \sec (c+d x)+5 B \tan (c+d x)+5 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 b^3 d \sqrt{b \cos (c+d x)}}","-\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^4 d \sqrt{\cos (c+d x)}}+\frac{6 A \sin (c+d x)}{5 b^3 d \sqrt{b \cos (c+d x)}}+\frac{2 A \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^3 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x)}{3 b^2 d (b \cos (c+d x))^{3/2}}",1,"(2*(-9*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 5*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 9*A*Sin[c + d*x] + 5*B*Tan[c + d*x] + 3*A*Sec[c + d*x]*Tan[c + d*x]))/(15*b^3*d*Sqrt[b*Cos[c + d*x]])","A",1
842,1,81,172,0.1738061,"\int \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{\sqrt{b \cos (c+d x)} (72 A \sin (c+d x)+8 A \sin (3 (c+d x))+24 B \sin (2 (c+d x))+3 B \sin (4 (c+d x))+36 B c+36 B d x)}{96 d \sqrt{\cos (c+d x)}}","-\frac{A \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{A \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{3 B x \sqrt{b \cos (c+d x)}}{8 \sqrt{\cos (c+d x)}}+\frac{B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{4 d}+\frac{3 B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{8 d}",1,"(Sqrt[b*Cos[c + d*x]]*(36*B*c + 36*B*d*x + 72*A*Sin[c + d*x] + 24*B*Sin[2*(c + d*x)] + 8*A*Sin[3*(c + d*x)] + 3*B*Sin[4*(c + d*x)]))/(96*d*Sqrt[Cos[c + d*x]])","A",1
843,1,69,136,0.1286801,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{\sqrt{b \cos (c+d x)} (3 A \sin (2 (c+d x))+6 A c+6 A d x+9 B \sin (c+d x)+B \sin (3 (c+d x)))}{12 d \sqrt{\cos (c+d x)}}","\frac{A x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{A \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}-\frac{B \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(Sqrt[b*Cos[c + d*x]]*(6*A*c + 6*A*d*x + 9*B*Sin[c + d*x] + 3*A*Sin[2*(c + d*x)] + B*Sin[3*(c + d*x)]))/(12*d*Sqrt[Cos[c + d*x]])","A",1
844,1,57,98,0.1164862,"\int \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x]),x]","\frac{\sqrt{b \cos (c+d x)} (4 A \sin (c+d x)+B (2 (c+d x)+\sin (2 (c+d x))))}{4 d \sqrt{\cos (c+d x)}}","\frac{A \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{B x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}",1,"(Sqrt[b*Cos[c + d*x]]*(4*A*Sin[c + d*x] + B*(2*(c + d*x) + Sin[2*(c + d*x)])))/(4*d*Sqrt[Cos[c + d*x]])","A",1
845,1,42,59,0.0532609,"\int \frac{\sqrt{b \cos (c+d x)} (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{\sqrt{b \cos (c+d x)} (A (c+d x)+B \sin (c+d x))}{d \sqrt{\cos (c+d x)}}","\frac{A x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(Sqrt[b*Cos[c + d*x]]*(A*(c + d*x) + B*Sin[c + d*x]))/(d*Sqrt[Cos[c + d*x]])","A",1
846,1,40,60,0.0348441,"\int \frac{\sqrt{b \cos (c+d x)} (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{\sqrt{b \cos (c+d x)} \left(A \tanh ^{-1}(\sin (c+d x))+B d x\right)}{d \sqrt{\cos (c+d x)}}","\frac{A \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}+\frac{B x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}",1,"((B*d*x + A*ArcTanh[Sin[c + d*x]])*Sqrt[b*Cos[c + d*x]])/(d*Sqrt[Cos[c + d*x]])","A",1
847,1,50,68,0.0512558,"\int \frac{\sqrt{b \cos (c+d x)} (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{\sqrt{b \cos (c+d x)} \left(A \sin (c+d x)+B \cos (c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{d \cos ^{\frac{3}{2}}(c+d x)}","\frac{A \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}",1,"(Sqrt[b*Cos[c + d*x]]*(B*ArcTanh[Sin[c + d*x]]*Cos[c + d*x] + A*Sin[c + d*x]))/(d*Cos[c + d*x]^(3/2))","A",1
848,1,65,107,0.1127456,"\int \frac{\sqrt{b \cos (c+d x)} (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","\frac{\sqrt{b \cos (c+d x)} \left(\sin (c+d x) (A+2 B \cos (c+d x))+A \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{2 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{A \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{A \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[b*Cos[c + d*x]]*(A*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + (A + 2*B*Cos[c + d*x])*Sin[c + d*x]))/(2*d*Cos[c + d*x]^(5/2))","A",1
849,1,76,145,0.3181114,"\int \frac{\sqrt{b \cos (c+d x)} (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(9/2),x]","\frac{\sqrt{b \cos (c+d x)} \left(2 A (\cos (2 (c+d x))+2) \tan (c+d x)+3 B \sin (c+d x)+3 B \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{6 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{A \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{A \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{B \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}",1,"(Sqrt[b*Cos[c + d*x]]*(3*B*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + 3*B*Sin[c + d*x] + 2*A*(2 + Cos[2*(c + d*x)])*Tan[c + d*x]))/(6*d*Cos[c + d*x]^(5/2))","A",1
850,1,81,177,0.1633528,"\int \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{(b \cos (c+d x))^{3/2} (72 A \sin (c+d x)+8 A \sin (3 (c+d x))+24 B \sin (2 (c+d x))+3 B \sin (4 (c+d x))+36 B c+36 B d x)}{96 d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{A b \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{3 b B x \sqrt{b \cos (c+d x)}}{8 \sqrt{\cos (c+d x)}}+\frac{b B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{4 d}+\frac{3 b B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{8 d}",1,"((b*Cos[c + d*x])^(3/2)*(36*B*c + 36*B*d*x + 72*A*Sin[c + d*x] + 24*B*Sin[2*(c + d*x)] + 8*A*Sin[3*(c + d*x)] + 3*B*Sin[4*(c + d*x)]))/(96*d*Cos[c + d*x]^(3/2))","A",1
851,1,69,140,0.1363579,"\int \sqrt{\cos (c+d x)} (b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]),x]","\frac{(b \cos (c+d x))^{3/2} (3 A \sin (2 (c+d x))+6 A c+6 A d x+9 B \sin (c+d x)+B \sin (3 (c+d x)))}{12 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{A b x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{A b \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}-\frac{b B \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"((b*Cos[c + d*x])^(3/2)*(6*A*c + 6*A*d*x + 9*B*Sin[c + d*x] + 3*A*Sin[2*(c + d*x)] + B*Sin[3*(c + d*x)]))/(12*d*Cos[c + d*x]^(3/2))","A",1
852,1,58,101,0.0454733,"\int \frac{(b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{b \sqrt{b \cos (c+d x)} (4 A \sin (c+d x)+B (2 (c+d x)+\sin (2 (c+d x))))}{4 d \sqrt{\cos (c+d x)}}","\frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{b B x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{b B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}",1,"(b*Sqrt[b*Cos[c + d*x]]*(4*A*Sin[c + d*x] + B*(2*(c + d*x) + Sin[2*(c + d*x)])))/(4*d*Sqrt[Cos[c + d*x]])","A",1
853,1,42,61,0.0687061,"\int \frac{(b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{(b \cos (c+d x))^{3/2} (A (c+d x)+B \sin (c+d x))}{d \cos ^{\frac{3}{2}}(c+d x)}","\frac{A b x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"((b*Cos[c + d*x])^(3/2)*(A*(c + d*x) + B*Sin[c + d*x]))/(d*Cos[c + d*x]^(3/2))","A",1
854,1,40,62,0.0452965,"\int \frac{(b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{(b \cos (c+d x))^{3/2} \left(A \tanh ^{-1}(\sin (c+d x))+B d x\right)}{d \cos ^{\frac{3}{2}}(c+d x)}","\frac{A b \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}+\frac{b B x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}",1,"((B*d*x + A*ArcTanh[Sin[c + d*x]])*(b*Cos[c + d*x])^(3/2))/(d*Cos[c + d*x]^(3/2))","A",1
855,1,50,70,0.0558236,"\int \frac{(b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","\frac{(b \cos (c+d x))^{3/2} \left(A \sin (c+d x)+B \cos (c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{d \cos ^{\frac{5}{2}}(c+d x)}","\frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{b B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}",1,"((b*Cos[c + d*x])^(3/2)*(B*ArcTanh[Sin[c + d*x]]*Cos[c + d*x] + A*Sin[c + d*x]))/(d*Cos[c + d*x]^(5/2))","A",1
856,1,65,110,0.094897,"\int \frac{(b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(9/2),x]","\frac{(b \cos (c+d x))^{3/2} \left(\sin (c+d x) (A+2 B \cos (c+d x))+A \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{2 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{A b \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}+\frac{b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}",1,"((b*Cos[c + d*x])^(3/2)*(A*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + (A + 2*B*Cos[c + d*x])*Sin[c + d*x]))/(2*d*Cos[c + d*x]^(7/2))","A",1
857,1,77,149,0.0450862,"\int \frac{(b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(11/2),x]","\frac{b \sqrt{b \cos (c+d x)} \left(2 A (\cos (2 (c+d x))+2) \tan (c+d x)+3 B \sin (c+d x)+3 B \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{6 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{A b \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{b B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}",1,"(b*Sqrt[b*Cos[c + d*x]]*(3*B*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + 3*B*Sin[c + d*x] + 2*A*(2 + Cos[2*(c + d*x)])*Tan[c + d*x]))/(6*d*Cos[c + d*x]^(5/2))","A",1
858,1,81,187,0.1976919,"\int \sqrt{\cos (c+d x)} (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]),x]","\frac{(b \cos (c+d x))^{5/2} (72 A \sin (c+d x)+8 A \sin (3 (c+d x))+24 B \sin (2 (c+d x))+3 B \sin (4 (c+d x))+36 B c+36 B d x)}{96 d \cos ^{\frac{5}{2}}(c+d x)}","-\frac{A b^2 \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{3 b^2 B x \sqrt{b \cos (c+d x)}}{8 \sqrt{\cos (c+d x)}}+\frac{b^2 B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{4 d}+\frac{3 b^2 B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{8 d}",1,"((b*Cos[c + d*x])^(5/2)*(36*B*c + 36*B*d*x + 72*A*Sin[c + d*x] + 24*B*Sin[2*(c + d*x)] + 8*A*Sin[3*(c + d*x)] + 3*B*Sin[4*(c + d*x)]))/(96*d*Cos[c + d*x]^(5/2))","A",1
859,1,69,148,0.1565792,"\int \frac{(b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{(b \cos (c+d x))^{5/2} (3 A \sin (2 (c+d x))+6 A c+6 A d x+9 B \sin (c+d x)+B \sin (3 (c+d x)))}{12 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{A b^2 x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{A b^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}-\frac{b^2 B \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"((b*Cos[c + d*x])^(5/2)*(6*A*c + 6*A*d*x + 9*B*Sin[c + d*x] + 3*A*Sin[2*(c + d*x)] + B*Sin[3*(c + d*x)]))/(12*d*Cos[c + d*x]^(5/2))","A",1
860,1,57,107,0.1470592,"\int \frac{(b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{(b \cos (c+d x))^{5/2} (4 A \sin (c+d x)+B (2 (c+d x)+\sin (2 (c+d x))))}{4 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{b^2 B x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{b^2 B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}",1,"((b*Cos[c + d*x])^(5/2)*(4*A*Sin[c + d*x] + B*(2*(c + d*x) + Sin[2*(c + d*x)])))/(4*d*Cos[c + d*x]^(5/2))","A",1
861,1,42,65,0.0825349,"\int \frac{(b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{(b \cos (c+d x))^{5/2} (A (c+d x)+B \sin (c+d x))}{d \cos ^{\frac{5}{2}}(c+d x)}","\frac{A b^2 x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"((b*Cos[c + d*x])^(5/2)*(A*(c + d*x) + B*Sin[c + d*x]))/(d*Cos[c + d*x]^(5/2))","A",1
862,1,40,66,0.0652592,"\int \frac{(b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","\frac{(b \cos (c+d x))^{5/2} \left(A \tanh ^{-1}(\sin (c+d x))+B d x\right)}{d \cos ^{\frac{5}{2}}(c+d x)}","\frac{A b^2 \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}+\frac{b^2 B x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}",1,"((B*d*x + A*ArcTanh[Sin[c + d*x]])*(b*Cos[c + d*x])^(5/2))/(d*Cos[c + d*x]^(5/2))","A",1
863,1,50,74,0.092345,"\int \frac{(b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(9/2),x]","\frac{(b \cos (c+d x))^{5/2} \left(A \sin (c+d x)+B \cos (c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{d \cos ^{\frac{7}{2}}(c+d x)}","\frac{A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{b^2 B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}",1,"((b*Cos[c + d*x])^(5/2)*(B*ArcTanh[Sin[c + d*x]]*Cos[c + d*x] + A*Sin[c + d*x]))/(d*Cos[c + d*x]^(7/2))","A",1
864,1,65,116,0.153195,"\int \frac{(b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(11/2),x]","\frac{(b \cos (c+d x))^{5/2} \left(\sin (c+d x) (A+2 B \cos (c+d x))+A \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{2 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{A b^2 \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}+\frac{b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}",1,"((b*Cos[c + d*x])^(5/2)*(A*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + (A + 2*B*Cos[c + d*x])*Sin[c + d*x]))/(2*d*Cos[c + d*x]^(9/2))","A",1
865,1,76,157,0.1945833,"\int \frac{(b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(13/2),x]","\frac{(b \cos (c+d x))^{5/2} \left(2 A (\cos (2 (c+d x))+2) \tan (c+d x)+3 B \sin (c+d x)+3 B \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{6 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{A b^2 \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{b^2 B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}",1,"((b*Cos[c + d*x])^(5/2)*(3*B*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + 3*B*Sin[c + d*x] + 2*A*(2 + Cos[2*(c + d*x)])*Tan[c + d*x]))/(6*d*Cos[c + d*x]^(9/2))","A",1
866,1,69,136,0.1099973,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]))/Sqrt[b*Cos[c + d*x]],x]","\frac{\sqrt{\cos (c+d x)} (3 A \sin (2 (c+d x))+6 A c+6 A d x+9 B \sin (c+d x)+B \sin (3 (c+d x)))}{12 d \sqrt{b \cos (c+d x)}}","\frac{A x \sqrt{\cos (c+d x)}}{2 \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{b \cos (c+d x)}}-\frac{B \sin ^3(c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(6*A*c + 6*A*d*x + 9*B*Sin[c + d*x] + 3*A*Sin[2*(c + d*x)] + B*Sin[3*(c + d*x)]))/(12*d*Sqrt[b*Cos[c + d*x]])","A",1
867,1,57,98,0.0920178,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/Sqrt[b*Cos[c + d*x]],x]","\frac{\sqrt{\cos (c+d x)} (4 A \sin (c+d x)+B (2 (c+d x)+\sin (2 (c+d x))))}{4 d \sqrt{b \cos (c+d x)}}","\frac{A \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{2 \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(4*A*Sin[c + d*x] + B*(2*(c + d*x) + Sin[2*(c + d*x)])))/(4*d*Sqrt[b*Cos[c + d*x]])","A",1
868,1,42,59,0.0463444,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/Sqrt[b*Cos[c + d*x]],x]","\frac{\sqrt{\cos (c+d x)} (A (c+d x)+B \sin (c+d x))}{d \sqrt{b \cos (c+d x)}}","\frac{A x \sqrt{\cos (c+d x)}}{\sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(A*(c + d*x) + B*Sin[c + d*x]))/(d*Sqrt[b*Cos[c + d*x]])","A",1
869,1,40,60,0.0365935,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]),x]","\frac{\sqrt{\cos (c+d x)} \left(A \tanh ^{-1}(\sin (c+d x))+B d x\right)}{d \sqrt{b \cos (c+d x)}}","\frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{\sqrt{b \cos (c+d x)}}",1,"((B*d*x + A*ArcTanh[Sin[c + d*x]])*Sqrt[Cos[c + d*x]])/(d*Sqrt[b*Cos[c + d*x]])","A",1
870,1,50,68,0.051,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]),x]","\frac{A \sin (c+d x)+B \cos (c+d x) \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}","\frac{A \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{b \cos (c+d x)}}",1,"(B*ArcTanh[Sin[c + d*x]]*Cos[c + d*x] + A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])","A",1
871,1,65,107,0.0744094,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/(Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]),x]","\frac{\sin (c+d x) (A+2 B \cos (c+d x))+A \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))}{2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}","\frac{A \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}",1,"(A*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + (A + 2*B*Cos[c + d*x])*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]])","A",1
872,1,76,145,0.09005,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{b \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x])/(Cos[c + d*x]^(7/2)*Sqrt[b*Cos[c + d*x]]),x]","\frac{2 A (\cos (2 (c+d x))+2) \tan (c+d x)+3 B \sin (c+d x)+3 B \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))}{6 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}","\frac{A \sin ^3(c+d x)}{3 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{b \cos (c+d x)}}",1,"(3*B*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + 3*B*Sin[c + d*x] + 2*A*(2 + Cos[2*(c + d*x)])*Tan[c + d*x])/(6*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]])","A",1
873,1,69,148,0.0863431,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(3/2),x]","\frac{\cos ^{\frac{3}{2}}(c+d x) (3 A \sin (2 (c+d x))+6 A c+6 A d x+9 B \sin (c+d x)+B \sin (3 (c+d x)))}{12 d (b \cos (c+d x))^{3/2}}","\frac{A x \sqrt{\cos (c+d x)}}{2 b \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b d \sqrt{b \cos (c+d x)}}-\frac{B \sin ^3(c+d x) \sqrt{\cos (c+d x)}}{3 b d \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \sqrt{b \cos (c+d x)}}",1,"(Cos[c + d*x]^(3/2)*(6*A*c + 6*A*d*x + 9*B*Sin[c + d*x] + 3*A*Sin[2*(c + d*x)] + B*Sin[3*(c + d*x)]))/(12*d*(b*Cos[c + d*x])^(3/2))","A",1
874,1,57,107,0.1025024,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(3/2),x]","\frac{\cos ^{\frac{3}{2}}(c+d x) (4 A \sin (c+d x)+B (2 (c+d x)+\sin (2 (c+d x))))}{4 d (b \cos (c+d x))^{3/2}}","\frac{A \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{2 b \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b d \sqrt{b \cos (c+d x)}}",1,"(Cos[c + d*x]^(3/2)*(4*A*Sin[c + d*x] + B*(2*(c + d*x) + Sin[2*(c + d*x)])))/(4*d*(b*Cos[c + d*x])^(3/2))","A",1
875,1,42,65,0.0478878,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(3/2),x]","\frac{\cos ^{\frac{3}{2}}(c+d x) (A (c+d x)+B \sin (c+d x))}{d (b \cos (c+d x))^{3/2}}","\frac{A x \sqrt{\cos (c+d x)}}{b \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \sqrt{b \cos (c+d x)}}",1,"(Cos[c + d*x]^(3/2)*(A*(c + d*x) + B*Sin[c + d*x]))/(d*(b*Cos[c + d*x])^(3/2))","A",1
876,1,40,66,0.0471224,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(3/2),x]","\frac{\cos ^{\frac{3}{2}}(c+d x) \left(A \tanh ^{-1}(\sin (c+d x))+B d x\right)}{d (b \cos (c+d x))^{3/2}}","\frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{b \sqrt{b \cos (c+d x)}}",1,"((B*d*x + A*ArcTanh[Sin[c + d*x]])*Cos[c + d*x]^(3/2))/(d*(b*Cos[c + d*x])^(3/2))","A",1
877,1,50,74,0.0626279,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(3/2)),x]","\frac{\sqrt{\cos (c+d x)} \left(A \sin (c+d x)+B \cos (c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{d (b \cos (c+d x))^{3/2}}","\frac{A \sin (c+d x)}{b d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(B*ArcTanh[Sin[c + d*x]]*Cos[c + d*x] + A*Sin[c + d*x]))/(d*(b*Cos[c + d*x])^(3/2))","A",1
878,1,65,116,0.072325,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + B*Cos[c + d*x])/(Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^(3/2)),x]","\frac{\sin (c+d x) (A+2 B \cos (c+d x))+A \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)} (b \cos (c+d x))^{3/2}}","\frac{A \sin (c+d x)}{2 b d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 b d \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{b d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}",1,"(A*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + (A + 2*B*Cos[c + d*x])*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(3/2))","A",1
879,1,76,157,0.1055375,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + B*Cos[c + d*x])/(Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^(3/2)),x]","\frac{2 A (\cos (2 (c+d x))+2) \tan (c+d x)+3 B \sin (c+d x)+3 B \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))}{6 d \sqrt{\cos (c+d x)} (b \cos (c+d x))^{3/2}}","\frac{A \sin ^3(c+d x)}{3 b d \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{b d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{2 b d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 b d \sqrt{b \cos (c+d x)}}",1,"(3*B*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + 3*B*Sin[c + d*x] + 2*A*(2 + Cos[2*(c + d*x)])*Tan[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(3/2))","A",1
880,1,72,148,0.0744572,"\int \frac{\cos ^{\frac{9}{2}}(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(9/2)*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{\cos (c+d x)} (3 A \sin (2 (c+d x))+6 A c+6 A d x+9 B \sin (c+d x)+B \sin (3 (c+d x)))}{12 b^2 d \sqrt{b \cos (c+d x)}}","\frac{A x \sqrt{\cos (c+d x)}}{2 b^2 \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b^2 d \sqrt{b \cos (c+d x)}}-\frac{B \sin ^3(c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{b^2 d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(6*A*c + 6*A*d*x + 9*B*Sin[c + d*x] + 3*A*Sin[2*(c + d*x)] + B*Sin[3*(c + d*x)]))/(12*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
881,1,60,107,0.0603027,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{\cos (c+d x)} (4 A \sin (c+d x)+B (2 (c+d x)+\sin (2 (c+d x))))}{4 b^2 d \sqrt{b \cos (c+d x)}}","\frac{A \sin (c+d x) \sqrt{\cos (c+d x)}}{b^2 d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{2 b^2 \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b^2 d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(4*A*Sin[c + d*x] + B*(2*(c + d*x) + Sin[2*(c + d*x)])))/(4*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
882,1,45,65,0.0509742,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{\cos (c+d x)} (A (c+d x)+B \sin (c+d x))}{b^2 d \sqrt{b \cos (c+d x)}}","\frac{A x \sqrt{\cos (c+d x)}}{b^2 \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{b^2 d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(A*(c + d*x) + B*Sin[c + d*x]))/(b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
883,1,43,66,0.0373845,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{\cos (c+d x)} \left(A \tanh ^{-1}(\sin (c+d x))+B d x\right)}{b^2 d \sqrt{b \cos (c+d x)}}","\frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b^2 d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{b^2 \sqrt{b \cos (c+d x)}}",1,"((B*d*x + A*ArcTanh[Sin[c + d*x]])*Sqrt[Cos[c + d*x]])/(b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
884,1,50,74,0.0538459,"\int \frac{\sqrt{\cos (c+d x)} (A+B \cos (c+d x))}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(5/2),x]","\frac{\cos ^{\frac{3}{2}}(c+d x) \left(A \sin (c+d x)+B \cos (c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{d (b \cos (c+d x))^{5/2}}","\frac{A \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b^2 d \sqrt{b \cos (c+d x)}}",1,"(Cos[c + d*x]^(3/2)*(B*ArcTanh[Sin[c + d*x]]*Cos[c + d*x] + A*Sin[c + d*x]))/(d*(b*Cos[c + d*x])^(5/2))","A",1
885,1,65,116,0.0784708,"\int \frac{A+B \cos (c+d x)}{\sqrt{\cos (c+d x)} (b \cos (c+d x))^{5/2}} \, dx","Integrate[(A + B*Cos[c + d*x])/(Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(5/2)),x]","\frac{\sqrt{\cos (c+d x)} \left(\sin (c+d x) (A+2 B \cos (c+d x))+A \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{2 d (b \cos (c+d x))^{5/2}}","\frac{A \sin (c+d x)}{2 b^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 b^2 d \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(A*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + (A + 2*B*Cos[c + d*x])*Sin[c + d*x]))/(2*d*(b*Cos[c + d*x])^(5/2))","A",1
886,1,76,157,0.1056641,"\int \frac{A+B \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^{5/2}} \, dx","Integrate[(A + B*Cos[c + d*x])/(Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^(5/2)),x]","\frac{\sqrt{\cos (c+d x)} \left(2 A (\cos (2 (c+d x))+2) \tan (c+d x)+3 B \sin (c+d x)+3 B \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{6 d (b \cos (c+d x))^{5/2}}","\frac{A \sin ^3(c+d x)}{3 b^2 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{2 b^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 b^2 d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(3*B*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + 3*B*Sin[c + d*x] + 2*A*(2 + Cos[2*(c + d*x)])*Tan[c + d*x]))/(6*d*(b*Cos[c + d*x])^(5/2))","A",1
887,1,94,119,0.1797971,"\int \cos ^2(c+d x) \sqrt[3]{b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x]),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cos ^2(c+d x) \cot (c+d x) \sqrt[3]{b \cos (c+d x)} \left(13 A \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)+10 B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{13}{6};\frac{19}{6};\cos ^2(c+d x)\right)\right)}{130 d}","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{10/3} \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)}{10 b^3 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{13/3} \, _2F_1\left(\frac{1}{2},\frac{13}{6};\frac{19}{6};\cos ^2(c+d x)\right)}{13 b^4 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*Cos[c + d*x]^2*(b*Cos[c + d*x])^(1/3)*Cot[c + d*x]*(13*A*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2] + 10*B*Cos[c + d*x]*Hypergeometric2F1[1/2, 13/6, 19/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(130*d)","A",1
888,1,89,119,0.147092,"\int \cos (c+d x) \sqrt[3]{b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]*(b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x]),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^{4/3} \left(10 A \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)+7 B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)\right)}{70 b d}","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)}{7 b^2 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{10/3} \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)}{10 b^3 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*(b*Cos[c + d*x])^(4/3)*Cot[c + d*x]*(10*A*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2] + 7*B*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(70*b*d)","A",1
889,1,86,119,0.0871141,"\int \sqrt[3]{b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[(b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x]),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) \sqrt[3]{b \cos (c+d x)} \left(7 A \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)+4 B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)\right)}{28 d}","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{4 b d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)}{7 b^2 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*(b*Cos[c + d*x])^(1/3)*Cot[c + d*x]*(7*A*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2] + 4*B*Cos[c + d*x]*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(28*d)","A",1
890,1,86,114,0.1024752,"\int \sqrt[3]{b \cos (c+d x)} (A+B \cos (c+d x)) \sec (c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","-\frac{3 b \sqrt{\sin ^2(c+d x)} \cot (c+d x) \left(4 A \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)+B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)\right)}{4 d (b \cos (c+d x))^{2/3}}","-\frac{3 A \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{4 b d \sqrt{\sin ^2(c+d x)}}",1,"(-3*b*Cot[c + d*x]*(4*A*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2] + B*Cos[c + d*x]*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(4*d*(b*Cos[c + d*x])^(2/3))","A",1
891,1,86,112,0.1308967,"\int \sqrt[3]{b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","\frac{3 b \sqrt{\sin ^2(c+d x)} \csc (c+d x) \left(A \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)-2 B \cos (c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)\right)}{2 d (b \cos (c+d x))^{2/3}}","\frac{3 A b \sin (c+d x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{2 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}-\frac{3 B \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{d \sqrt{\sin ^2(c+d x)}}",1,"(3*b*Csc[c + d*x]*(A*Hypergeometric2F1[-1/3, 1/2, 2/3, Cos[c + d*x]^2] - 2*B*Cos[c + d*x]*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(2*d*(b*Cos[c + d*x])^(2/3))","A",1
892,1,94,117,0.0935753,"\int \sqrt[3]{b \cos (c+d x)} (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \sec ^2(c+d x) \sqrt[3]{b \cos (c+d x)} \left(2 A \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right)+5 B \cos (c+d x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)\right)}{10 d}","\frac{3 A b^2 \sin (c+d x) \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right)}{5 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{5/3}}+\frac{3 b B \sin (c+d x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{2 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}",1,"(3*(b*Cos[c + d*x])^(1/3)*Csc[c + d*x]*(2*A*Hypergeometric2F1[-5/6, 1/2, 1/6, Cos[c + d*x]^2] + 5*B*Cos[c + d*x]*Hypergeometric2F1[-1/3, 1/2, 2/3, Cos[c + d*x]^2])*Sec[c + d*x]^2*Sqrt[Sin[c + d*x]^2])/(10*d)","A",1
893,1,94,119,0.2135637,"\int \cos ^2(c+d x) (b \cos (c+d x))^{4/3} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x]),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cos ^2(c+d x) \cot (c+d x) (b \cos (c+d x))^{4/3} \left(16 A \, _2F_1\left(\frac{1}{2},\frac{13}{6};\frac{19}{6};\cos ^2(c+d x)\right)+13 B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{8}{3};\frac{11}{3};\cos ^2(c+d x)\right)\right)}{208 d}","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{13/3} \, _2F_1\left(\frac{1}{2},\frac{13}{6};\frac{19}{6};\cos ^2(c+d x)\right)}{13 b^3 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{16/3} \, _2F_1\left(\frac{1}{2},\frac{8}{3};\frac{11}{3};\cos ^2(c+d x)\right)}{16 b^4 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*Cos[c + d*x]^2*(b*Cos[c + d*x])^(4/3)*Cot[c + d*x]*(16*A*Hypergeometric2F1[1/2, 13/6, 19/6, Cos[c + d*x]^2] + 13*B*Cos[c + d*x]*Hypergeometric2F1[1/2, 8/3, 11/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(208*d)","A",1
894,1,89,119,0.1949348,"\int \cos (c+d x) (b \cos (c+d x))^{4/3} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]*(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x]),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^{7/3} \left(13 A \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)+10 B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{13}{6};\frac{19}{6};\cos ^2(c+d x)\right)\right)}{130 b d}","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{10/3} \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)}{10 b^2 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{13/3} \, _2F_1\left(\frac{1}{2},\frac{13}{6};\frac{19}{6};\cos ^2(c+d x)\right)}{13 b^3 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*(b*Cos[c + d*x])^(7/3)*Cot[c + d*x]*(13*A*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2] + 10*B*Cos[c + d*x]*Hypergeometric2F1[1/2, 13/6, 19/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(130*b*d)","A",1
895,1,86,119,0.0488616,"\int (b \cos (c+d x))^{4/3} (A+B \cos (c+d x)) \, dx","Integrate[(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x]),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^{4/3} \left(10 A \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)+7 B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)\right)}{70 d}","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)}{7 b d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{10/3} \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)}{10 b^2 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*(b*Cos[c + d*x])^(4/3)*Cot[c + d*x]*(10*A*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2] + 7*B*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(70*d)","A",1
896,1,87,116,0.0140856,"\int (b \cos (c+d x))^{4/3} (A+B \cos (c+d x)) \sec (c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","-\frac{3 b \sqrt{\sin ^2(c+d x)} \cot (c+d x) \sqrt[3]{b \cos (c+d x)} \left(7 A \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)+4 B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)\right)}{28 d}","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{4 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)}{7 b d \sqrt{\sin ^2(c+d x)}}",1,"(-3*b*(b*Cos[c + d*x])^(1/3)*Cot[c + d*x]*(7*A*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2] + 4*B*Cos[c + d*x]*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(28*d)","A",1
897,1,88,112,0.0256698,"\int (b \cos (c+d x))^{4/3} (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","-\frac{3 b^2 \sqrt{\sin ^2(c+d x)} \cot (c+d x) \left(4 A \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)+B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)\right)}{4 d (b \cos (c+d x))^{2/3}}","-\frac{3 A b \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{4 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*b^2*Cot[c + d*x]*(4*A*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2] + B*Cos[c + d*x]*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(4*d*(b*Cos[c + d*x])^(2/3))","A",1
898,1,88,115,0.1146322,"\int (b \cos (c+d x))^{4/3} (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","\frac{3 b^2 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \left(A \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)-2 B \cos (c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)\right)}{2 d (b \cos (c+d x))^{2/3}}","\frac{3 A b^2 \sin (c+d x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{2 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}-\frac{3 b B \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{d \sqrt{\sin ^2(c+d x)}}",1,"(3*b^2*Csc[c + d*x]*(A*Hypergeometric2F1[-1/3, 1/2, 2/3, Cos[c + d*x]^2] - 2*B*Cos[c + d*x]*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(2*d*(b*Cos[c + d*x])^(2/3))","A",1
899,1,94,119,0.1822004,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{2/3}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(2/3),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cos ^2(c+d x) \cot (c+d x) \left(10 A \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)+7 B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)\right)}{70 d (b \cos (c+d x))^{2/3}}","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)}{7 b^3 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{10/3} \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)}{10 b^4 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*Cos[c + d*x]^2*Cot[c + d*x]*(10*A*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2] + 7*B*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(70*d*(b*Cos[c + d*x])^(2/3))","A",1
900,1,89,119,0.0225894,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{2/3}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(2/3),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) \sqrt[3]{b \cos (c+d x)} \left(7 A \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)+4 B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)\right)}{28 b d}","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{4 b^2 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)}{7 b^3 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*(b*Cos[c + d*x])^(1/3)*Cot[c + d*x]*(7*A*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2] + 4*B*Cos[c + d*x]*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(28*b*d)","A",1
901,1,85,117,0.0161174,"\int \frac{A+B \cos (c+d x)}{(b \cos (c+d x))^{2/3}} \, dx","Integrate[(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(2/3),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) \left(4 A \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)+B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)\right)}{4 d (b \cos (c+d x))^{2/3}}","-\frac{3 A \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{b d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{4 b^2 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*Cot[c + d*x]*(4*A*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2] + B*Cos[c + d*x]*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(4*d*(b*Cos[c + d*x])^(2/3))","A",1
902,1,85,114,0.0716881,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(b \cos (c+d x))^{2/3}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x])/(b*Cos[c + d*x])^(2/3),x]","\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \left(A \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)-2 B \cos (c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)\right)}{2 d (b \cos (c+d x))^{2/3}}","\frac{3 A \sin (c+d x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{2 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}-\frac{3 B \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{b d \sqrt{\sin ^2(c+d x)}}",1,"(3*Csc[c + d*x]*(A*Hypergeometric2F1[-1/3, 1/2, 2/3, Cos[c + d*x]^2] - 2*B*Cos[c + d*x]*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(2*d*(b*Cos[c + d*x])^(2/3))","A",1
903,1,89,114,0.1555718,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(b*Cos[c + d*x])^(2/3),x]","\frac{3 b^2 \sqrt{\sin ^2(c+d x)} \cot (c+d x) \left(2 A \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right)+5 B \cos (c+d x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)\right)}{10 d (b \cos (c+d x))^{8/3}}","\frac{3 A b \sin (c+d x) \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right)}{5 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{5/3}}+\frac{3 B \sin (c+d x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{2 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}",1,"(3*b^2*Cot[c + d*x]*(2*A*Hypergeometric2F1[-5/6, 1/2, 1/6, Cos[c + d*x]^2] + 5*B*Cos[c + d*x]*Hypergeometric2F1[-1/3, 1/2, 2/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(10*d*(b*Cos[c + d*x])^(8/3))","A",1
904,1,89,117,0.1497051,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(b*Cos[c + d*x])^(2/3),x]","\frac{3 b^2 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \left(5 A \, _2F_1\left(-\frac{4}{3},\frac{1}{2};-\frac{1}{3};\cos ^2(c+d x)\right)+8 B \cos (c+d x) \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right)\right)}{40 d (b \cos (c+d x))^{8/3}}","\frac{3 A b^2 \sin (c+d x) \, _2F_1\left(-\frac{4}{3},\frac{1}{2};-\frac{1}{3};\cos ^2(c+d x)\right)}{8 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{8/3}}+\frac{3 b B \sin (c+d x) \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right)}{5 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{5/3}}",1,"(3*b^2*Csc[c + d*x]*(5*A*Hypergeometric2F1[-4/3, 1/2, -1/3, Cos[c + d*x]^2] + 8*B*Cos[c + d*x]*Hypergeometric2F1[-5/6, 1/2, 1/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(40*d*(b*Cos[c + d*x])^(8/3))","A",1
905,1,94,119,0.1830659,"\int \frac{\cos ^2(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{4/3}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(4/3),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cos ^2(c+d x) \cot (c+d x) \left(8 A \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)+5 B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right)\right)}{40 d (b \cos (c+d x))^{4/3}}","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{5 b^3 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{8/3} \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right)}{8 b^4 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*Cos[c + d*x]^2*Cot[c + d*x]*(8*A*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2] + 5*B*Cos[c + d*x]*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(40*d*(b*Cos[c + d*x])^(4/3))","A",1
906,1,89,119,0.0918339,"\int \frac{\cos (c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{4/3}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(4/3),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) \left(5 A \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)+2 B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)\right)}{10 b d \sqrt[3]{b \cos (c+d x)}}","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{2 b^2 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{5 b^3 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*Cot[c + d*x]*(5*A*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2] + 2*B*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(10*b*d*(b*Cos[c + d*x])^(1/3))","A",1
907,1,85,117,0.1262089,"\int \frac{A+B \cos (c+d x)}{(b \cos (c+d x))^{4/3}} \, dx","Integrate[(A + B*Cos[c + d*x])/(b*Cos[c + d*x])^(4/3),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) \left(B \cos (c+d x) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)-2 A \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)\right)}{2 d (b \cos (c+d x))^{4/3}}","\frac{3 A \sin (c+d x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)}{b d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{2 b^2 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*Cot[c + d*x]*(-2*A*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2] + B*Cos[c + d*x]*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(2*d*(b*Cos[c + d*x])^(4/3))","A",1
908,1,86,114,0.153332,"\int \frac{(A+B \cos (c+d x)) \sec (c+d x)}{(b \cos (c+d x))^{4/3}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x])/(b*Cos[c + d*x])^(4/3),x]","\frac{3 b \sqrt{\sin ^2(c+d x)} \cot (c+d x) \left(A \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right)+4 B \cos (c+d x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)\right)}{4 d (b \cos (c+d x))^{7/3}}","\frac{3 A \sin (c+d x) \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right)}{4 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{4/3}}+\frac{3 B \sin (c+d x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)}{b d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}",1,"(3*b*Cot[c + d*x]*(A*Hypergeometric2F1[-2/3, 1/2, 1/3, Cos[c + d*x]^2] + 4*B*Cos[c + d*x]*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(4*d*(b*Cos[c + d*x])^(7/3))","A",1
909,1,89,114,0.2027979,"\int \frac{(A+B \cos (c+d x)) \sec ^2(c+d x)}{(b \cos (c+d x))^{4/3}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^2)/(b*Cos[c + d*x])^(4/3),x]","\frac{3 b^2 \sqrt{\sin ^2(c+d x)} \cot (c+d x) \left(4 A \, _2F_1\left(-\frac{7}{6},\frac{1}{2};-\frac{1}{6};\cos ^2(c+d x)\right)+7 B \cos (c+d x) \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right)\right)}{28 d (b \cos (c+d x))^{10/3}}","\frac{3 A b \sin (c+d x) \, _2F_1\left(-\frac{7}{6},\frac{1}{2};-\frac{1}{6};\cos ^2(c+d x)\right)}{7 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{7/3}}+\frac{3 B \sin (c+d x) \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right)}{4 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{4/3}}",1,"(3*b^2*Cot[c + d*x]*(4*A*Hypergeometric2F1[-7/6, 1/2, -1/6, Cos[c + d*x]^2] + 7*B*Cos[c + d*x]*Hypergeometric2F1[-2/3, 1/2, 1/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(28*d*(b*Cos[c + d*x])^(10/3))","A",1
910,1,89,117,0.1957161,"\int \frac{(A+B \cos (c+d x)) \sec ^3(c+d x)}{(b \cos (c+d x))^{4/3}} \, dx","Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(b*Cos[c + d*x])^(4/3),x]","\frac{3 b^2 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \left(7 A \, _2F_1\left(-\frac{5}{3},\frac{1}{2};-\frac{2}{3};\cos ^2(c+d x)\right)+10 B \cos (c+d x) \, _2F_1\left(-\frac{7}{6},\frac{1}{2};-\frac{1}{6};\cos ^2(c+d x)\right)\right)}{70 d (b \cos (c+d x))^{10/3}}","\frac{3 A b^2 \sin (c+d x) \, _2F_1\left(-\frac{5}{3},\frac{1}{2};-\frac{2}{3};\cos ^2(c+d x)\right)}{10 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{10/3}}+\frac{3 b B \sin (c+d x) \, _2F_1\left(-\frac{7}{6},\frac{1}{2};-\frac{1}{6};\cos ^2(c+d x)\right)}{7 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{7/3}}",1,"(3*b^2*Csc[c + d*x]*(7*A*Hypergeometric2F1[-5/3, 1/2, -2/3, Cos[c + d*x]^2] + 10*B*Cos[c + d*x]*Hypergeometric2F1[-7/6, 1/2, -1/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(70*d*(b*Cos[c + d*x])^(10/3))","A",1
911,1,130,157,0.2682892,"\int \cos ^m(c+d x) (b \cos (c+d x))^n (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^m*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]),x]","-\frac{\sqrt{\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+1}(c+d x) (b \cos (c+d x))^n \left(A (m+n+2) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\cos ^2(c+d x)\right)+B (m+n+1) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m+n+2);\frac{1}{2} (m+n+4);\cos ^2(c+d x)\right)\right)}{d (m+n+1) (m+n+2)}","-\frac{A \sin (c+d x) \cos ^{m+1}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\cos ^2(c+d x)\right)}{d (m+n+1) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) \cos ^{m+2}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m+n+2);\frac{1}{2} (m+n+4);\cos ^2(c+d x)\right)}{d (m+n+2) \sqrt{\sin ^2(c+d x)}}",1,"-((Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^n*Csc[c + d*x]*(A*(2 + m + n)*Hypergeometric2F1[1/2, (1 + m + n)/2, (3 + m + n)/2, Cos[c + d*x]^2] + B*(1 + m + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (2 + m + n)/2, (4 + m + n)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(1 + m + n)*(2 + m + n)))","A",1
912,1,120,141,0.3141537,"\int \cos ^2(c+d x) (b \cos (c+d x))^n (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]),x]","-\frac{\sqrt{\sin ^2(c+d x)} \cos ^2(c+d x) \cot (c+d x) (b \cos (c+d x))^n \left(A (n+4) \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(c+d x)\right)+B (n+3) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+4}{2};\frac{n+6}{2};\cos ^2(c+d x)\right)\right)}{d (n+3) (n+4)}","-\frac{A \sin (c+d x) (b \cos (c+d x))^{n+3} \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(c+d x)\right)}{b^3 d (n+3) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^{n+4} \, _2F_1\left(\frac{1}{2},\frac{n+4}{2};\frac{n+6}{2};\cos ^2(c+d x)\right)}{b^4 d (n+4) \sqrt{\sin ^2(c+d x)}}",1,"-((Cos[c + d*x]^2*(b*Cos[c + d*x])^n*Cot[c + d*x]*(A*(4 + n)*Hypergeometric2F1[1/2, (3 + n)/2, (5 + n)/2, Cos[c + d*x]^2] + B*(3 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (4 + n)/2, (6 + n)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(3 + n)*(4 + n)))","A",1
913,1,118,141,0.2518623,"\int \cos (c+d x) (b \cos (c+d x))^n (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]),x]","-\frac{\sqrt{\sin ^2(c+d x)} \cos (c+d x) \cot (c+d x) (b \cos (c+d x))^n \left(A (n+3) \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right)+B (n+2) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(c+d x)\right)\right)}{d (n+2) (n+3)}","-\frac{A \sin (c+d x) (b \cos (c+d x))^{n+2} \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right)}{b^2 d (n+2) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^{n+3} \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(c+d x)\right)}{b^3 d (n+3) \sqrt{\sin ^2(c+d x)}}",1,"-((Cos[c + d*x]*(b*Cos[c + d*x])^n*Cot[c + d*x]*(A*(3 + n)*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Cos[c + d*x]^2] + B*(2 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (3 + n)/2, (5 + n)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(2 + n)*(3 + n)))","A",1
914,1,112,141,0.1768818,"\int (b \cos (c+d x))^n (A+B \cos (c+d x)) \, dx","Integrate[(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]),x]","-\frac{\sqrt{\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^n \left(A (n+2) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(c+d x)\right)+B (n+1) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right)\right)}{d (n+1) (n+2)}","-\frac{A \sin (c+d x) (b \cos (c+d x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(c+d x)\right)}{b d (n+1) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^{n+2} \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right)}{b^2 d (n+2) \sqrt{\sin ^2(c+d x)}}",1,"-(((b*Cos[c + d*x])^n*Cot[c + d*x]*(A*(2 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[c + d*x]^2] + B*(1 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(1 + n)*(2 + n)))","A",1
915,1,109,132,0.1987477,"\int (b \cos (c+d x))^n (A+B \cos (c+d x)) \sec (c+d x) \, dx","Integrate[(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x])*Sec[c + d*x],x]","-\frac{b \sqrt{\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^{n-1} \left(A (n+1) \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right)+B n \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(c+d x)\right)\right)}{d n (n+1)}","-\frac{A \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right)}{d n \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(c+d x)\right)}{b d (n+1) \sqrt{\sin ^2(c+d x)}}",1,"-((b*(b*Cos[c + d*x])^(-1 + n)*Cot[c + d*x]*(A*(1 + n)*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[c + d*x]^2] + B*n*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*n*(1 + n)))","A",1
916,1,109,131,0.1924244,"\int (b \cos (c+d x))^n (A+B \cos (c+d x)) \sec ^2(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x])*Sec[c + d*x]^2,x]","-\frac{b \sqrt{\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^{n-1} \left(A n \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\cos ^2(c+d x)\right)+B (n-1) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right)\right)}{d (n-1) n}","\frac{A b \sin (c+d x) (b \cos (c+d x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\cos ^2(c+d x)\right)}{d (1-n) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right)}{d n \sqrt{\sin ^2(c+d x)}}",1,"-((b*(b*Cos[c + d*x])^(-1 + n)*Csc[c + d*x]*(A*n*Hypergeometric2F1[1/2, (-1 + n)/2, (1 + n)/2, Cos[c + d*x]^2] + B*(-1 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(-1 + n)*n))","A",1
917,1,118,139,0.1799833,"\int (b \cos (c+d x))^n (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x])*Sec[c + d*x]^3,x]","-\frac{\sqrt{\sin ^2(c+d x)} \csc (c+d x) \sec ^2(c+d x) (b \cos (c+d x))^n \left(A (n-1) \, _2F_1\left(\frac{1}{2},\frac{n-2}{2};\frac{n}{2};\cos ^2(c+d x)\right)+B (n-2) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\cos ^2(c+d x)\right)\right)}{d (n-2) (n-1)}","\frac{A b^2 \sin (c+d x) (b \cos (c+d x))^{n-2} \, _2F_1\left(\frac{1}{2},\frac{n-2}{2};\frac{n}{2};\cos ^2(c+d x)\right)}{d (2-n) \sqrt{\sin ^2(c+d x)}}+\frac{b B \sin (c+d x) (b \cos (c+d x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\cos ^2(c+d x)\right)}{d (1-n) \sqrt{\sin ^2(c+d x)}}",1,"-(((b*Cos[c + d*x])^n*Csc[c + d*x]*(A*(-1 + n)*Hypergeometric2F1[1/2, (-2 + n)/2, n/2, Cos[c + d*x]^2] + B*(-2 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (-1 + n)/2, (1 + n)/2, Cos[c + d*x]^2])*Sec[c + d*x]^2*Sqrt[Sin[c + d*x]^2])/(d*(-2 + n)*(-1 + n)))","A",1
918,1,118,141,0.1791384,"\int (b \cos (c+d x))^n (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]","-\frac{\sqrt{\sin ^2(c+d x)} \csc (c+d x) \sec ^3(c+d x) (b \cos (c+d x))^n \left(A (n-2) \, _2F_1\left(\frac{1}{2},\frac{n-3}{2};\frac{n-1}{2};\cos ^2(c+d x)\right)+B (n-3) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n-2}{2};\frac{n}{2};\cos ^2(c+d x)\right)\right)}{d (n-3) (n-2)}","\frac{A b^3 \sin (c+d x) (b \cos (c+d x))^{n-3} \, _2F_1\left(\frac{1}{2},\frac{n-3}{2};\frac{n-1}{2};\cos ^2(c+d x)\right)}{d (3-n) \sqrt{\sin ^2(c+d x)}}+\frac{b^2 B \sin (c+d x) (b \cos (c+d x))^{n-2} \, _2F_1\left(\frac{1}{2},\frac{n-2}{2};\frac{n}{2};\cos ^2(c+d x)\right)}{d (2-n) \sqrt{\sin ^2(c+d x)}}",1,"-(((b*Cos[c + d*x])^n*Csc[c + d*x]*(A*(-2 + n)*Hypergeometric2F1[1/2, (-3 + n)/2, (-1 + n)/2, Cos[c + d*x]^2] + B*(-3 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (-2 + n)/2, n/2, Cos[c + d*x]^2])*Sec[c + d*x]^3*Sqrt[Sin[c + d*x]^2])/(d*(-3 + n)*(-2 + n)))","A",1
919,1,138,163,0.452483,"\int \cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^n (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]),x]","-\frac{2 \sqrt{\sin ^2(c+d x)} \cos ^{\frac{7}{2}}(c+d x) \csc (c+d x) (b \cos (c+d x))^n \left(A (2 n+9) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+7);\frac{1}{4} (2 n+11);\cos ^2(c+d x)\right)+B (2 n+7) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+9);\frac{1}{4} (2 n+13);\cos ^2(c+d x)\right)\right)}{d (2 n+7) (2 n+9)}","-\frac{2 A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+7);\frac{1}{4} (2 n+11);\cos ^2(c+d x)\right)}{d (2 n+7) \sqrt{\sin ^2(c+d x)}}-\frac{2 B \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+9);\frac{1}{4} (2 n+13);\cos ^2(c+d x)\right)}{d (2 n+9) \sqrt{\sin ^2(c+d x)}}",1,"(-2*Cos[c + d*x]^(7/2)*(b*Cos[c + d*x])^n*Csc[c + d*x]*(A*(9 + 2*n)*Hypergeometric2F1[1/2, (7 + 2*n)/4, (11 + 2*n)/4, Cos[c + d*x]^2] + B*(7 + 2*n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (9 + 2*n)/4, (13 + 2*n)/4, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(7 + 2*n)*(9 + 2*n))","A",1
920,1,138,163,0.3579435,"\int \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^n (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]),x]","-\frac{2 \sqrt{\sin ^2(c+d x)} \cos ^{\frac{5}{2}}(c+d x) \csc (c+d x) (b \cos (c+d x))^n \left(A (2 n+7) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);\cos ^2(c+d x)\right)+B (2 n+5) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+7);\frac{1}{4} (2 n+11);\cos ^2(c+d x)\right)\right)}{d (2 n+5) (2 n+7)}","-\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);\cos ^2(c+d x)\right)}{d (2 n+5) \sqrt{\sin ^2(c+d x)}}-\frac{2 B \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+7);\frac{1}{4} (2 n+11);\cos ^2(c+d x)\right)}{d (2 n+7) \sqrt{\sin ^2(c+d x)}}",1,"(-2*Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*Csc[c + d*x]*(A*(7 + 2*n)*Hypergeometric2F1[1/2, (5 + 2*n)/4, (9 + 2*n)/4, Cos[c + d*x]^2] + B*(5 + 2*n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (7 + 2*n)/4, (11 + 2*n)/4, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(5 + 2*n)*(7 + 2*n))","A",1
921,1,138,163,0.259034,"\int \sqrt{\cos (c+d x)} (b \cos (c+d x))^n (A+B \cos (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]),x]","-\frac{2 \sqrt{\sin ^2(c+d x)} \cos ^{\frac{3}{2}}(c+d x) \csc (c+d x) (b \cos (c+d x))^n \left(A (2 n+5) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);\cos ^2(c+d x)\right)+B (2 n+3) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);\cos ^2(c+d x)\right)\right)}{d (2 n+3) (2 n+5)}","-\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);\cos ^2(c+d x)\right)}{d (2 n+3) \sqrt{\sin ^2(c+d x)}}-\frac{2 B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);\cos ^2(c+d x)\right)}{d (2 n+5) \sqrt{\sin ^2(c+d x)}}",1,"(-2*Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*Csc[c + d*x]*(A*(5 + 2*n)*Hypergeometric2F1[1/2, (3 + 2*n)/4, (7 + 2*n)/4, Cos[c + d*x]^2] + B*(3 + 2*n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (5 + 2*n)/4, (9 + 2*n)/4, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(3 + 2*n)*(5 + 2*n))","A",1
922,1,138,163,0.2364286,"\int \frac{(b \cos (c+d x))^n (A+B \cos (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]))/Sqrt[Cos[c + d*x]],x]","-\frac{2 \sqrt{\sin ^2(c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) (b \cos (c+d x))^n \left(A (2 n+3) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\cos ^2(c+d x)\right)+B (2 n+1) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);\cos ^2(c+d x)\right)\right)}{d (2 n+1) (2 n+3)}","-\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\cos ^2(c+d x)\right)}{d (2 n+1) \sqrt{\sin ^2(c+d x)}}-\frac{2 B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);\cos ^2(c+d x)\right)}{d (2 n+3) \sqrt{\sin ^2(c+d x)}}",1,"(-2*Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*Csc[c + d*x]*(A*(3 + 2*n)*Hypergeometric2F1[1/2, (1 + 2*n)/4, (5 + 2*n)/4, Cos[c + d*x]^2] + B*(1 + 2*n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (3 + 2*n)/4, (7 + 2*n)/4, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(1 + 2*n)*(3 + 2*n))","A",1
923,1,133,163,0.2665565,"\int \frac{(b \cos (c+d x))^n (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(3/2),x]","-\frac{2 \sqrt{\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^n \left(A (2 n+1) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\cos ^2(c+d x)\right)+B (2 n-1) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\cos ^2(c+d x)\right)\right)}{d \left(4 n^2-1\right) \sqrt{\cos (c+d x)}}","\frac{2 A \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\cos ^2(c+d x)\right)}{d (1-2 n) \sqrt{\sin ^2(c+d x)} \sqrt{\cos (c+d x)}}-\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)} (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\cos ^2(c+d x)\right)}{d (2 n+1) \sqrt{\sin ^2(c+d x)}}",1,"(-2*(b*Cos[c + d*x])^n*Csc[c + d*x]*(A*(1 + 2*n)*Hypergeometric2F1[1/2, (-1 + 2*n)/4, (3 + 2*n)/4, Cos[c + d*x]^2] + B*(-1 + 2*n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + 2*n)/4, (5 + 2*n)/4, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(-1 + 4*n^2)*Sqrt[Cos[c + d*x]])","A",1
924,1,138,163,0.2323064,"\int \frac{(b \cos (c+d x))^n (A+B \cos (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(5/2),x]","-\frac{2 \sqrt{\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^n \left(A (2 n-1) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-3);\frac{1}{4} (2 n+1);\cos ^2(c+d x)\right)+B (2 n-3) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\cos ^2(c+d x)\right)\right)}{d (2 n-3) (2 n-1) \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 A \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-3);\frac{1}{4} (2 n+1);\cos ^2(c+d x)\right)}{d (3-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\cos ^2(c+d x)\right)}{d (1-2 n) \sqrt{\sin ^2(c+d x)} \sqrt{\cos (c+d x)}}",1,"(-2*(b*Cos[c + d*x])^n*Csc[c + d*x]*(A*(-1 + 2*n)*Hypergeometric2F1[1/2, (-3 + 2*n)/4, (1 + 2*n)/4, Cos[c + d*x]^2] + B*(-3 + 2*n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (-1 + 2*n)/4, (3 + 2*n)/4, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(-3 + 2*n)*(-1 + 2*n)*Cos[c + d*x]^(3/2))","A",1
925,1,138,163,0.2370457,"\int \frac{(b \cos (c+d x))^n (A+B \cos (c+d x))}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(7/2),x]","-\frac{2 \sqrt{\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^n \left(A (2 n-3) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-5);\frac{1}{4} (2 n-1);\cos ^2(c+d x)\right)+B (2 n-5) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-3);\frac{1}{4} (2 n+1);\cos ^2(c+d x)\right)\right)}{d (2 n-5) (2 n-3) \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 A \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-5);\frac{1}{4} (2 n-1);\cos ^2(c+d x)\right)}{d (5-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-3);\frac{1}{4} (2 n+1);\cos ^2(c+d x)\right)}{d (3-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{3}{2}}(c+d x)}",1,"(-2*(b*Cos[c + d*x])^n*Csc[c + d*x]*(A*(-3 + 2*n)*Hypergeometric2F1[1/2, (-5 + 2*n)/4, (-1 + 2*n)/4, Cos[c + d*x]^2] + B*(-5 + 2*n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (-3 + 2*n)/4, (1 + 2*n)/4, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(-5 + 2*n)*(-3 + 2*n)*Cos[c + d*x]^(5/2))","A",1
926,1,138,163,0.2362553,"\int \frac{(b \cos (c+d x))^n (A+B \cos (c+d x))}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^n*(A + B*Cos[c + d*x]))/Cos[c + d*x]^(9/2),x]","-\frac{2 \sqrt{\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^n \left(A (2 n-5) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-7);\frac{1}{4} (2 n-3);\cos ^2(c+d x)\right)+B (2 n-7) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-5);\frac{1}{4} (2 n-1);\cos ^2(c+d x)\right)\right)}{d (2 n-7) (2 n-5) \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 A \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-7);\frac{1}{4} (2 n-3);\cos ^2(c+d x)\right)}{d (7-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-5);\frac{1}{4} (2 n-1);\cos ^2(c+d x)\right)}{d (5-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{5}{2}}(c+d x)}",1,"(-2*(b*Cos[c + d*x])^n*Csc[c + d*x]*(A*(-5 + 2*n)*Hypergeometric2F1[1/2, (-7 + 2*n)/4, (-3 + 2*n)/4, Cos[c + d*x]^2] + B*(-7 + 2*n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (-5 + 2*n)/4, (-1 + 2*n)/4, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(-7 + 2*n)*(-5 + 2*n)*Cos[c + d*x]^(7/2))","A",1
927,1,140,169,0.5147755,"\int \cos ^m(c+d x) (b \cos (c+d x))^{4/3} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^m*(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x]),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^{4/3} \cos ^{m+1}(c+d x) \left(A (3 m+10) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+7);\frac{1}{6} (3 m+13);\cos ^2(c+d x)\right)+B (3 m+7) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m}{2}+\frac{5}{3};\frac{m}{2}+\frac{8}{3};\cos ^2(c+d x)\right)\right)}{d (3 m+7) (3 m+10)}","-\frac{3 A b \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+7);\frac{1}{6} (3 m+13);\cos ^2(c+d x)\right)}{d (3 m+7) \sqrt{\sin ^2(c+d x)}}-\frac{3 b B \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+3}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+10);\frac{1}{6} (3 m+16);\cos ^2(c+d x)\right)}{d (3 m+10) \sqrt{\sin ^2(c+d x)}}",1,"(-3*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(4/3)*Csc[c + d*x]*(B*(7 + 3*m)*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/3 + m/2, 8/3 + m/2, Cos[c + d*x]^2] + A*(10 + 3*m)*Hypergeometric2F1[1/2, (7 + 3*m)/6, (13 + 3*m)/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(7 + 3*m)*(10 + 3*m))","A",1
928,1,140,167,0.3209245,"\int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^m*(b*Cos[c + d*x])^(2/3)*(A + B*Cos[c + d*x]),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x) \left(A (3 m+8) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right)+B (3 m+5) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+8);\frac{m}{2}+\frac{7}{3};\cos ^2(c+d x)\right)\right)}{d (3 m+5) (3 m+8)}","-\frac{3 A \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right)}{d (3 m+5) \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+8);\frac{1}{6} (3 m+14);\cos ^2(c+d x)\right)}{d (3 m+8) \sqrt{\sin ^2(c+d x)}}",1,"(-3*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(2/3)*Csc[c + d*x]*(A*(8 + 3*m)*Hypergeometric2F1[1/2, (5 + 3*m)/6, (11 + 3*m)/6, Cos[c + d*x]^2] + B*(5 + 3*m)*Cos[c + d*x]*Hypergeometric2F1[1/2, (8 + 3*m)/6, 7/3 + m/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(5 + 3*m)*(8 + 3*m))","A",1
929,1,140,167,0.3074334,"\int \cos ^m(c+d x) \sqrt[3]{b \cos (c+d x)} (A+B \cos (c+d x)) \, dx","Integrate[Cos[c + d*x]^m*(b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x]),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+1}(c+d x) \left(A (3 m+7) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+4);\frac{m}{2}+\frac{5}{3};\cos ^2(c+d x)\right)+B (3 m+4) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+7);\frac{1}{6} (3 m+13);\cos ^2(c+d x)\right)\right)}{d (3 m+4) (3 m+7)}","-\frac{3 A \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+4);\frac{1}{6} (3 m+10);\cos ^2(c+d x)\right)}{d (3 m+4) \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+7);\frac{1}{6} (3 m+13);\cos ^2(c+d x)\right)}{d (3 m+7) \sqrt{\sin ^2(c+d x)}}",1,"(-3*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(1/3)*Csc[c + d*x]*(A*(7 + 3*m)*Hypergeometric2F1[1/2, (4 + 3*m)/6, 5/3 + m/2, Cos[c + d*x]^2] + B*(4 + 3*m)*Cos[c + d*x]*Hypergeometric2F1[1/2, (7 + 3*m)/6, (13 + 3*m)/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(4 + 3*m)*(7 + 3*m))","A",1
930,1,140,167,0.3198813,"\int \frac{\cos ^m(c+d x) (A+B \cos (c+d x))}{\sqrt[3]{b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^m*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(1/3),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+1}(c+d x) \left(A (3 m+5) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\cos ^2(c+d x)\right)+B (3 m+2) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right)\right)}{d (3 m+2) (3 m+5) \sqrt[3]{b \cos (c+d x)}}","-\frac{3 A \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\cos ^2(c+d x)\right)}{d (3 m+2) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}-\frac{3 B \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right)}{d (3 m+5) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}",1,"(-3*Cos[c + d*x]^(1 + m)*Csc[c + d*x]*(A*(5 + 3*m)*Hypergeometric2F1[1/2, (2 + 3*m)/6, (8 + 3*m)/6, Cos[c + d*x]^2] + B*(2 + 3*m)*Cos[c + d*x]*Hypergeometric2F1[1/2, (5 + 3*m)/6, (11 + 3*m)/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(2 + 3*m)*(5 + 3*m)*(b*Cos[c + d*x])^(1/3))","A",1
931,1,140,167,0.3104736,"\int \frac{\cos ^m(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{2/3}} \, dx","Integrate[(Cos[c + d*x]^m*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(2/3),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+1}(c+d x) \left(A (3 m+4) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+1);\frac{1}{6} (3 m+7);\cos ^2(c+d x)\right)+B (3 m+1) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+4);\frac{m}{2}+\frac{5}{3};\cos ^2(c+d x)\right)\right)}{d (3 m+1) (3 m+4) (b \cos (c+d x))^{2/3}}","-\frac{3 A \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+1);\frac{1}{6} (3 m+7);\cos ^2(c+d x)\right)}{d (3 m+1) \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}-\frac{3 B \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+4);\frac{1}{6} (3 m+10);\cos ^2(c+d x)\right)}{d (3 m+4) \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}",1,"(-3*Cos[c + d*x]^(1 + m)*Csc[c + d*x]*(A*(4 + 3*m)*Hypergeometric2F1[1/2, (1 + 3*m)/6, (7 + 3*m)/6, Cos[c + d*x]^2] + B*(1 + 3*m)*Cos[c + d*x]*Hypergeometric2F1[1/2, (4 + 3*m)/6, 5/3 + m/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(1 + 3*m)*(4 + 3*m)*(b*Cos[c + d*x])^(2/3))","A",1
932,1,140,171,0.3436328,"\int \frac{\cos ^m(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{4/3}} \, dx","Integrate[(Cos[c + d*x]^m*(A + B*Cos[c + d*x]))/(b*Cos[c + d*x])^(4/3),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+1}(c+d x) \left(A (3 m+2) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m-1);\frac{1}{6} (3 m+5);\cos ^2(c+d x)\right)+B (3 m-1) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\cos ^2(c+d x)\right)\right)}{d (3 m-1) (3 m+2) (b \cos (c+d x))^{4/3}}","\frac{3 A \sin (c+d x) \cos ^m(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m-1);\frac{1}{6} (3 m+5);\cos ^2(c+d x)\right)}{b d (1-3 m) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}-\frac{3 B \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\cos ^2(c+d x)\right)}{b d (3 m+2) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}",1,"(-3*Cos[c + d*x]^(1 + m)*Csc[c + d*x]*(A*(2 + 3*m)*Hypergeometric2F1[1/2, (-1 + 3*m)/6, (5 + 3*m)/6, Cos[c + d*x]^2] + B*(-1 + 3*m)*Cos[c + d*x]*Hypergeometric2F1[1/2, (2 + 3*m)/6, (8 + 3*m)/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(-1 + 3*m)*(2 + 3*m)*(b*Cos[c + d*x])^(4/3))","A",1