1,1,25,31,0.041013,"\int \frac{\sin ^4(x)}{a+a \cos (x)} \, dx","Integrate[Sin[x]^4/(a + a*Cos[x]),x]","\frac{6 x-3 \sin (x)-3 \sin (2 x)+\sin (3 x)}{12 a}","\frac{x}{2 a}-\frac{\sin ^3(x)}{3 a}-\frac{\sin (x) \cos (x)}{2 a}",1,"(6*x - 3*Sin[x] - 3*Sin[2*x] + Sin[3*x])/(12*a)","A",1
2,1,13,19,0.011791,"\int \frac{\sin ^3(x)}{a+a \cos (x)} \, dx","Integrate[Sin[x]^3/(a + a*Cos[x]),x]","\frac{2 \sin ^4\left(\frac{x}{2}\right)}{a}","\frac{\cos ^2(x)}{2 a}-\frac{\cos (x)}{a}",1,"(2*Sin[x/2]^4)/a","A",1
3,1,17,13,0.0084086,"\int \frac{\sin ^2(x)}{a+a \cos (x)} \, dx","Integrate[Sin[x]^2/(a + a*Cos[x]),x]","\frac{2 \left(\frac{x}{2}-\frac{\sin (x)}{2}\right)}{a}","\frac{x}{a}-\frac{\sin (x)}{a}",1,"(2*(x/2 - Sin[x]/2))/a","A",1
4,1,12,10,0.0061282,"\int \frac{\sin (x)}{a+a \cos (x)} \, dx","Integrate[Sin[x]/(a + a*Cos[x]),x]","-\frac{2 \log \left(\cos \left(\frac{x}{2}\right)\right)}{a}","-\frac{\log (\cos (x)+1)}{a}",1,"(-2*Log[Cos[x/2]])/a","A",1
5,1,10,11,0.0047496,"\int \frac{1}{a+a \cos (x)} \, dx","Integrate[(a + a*Cos[x])^(-1),x]","\frac{\tan \left(\frac{x}{2}\right)}{a}","\frac{\sin (x)}{a \cos (x)+a}",1,"Tan[x/2]/a","A",1
6,1,42,23,0.0337284,"\int \frac{\csc (x)}{a+a \cos (x)} \, dx","Integrate[Csc[x]/(a + a*Cos[x]),x]","\frac{1-2 \cos ^2\left(\frac{x}{2}\right) \left(\log \left(\cos \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)\right)\right)}{2 a (\cos (x)+1)}","\frac{1}{2 (a \cos (x)+a)}-\frac{\tanh ^{-1}(\cos (x))}{2 a}",1,"(1 - 2*Cos[x/2]^2*(Log[Cos[x/2]] - Log[Sin[x/2]]))/(2*a*(1 + Cos[x]))","A",1
7,1,30,24,0.0490928,"\int \frac{\csc ^2(x)}{a+a \cos (x)} \, dx","Integrate[Csc[x]^2/(a + a*Cos[x]),x]","-\frac{(2 \cos (x)+\cos (2 x)) \csc \left(\frac{x}{2}\right) \sec ^3\left(\frac{x}{2}\right)}{12 a}","\frac{\csc (x)}{3 (a \cos (x)+a)}-\frac{2 \cot (x)}{3 a}",1,"-1/12*((2*Cos[x] + Cos[2*x])*Csc[x/2]*Sec[x/2]^3)/a","A",1
8,1,60,49,0.1118852,"\int \frac{\csc ^3(x)}{a+a \cos (x)} \, dx","Integrate[Csc[x]^3/(a + a*Cos[x]),x]","\frac{-2 \cot ^2\left(\frac{x}{2}\right)+\sec ^2\left(\frac{x}{2}\right)-12 \cos ^2\left(\frac{x}{2}\right) \left(\log \left(\cos \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)\right)\right)+4}{16 a (\cos (x)+1)}","\frac{a}{8 (a \cos (x)+a)^2}-\frac{1}{8 (a-a \cos (x))}+\frac{1}{4 (a \cos (x)+a)}-\frac{3 \tanh ^{-1}(\cos (x))}{8 a}",1,"(4 - 2*Cot[x/2]^2 - 12*Cos[x/2]^2*(Log[Cos[x/2]] - Log[Sin[x/2]]) + Sec[x/2]^2)/(16*a*(1 + Cos[x]))","A",1
9,1,38,37,0.0581316,"\int \frac{\csc ^4(x)}{a+a \cos (x)} \, dx","Integrate[Csc[x]^4/(a + a*Cos[x]),x]","\frac{(-6 \cos (x)-2 \cos (2 x)+2 \cos (3 x)+\cos (4 x)) \csc ^3(x)}{15 a (\cos (x)+1)}","-\frac{4 \cot ^3(x)}{15 a}-\frac{4 \cot (x)}{5 a}+\frac{\csc ^3(x)}{5 (a \cos (x)+a)}",1,"((-6*Cos[x] - 2*Cos[2*x] + 2*Cos[3*x] + Cos[4*x])*Csc[x]^3)/(15*a*(1 + Cos[x]))","A",1
10,1,5,5,0.0029044,"\int \frac{\sin (2 x)}{1+\cos (2 x)} \, dx","Integrate[Sin[2*x]/(1 + Cos[2*x]),x]","-\log (\cos (x))","-\log (\cos (x))",1,"-Log[Cos[x]]","A",1
11,1,3,3,0.0044291,"\int \frac{\sin (2 x)}{1-\cos (2 x)} \, dx","Integrate[Sin[2*x]/(1 - Cos[2*x]),x]","\log (\sin (x))","\log (\sin (x))",1,"Log[Sin[x]]","A",1
12,1,12,6,0.008341,"\int \frac{\sin (x)}{(1+\cos (x))^2} \, dx","Integrate[Sin[x]/(1 + Cos[x])^2,x]","\frac{1}{2} \sec ^2\left(\frac{x}{2}\right)","\frac{1}{\cos (x)+1}",1,"Sec[x/2]^2/2","A",1
13,1,12,10,0.0094674,"\int \frac{\sin (x)}{(1-\cos (x))^2} \, dx","Integrate[Sin[x]/(1 - Cos[x])^2,x]","-\frac{1}{2} \csc ^2\left(\frac{x}{2}\right)","-\frac{1}{1-\cos (x)}",1,"-1/2*Csc[x/2]^2","A",1
14,1,18,14,0.0056214,"\int \frac{\sin ^2(x)}{(1+\cos (x))^2} \, dx","Integrate[Sin[x]^2/(1 + Cos[x])^2,x]","2 \tan \left(\frac{x}{2}\right)-2 \tan ^{-1}\left(\tan \left(\frac{x}{2}\right)\right)","\frac{2 \sin (x)}{\cos (x)+1}-x",1,"-2*ArcTan[Tan[x/2]] + 2*Tan[x/2]","A",1
15,1,26,16,0.0107171,"\int \frac{\sin ^2(x)}{(1-\cos (x))^2} \, dx","Integrate[Sin[x]^2/(1 - Cos[x])^2,x]","-2 \cot \left(\frac{x}{2}\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2\left(\frac{x}{2}\right)\right)","-x-\frac{2 \sin (x)}{1-\cos (x)}",1,"-2*Cot[x/2]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[x/2]^2]","C",1
16,1,13,10,0.0186084,"\int \frac{\sin ^3(x)}{(1+\cos (x))^2} \, dx","Integrate[Sin[x]^3/(1 + Cos[x])^2,x]","\cos (x)-4 \log \left(\cos \left(\frac{x}{2}\right)\right)-1","\cos (x)-2 \log (\cos (x)+1)",1,"-1 + Cos[x] - 4*Log[Cos[x/2]]","A",1
17,1,13,12,0.0169097,"\int \frac{\sin ^3(x)}{(1-\cos (x))^2} \, dx","Integrate[Sin[x]^3/(1 - Cos[x])^2,x]","\cos (x)+4 \log \left(\sin \left(\frac{x}{2}\right)\right)-1","\cos (x)+2 \log (1-\cos (x))",1,"-1 + Cos[x] + 4*Log[Sin[x/2]]","A",1
18,1,12,10,0.0076465,"\int \frac{\sin (x)}{(1+\cos (x))^3} \, dx","Integrate[Sin[x]/(1 + Cos[x])^3,x]","\frac{1}{8} \sec ^4\left(\frac{x}{2}\right)","\frac{1}{2 (\cos (x)+1)^2}",1,"Sec[x/2]^4/8","A",1
19,1,12,12,0.010622,"\int \frac{\sin (x)}{(1-\cos (x))^3} \, dx","Integrate[Sin[x]/(1 - Cos[x])^3,x]","-\frac{1}{8} \csc ^4\left(\frac{x}{2}\right)","-\frac{1}{2 (1-\cos (x))^2}",1,"-1/8*Csc[x/2]^4","A",1
20,1,12,14,0.0270436,"\int \frac{\sin ^2(x)}{(1+\cos (x))^3} \, dx","Integrate[Sin[x]^2/(1 + Cos[x])^3,x]","\frac{1}{3} \tan ^3\left(\frac{x}{2}\right)","\frac{\sin ^3(x)}{3 (\cos (x)+1)^3}",1,"Tan[x/2]^3/3","A",1
21,1,12,16,0.0316004,"\int \frac{\sin ^2(x)}{(1-\cos (x))^3} \, dx","Integrate[Sin[x]^2/(1 - Cos[x])^3,x]","-\frac{1}{3} \cot ^3\left(\frac{x}{2}\right)","-\frac{\sin ^3(x)}{3 (1-\cos (x))^3}",1,"-1/3*Cot[x/2]^3","A",1
22,1,18,14,0.0081924,"\int \frac{\sin ^3(x)}{(1+\cos (x))^3} \, dx","Integrate[Sin[x]^3/(1 + Cos[x])^3,x]","\tan ^2\left(\frac{x}{2}\right)+2 \log \left(\cos \left(\frac{x}{2}\right)\right)","\frac{2}{\cos (x)+1}+\log (\cos (x)+1)",1,"2*Log[Cos[x/2]] + Tan[x/2]^2","A",1
23,1,29,20,0.0105279,"\int \frac{\sin ^3(x)}{(1-\cos (x))^3} \, dx","Integrate[Sin[x]^3/(1 - Cos[x])^3,x]","-\cot ^2\left(\frac{x}{2}\right)-2 \log \left(\tan \left(\frac{x}{2}\right)\right)-2 \log \left(\cos \left(\frac{x}{2}\right)\right)","-\frac{2}{1-\cos (x)}-\log (1-\cos (x))",1,"-Cot[x/2]^2 - 2*Log[Cos[x/2]] - 2*Log[Tan[x/2]]","A",1
24,1,96,104,0.2097281,"\int \frac{\sin ^4(x)}{a+b \cos (x)} \, dx","Integrate[Sin[x]^4/(a + b*Cos[x]),x]","\frac{-12 a^3 x+3 b \left(4 a^2-5 b^2\right) \sin (x)-24 \left(b^2-a^2\right)^{3/2} \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)+18 a b^2 x-3 a b^2 \sin (2 x)+b^3 \sin (3 x)}{12 b^4}","-\frac{a x \left(2 a^2-3 b^2\right)}{2 b^4}+\frac{\sin (x) \left(2 \left(a^2-b^2\right)-a b \cos (x)\right)}{2 b^3}+\frac{2 (a-b)^{3/2} (a+b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{b^4}-\frac{\sin ^3(x)}{3 b}",1,"(-12*a^3*x + 18*a*b^2*x - 24*(-a^2 + b^2)^(3/2)*ArcTanh[((a - b)*Tan[x/2])/Sqrt[-a^2 + b^2]] + 3*b*(4*a^2 - 5*b^2)*Sin[x] - 3*a*b^2*Sin[2*x] + b^3*Sin[3*x])/(12*b^4)","A",1
25,1,40,40,0.0561483,"\int \frac{\sin ^3(x)}{a+b \cos (x)} \, dx","Integrate[Sin[x]^3/(a + b*Cos[x]),x]","\frac{\left(a^2-b^2\right) \log (a+b \cos (x))}{b^3}-\frac{a \cos (x)}{b^2}+\frac{\cos (2 x)}{4 b}","\frac{\left(a^2-b^2\right) \log (a+b \cos (x))}{b^3}-\frac{a \cos (x)}{b^2}+\frac{\cos ^2(x)}{2 b}",1,"-((a*Cos[x])/b^2) + Cos[2*x]/(4*b) + ((a^2 - b^2)*Log[a + b*Cos[x]])/b^3","A",1
26,1,54,59,0.0805146,"\int \frac{\sin ^2(x)}{a+b \cos (x)} \, dx","Integrate[Sin[x]^2/(a + b*Cos[x]),x]","\frac{-2 \sqrt{b^2-a^2} \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)+a x-b \sin (x)}{b^2}","\frac{a x}{b^2}-\frac{2 \sqrt{a-b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{b^2}-\frac{\sin (x)}{b}",1,"(a*x - 2*Sqrt[-a^2 + b^2]*ArcTanh[((a - b)*Tan[x/2])/Sqrt[-a^2 + b^2]] - b*Sin[x])/b^2","A",1
27,1,12,12,0.0152093,"\int \frac{\sin (x)}{a+b \cos (x)} \, dx","Integrate[Sin[x]/(a + b*Cos[x]),x]","-\frac{\log (a+b \cos (x))}{b}","-\frac{\log (a+b \cos (x))}{b}",1,"-(Log[a + b*Cos[x]]/b)","A",1
28,1,41,42,0.0233407,"\int \frac{1}{a+b \cos (x)} \, dx","Integrate[(a + b*Cos[x])^(-1),x]","-\frac{2 \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{\sqrt{a-b} \sqrt{a+b}}",1,"(-2*ArcTanh[((a - b)*Tan[x/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2]","A",1
29,1,50,53,0.0426695,"\int \frac{\csc (x)}{a+b \cos (x)} \, dx","Integrate[Csc[x]/(a + b*Cos[x]),x]","\frac{(a-b) \log (1-\cos (x))-(a+b) \log (\cos (x)+1)+2 b \log (a+b \cos (x))}{2 (a-b) (a+b)}","\frac{b \log (a+b \cos (x))}{a^2-b^2}+\frac{\log (1-\cos (x))}{2 (a+b)}-\frac{\log (\cos (x)+1)}{2 (a-b)}",1,"((a - b)*Log[1 - Cos[x]] - (a + b)*Log[1 + Cos[x]] + 2*b*Log[a + b*Cos[x]])/(2*(a - b)*(a + b))","A",1
30,1,66,67,0.322709,"\int \frac{\csc ^2(x)}{a+b \cos (x)} \, dx","Integrate[Csc[x]^2/(a + b*Cos[x]),x]","\frac{\csc (x) (b-a \cos (x))}{a^2-b^2}-\frac{2 b^2 \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}","\frac{\csc (x) (b-a \cos (x))}{a^2-b^2}-\frac{2 b^2 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{(a-b)^{3/2} (a+b)^{3/2}}",1,"(-2*b^2*ArcTanh[((a - b)*Tan[x/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + ((b - a*Cos[x])*Csc[x])/(a^2 - b^2)","A",1
31,1,99,92,0.4293021,"\int \frac{\csc ^3(x)}{a+b \cos (x)} \, dx","Integrate[Csc[x]^3/(a + b*Cos[x]),x]","\frac{1}{8} \left(-\frac{8 b^3 \log (a+b \cos (x))}{\left(a^2-b^2\right)^2}-\frac{\csc ^2\left(\frac{x}{2}\right)}{a+b}+\frac{\sec ^2\left(\frac{x}{2}\right)}{a-b}+\frac{4 (a+2 b) \log \left(\sin \left(\frac{x}{2}\right)\right)}{(a+b)^2}-\frac{4 (a-2 b) \log \left(\cos \left(\frac{x}{2}\right)\right)}{(a-b)^2}\right)","\frac{\csc ^2(x) (b-a \cos (x))}{2 \left(a^2-b^2\right)}-\frac{b^3 \log (a+b \cos (x))}{\left(a^2-b^2\right)^2}+\frac{(a+2 b) \log (1-\cos (x))}{4 (a+b)^2}-\frac{(a-2 b) \log (\cos (x)+1)}{4 (a-b)^2}",1,"(-(Csc[x/2]^2/(a + b)) - (4*(a - 2*b)*Log[Cos[x/2]])/(a - b)^2 - (8*b^3*Log[a + b*Cos[x]])/(a^2 - b^2)^2 + (4*(a + 2*b)*Log[Sin[x/2]])/(a + b)^2 + Sec[x/2]^2/(a - b))/8","A",1
32,1,112,110,0.637289,"\int \frac{\csc ^4(x)}{a+b \cos (x)} \, dx","Integrate[Csc[x]^4/(a + b*Cos[x]),x]","\frac{\csc ^3(x) \left(\left(9 a b^2-6 a^3\right) \cos (x)+\left(2 a^2-5 b^2\right) (a \cos (3 x)+2 b)+6 b^3 \cos (2 x)\right)}{12 (a-b)^2 (a+b)^2}-\frac{2 b^4 \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}","\frac{\csc ^3(x) (b-a \cos (x))}{3 \left(a^2-b^2\right)}-\frac{\csc (x) \left(a \left(2 a^2-5 b^2\right) \cos (x)+3 b^3\right)}{3 \left(a^2-b^2\right)^2}+\frac{2 b^4 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{(a-b)^{5/2} (a+b)^{5/2}}",1,"(-2*b^4*ArcTanh[((a - b)*Tan[x/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (((-6*a^3 + 9*a*b^2)*Cos[x] + 6*b^3*Cos[2*x] + (2*a^2 - 5*b^2)*(2*b + a*Cos[3*x]))*Csc[x]^3)/(12*(a - b)^2*(a + b)^2)","A",1
33,1,108,129,0.8519499,"\int (a+b \cos (c+d x)) (e \sin (c+d x))^{7/2} \, dx","Integrate[(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(7/2),x]","\frac{e^3 \sqrt{e \sin (c+d x)} \left(\sqrt{\sin (c+d x)} (-138 a \cos (c+d x)+18 a \cos (3 (c+d x))-28 b \cos (2 (c+d x))+7 b \cos (4 (c+d x))+21 b)-120 a F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)\right)}{252 d \sqrt{\sin (c+d x)}}","\frac{10 a e^4 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 d \sqrt{e \sin (c+d x)}}-\frac{10 a e^3 \cos (c+d x) \sqrt{e \sin (c+d x)}}{21 d}-\frac{2 a e \cos (c+d x) (e \sin (c+d x))^{5/2}}{7 d}+\frac{2 b (e \sin (c+d x))^{9/2}}{9 d e}",1,"(e^3*(-120*a*EllipticF[(-2*c + Pi - 2*d*x)/4, 2] + (21*b - 138*a*Cos[c + d*x] - 28*b*Cos[2*(c + d*x)] + 18*a*Cos[3*(c + d*x)] + 7*b*Cos[4*(c + d*x)])*Sqrt[Sin[c + d*x]])*Sqrt[e*Sin[c + d*x]])/(252*d*Sqrt[Sin[c + d*x]])","A",1
34,1,80,100,0.5163211,"\int (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2} \, dx","Integrate[(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2),x]","\frac{2 (e \sin (c+d x))^{5/2} \left(\sin ^{\frac{3}{2}}(c+d x) \left(5 b \sin ^2(c+d x)-7 a \cos (c+d x)\right)-21 a E\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)\right)}{35 d \sin ^{\frac{5}{2}}(c+d x)}","\frac{6 a e^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 d \sqrt{\sin (c+d x)}}-\frac{2 a e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}+\frac{2 b (e \sin (c+d x))^{7/2}}{7 d e}",1,"(2*(e*Sin[c + d*x])^(5/2)*(-21*a*EllipticE[(-2*c + Pi - 2*d*x)/4, 2] + Sin[c + d*x]^(3/2)*(-7*a*Cos[c + d*x] + 5*b*Sin[c + d*x]^2)))/(35*d*Sin[c + d*x]^(5/2))","A",1
35,1,80,100,0.4622366,"\int (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2} \, dx","Integrate[(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2),x]","\frac{2 (e \sin (c+d x))^{3/2} \left(\sqrt{\sin (c+d x)} \left(3 b \sin ^2(c+d x)-5 a \cos (c+d x)\right)-5 a F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)\right)}{15 d \sin ^{\frac{3}{2}}(c+d x)}","\frac{2 a e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d \sqrt{e \sin (c+d x)}}-\frac{2 a e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 d}+\frac{2 b (e \sin (c+d x))^{5/2}}{5 d e}",1,"(2*(e*Sin[c + d*x])^(3/2)*(-5*a*EllipticF[(-2*c + Pi - 2*d*x)/4, 2] + Sqrt[Sin[c + d*x]]*(-5*a*Cos[c + d*x] + 3*b*Sin[c + d*x]^2)))/(15*d*Sin[c + d*x]^(3/2))","A",1
36,1,60,68,0.0982228,"\int (a+b \cos (c+d x)) \sqrt{e \sin (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]],x]","\frac{2 \sqrt{e \sin (c+d x)} \left(b \sin ^{\frac{3}{2}}(c+d x)-3 a E\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)\right)}{3 d \sqrt{\sin (c+d x)}}","\frac{2 a E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{d \sqrt{\sin (c+d x)}}+\frac{2 b (e \sin (c+d x))^{3/2}}{3 d e}",1,"(2*Sqrt[e*Sin[c + d*x]]*(-3*a*EllipticE[(-2*c + Pi - 2*d*x)/4, 2] + b*Sin[c + d*x]^(3/2)))/(3*d*Sqrt[Sin[c + d*x]])","A",1
37,1,54,66,0.2018642,"\int \frac{a+b \cos (c+d x)}{\sqrt{e \sin (c+d x)}} \, dx","Integrate[(a + b*Cos[c + d*x])/Sqrt[e*Sin[c + d*x]],x]","\frac{2 \left(b \sin (c+d x)-a \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)\right)}{d \sqrt{e \sin (c+d x)}}","\frac{2 a \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d \sqrt{e \sin (c+d x)}}+\frac{2 b \sqrt{e \sin (c+d x)}}{d e}",1,"(2*(-(a*EllipticF[(-2*c + Pi - 2*d*x)/4, 2]*Sqrt[Sin[c + d*x]]) + b*Sin[c + d*x]))/(d*Sqrt[e*Sin[c + d*x]])","A",1
38,1,58,96,0.1174464,"\int \frac{a+b \cos (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx","Integrate[(a + b*Cos[c + d*x])/(e*Sin[c + d*x])^(3/2),x]","-\frac{2 \left(a \cos (c+d x)-a \sqrt{\sin (c+d x)} E\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)+b\right)}{d e \sqrt{e \sin (c+d x)}}","-\frac{2 a E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{d e^2 \sqrt{\sin (c+d x)}}-\frac{2 a \cos (c+d x)}{d e \sqrt{e \sin (c+d x)}}-\frac{2 b}{d e \sqrt{e \sin (c+d x)}}",1,"(-2*(b + a*Cos[c + d*x] - a*EllipticE[(-2*c + Pi - 2*d*x)/4, 2]*Sqrt[Sin[c + d*x]]))/(d*e*Sqrt[e*Sin[c + d*x]])","A",1
39,1,59,102,0.15194,"\int \frac{a+b \cos (c+d x)}{(e \sin (c+d x))^{5/2}} \, dx","Integrate[(a + b*Cos[c + d*x])/(e*Sin[c + d*x])^(5/2),x]","-\frac{2 \left(a \cos (c+d x)+a \sin ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)+b\right)}{3 d e (e \sin (c+d x))^{3/2}}","\frac{2 a \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d e^2 \sqrt{e \sin (c+d x)}}-\frac{2 a \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}-\frac{2 b}{3 d e (e \sin (c+d x))^{3/2}}",1,"(-2*(b + a*Cos[c + d*x] + a*EllipticF[(-2*c + Pi - 2*d*x)/4, 2]*Sin[c + d*x]^(3/2)))/(3*d*e*(e*Sin[c + d*x])^(3/2))","A",1
40,1,74,131,0.2445088,"\int \frac{a+b \cos (c+d x)}{(e \sin (c+d x))^{7/2}} \, dx","Integrate[(a + b*Cos[c + d*x])/(e*Sin[c + d*x])^(7/2),x]","\frac{-7 a \cos (c+d x)+3 a \cos (3 (c+d x))+12 a \sin ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)-4 b}{10 d e (e \sin (c+d x))^{5/2}}","-\frac{6 a E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 d e^4 \sqrt{\sin (c+d x)}}-\frac{6 a \cos (c+d x)}{5 d e^3 \sqrt{e \sin (c+d x)}}-\frac{2 a \cos (c+d x)}{5 d e (e \sin (c+d x))^{5/2}}-\frac{2 b}{5 d e (e \sin (c+d x))^{5/2}}",1,"(-4*b - 7*a*Cos[c + d*x] + 3*a*Cos[3*(c + d*x)] + 12*a*EllipticE[(-2*c + Pi - 2*d*x)/4, 2]*Sin[c + d*x]^(5/2))/(10*d*e*(e*Sin[c + d*x])^(5/2))","A",1
41,1,157,193,1.6674595,"\int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{7/2} \, dx","Integrate[(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(7/2),x]","\frac{(e \sin (c+d x))^{7/2} \left(\frac{1}{6} \csc ^3(c+d x) \left(-6 \left(506 a^2+71 b^2\right) \cos (c+d x)+396 a^2 \cos (3 (c+d x))-1232 a b \cos (2 (c+d x))+308 a b \cos (4 (c+d x))+924 a b-117 b^2 \cos (3 (c+d x))+63 b^2 \cos (5 (c+d x))\right)-\frac{40 \left(11 a^2+2 b^2\right) F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)}{\sin ^{\frac{7}{2}}(c+d x)}\right)}{924 d}","\frac{10 e^4 \left(11 a^2+2 b^2\right) \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{231 d \sqrt{e \sin (c+d x)}}-\frac{10 e^3 \left(11 a^2+2 b^2\right) \cos (c+d x) \sqrt{e \sin (c+d x)}}{231 d}-\frac{2 e \left(11 a^2+2 b^2\right) \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac{26 a b (e \sin (c+d x))^{9/2}}{99 d e}+\frac{2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e}",1,"((((924*a*b - 6*(506*a^2 + 71*b^2)*Cos[c + d*x] - 1232*a*b*Cos[2*(c + d*x)] + 396*a^2*Cos[3*(c + d*x)] - 117*b^2*Cos[3*(c + d*x)] + 308*a*b*Cos[4*(c + d*x)] + 63*b^2*Cos[5*(c + d*x)])*Csc[c + d*x]^3)/6 - (40*(11*a^2 + 2*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, 2])/Sin[c + d*x]^(7/2))*(e*Sin[c + d*x])^(7/2))/(924*d)","A",1
42,1,116,154,0.801942,"\int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2} \, dx","Integrate[(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(5/2),x]","-\frac{(e \sin (c+d x))^{5/2} \left(84 \left(9 a^2+2 b^2\right) E\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)+\sin ^{\frac{3}{2}}(c+d x) \left(21 \left(12 a^2+b^2\right) \cos (c+d x)+5 b (36 a \cos (2 (c+d x))-36 a+7 b \cos (3 (c+d x)))\right)\right)}{630 d \sin ^{\frac{5}{2}}(c+d x)}","\frac{2 e^2 \left(9 a^2+2 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{15 d \sqrt{\sin (c+d x)}}-\frac{2 e \left(9 a^2+2 b^2\right) \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 d}+\frac{22 a b (e \sin (c+d x))^{7/2}}{63 d e}+\frac{2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))}{9 d e}",1,"-1/630*((e*Sin[c + d*x])^(5/2)*(84*(9*a^2 + 2*b^2)*EllipticE[(-2*c + Pi - 2*d*x)/4, 2] + (21*(12*a^2 + b^2)*Cos[c + d*x] + 5*b*(-36*a + 36*a*Cos[2*(c + d*x)] + 7*b*Cos[3*(c + d*x)]))*Sin[c + d*x]^(3/2)))/(d*Sin[c + d*x]^(5/2))","A",1
43,1,117,154,0.8162338,"\int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2} \, dx","Integrate[(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(3/2),x]","\frac{(e \sin (c+d x))^{3/2} \left(-\frac{1}{2} \csc (c+d x) \left(5 \left(28 a^2+5 b^2\right) \cos (c+d x)+3 b (28 a \cos (2 (c+d x))-28 a+5 b \cos (3 (c+d x)))\right)-\frac{10 \left(7 a^2+2 b^2\right) F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)}{\sin ^{\frac{3}{2}}(c+d x)}\right)}{105 d}","\frac{2 e^2 \left(7 a^2+2 b^2\right) \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 d \sqrt{e \sin (c+d x)}}-\frac{2 e \left(7 a^2+2 b^2\right) \cos (c+d x) \sqrt{e \sin (c+d x)}}{21 d}+\frac{18 a b (e \sin (c+d x))^{5/2}}{35 d e}+\frac{2 b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}{7 d e}",1,"((-1/2*((5*(28*a^2 + 5*b^2)*Cos[c + d*x] + 3*b*(-28*a + 28*a*Cos[2*(c + d*x)] + 5*b*Cos[3*(c + d*x)]))*Csc[c + d*x]) - (10*(7*a^2 + 2*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, 2])/Sin[c + d*x]^(3/2))*(e*Sin[c + d*x])^(3/2))/(105*d)","A",1
44,1,83,114,0.2688369,"\int (a+b \cos (c+d x))^2 \sqrt{e \sin (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^2*Sqrt[e*Sin[c + d*x]],x]","\frac{2 \sqrt{e \sin (c+d x)} \left(b \sin ^{\frac{3}{2}}(c+d x) (10 a+3 b \cos (c+d x))-3 \left(5 a^2+2 b^2\right) E\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)\right)}{15 d \sqrt{\sin (c+d x)}}","\frac{2 \left(5 a^2+2 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 d \sqrt{\sin (c+d x)}}+\frac{14 a b (e \sin (c+d x))^{3/2}}{15 d e}+\frac{2 b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}{5 d e}",1,"(2*Sqrt[e*Sin[c + d*x]]*(-3*(5*a^2 + 2*b^2)*EllipticE[(-2*c + Pi - 2*d*x)/4, 2] + b*(10*a + 3*b*Cos[c + d*x])*Sin[c + d*x]^(3/2)))/(15*d*Sqrt[Sin[c + d*x]])","A",1
45,1,79,114,0.3788746,"\int \frac{(a+b \cos (c+d x))^2}{\sqrt{e \sin (c+d x)}} \, dx","Integrate[(a + b*Cos[c + d*x])^2/Sqrt[e*Sin[c + d*x]],x]","\frac{2 b \sin (c+d x) (6 a+b \cos (c+d x))-2 \left(3 a^2+2 b^2\right) \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)}{3 d \sqrt{e \sin (c+d x)}}","\frac{2 \left(3 a^2+2 b^2\right) \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d \sqrt{e \sin (c+d x)}}+\frac{10 a b \sqrt{e \sin (c+d x)}}{3 d e}+\frac{2 b \sqrt{e \sin (c+d x)} (a+b \cos (c+d x))}{3 d e}",1,"(-2*(3*a^2 + 2*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, 2]*Sqrt[Sin[c + d*x]] + 2*b*(6*a + b*Cos[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[e*Sin[c + d*x]])","A",1
46,1,75,118,0.235087,"\int \frac{(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx","Integrate[(a + b*Cos[c + d*x])^2/(e*Sin[c + d*x])^(3/2),x]","\frac{-2 \left(a^2+b^2\right) \cos (c+d x)+2 \left(a^2+2 b^2\right) \sqrt{\sin (c+d x)} E\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)-4 a b}{d e \sqrt{e \sin (c+d x)}}","-\frac{2 \left(a^2+2 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{d e^2 \sqrt{\sin (c+d x)}}-\frac{2 a b (e \sin (c+d x))^{3/2}}{d e^3}-\frac{2 (a \cos (c+d x)+b) (a+b \cos (c+d x))}{d e \sqrt{e \sin (c+d x)}}",1,"(-4*a*b - 2*(a^2 + b^2)*Cos[c + d*x] + 2*(a^2 + 2*b^2)*EllipticE[(-2*c + Pi - 2*d*x)/4, 2]*Sqrt[Sin[c + d*x]])/(d*e*Sqrt[e*Sin[c + d*x]])","A",1
47,1,76,124,0.2644057,"\int \frac{(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{5/2}} \, dx","Integrate[(a + b*Cos[c + d*x])^2/(e*Sin[c + d*x])^(5/2),x]","-\frac{2 \left(\left(a^2+b^2\right) \cos (c+d x)+\left(a^2-2 b^2\right) \sin ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)+2 a b\right)}{3 d e (e \sin (c+d x))^{3/2}}","\frac{2 \left(a^2-2 b^2\right) \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d e^2 \sqrt{e \sin (c+d x)}}-\frac{2 a b \sqrt{e \sin (c+d x)}}{3 d e^3}-\frac{2 (a \cos (c+d x)+b) (a+b \cos (c+d x))}{3 d e (e \sin (c+d x))^{3/2}}",1,"(-2*(2*a*b + (a^2 + b^2)*Cos[c + d*x] + (a^2 - 2*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, 2]*Sin[c + d*x]^(3/2)))/(3*d*e*(e*Sin[c + d*x])^(3/2))","A",1
48,1,109,165,0.4738261,"\int \frac{(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{7/2}} \, dx","Integrate[(a + b*Cos[c + d*x])^2/(e*Sin[c + d*x])^(7/2),x]","-\frac{\left(7 a^2+2 b^2\right) \cos (c+d x)-4 \left(3 a^2-2 b^2\right) \sin ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)-3 a^2 \cos (3 (c+d x))+8 a b+2 b^2 \cos (3 (c+d x))}{10 d e (e \sin (c+d x))^{5/2}}","-\frac{2 \left(3 a^2-2 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 d e^4 \sqrt{\sin (c+d x)}}-\frac{2 \left(3 a^2-2 b^2\right) \cos (c+d x)}{5 d e^3 \sqrt{e \sin (c+d x)}}-\frac{2 a b}{5 d e^3 \sqrt{e \sin (c+d x)}}-\frac{2 (a \cos (c+d x)+b) (a+b \cos (c+d x))}{5 d e (e \sin (c+d x))^{5/2}}",1,"-1/10*(8*a*b + (7*a^2 + 2*b^2)*Cos[c + d*x] - 3*a^2*Cos[3*(c + d*x)] + 2*b^2*Cos[3*(c + d*x)] - 4*(3*a^2 - 2*b^2)*EllipticE[(-2*c + Pi - 2*d*x)/4, 2]*Sin[c + d*x]^(5/2))/(d*e*(e*Sin[c + d*x])^(5/2))","A",1
49,1,205,242,2.4850213,"\int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{7/2} \, dx","Integrate[(a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(7/2),x]","\frac{(e \sin (c+d x))^{7/2} \left(154 b \left(78 a^2+11 b^2\right) \csc ^3(c+d x)-\frac{2080 a \left(11 a^2+6 b^2\right) F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)}{\sin ^{\frac{7}{2}}(c+d x)}+\frac{1}{3} \csc ^3(c+d x) \left(-77 b \left(624 a^2+73 b^2\right) \cos (2 (c+d x))-154 b \left(b^2-78 a^2\right) \cos (4 (c+d x))-156 a \left(506 a^2+213 b^2\right) \cos (c+d x)+234 a \left(44 a^2-39 b^2\right) \cos (3 (c+d x))+4914 a b^2 \cos (5 (c+d x))+693 b^3 \cos (6 (c+d x))\right)\right)}{48048 d}","\frac{10 a e^4 \left(11 a^2+6 b^2\right) \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{231 d \sqrt{e \sin (c+d x)}}-\frac{10 a e^3 \left(11 a^2+6 b^2\right) \cos (c+d x) \sqrt{e \sin (c+d x)}}{231 d}+\frac{2 b \left(177 a^2+44 b^2\right) (e \sin (c+d x))^{9/2}}{1287 d e}-\frac{2 a e \left(11 a^2+6 b^2\right) \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac{2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))^2}{13 d e}+\frac{34 a b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{143 d e}",1,"((154*b*(78*a^2 + 11*b^2)*Csc[c + d*x]^3 + ((-156*a*(506*a^2 + 213*b^2)*Cos[c + d*x] - 77*b*(624*a^2 + 73*b^2)*Cos[2*(c + d*x)] + 234*a*(44*a^2 - 39*b^2)*Cos[3*(c + d*x)] - 154*b*(-78*a^2 + b^2)*Cos[4*(c + d*x)] + 4914*a*b^2*Cos[5*(c + d*x)] + 693*b^3*Cos[6*(c + d*x)])*Csc[c + d*x]^3)/3 - (2080*a*(11*a^2 + 6*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, 2])/Sin[c + d*x]^(7/2))*(e*Sin[c + d*x])^(7/2))/(48048*d)","A",1
50,1,149,202,1.3887239,"\int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{5/2} \, dx","Integrate[(a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(5/2),x]","-\frac{(e \sin (c+d x))^{5/2} \left(1848 \left(3 a^3+2 a b^2\right) E\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)+\sin ^{\frac{3}{2}}(c+d x) \left(462 a \left(4 a^2+b^2\right) \cos (c+d x)+5 b \left(12 \left(33 a^2+4 b^2\right) \cos (2 (c+d x))-396 a^2+154 a b \cos (3 (c+d x))+21 b^2 \cos (4 (c+d x))-69 b^2\right)\right)\right)}{4620 d \sin ^{\frac{5}{2}}(c+d x)}","\frac{2 a e^2 \left(3 a^2+2 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 d \sqrt{\sin (c+d x)}}+\frac{2 b \left(43 a^2+12 b^2\right) (e \sin (c+d x))^{7/2}}{231 d e}-\frac{2 a e \left(3 a^2+2 b^2\right) \cos (c+d x) (e \sin (c+d x))^{3/2}}{15 d}+\frac{2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))^2}{11 d e}+\frac{10 a b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))}{33 d e}",1,"-1/4620*((e*Sin[c + d*x])^(5/2)*(1848*(3*a^3 + 2*a*b^2)*EllipticE[(-2*c + Pi - 2*d*x)/4, 2] + (462*a*(4*a^2 + b^2)*Cos[c + d*x] + 5*b*(-396*a^2 - 69*b^2 + 12*(33*a^2 + 4*b^2)*Cos[2*(c + d*x)] + 154*a*b*Cos[3*(c + d*x)] + 21*b^2*Cos[4*(c + d*x)]))*Sin[c + d*x]^(3/2)))/(d*Sin[c + d*x]^(5/2))","A",1
51,1,147,202,1.2001159,"\int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2} \, dx","Integrate[(a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(3/2),x]","\frac{(e \sin (c+d x))^{3/2} \left(-20 a \left(28 a^2+15 b^2\right) \cot (c+d x)-\frac{2}{3} b \csc (c+d x) \left(28 \left(27 a^2+4 b^2\right) \cos (2 (c+d x))-756 a^2+270 a b \cos (3 (c+d x))+35 b^2 \cos (4 (c+d x))-147 b^2\right)-\frac{80 a \left(7 a^2+6 b^2\right) F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)}{\sin ^{\frac{3}{2}}(c+d x)}\right)}{840 d}","\frac{2 a e^2 \left(7 a^2+6 b^2\right) \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 d \sqrt{e \sin (c+d x)}}+\frac{2 b \left(89 a^2+28 b^2\right) (e \sin (c+d x))^{5/2}}{315 d e}-\frac{2 a e \left(7 a^2+6 b^2\right) \cos (c+d x) \sqrt{e \sin (c+d x)}}{21 d}+\frac{2 b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))^2}{9 d e}+\frac{26 a b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}{63 d e}",1,"((-20*a*(28*a^2 + 15*b^2)*Cot[c + d*x] - (2*b*(-756*a^2 - 147*b^2 + 28*(27*a^2 + 4*b^2)*Cos[2*(c + d*x)] + 270*a*b*Cos[3*(c + d*x)] + 35*b^2*Cos[4*(c + d*x)])*Csc[c + d*x])/3 - (80*a*(7*a^2 + 6*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, 2])/Sin[c + d*x]^(3/2))*(e*Sin[c + d*x])^(3/2))/(840*d)","A",1
52,1,105,161,0.5512259,"\int (a+b \cos (c+d x))^3 \sqrt{e \sin (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^3*Sqrt[e*Sin[c + d*x]],x]","\frac{\sqrt{e \sin (c+d x)} \left(b \sin ^{\frac{3}{2}}(c+d x) \left(210 a^2+126 a b \cos (c+d x)+15 b^2 \cos (2 (c+d x))+55 b^2\right)-42 \left(5 a^3+6 a b^2\right) E\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)\right)}{105 d \sqrt{\sin (c+d x)}}","\frac{2 b \left(57 a^2+20 b^2\right) (e \sin (c+d x))^{3/2}}{105 d e}+\frac{2 a \left(5 a^2+6 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 d \sqrt{\sin (c+d x)}}+\frac{2 b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))^2}{7 d e}+\frac{22 a b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}{35 d e}",1,"(Sqrt[e*Sin[c + d*x]]*(-42*(5*a^3 + 6*a*b^2)*EllipticE[(-2*c + Pi - 2*d*x)/4, 2] + b*(210*a^2 + 55*b^2 + 126*a*b*Cos[c + d*x] + 15*b^2*Cos[2*(c + d*x)])*Sin[c + d*x]^(3/2)))/(105*d*Sqrt[Sin[c + d*x]])","A",1
53,1,98,157,0.7106741,"\int \frac{(a+b \cos (c+d x))^3}{\sqrt{e \sin (c+d x)}} \, dx","Integrate[(a + b*Cos[c + d*x])^3/Sqrt[e*Sin[c + d*x]],x]","\frac{b \sin (c+d x) \left(30 a^2+10 a b \cos (c+d x)+b^2 \cos (2 (c+d x))+9 b^2\right)-10 a \left(a^2+2 b^2\right) \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)}{5 d \sqrt{e \sin (c+d x)}}","\frac{2 b \left(11 a^2+4 b^2\right) \sqrt{e \sin (c+d x)}}{5 d e}+\frac{2 a \left(a^2+2 b^2\right) \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d \sqrt{e \sin (c+d x)}}+\frac{2 b \sqrt{e \sin (c+d x)} (a+b \cos (c+d x))^2}{5 d e}+\frac{6 a b \sqrt{e \sin (c+d x)} (a+b \cos (c+d x))}{5 d e}",1,"(-10*a*(a^2 + 2*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, 2]*Sqrt[Sin[c + d*x]] + b*(30*a^2 + 9*b^2 + 10*a*b*Cos[c + d*x] + b^2*Cos[2*(c + d*x)])*Sin[c + d*x])/(5*d*Sqrt[e*Sin[c + d*x]])","A",1
54,1,101,165,0.3188432,"\int \frac{(a+b \cos (c+d x))^3}{(e \sin (c+d x))^{3/2}} \, dx","Integrate[(a + b*Cos[c + d*x])^3/(e*Sin[c + d*x])^(3/2),x]","-\frac{2 \left(3 a \left(a^2+3 b^2\right) \cos (c+d x)-3 a \left(a^2+6 b^2\right) \sqrt{\sin (c+d x)} E\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)+9 a^2 b+b^3 \sin ^2(c+d x)+3 b^3\right)}{3 d e \sqrt{e \sin (c+d x)}}","-\frac{2 b \left(3 a^2+4 b^2\right) (e \sin (c+d x))^{3/2}}{3 d e^3}-\frac{2 a \left(a^2+6 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{d e^2 \sqrt{\sin (c+d x)}}-\frac{2 a b (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}{d e^3}-\frac{2 (a \cos (c+d x)+b) (a+b \cos (c+d x))^2}{d e \sqrt{e \sin (c+d x)}}",1,"(-2*(9*a^2*b + 3*b^3 + 3*a*(a^2 + 3*b^2)*Cos[c + d*x] - 3*a*(a^2 + 6*b^2)*EllipticE[(-2*c + Pi - 2*d*x)/4, 2]*Sqrt[Sin[c + d*x]] + b^3*Sin[c + d*x]^2))/(3*d*e*Sqrt[e*Sin[c + d*x]])","A",1
55,1,102,169,0.823978,"\int \frac{(a+b \cos (c+d x))^3}{(e \sin (c+d x))^{5/2}} \, dx","Integrate[(a + b*Cos[c + d*x])^3/(e*Sin[c + d*x])^(5/2),x]","-\frac{2 a \left(a^2+3 b^2\right) \cos (c+d x)+2 a \left(a^2-6 b^2\right) \sin ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)+6 a^2 b-3 b^3 \cos (2 (c+d x))+5 b^3}{3 d e (e \sin (c+d x))^{3/2}}","-\frac{2 b \left(a^2+4 b^2\right) \sqrt{e \sin (c+d x)}}{3 d e^3}+\frac{2 a \left(a^2-6 b^2\right) \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d e^2 \sqrt{e \sin (c+d x)}}-\frac{2 a b \sqrt{e \sin (c+d x)} (a+b \cos (c+d x))}{3 d e^3}-\frac{2 (a \cos (c+d x)+b) (a+b \cos (c+d x))^2}{3 d e (e \sin (c+d x))^{3/2}}",1,"-1/3*(6*a^2*b + 5*b^3 + 2*a*(a^2 + 3*b^2)*Cos[c + d*x] - 3*b^3*Cos[2*(c + d*x)] + 2*a*(a^2 - 6*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, 2]*Sin[c + d*x]^(3/2))/(d*e*(e*Sin[c + d*x])^(3/2))","A",1
56,1,130,192,0.6103567,"\int \frac{(a+b \cos (c+d x))^3}{(e \sin (c+d x))^{7/2}} \, dx","Integrate[(a + b*Cos[c + d*x])^3/(e*Sin[c + d*x])^(7/2),x]","-\frac{-3 a^3 \cos (3 (c+d x))+a \left(7 a^2+6 b^2\right) \cos (c+d x)-12 a \left(a^2-2 b^2\right) \sin ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)+12 a^2 b+6 a b^2 \cos (3 (c+d x))+10 b^3 \cos (2 (c+d x))-6 b^3}{10 d e (e \sin (c+d x))^{5/2}}","-\frac{2 b \left(3 a^2-4 b^2\right) (e \sin (c+d x))^{3/2}}{5 d e^5}-\frac{6 a \left(a^2-2 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 d e^4 \sqrt{\sin (c+d x)}}+\frac{2 \left(a b-\left(3 a^2-4 b^2\right) \cos (c+d x)\right) (a+b \cos (c+d x))}{5 d e^3 \sqrt{e \sin (c+d x)}}-\frac{2 (a \cos (c+d x)+b) (a+b \cos (c+d x))^2}{5 d e (e \sin (c+d x))^{5/2}}",1,"-1/10*(12*a^2*b - 6*b^3 + a*(7*a^2 + 6*b^2)*Cos[c + d*x] + 10*b^3*Cos[2*(c + d*x)] - 3*a^3*Cos[3*(c + d*x)] + 6*a*b^2*Cos[3*(c + d*x)] - 12*a*(a^2 - 2*b^2)*EllipticE[(-2*c + Pi - 2*d*x)/4, 2]*Sin[c + d*x]^(5/2))/(d*e*(e*Sin[c + d*x])^(5/2))","A",1
57,1,144,193,0.6222003,"\int \frac{(a+b \cos (c+d x))^3}{(e \sin (c+d x))^{9/2}} \, dx","Integrate[(a + b*Cos[c + d*x])^3/(e*Sin[c + d*x])^(9/2),x]","-\frac{2 \csc ^4(c+d x) \sqrt{e \sin (c+d x)} \left(a \left(5 a^2-6 b^2\right) \sin ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)+\frac{1}{4} \left(-5 a^3 \cos (3 (c+d x))+a \left(17 a^2+30 b^2\right) \cos (c+d x)+36 a^2 b+6 a b^2 \cos (3 (c+d x))+14 b^3 \cos (2 (c+d x))-2 b^3\right)\right)}{21 d e^5}","-\frac{2 b \left(5 a^2-4 b^2\right) \sqrt{e \sin (c+d x)}}{21 d e^5}+\frac{2 a \left(5 a^2-6 b^2\right) \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 d e^4 \sqrt{e \sin (c+d x)}}-\frac{2 \left(\left(5 a^2-4 b^2\right) \cos (c+d x)+a b\right) (a+b \cos (c+d x))}{21 d e^3 (e \sin (c+d x))^{3/2}}-\frac{2 (a \cos (c+d x)+b) (a+b \cos (c+d x))^2}{7 d e (e \sin (c+d x))^{7/2}}",1,"(-2*Csc[c + d*x]^4*Sqrt[e*Sin[c + d*x]]*((36*a^2*b - 2*b^3 + a*(17*a^2 + 30*b^2)*Cos[c + d*x] + 14*b^3*Cos[2*(c + d*x)] - 5*a^3*Cos[3*(c + d*x)] + 6*a*b^2*Cos[3*(c + d*x)])/4 + a*(5*a^2 - 6*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, 2]*Sin[c + d*x]^(7/2)))/(21*d*e^5)","A",1
58,1,2035,544,17.6283539,"\int \frac{(e \sin (c+d x))^{11/2}}{a+b \cos (c+d x)} \, dx","Integrate[(e*Sin[c + d*x])^(11/2)/(a + b*Cos[c + d*x]),x]","\text{Result too large to show}","\frac{e^{11/2} \left(b^2-a^2\right)^{9/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{11/2} d}+\frac{e^{11/2} \left(b^2-a^2\right)^{9/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{11/2} d}-\frac{a e^6 \left(a^2-b^2\right)^3 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{b^6 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \sin (c+d x)}}-\frac{a e^6 \left(a^2-b^2\right)^3 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{b^6 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \sin (c+d x)}}-\frac{2 e^5 \sqrt{e \sin (c+d x)} \left(21 \left(a^2-b^2\right)^2-a b \left(7 a^2-12 b^2\right) \cos (c+d x)\right)}{21 b^5 d}+\frac{2 e^3 (e \sin (c+d x))^{5/2} \left(7 \left(a^2-b^2\right)-5 a b \cos (c+d x)\right)}{35 b^3 d}+\frac{2 a e^6 \left(21 a^4-49 a^2 b^2+33 b^4\right) \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 b^6 d \sqrt{e \sin (c+d x)}}-\frac{2 e (e \sin (c+d x))^{9/2}}{9 b d}",1,"(((a*(28*a^2 - 51*b^2)*Cos[c + d*x])/(42*b^4) + ((-9*a^2 + 14*b^2)*Cos[2*(c + d*x)])/(45*b^3) + (a*Cos[3*(c + d*x)])/(14*b^2) - Cos[4*(c + d*x)]/(36*b))*Csc[c + d*x]^5*(e*Sin[c + d*x])^(11/2))/d - ((e*Sin[c + d*x])^(11/2)*((2*(392*a^3*b - 722*a*b^3)*Cos[c + d*x]^2*(a + b*Sqrt[1 - Sin[c + d*x]^2])*((a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)) + (5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(-280*a^4 + 636*a^2*b^2 - 721*b^4)*Cos[c + d*x]*(a + b*Sqrt[1 - Sin[c + d*x]^2])*(((-1/8 + I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(-a^2 + b^2)^(3/4) + (5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2]) + ((840*a^4 - 1764*a^2*b^2 + 959*b^4)*Cos[c + d*x]*Cos[2*(c + d*x)]*(a + b*Sqrt[1 - Sin[c + d*x]^2])*(((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + (4*Sqrt[Sin[c + d*x]])/b - (4*a*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(5/2))/(5*(a^2 - b^2)) + (10*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*(1 - 2*Sin[c + d*x]^2)*Sqrt[1 - Sin[c + d*x]^2])))/(1680*b^4*d*Sin[c + d*x]^(11/2))","C",0
59,1,834,461,14.9920431,"\int \frac{(e \sin (c+d x))^{9/2}}{a+b \cos (c+d x)} \, dx","Integrate[(e*Sin[c + d*x])^(9/2)/(a + b*Cos[c + d*x]),x]","\frac{\csc ^4(c+d x) (e \sin (c+d x))^{9/2} \left(-\frac{\left(37 b^2-28 a^2\right) \sin (c+d x)}{42 b^3}-\frac{a \sin (2 (c+d x))}{5 b^2}+\frac{\sin (3 (c+d x))}{14 b}\right)}{d}-\frac{(e \sin (c+d x))^{9/2} \left(\frac{\left(5 a^3-8 a b^2\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sin ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \sin (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \sin (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \cos ^2(c+d x)}{12 b^{3/2} \left(b^2-a^2\right) (a+b \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{2 \left(2 a^2 b-5 b^3\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sin ^{\frac{3}{2}}(c+d x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \sin (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \sin (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right)}{5 b^3 d \sin ^{\frac{9}{2}}(c+d x)}","-\frac{e^{9/2} \left(b^2-a^2\right)^{7/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{9/2} d}+\frac{e^{9/2} \left(b^2-a^2\right)^{7/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{9/2} d}+\frac{a e^5 \left(a^2-b^2\right)^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{b^5 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \sin (c+d x)}}+\frac{a e^5 \left(a^2-b^2\right)^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{b^5 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \sin (c+d x)}}-\frac{2 a e^4 \left(5 a^2-8 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 b^4 d \sqrt{\sin (c+d x)}}+\frac{2 e^3 (e \sin (c+d x))^{3/2} \left(5 \left(a^2-b^2\right)-3 a b \cos (c+d x)\right)}{15 b^3 d}-\frac{2 e (e \sin (c+d x))^{7/2}}{7 b d}",1,"-1/5*((e*Sin[c + d*x])^(9/2)*(((5*a^3 - 8*a*b^2)*Cos[c + d*x]^2*(3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]) + 8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/(12*b^(3/2)*(-a^2 + b^2)*(a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(2*a^2*b - 5*b^3)*Cos[c + d*x]*(((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)) + (a*AppellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))/(3*(a^2 - b^2)))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(b^3*d*Sin[c + d*x]^(9/2)) + (Csc[c + d*x]^4*(e*Sin[c + d*x])^(9/2)*(-1/42*((-28*a^2 + 37*b^2)*Sin[c + d*x])/b^3 - (a*Sin[2*(c + d*x)])/(5*b^2) + Sin[3*(c + d*x)]/(14*b)))/d","C",0
60,1,1955,474,15.5256693,"\int \frac{(e \sin (c+d x))^{7/2}}{a+b \cos (c+d x)} \, dx","Integrate[(e*Sin[c + d*x])^(7/2)/(a + b*Cos[c + d*x]),x]","\frac{\left(\frac{\cos (2 (c+d x))}{5 b}-\frac{2 a \cos (c+d x)}{3 b^2}\right) \csc ^3(c+d x) (e \sin (c+d x))^{7/2}}{d}+\frac{\left(\frac{28 a b \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \sqrt{1-\sin ^2(c+d x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \sin ^2(c+d x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\sin ^2(c+d x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \sin (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \sin (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \cos ^2(c+d x)}{(a+b \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{2 \left(27 b^2-10 a^2\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sqrt{\sin (c+d x)}}{\sqrt{1-\sin ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \sin ^2(c+d x)\right) \left(a^2+b^2 \left(\sin ^2(c+d x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \sin (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \sin (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}+\frac{\left(30 a^2-33 b^2\right) \cos (2 (c+d x)) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(-\frac{4 a F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sin ^{\frac{5}{2}}(c+d x)}{5 \left(a^2-b^2\right)}+\frac{4 \sqrt{\sin (c+d x)}}{b}+\frac{10 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sqrt{\sin (c+d x)}}{\sqrt{1-\sin ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \sin ^2(c+d x)\right) \left(a^2+b^2 \left(\sin ^2(c+d x)-1\right)\right)}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}+\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \sin (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \sin (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}\right) \cos (c+d x)}{(a+b \cos (c+d x)) \left(1-2 \sin ^2(c+d x)\right) \sqrt{1-\sin ^2(c+d x)}}\right) (e \sin (c+d x))^{7/2}}{60 b^2 d \sin ^{\frac{7}{2}}(c+d x)}","\frac{e^{7/2} \left(b^2-a^2\right)^{5/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{7/2} d}+\frac{e^{7/2} \left(b^2-a^2\right)^{5/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{7/2} d}-\frac{2 a e^4 \left(3 a^2-4 b^2\right) \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 b^4 d \sqrt{e \sin (c+d x)}}+\frac{a e^4 \left(a^2-b^2\right)^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{b^4 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \sin (c+d x)}}+\frac{a e^4 \left(a^2-b^2\right)^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{b^4 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \sin (c+d x)}}+\frac{2 e^3 \sqrt{e \sin (c+d x)} \left(3 \left(a^2-b^2\right)-a b \cos (c+d x)\right)}{3 b^3 d}-\frac{2 e (e \sin (c+d x))^{5/2}}{5 b d}",1,"(((-2*a*Cos[c + d*x])/(3*b^2) + Cos[2*(c + d*x)]/(5*b))*Csc[c + d*x]^3*(e*Sin[c + d*x])^(7/2))/d + ((e*Sin[c + d*x])^(7/2)*((28*a*b*Cos[c + d*x]^2*(a + b*Sqrt[1 - Sin[c + d*x]^2])*((a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)) + (5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(-10*a^2 + 27*b^2)*Cos[c + d*x]*(a + b*Sqrt[1 - Sin[c + d*x]^2])*(((-1/8 + I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(-a^2 + b^2)^(3/4) + (5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2]) + ((30*a^2 - 33*b^2)*Cos[c + d*x]*Cos[2*(c + d*x)]*(a + b*Sqrt[1 - Sin[c + d*x]^2])*(((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + (4*Sqrt[Sin[c + d*x]])/b - (4*a*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(5/2))/(5*(a^2 - b^2)) + (10*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*(1 - 2*Sin[c + d*x]^2)*Sqrt[1 - Sin[c + d*x]^2])))/(60*b^2*d*Sin[c + d*x]^(7/2))","C",0
61,1,757,399,14.5293165,"\int \frac{(e \sin (c+d x))^{5/2}}{a+b \cos (c+d x)} \, dx","Integrate[(e*Sin[c + d*x])^(5/2)/(a + b*Cos[c + d*x]),x]","-\frac{2 \csc (c+d x) (e \sin (c+d x))^{5/2}}{3 b d}+\frac{(e \sin (c+d x))^{5/2} \left(\frac{a \cos ^2(c+d x) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(8 b^{5/2} \sin ^{\frac{3}{2}}(c+d x) F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(-\log \left(-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}+b \sin (c+d x)\right)+\log \left(\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}+b \sin (c+d x)\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)\right)\right)}{12 b^{3/2} \left(b^2-a^2\right) \left(1-\sin ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{2 b \cos (c+d x) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(\frac{a \sin ^{\frac{3}{2}}(c+d x) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(-\log \left(-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}+i b \sin (c+d x)\right)+\log \left((1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}+i b \sin (c+d x)\right)+2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(1+\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{1-\sin ^2(c+d x)} (a+b \cos (c+d x))}\right)}{b d \sin ^{\frac{5}{2}}(c+d x)}","-\frac{e^{5/2} \left(b^2-a^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{5/2} d}+\frac{e^{5/2} \left(b^2-a^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{5/2} d}-\frac{a e^3 \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{b^3 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \sin (c+d x)}}-\frac{a e^3 \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{b^3 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \sin (c+d x)}}+\frac{2 a e^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{b^2 d \sqrt{\sin (c+d x)}}-\frac{2 e (e \sin (c+d x))^{3/2}}{3 b d}",1,"(-2*Csc[c + d*x]*(e*Sin[c + d*x])^(5/2))/(3*b*d) + ((e*Sin[c + d*x])^(5/2)*((a*Cos[c + d*x]^2*(3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]) + 8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/(12*b^(3/2)*(-a^2 + b^2)*(a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*b*Cos[c + d*x]*(((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)) + (a*AppellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))/(3*(a^2 - b^2)))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(b*d*Sin[c + d*x]^(5/2))","C",0
62,1,434,410,5.8179708,"\int \frac{(e \sin (c+d x))^{3/2}}{a+b \cos (c+d x)} \, dx","Integrate[(e*Sin[c + d*x])^(3/2)/(a + b*Cos[c + d*x]),x]","-\frac{\left(\frac{1}{20}-\frac{i}{20}\right) \cos (c+d x) (e \sin (c+d x))^{3/2} \left(a+b \sqrt{\cos ^2(c+d x)}\right) \left((4+4 i) a b^{3/2} \sin ^{\frac{5}{2}}(c+d x) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)-5 \left(a^2-b^2\right) \left(\sqrt[4]{b^2-a^2} \log \left(-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}+i b \sin (c+d x)\right)-\sqrt[4]{b^2-a^2} \log \left((1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}+i b \sin (c+d x)\right)+2 \sqrt[4]{b^2-a^2} \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \sqrt[4]{b^2-a^2} \tan ^{-1}\left(1+\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)+(4+4 i) \sqrt{b} \sqrt{\sin (c+d x)}\right)\right)}{b^{3/2} d \left(b^2-a^2\right) \sin ^{\frac{3}{2}}(c+d x) \sqrt{\cos ^2(c+d x)} (a+b \cos (c+d x))}","-\frac{a e^2 \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{b^2 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \sin (c+d x)}}-\frac{a e^2 \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{b^2 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \sin (c+d x)}}+\frac{e^{3/2} \sqrt[4]{b^2-a^2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{3/2} d}+\frac{e^{3/2} \sqrt[4]{b^2-a^2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{3/2} d}+\frac{2 a e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{b^2 d \sqrt{e \sin (c+d x)}}-\frac{2 e \sqrt{e \sin (c+d x)}}{b d}",1,"((-1/20 + I/20)*Cos[c + d*x]*(a + b*Sqrt[Cos[c + d*x]^2])*(e*Sin[c + d*x])^(3/2)*(-5*(a^2 - b^2)*(2*(-a^2 + b^2)^(1/4)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*(-a^2 + b^2)^(1/4)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + (-a^2 + b^2)^(1/4)*Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] - (-a^2 + b^2)^(1/4)*Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] + (4 + 4*I)*Sqrt[b]*Sqrt[Sin[c + d*x]]) + (4 + 4*I)*a*b^(3/2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(5/2)))/(b^(3/2)*(-a^2 + b^2)*d*Sqrt[Cos[c + d*x]^2]*(a + b*Cos[c + d*x])*Sin[c + d*x]^(3/2))","C",0
63,1,361,302,1.8015623,"\int \frac{\sqrt{e \sin (c+d x)}}{a+b \cos (c+d x)} \, dx","Integrate[Sqrt[e*Sin[c + d*x]]/(a + b*Cos[c + d*x]),x]","\frac{2 \cos (c+d x) \sqrt{e \sin (c+d x)} \left(a+b \sqrt{\cos ^2(c+d x)}\right) \left(\frac{a \sin ^{\frac{3}{2}}(c+d x) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(-\log \left(-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}+i b \sin (c+d x)\right)+\log \left((1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}+i b \sin (c+d x)\right)+2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(1+\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right)}{d \sqrt{\sin (c+d x)} \sqrt{\cos ^2(c+d x)} (a+b \cos (c+d x))}","-\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{b} d \sqrt[4]{b^2-a^2}}+\frac{\sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{b} d \sqrt[4]{b^2-a^2}}+\frac{a e \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{b d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \sin (c+d x)}}+\frac{a e \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{b d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \sin (c+d x)}}",1,"(2*Cos[c + d*x]*(a + b*Sqrt[Cos[c + d*x]^2])*Sqrt[e*Sin[c + d*x]]*(((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)) + (a*AppellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))/(3*(a^2 - b^2))))/(d*Sqrt[Cos[c + d*x]^2]*(a + b*Cos[c + d*x])*Sqrt[Sin[c + d*x]])","C",0
64,1,261,307,1.6046676,"\int \frac{1}{(a+b \cos (c+d x)) \sqrt{e \sin (c+d x)}} \, dx","Integrate[1/((a + b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]]),x]","\frac{10 (a+b) \sqrt{e \sin (c+d x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(b-a) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right)}{d e (a+b \cos (c+d x)) \left(2 \tan ^2\left(\frac{1}{2} (c+d x)\right) \left((a+b) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(b-a) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right)-2 (a-b) F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(b-a) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right)\right)+5 (a+b) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(b-a) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right)\right)}","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d \sqrt{e} \left(b^2-a^2\right)^{3/4}}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d \sqrt{e} \left(b^2-a^2\right)^{3/4}}+\frac{a \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \sin (c+d x)}}+\frac{a \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \sin (c+d x)}}",1,"(10*(a + b)*AppellF1[1/4, -1/2, 1, 5/4, -Tan[(c + d*x)/2]^2, ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]*Sqrt[e*Sin[c + d*x]])/(d*e*(a + b*Cos[c + d*x])*(5*(a + b)*AppellF1[1/4, -1/2, 1, 5/4, -Tan[(c + d*x)/2]^2, ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)] + 2*(-2*(a - b)*AppellF1[5/4, -1/2, 2, 9/4, -Tan[(c + d*x)/2]^2, ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*AppellF1[5/4, 1/2, 1, 9/4, -Tan[(c + d*x)/2]^2, ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])*Tan[(c + d*x)/2]^2))","C",0
65,1,791,426,14.7465316,"\int \frac{1}{(a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}} \, dx","Integrate[1/((a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2)),x]","-\frac{2 \sin (c+d x) (a \cos (c+d x)-b)}{d \left(a^2-b^2\right) (e \sin (c+d x))^{3/2}}-\frac{\sin ^{\frac{3}{2}}(c+d x) \left(\frac{a \cos ^2(c+d x) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(8 b^{5/2} \sin ^{\frac{3}{2}}(c+d x) F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(-\log \left(-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}+b \sin (c+d x)\right)+\log \left(\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}+b \sin (c+d x)\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)\right)\right)}{12 \sqrt{b} \left(b^2-a^2\right) \left(1-\sin ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{2 \left(a^2+b^2\right) \cos (c+d x) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(\frac{a \sin ^{\frac{3}{2}}(c+d x) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(-\log \left(-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}+i b \sin (c+d x)\right)+\log \left((1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}+i b \sin (c+d x)\right)+2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(1+\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{1-\sin ^2(c+d x)} (a+b \cos (c+d x))}\right)}{d (a-b) (a+b) (e \sin (c+d x))^{3/2}}","-\frac{2 a E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{d e^2 \left(a^2-b^2\right) \sqrt{\sin (c+d x)}}+\frac{2 (b-a \cos (c+d x))}{d e \left(a^2-b^2\right) \sqrt{e \sin (c+d x)}}-\frac{a b \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d e \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \sin (c+d x)}}-\frac{a b \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d e \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \sin (c+d x)}}-\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d e^{3/2} \left(b^2-a^2\right)^{5/4}}+\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d e^{3/2} \left(b^2-a^2\right)^{5/4}}",1,"(-2*(-b + a*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)*d*(e*Sin[c + d*x])^(3/2)) - (Sin[c + d*x]^(3/2)*((a*Cos[c + d*x]^2*(3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]) + 8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/(12*Sqrt[b]*(-a^2 + b^2)*(a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(a^2 + b^2)*Cos[c + d*x]*(((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)) + (a*AppellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))/(3*(a^2 - b^2)))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/((a - b)*(a + b)*d*(e*Sin[c + d*x])^(3/2))","C",0
66,1,1192,447,11.2515038,"\int \frac{1}{(a+b \cos (c+d x)) (e \sin (c+d x))^{5/2}} \, dx","Integrate[1/((a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2)),x]","\frac{\sin ^{\frac{5}{2}}(c+d x) \left(\frac{2 a b \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \sqrt{1-\sin ^2(c+d x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \sin ^2(c+d x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\sin ^2(c+d x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \sin (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \sin (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \cos ^2(c+d x)}{(a+b \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{2 \left(a^2-3 b^2\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sqrt{\sin (c+d x)}}{\sqrt{1-\sin ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \sin ^2(c+d x)\right) \left(a^2+b^2 \left(\sin ^2(c+d x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \sin (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \sin (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right)}{3 (a-b) (a+b) d (e \sin (c+d x))^{5/2}}-\frac{2 (a \cos (c+d x)-b) \sin (c+d x)}{3 \left(a^2-b^2\right) d (e \sin (c+d x))^{5/2}}","\frac{2 a \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d e^2 \left(a^2-b^2\right) \sqrt{e \sin (c+d x)}}-\frac{a b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d e^2 \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \sin (c+d x)}}-\frac{a b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d e^2 \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \sin (c+d x)}}+\frac{2 (b-a \cos (c+d x))}{3 d e \left(a^2-b^2\right) (e \sin (c+d x))^{3/2}}+\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d e^{5/2} \left(b^2-a^2\right)^{7/4}}+\frac{b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d e^{5/2} \left(b^2-a^2\right)^{7/4}}",1,"(-2*(-b + a*Cos[c + d*x])*Sin[c + d*x])/(3*(a^2 - b^2)*d*(e*Sin[c + d*x])^(5/2)) + (Sin[c + d*x]^(5/2)*((2*a*b*Cos[c + d*x]^2*(a + b*Sqrt[1 - Sin[c + d*x]^2])*((a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)) + (5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(a^2 - 3*b^2)*Cos[c + d*x]*(a + b*Sqrt[1 - Sin[c + d*x]^2])*(((-1/8 + I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(-a^2 + b^2)^(3/4) + (5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(3*(a - b)*(a + b)*d*(e*Sin[c + d*x])^(5/2))","C",0
67,1,881,501,6.5759983,"\int \frac{1}{(a+b \cos (c+d x)) (e \sin (c+d x))^{7/2}} \, dx","Integrate[1/((a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(7/2)),x]","\frac{\left(-\frac{2 (a \cos (c+d x)-b) \csc ^3(c+d x)}{5 \left(a^2-b^2\right)}-\frac{2 \left(3 \cos (c+d x) a^3-8 b^2 \cos (c+d x) a+5 b^3\right) \csc (c+d x)}{5 \left(a^2-b^2\right)^2}\right) \sin ^4(c+d x)}{d (e \sin (c+d x))^{7/2}}-\frac{\sin ^{\frac{7}{2}}(c+d x) \left(\frac{\left(3 a^3 b-8 a b^3\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sin ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \sin (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \sin (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \cos ^2(c+d x)}{12 b^{3/2} \left(b^2-a^2\right) (a+b \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{2 \left(3 a^4-8 b^2 a^2-5 b^4\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sin ^{\frac{3}{2}}(c+d x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \sin (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \sin (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right)}{5 (a-b)^2 (a+b)^2 d (e \sin (c+d x))^{7/2}}","-\frac{2 a \left(3 a^2-8 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 d e^4 \left(a^2-b^2\right)^2 \sqrt{\sin (c+d x)}}+\frac{2 (b-a \cos (c+d x))}{5 d e \left(a^2-b^2\right) (e \sin (c+d x))^{5/2}}-\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d e^{7/2} \left(b^2-a^2\right)^{9/4}}+\frac{b^{7/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d e^{7/2} \left(b^2-a^2\right)^{9/4}}-\frac{2 \left(a \left(3 a^2-8 b^2\right) \cos (c+d x)+5 b^3\right)}{5 d e^3 \left(a^2-b^2\right)^2 \sqrt{e \sin (c+d x)}}+\frac{a b^3 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d e^3 \left(a^2-b^2\right)^2 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \sin (c+d x)}}+\frac{a b^3 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d e^3 \left(a^2-b^2\right)^2 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \sin (c+d x)}}",1,"(((-2*(5*b^3 + 3*a^3*Cos[c + d*x] - 8*a*b^2*Cos[c + d*x])*Csc[c + d*x])/(5*(a^2 - b^2)^2) - (2*(-b + a*Cos[c + d*x])*Csc[c + d*x]^3)/(5*(a^2 - b^2)))*Sin[c + d*x]^4)/(d*(e*Sin[c + d*x])^(7/2)) - (Sin[c + d*x]^(7/2)*(((3*a^3*b - 8*a*b^3)*Cos[c + d*x]^2*(3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]) + 8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/(12*b^(3/2)*(-a^2 + b^2)*(a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(3*a^4 - 8*a^2*b^2 - 5*b^4)*Cos[c + d*x]*(((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)) + (a*AppellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))/(3*(a^2 - b^2)))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(5*(a - b)^2*(a + b)^2*d*(e*Sin[c + d*x])^(7/2))","C",0
68,1,2029,557,15.3651858,"\int \frac{(e \sin (c+d x))^{11/2}}{(a+b \cos (c+d x))^2} \, dx","Integrate[(e*Sin[c + d*x])^(11/2)/(a + b*Cos[c + d*x])^2,x]","\text{Result too large to show}","\frac{9 a e^{11/2} \left(b^2-a^2\right)^{5/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{11/2} d}+\frac{9 a e^{11/2} \left(b^2-a^2\right)^{5/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{11/2} d}+\frac{9 a^2 e^6 \left(a^2-b^2\right)^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{2 b^6 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \sin (c+d x)}}+\frac{9 a^2 e^6 \left(a^2-b^2\right)^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{2 b^6 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \sin (c+d x)}}+\frac{3 e^5 \sqrt{e \sin (c+d x)} \left(21 a \left(a^2-b^2\right)-b \left(7 a^2-5 b^2\right) \cos (c+d x)\right)}{7 b^5 d}-\frac{3 e^6 \left(21 a^4-28 a^2 b^2+5 b^4\right) \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{7 b^6 d \sqrt{e \sin (c+d x)}}-\frac{9 e^3 (e \sin (c+d x))^{5/2} (7 a-5 b \cos (c+d x))}{35 b^3 d}+\frac{e (e \sin (c+d x))^{9/2}}{b d (a+b \cos (c+d x))}",1,"((((-28*a^2 + 17*b^2)*Cos[c + d*x])/(14*b^4) + (-a^2 + b^2)^2/(b^5*(a + b*Cos[c + d*x])) + (2*a*Cos[2*(c + d*x)])/(5*b^3) - Cos[3*(c + d*x)]/(14*b^2))*Csc[c + d*x]^5*(e*Sin[c + d*x])^(11/2))/d - ((e*Sin[c + d*x])^(11/2)*((2*(35*a^4 - 126*a^2*b^2 + 75*b^4)*Cos[c + d*x]^2*(a + b*Sqrt[1 - Sin[c + d*x]^2])*((a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)) + (5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(70*a^3*b - 93*a*b^3)*Cos[c + d*x]*(a + b*Sqrt[1 - Sin[c + d*x]^2])*(((-1/8 + I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(-a^2 + b^2)^(3/4) + (5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2]) + ((-140*a^3*b + 147*a*b^3)*Cos[c + d*x]*Cos[2*(c + d*x)]*(a + b*Sqrt[1 - Sin[c + d*x]^2])*(((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + (4*Sqrt[Sin[c + d*x]])/b - (4*a*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(5/2))/(5*(a^2 - b^2)) + (10*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*(1 - 2*Sin[c + d*x]^2)*Sqrt[1 - Sin[c + d*x]^2])))/(70*b^5*d*Sin[c + d*x]^(11/2))","C",0
69,1,835,473,14.707719,"\int \frac{(e \sin (c+d x))^{9/2}}{(a+b \cos (c+d x))^2} \, dx","Integrate[(e*Sin[c + d*x])^(9/2)/(a + b*Cos[c + d*x])^2,x]","\frac{7 \left(\frac{\left(5 a^2-3 b^2\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sin ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \sin (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \sin (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \cos ^2(c+d x)}{12 b^{3/2} \left(b^2-a^2\right) (a+b \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{4 a b \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sin ^{\frac{3}{2}}(c+d x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \sin (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \sin (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right) (e \sin (c+d x))^{9/2}}{10 b^3 d \sin ^{\frac{9}{2}}(c+d x)}+\frac{\csc ^4(c+d x) \left(-\frac{4 a \sin (c+d x)}{3 b^3}+\frac{b^2 \sin (c+d x)-a^2 \sin (c+d x)}{b^3 (a+b \cos (c+d x))}+\frac{\sin (2 (c+d x))}{5 b^2}\right) (e \sin (c+d x))^{9/2}}{d}","-\frac{7 a e^{9/2} \left(b^2-a^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{9/2} d}+\frac{7 a e^{9/2} \left(b^2-a^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{9/2} d}-\frac{7 a^2 e^5 \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{2 b^5 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \sin (c+d x)}}-\frac{7 a^2 e^5 \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{2 b^5 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \sin (c+d x)}}+\frac{7 e^4 \left(5 a^2-3 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 b^4 d \sqrt{\sin (c+d x)}}-\frac{7 e^3 (e \sin (c+d x))^{3/2} (5 a-3 b \cos (c+d x))}{15 b^3 d}+\frac{e (e \sin (c+d x))^{7/2}}{b d (a+b \cos (c+d x))}",1,"(7*(e*Sin[c + d*x])^(9/2)*(((5*a^2 - 3*b^2)*Cos[c + d*x]^2*(3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]) + 8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/(12*b^(3/2)*(-a^2 + b^2)*(a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (4*a*b*Cos[c + d*x]*(((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)) + (a*AppellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))/(3*(a^2 - b^2)))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(10*b^3*d*Sin[c + d*x]^(9/2)) + (Csc[c + d*x]^4*(e*Sin[c + d*x])^(9/2)*((-4*a*Sin[c + d*x])/(3*b^3) + (-(a^2*Sin[c + d*x]) + b^2*Sin[c + d*x])/(b^3*(a + b*Cos[c + d*x])) + Sin[2*(c + d*x)]/(5*b^2)))/d","C",0
70,1,1956,487,14.5970238,"\int \frac{(e \sin (c+d x))^{7/2}}{(a+b \cos (c+d x))^2} \, dx","Integrate[(e*Sin[c + d*x])^(7/2)/(a + b*Cos[c + d*x])^2,x]","\frac{\left(\frac{b^2-a^2}{b^3 (a+b \cos (c+d x))}+\frac{2 \cos (c+d x)}{3 b^2}\right) \csc ^3(c+d x) (e \sin (c+d x))^{7/2}}{d}+\frac{\left(\frac{2 \left(3 a^2-5 b^2\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \sqrt{1-\sin ^2(c+d x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \sin ^2(c+d x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\sin ^2(c+d x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \sin (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \sin (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \cos ^2(c+d x)}{(a+b \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{8 a b \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sqrt{\sin (c+d x)}}{\sqrt{1-\sin ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \sin ^2(c+d x)\right) \left(a^2+b^2 \left(\sin ^2(c+d x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \sin (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \sin (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}-\frac{6 a b \cos (2 (c+d x)) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(-\frac{4 a F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sin ^{\frac{5}{2}}(c+d x)}{5 \left(a^2-b^2\right)}+\frac{4 \sqrt{\sin (c+d x)}}{b}+\frac{10 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sqrt{\sin (c+d x)}}{\sqrt{1-\sin ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \sin ^2(c+d x)\right) \left(a^2+b^2 \left(\sin ^2(c+d x)-1\right)\right)}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}+\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \sin (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \sin (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}\right) \cos (c+d x)}{(a+b \cos (c+d x)) \left(1-2 \sin ^2(c+d x)\right) \sqrt{1-\sin ^2(c+d x)}}\right) (e \sin (c+d x))^{7/2}}{6 b^3 d \sin ^{\frac{7}{2}}(c+d x)}","\frac{5 a e^{7/2} \sqrt[4]{b^2-a^2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{7/2} d}+\frac{5 a e^{7/2} \sqrt[4]{b^2-a^2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{7/2} d}+\frac{5 e^4 \left(3 a^2-b^2\right) \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 b^4 d \sqrt{e \sin (c+d x)}}-\frac{5 a^2 e^4 \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{2 b^4 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \sin (c+d x)}}-\frac{5 a^2 e^4 \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{2 b^4 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \sin (c+d x)}}-\frac{5 e^3 \sqrt{e \sin (c+d x)} (3 a-b \cos (c+d x))}{3 b^3 d}+\frac{e (e \sin (c+d x))^{5/2}}{b d (a+b \cos (c+d x))}",1,"(((2*Cos[c + d*x])/(3*b^2) + (-a^2 + b^2)/(b^3*(a + b*Cos[c + d*x])))*Csc[c + d*x]^3*(e*Sin[c + d*x])^(7/2))/d + ((e*Sin[c + d*x])^(7/2)*((2*(3*a^2 - 5*b^2)*Cos[c + d*x]^2*(a + b*Sqrt[1 - Sin[c + d*x]^2])*((a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)) + (5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (8*a*b*Cos[c + d*x]*(a + b*Sqrt[1 - Sin[c + d*x]^2])*(((-1/8 + I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(-a^2 + b^2)^(3/4) + (5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2]) - (6*a*b*Cos[c + d*x]*Cos[2*(c + d*x)]*(a + b*Sqrt[1 - Sin[c + d*x]^2])*(((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + (4*Sqrt[Sin[c + d*x]])/b - (4*a*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(5/2))/(5*(a^2 - b^2)) + (10*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*(1 - 2*Sin[c + d*x]^2)*Sqrt[1 - Sin[c + d*x]^2])))/(6*b^3*d*Sin[c + d*x]^(7/2))","C",0
71,1,366,404,20.2002152,"\int \frac{(e \sin (c+d x))^{5/2}}{(a+b \cos (c+d x))^2} \, dx","Integrate[(e*Sin[c + d*x])^(5/2)/(a + b*Cos[c + d*x])^2,x]","\frac{(e \sin (c+d x))^{5/2} \left(\frac{\left(a+b \sqrt{\cos ^2(c+d x)}\right) \left(8 b^{5/2} \sin ^{\frac{3}{2}}(c+d x) F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(-\log \left(-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}+b \sin (c+d x)\right)+\log \left(\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}+b \sin (c+d x)\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)\right)\right)}{\left(a^2-b^2\right) \sin ^{\frac{5}{2}}(c+d x)}+8 b^{3/2} \csc (c+d x)\right)}{8 b^{5/2} d (a+b \cos (c+d x))}","-\frac{3 a e^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{5/2} d \sqrt[4]{b^2-a^2}}+\frac{3 a e^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{5/2} d \sqrt[4]{b^2-a^2}}+\frac{3 a^2 e^3 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{2 b^3 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \sin (c+d x)}}+\frac{3 a^2 e^3 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{2 b^3 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \sin (c+d x)}}+\frac{e (e \sin (c+d x))^{3/2}}{b d (a+b \cos (c+d x))}-\frac{3 e^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{b^2 d \sqrt{\sin (c+d x)}}",1,"((e*Sin[c + d*x])^(5/2)*(8*b^(3/2)*Csc[c + d*x] + ((a + b*Sqrt[Cos[c + d*x]^2])*(3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]) + 8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2)))/((a^2 - b^2)*Sin[c + d*x]^(5/2))))/(8*b^(5/2)*d*(a + b*Cos[c + d*x]))","C",0
72,1,614,418,9.2213167,"\int \frac{(e \sin (c+d x))^{3/2}}{(a+b \cos (c+d x))^2} \, dx","Integrate[(e*Sin[c + d*x])^(3/2)/(a + b*Cos[c + d*x])^2,x]","\frac{\csc (c+d x) (e \sin (c+d x))^{3/2}}{b d (a+b \cos (c+d x))}-\frac{\cos ^2(c+d x) (e \sin (c+d x))^{3/2} \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \sqrt{1-\sin ^2(c+d x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)}{\left(a^2+b^2 \left(\sin ^2(c+d x)-1\right)\right) \left(2 \sin ^2(c+d x) \left(2 b^2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right)}+\frac{a \left(-\log \left(-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}+b \sin (c+d x)\right)+\log \left(\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}+b \sin (c+d x)\right)-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right)}{b d \sin ^{\frac{3}{2}}(c+d x) \left(1-\sin ^2(c+d x)\right) (a+b \cos (c+d x))}","\frac{a^2 e^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{2 b^2 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \sin (c+d x)}}+\frac{a^2 e^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{2 b^2 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \sin (c+d x)}}+\frac{a e^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{3/2} d \left(b^2-a^2\right)^{3/4}}+\frac{a e^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{3/2} d \left(b^2-a^2\right)^{3/4}}+\frac{e \sqrt{e \sin (c+d x)}}{b d (a+b \cos (c+d x))}-\frac{e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{b^2 d \sqrt{e \sin (c+d x)}}",1,"(Csc[c + d*x]*(e*Sin[c + d*x])^(3/2))/(b*d*(a + b*Cos[c + d*x])) - (Cos[c + d*x]^2*(e*Sin[c + d*x])^(3/2)*(a + b*Sqrt[1 - Sin[c + d*x]^2])*((a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)) + (5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/(b*d*(a + b*Cos[c + d*x])*Sin[c + d*x]^(3/2)*(1 - Sin[c + d*x]^2))","C",0
73,1,786,438,13.9382404,"\int \frac{\sqrt{e \sin (c+d x)}}{(a+b \cos (c+d x))^2} \, dx","Integrate[Sqrt[e*Sin[c + d*x]]/(a + b*Cos[c + d*x])^2,x]","\frac{b \sin (c+d x) \sqrt{e \sin (c+d x)}}{d \left(b^2-a^2\right) (a+b \cos (c+d x))}+\frac{\sqrt{e \sin (c+d x)} \left(\frac{\cos ^2(c+d x) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(8 b^{5/2} \sin ^{\frac{3}{2}}(c+d x) F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(-\log \left(-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}+b \sin (c+d x)\right)+\log \left(\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}+b \sin (c+d x)\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)\right)\right)}{12 \sqrt{b} \left(b^2-a^2\right) \left(1-\sin ^2(c+d x)\right) (a+b \cos (c+d x))}+\frac{4 a \cos (c+d x) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(\frac{a \sin ^{\frac{3}{2}}(c+d x) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(-\log \left(-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}+i b \sin (c+d x)\right)+\log \left((1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}+i b \sin (c+d x)\right)+2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(1+\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{1-\sin ^2(c+d x)} (a+b \cos (c+d x))}\right)}{2 d (a-b) (a+b) \sqrt{\sin (c+d x)}}","\frac{a \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 \sqrt{b} d \left(b^2-a^2\right)^{5/4}}-\frac{a \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 \sqrt{b} d \left(b^2-a^2\right)^{5/4}}-\frac{b (e \sin (c+d x))^{3/2}}{d e \left(a^2-b^2\right) (a+b \cos (c+d x))}+\frac{E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{d \left(a^2-b^2\right) \sqrt{\sin (c+d x)}}+\frac{a^2 e \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{2 b d \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \sin (c+d x)}}+\frac{a^2 e \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{2 b d \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \sin (c+d x)}}",1,"(b*Sin[c + d*x]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)*d*(a + b*Cos[c + d*x])) + (Sqrt[e*Sin[c + d*x]]*((Cos[c + d*x]^2*(3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]) + 8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/(12*Sqrt[b]*(-a^2 + b^2)*(a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (4*a*Cos[c + d*x]*(((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)) + (a*AppellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))/(3*(a^2 - b^2)))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(2*(a - b)*(a + b)*d*Sqrt[Sin[c + d*x]])","C",0
74,1,1182,445,10.1215685,"\int \frac{1}{(a+b \cos (c+d x))^2 \sqrt{e \sin (c+d x)}} \, dx","Integrate[1/((a + b*Cos[c + d*x])^2*Sqrt[e*Sin[c + d*x]]),x]","\frac{\sqrt{\sin (c+d x)} \left(\frac{4 a \cos (c+d x) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sqrt{\sin (c+d x)}}{\sqrt{1-\sin ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \sin ^2(c+d x)\right) \left(a^2+b^2 \left(\sin ^2(c+d x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \sin (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \sin (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right)}{(a+b \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}-\frac{2 b \cos ^2(c+d x) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \sqrt{1-\sin ^2(c+d x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \sin ^2(c+d x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\sin ^2(c+d x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \sin (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \sin (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right)}{(a+b \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}\right)}{2 (a-b) (a+b) d \sqrt{e \sin (c+d x)}}-\frac{b \sin (c+d x)}{\left(a^2-b^2\right) d (a+b \cos (c+d x)) \sqrt{e \sin (c+d x)}}","-\frac{3 a \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d \sqrt{e} \left(b^2-a^2\right)^{7/4}}-\frac{3 a \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d \sqrt{e} \left(b^2-a^2\right)^{7/4}}-\frac{b \sqrt{e \sin (c+d x)}}{d e \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{\sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d \left(a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{3 a^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{2 d \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \sin (c+d x)}}+\frac{3 a^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{2 d \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \sin (c+d x)}}",1,"-((b*Sin[c + d*x])/((a^2 - b^2)*d*(a + b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])) + (Sqrt[Sin[c + d*x]]*((-2*b*Cos[c + d*x]^2*(a + b*Sqrt[1 - Sin[c + d*x]^2])*((a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)) + (5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (4*a*Cos[c + d*x]*(a + b*Sqrt[1 - Sin[c + d*x]^2])*(((-1/8 + I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(-a^2 + b^2)^(3/4) + (5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(2*(a - b)*(a + b)*d*Sqrt[e*Sin[c + d*x]])","C",0
75,1,865,507,6.4856269,"\int \frac{1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx","Integrate[1/((a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(3/2)),x]","\frac{\sin ^2(c+d x) \left(\frac{b^3 \sin (c+d x)}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{2 \left(\cos (c+d x) a^2-2 b a+b^2 \cos (c+d x)\right) \csc (c+d x)}{\left(a^2-b^2\right)^2}\right)}{d (e \sin (c+d x))^{3/2}}-\frac{\sin ^{\frac{3}{2}}(c+d x) \left(\frac{\left(3 b^3+2 a^2 b\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sin ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \sin (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \sin (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \cos ^2(c+d x)}{12 b^{3/2} \left(b^2-a^2\right) (a+b \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{2 \left(2 a^3+8 b^2 a\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sin ^{\frac{3}{2}}(c+d x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \sin (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \sin (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right)}{2 (a-b)^2 (a+b)^2 d (e \sin (c+d x))^{3/2}}","-\frac{\left(2 a^2+3 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{d e^2 \left(a^2-b^2\right)^2 \sqrt{\sin (c+d x)}}+\frac{5 a b-\left(2 a^2+3 b^2\right) \cos (c+d x)}{d e \left(a^2-b^2\right)^2 \sqrt{e \sin (c+d x)}}-\frac{b}{d e \left(a^2-b^2\right) \sqrt{e \sin (c+d x)} (a+b \cos (c+d x))}-\frac{5 a^2 b \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{2 d e \left(a^2-b^2\right)^2 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \sin (c+d x)}}-\frac{5 a^2 b \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{2 d e \left(a^2-b^2\right)^2 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \sin (c+d x)}}+\frac{5 a b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d e^{3/2} \left(b^2-a^2\right)^{9/4}}-\frac{5 a b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d e^{3/2} \left(b^2-a^2\right)^{9/4}}",1,"(Sin[c + d*x]^2*((-2*(-2*a*b + a^2*Cos[c + d*x] + b^2*Cos[c + d*x])*Csc[c + d*x])/(a^2 - b^2)^2 + (b^3*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/(d*(e*Sin[c + d*x])^(3/2)) - (Sin[c + d*x]^(3/2)*(((2*a^2*b + 3*b^3)*Cos[c + d*x]^2*(3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]) + 8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/(12*b^(3/2)*(-a^2 + b^2)*(a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(2*a^3 + 8*a*b^2)*Cos[c + d*x]*(((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)) + (a*AppellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))/(3*(a^2 - b^2)))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(2*(a - b)^2*(a + b)^2*d*(e*Sin[c + d*x])^(3/2))","C",0
76,1,1257,530,13.1686418,"\int \frac{1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2}} \, dx","Integrate[1/((a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(5/2)),x]","\frac{\left(\frac{b^3}{\left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{2 \left(\cos (c+d x) a^2-2 b a+b^2 \cos (c+d x)\right) \csc ^2(c+d x)}{3 \left(a^2-b^2\right)^2}\right) \sin ^3(c+d x)}{d (e \sin (c+d x))^{5/2}}+\frac{\left(\frac{2 \left(5 b^3+2 a^2 b\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \sqrt{1-\sin ^2(c+d x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \sin ^2(c+d x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\sin ^2(c+d x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \sin (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \sin (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \cos ^2(c+d x)}{(a+b \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{2 \left(2 a^3-16 a b^2\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sqrt{\sin (c+d x)}}{\sqrt{1-\sin ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \sin ^2(c+d x)\right) \left(a^2+b^2 \left(\sin ^2(c+d x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \sin (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \sin (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right) \sin ^{\frac{5}{2}}(c+d x)}{6 (a-b)^2 (a+b)^2 d (e \sin (c+d x))^{5/2}}","\frac{\left(2 a^2+5 b^2\right) \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d e^2 \left(a^2-b^2\right)^2 \sqrt{e \sin (c+d x)}}-\frac{7 a^2 b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{2 d e^2 \left(a^2-b^2\right)^2 \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \sin (c+d x)}}-\frac{7 a^2 b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{2 d e^2 \left(a^2-b^2\right)^2 \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \sin (c+d x)}}-\frac{b}{d e \left(a^2-b^2\right) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}+\frac{7 a b-\left(2 a^2+5 b^2\right) \cos (c+d x)}{3 d e \left(a^2-b^2\right)^2 (e \sin (c+d x))^{3/2}}-\frac{7 a b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d e^{5/2} \left(b^2-a^2\right)^{11/4}}-\frac{7 a b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d e^{5/2} \left(b^2-a^2\right)^{11/4}}",1,"((b^3/((a^2 - b^2)^2*(a + b*Cos[c + d*x])) - (2*(-2*a*b + a^2*Cos[c + d*x] + b^2*Cos[c + d*x])*Csc[c + d*x]^2)/(3*(a^2 - b^2)^2))*Sin[c + d*x]^3)/(d*(e*Sin[c + d*x])^(5/2)) + (Sin[c + d*x]^(5/2)*((2*(2*a^2*b + 5*b^3)*Cos[c + d*x]^2*(a + b*Sqrt[1 - Sin[c + d*x]^2])*((a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)) + (5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(2*a^3 - 16*a*b^2)*Cos[c + d*x]*(a + b*Sqrt[1 - Sin[c + d*x]^2])*(((-1/8 + I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(-a^2 + b^2)^(3/4) + (5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(6*(a - b)^2*(a + b)^2*d*(e*Sin[c + d*x])^(5/2))","C",0
77,1,950,590,6.6094699,"\int \frac{1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{7/2}} \, dx","Integrate[1/((a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(7/2)),x]","\frac{\sin ^4(c+d x) \left(-\frac{\sin (c+d x) b^5}{\left(a^2-b^2\right)^3 (a+b \cos (c+d x))}-\frac{2 \left(\cos (c+d x) a^2-2 b a+b^2 \cos (c+d x)\right) \csc ^3(c+d x)}{5 \left(a^2-b^2\right)^2}-\frac{2 \left(3 \cos (c+d x) a^4-15 b^2 \cos (c+d x) a^2+20 b^3 a-8 b^4 \cos (c+d x)\right) \csc (c+d x)}{5 \left(a^2-b^2\right)^3}\right)}{d (e \sin (c+d x))^{7/2}}-\frac{3 \sin ^{\frac{7}{2}}(c+d x) \left(\frac{\left(-7 b^5-10 a^2 b^3+2 a^4 b\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sin ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \sin (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \sin (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \cos ^2(c+d x)}{12 b^{3/2} \left(b^2-a^2\right) (a+b \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{2 \left(2 a^5-10 b^2 a^3-22 b^4 a\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sin ^{\frac{3}{2}}(c+d x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \sin (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \sin (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right)}{10 (a-b)^3 (a+b)^3 d (e \sin (c+d x))^{7/2}}","-\frac{b}{d e \left(a^2-b^2\right) (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}+\frac{9 a b-\left(2 a^2+7 b^2\right) \cos (c+d x)}{5 d e \left(a^2-b^2\right)^2 (e \sin (c+d x))^{5/2}}+\frac{9 a b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d e^{7/2} \left(b^2-a^2\right)^{13/4}}-\frac{9 a b^{7/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d e^{7/2} \left(b^2-a^2\right)^{13/4}}+\frac{9 a^2 b^3 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{2 d e^3 \left(a^2-b^2\right)^3 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \sin (c+d x)}}+\frac{9 a^2 b^3 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{2 d e^3 \left(a^2-b^2\right)^3 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \sin (c+d x)}}-\frac{3 \left(2 a^4-10 a^2 b^2-7 b^4\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 d e^4 \left(a^2-b^2\right)^3 \sqrt{\sin (c+d x)}}-\frac{3 \left(\left(2 a^4-10 a^2 b^2-7 b^4\right) \cos (c+d x)+15 a b^3\right)}{5 d e^3 \left(a^2-b^2\right)^3 \sqrt{e \sin (c+d x)}}",1,"(Sin[c + d*x]^4*((-2*(20*a*b^3 + 3*a^4*Cos[c + d*x] - 15*a^2*b^2*Cos[c + d*x] - 8*b^4*Cos[c + d*x])*Csc[c + d*x])/(5*(a^2 - b^2)^3) - (2*(-2*a*b + a^2*Cos[c + d*x] + b^2*Cos[c + d*x])*Csc[c + d*x]^3)/(5*(a^2 - b^2)^2) - (b^5*Sin[c + d*x])/((a^2 - b^2)^3*(a + b*Cos[c + d*x]))))/(d*(e*Sin[c + d*x])^(7/2)) - (3*Sin[c + d*x]^(7/2)*(((2*a^4*b - 10*a^2*b^3 - 7*b^5)*Cos[c + d*x]^2*(3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]) + 8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/(12*b^(3/2)*(-a^2 + b^2)*(a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(2*a^5 - 10*a^3*b^2 - 22*a*b^4)*Cos[c + d*x]*(((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)) + (a*AppellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))/(3*(a^2 - b^2)))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(10*(a - b)^3*(a + b)^3*d*(e*Sin[c + d*x])^(7/2))","C",0
78,1,930,590,15.0223612,"\int \frac{(e \sin (c+d x))^{13/2}}{(a+b \cos (c+d x))^3} \, dx","Integrate[(e*Sin[c + d*x])^(13/2)/(a + b*Cos[c + d*x])^3,x]","\frac{11 \left(\frac{\left(45 a^3-37 a b^2\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sin ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \sin (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \sin (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \cos ^2(c+d x)}{12 b^{3/2} \left(b^2-a^2\right) (a+b \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{2 \left(18 a^2 b-10 b^3\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sin ^{\frac{3}{2}}(c+d x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \sin (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \sin (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right) (e \sin (c+d x))^{13/2}}{40 b^5 d \sin ^{\frac{13}{2}}(c+d x)}+\frac{\csc ^6(c+d x) \left(\frac{\left(65 b^2-168 a^2\right) \sin (c+d x)}{42 b^5}-\frac{19 \left(a^3 \sin (c+d x)-a b^2 \sin (c+d x)\right)}{4 b^5 (a+b \cos (c+d x))}+\frac{\sin (c+d x) a^4-2 b^2 \sin (c+d x) a^2+b^4 \sin (c+d x)}{2 b^5 (a+b \cos (c+d x))^2}+\frac{3 a \sin (2 (c+d x))}{5 b^4}-\frac{\sin (3 (c+d x))}{14 b^3}\right) (e \sin (c+d x))^{13/2}}{d}","\frac{11 a e^6 \left(45 a^2-37 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{20 b^6 d \sqrt{\sin (c+d x)}}-\frac{11 e^5 (e \sin (c+d x))^{3/2} \left(5 \left(9 a^2-2 b^2\right)-27 a b \cos (c+d x)\right)}{60 b^5 d}+\frac{11 e^{13/2} \left(9 a^4-11 a^2 b^2+2 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{13/2} d \sqrt[4]{b^2-a^2}}-\frac{11 e^{13/2} \left(9 a^4-11 a^2 b^2+2 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{13/2} d \sqrt[4]{b^2-a^2}}-\frac{11 a e^7 \left(9 a^4-11 a^2 b^2+2 b^4\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{8 b^7 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \sin (c+d x)}}-\frac{11 a e^7 \left(9 a^4-11 a^2 b^2+2 b^4\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{8 b^7 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \sin (c+d x)}}+\frac{11 e^3 (e \sin (c+d x))^{7/2} (9 a+2 b \cos (c+d x))}{28 b^3 d (a+b \cos (c+d x))}+\frac{e (e \sin (c+d x))^{11/2}}{2 b d (a+b \cos (c+d x))^2}",1,"(11*(e*Sin[c + d*x])^(13/2)*(((45*a^3 - 37*a*b^2)*Cos[c + d*x]^2*(3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]) + 8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/(12*b^(3/2)*(-a^2 + b^2)*(a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(18*a^2*b - 10*b^3)*Cos[c + d*x]*(((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)) + (a*AppellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))/(3*(a^2 - b^2)))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(40*b^5*d*Sin[c + d*x]^(13/2)) + (Csc[c + d*x]^6*(e*Sin[c + d*x])^(13/2)*(((-168*a^2 + 65*b^2)*Sin[c + d*x])/(42*b^5) - (19*(a^3*Sin[c + d*x] - a*b^2*Sin[c + d*x]))/(4*b^5*(a + b*Cos[c + d*x])) + (a^4*Sin[c + d*x] - 2*a^2*b^2*Sin[c + d*x] + b^4*Sin[c + d*x])/(2*b^5*(a + b*Cos[c + d*x])^2) + (3*a*Sin[2*(c + d*x)])/(5*b^4) - Sin[3*(c + d*x)]/(14*b^3)))/d","C",0
79,1,2024,604,14.5767517,"\int \frac{(e \sin (c+d x))^{11/2}}{(a+b \cos (c+d x))^3} \, dx","Integrate[(e*Sin[c + d*x])^(11/2)/(a + b*Cos[c + d*x])^3,x]","\text{Result too large to show}","\frac{3 a e^6 \left(21 a^2-13 b^2\right) \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{4 b^6 d \sqrt{e \sin (c+d x)}}-\frac{3 e^5 \sqrt{e \sin (c+d x)} \left(3 \left(7 a^2-2 b^2\right)-7 a b \cos (c+d x)\right)}{4 b^5 d}-\frac{9 e^{11/2} \left(7 a^4-9 a^2 b^2+2 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{11/2} d \left(b^2-a^2\right)^{3/4}}-\frac{9 e^{11/2} \left(7 a^4-9 a^2 b^2+2 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{11/2} d \left(b^2-a^2\right)^{3/4}}-\frac{9 a e^6 \left(7 a^4-9 a^2 b^2+2 b^4\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{8 b^6 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \sin (c+d x)}}-\frac{9 a e^6 \left(7 a^4-9 a^2 b^2+2 b^4\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{8 b^6 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \sin (c+d x)}}+\frac{9 e^3 (e \sin (c+d x))^{5/2} (7 a+2 b \cos (c+d x))}{20 b^3 d (a+b \cos (c+d x))}+\frac{e (e \sin (c+d x))^{9/2}}{2 b d (a+b \cos (c+d x))^2}",1,"(((2*a*Cos[c + d*x])/b^4 + (-a^2 + b^2)^2/(2*b^5*(a + b*Cos[c + d*x])^2) - (17*a*(a^2 - b^2))/(4*b^5*(a + b*Cos[c + d*x])) - Cos[2*(c + d*x)]/(5*b^3))*Csc[c + d*x]^5*(e*Sin[c + d*x])^(11/2))/d + (3*(e*Sin[c + d*x])^(11/2)*((2*(25*a^3 - 37*a*b^2)*Cos[c + d*x]^2*(a + b*Sqrt[1 - Sin[c + d*x]^2])*((a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)) + (5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(30*a^2*b - 16*b^3)*Cos[c + d*x]*(a + b*Sqrt[1 - Sin[c + d*x]^2])*(((-1/8 + I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(-a^2 + b^2)^(3/4) + (5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2]) + ((-40*a^2*b + 14*b^3)*Cos[c + d*x]*Cos[2*(c + d*x)]*(a + b*Sqrt[1 - Sin[c + d*x]^2])*(((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + (4*Sqrt[Sin[c + d*x]])/b - (4*a*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(5/2))/(5*(a^2 - b^2)) + (10*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*(1 - 2*Sin[c + d*x]^2)*Sqrt[1 - Sin[c + d*x]^2])))/(40*b^5*d*Sin[c + d*x]^(11/2))","C",0
80,1,837,498,14.3629523,"\int \frac{(e \sin (c+d x))^{9/2}}{(a+b \cos (c+d x))^3} \, dx","Integrate[(e*Sin[c + d*x])^(9/2)/(a + b*Cos[c + d*x])^3,x]","\frac{\csc ^4(c+d x) (e \sin (c+d x))^{9/2} \left(\frac{11 a \sin (c+d x)}{4 b^3 (a+b \cos (c+d x))}+\frac{2 \sin (c+d x)}{3 b^3}+\frac{b^2 \sin (c+d x)-a^2 \sin (c+d x)}{2 b^3 (a+b \cos (c+d x))^2}\right)}{d}-\frac{7 (e \sin (c+d x))^{9/2} \left(\frac{5 a \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sin ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \sin (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \sin (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \cos ^2(c+d x)}{12 b^{3/2} \left(b^2-a^2\right) (a+b \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{4 b \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sin ^{\frac{3}{2}}(c+d x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \sin (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \sin (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right)}{8 b^3 d \sin ^{\frac{9}{2}}(c+d x)}","-\frac{7 e^{9/2} \left(5 a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{9/2} d \sqrt[4]{b^2-a^2}}+\frac{7 e^{9/2} \left(5 a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{9/2} d \sqrt[4]{b^2-a^2}}+\frac{7 a e^5 \left(5 a^2-2 b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{8 b^5 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \sin (c+d x)}}+\frac{7 a e^5 \left(5 a^2-2 b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{8 b^5 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \sin (c+d x)}}-\frac{35 a e^4 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{4 b^4 d \sqrt{\sin (c+d x)}}+\frac{7 e^3 (e \sin (c+d x))^{3/2} (5 a+2 b \cos (c+d x))}{12 b^3 d (a+b \cos (c+d x))}+\frac{e (e \sin (c+d x))^{7/2}}{2 b d (a+b \cos (c+d x))^2}",1,"(Csc[c + d*x]^4*(e*Sin[c + d*x])^(9/2)*((2*Sin[c + d*x])/(3*b^3) + (11*a*Sin[c + d*x])/(4*b^3*(a + b*Cos[c + d*x])) + (-(a^2*Sin[c + d*x]) + b^2*Sin[c + d*x])/(2*b^3*(a + b*Cos[c + d*x])^2)))/d - (7*(e*Sin[c + d*x])^(9/2)*((5*a*Cos[c + d*x]^2*(3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]) + 8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/(12*b^(3/2)*(-a^2 + b^2)*(a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (4*b*Cos[c + d*x]*(((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)) + (a*AppellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))/(3*(a^2 - b^2)))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(8*b^3*d*Sin[c + d*x]^(9/2))","C",0
81,1,1954,512,14.5558055,"\int \frac{(e \sin (c+d x))^{7/2}}{(a+b \cos (c+d x))^3} \, dx","Integrate[(e*Sin[c + d*x])^(7/2)/(a + b*Cos[c + d*x])^3,x]","\frac{\left(\frac{9 a}{4 b^3 (a+b \cos (c+d x))}+\frac{b^2-a^2}{2 b^3 (a+b \cos (c+d x))^2}\right) \csc ^3(c+d x) (e \sin (c+d x))^{7/2}}{d}-\frac{(e \sin (c+d x))^{7/2} \left(\frac{14 a \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \sqrt{1-\sin ^2(c+d x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \sin ^2(c+d x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\sin ^2(c+d x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \sin (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \sin (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \cos ^2(c+d x)}{(a+b \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{12 b \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sqrt{\sin (c+d x)}}{\sqrt{1-\sin ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \sin ^2(c+d x)\right) \left(a^2+b^2 \left(\sin ^2(c+d x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \sin (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \sin (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}-\frac{4 b \cos (2 (c+d x)) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(-\frac{4 a F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sin ^{\frac{5}{2}}(c+d x)}{5 \left(a^2-b^2\right)}+\frac{4 \sqrt{\sin (c+d x)}}{b}+\frac{10 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sqrt{\sin (c+d x)}}{\sqrt{1-\sin ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \sin ^2(c+d x)\right) \left(a^2+b^2 \left(\sin ^2(c+d x)-1\right)\right)}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}+\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \sin (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \sin (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}\right) \cos (c+d x)}{(a+b \cos (c+d x)) \left(1-2 \sin ^2(c+d x)\right) \sqrt{1-\sin ^2(c+d x)}}\right)}{8 b^3 d \sin ^{\frac{7}{2}}(c+d x)}","\frac{5 e^{7/2} \left(3 a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{7/2} d \left(b^2-a^2\right)^{3/4}}+\frac{5 e^{7/2} \left(3 a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{7/2} d \left(b^2-a^2\right)^{3/4}}+\frac{5 a e^4 \left(3 a^2-2 b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{8 b^4 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \sin (c+d x)}}+\frac{5 a e^4 \left(3 a^2-2 b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{8 b^4 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \sin (c+d x)}}-\frac{15 a e^4 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{4 b^4 d \sqrt{e \sin (c+d x)}}+\frac{5 e^3 \sqrt{e \sin (c+d x)} (3 a+2 b \cos (c+d x))}{4 b^3 d (a+b \cos (c+d x))}+\frac{e (e \sin (c+d x))^{5/2}}{2 b d (a+b \cos (c+d x))^2}",1,"(((-a^2 + b^2)/(2*b^3*(a + b*Cos[c + d*x])^2) + (9*a)/(4*b^3*(a + b*Cos[c + d*x])))*Csc[c + d*x]^3*(e*Sin[c + d*x])^(7/2))/d - ((e*Sin[c + d*x])^(7/2)*((14*a*Cos[c + d*x]^2*(a + b*Sqrt[1 - Sin[c + d*x]^2])*((a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)) + (5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (12*b*Cos[c + d*x]*(a + b*Sqrt[1 - Sin[c + d*x]^2])*(((-1/8 + I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(-a^2 + b^2)^(3/4) + (5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2]) - (4*b*Cos[c + d*x]*Cos[2*(c + d*x)]*(a + b*Sqrt[1 - Sin[c + d*x]^2])*(((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + (4*Sqrt[Sin[c + d*x]])/b - (4*a*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(5/2))/(5*(a^2 - b^2)) + (10*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*(1 - 2*Sin[c + d*x]^2)*Sqrt[1 - Sin[c + d*x]^2])))/(8*b^3*d*Sin[c + d*x]^(7/2))","C",0
82,1,831,520,14.3560787,"\int \frac{(e \sin (c+d x))^{5/2}}{(a+b \cos (c+d x))^3} \, dx","Integrate[(e*Sin[c + d*x])^(5/2)/(a + b*Cos[c + d*x])^3,x]","\frac{\csc ^2(c+d x) \left(\frac{3 a \sin (c+d x)}{4 b \left(b^2-a^2\right) (a+b \cos (c+d x))}+\frac{\sin (c+d x)}{2 b (a+b \cos (c+d x))^2}\right) (e \sin (c+d x))^{5/2}}{d}+\frac{3 \left(\frac{a \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sin ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \sin (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \sin (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \cos ^2(c+d x)}{12 b^{3/2} \left(b^2-a^2\right) (a+b \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{4 b \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sin ^{\frac{3}{2}}(c+d x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \sin (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \sin (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right) (e \sin (c+d x))^{5/2}}{8 (a-b) b (a+b) d \sin ^{\frac{5}{2}}(c+d x)}","\frac{3 a e^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{4 b^2 d \left(a^2-b^2\right) \sqrt{\sin (c+d x)}}-\frac{3 a e (e \sin (c+d x))^{3/2}}{4 b d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{3 e^{5/2} \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{5/2} d \left(b^2-a^2\right)^{5/4}}+\frac{3 e^{5/2} \left(a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{5/2} d \left(b^2-a^2\right)^{5/4}}-\frac{3 a e^3 \left(a^2-2 b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{8 b^3 d \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \sin (c+d x)}}-\frac{3 a e^3 \left(a^2-2 b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{8 b^3 d \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \sin (c+d x)}}+\frac{e (e \sin (c+d x))^{3/2}}{2 b d (a+b \cos (c+d x))^2}",1,"(Csc[c + d*x]^2*(e*Sin[c + d*x])^(5/2)*(Sin[c + d*x]/(2*b*(a + b*Cos[c + d*x])^2) + (3*a*Sin[c + d*x])/(4*b*(-a^2 + b^2)*(a + b*Cos[c + d*x]))))/d + (3*(e*Sin[c + d*x])^(5/2)*((a*Cos[c + d*x]^2*(3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]) + 8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/(12*b^(3/2)*(-a^2 + b^2)*(a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (4*b*Cos[c + d*x]*(((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)) + (a*AppellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))/(3*(a^2 - b^2)))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(8*(a - b)*b*(a + b)*d*Sin[c + d*x]^(5/2))","C",0
83,1,1211,534,10.3878856,"\int \frac{(e \sin (c+d x))^{3/2}}{(a+b \cos (c+d x))^3} \, dx","Integrate[(e*Sin[c + d*x])^(3/2)/(a + b*Cos[c + d*x])^3,x]","\frac{\left(\frac{a}{4 b \left(b^2-a^2\right) (a+b \cos (c+d x))}+\frac{1}{2 b (a+b \cos (c+d x))^2}\right) \csc (c+d x) (e \sin (c+d x))^{3/2}}{d}-\frac{(e \sin (c+d x))^{3/2} \left(\frac{2 a \cos ^2(c+d x) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \sqrt{1-\sin ^2(c+d x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \sin ^2(c+d x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\sin ^2(c+d x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \sin (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \sin (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right)}{(a+b \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}-\frac{4 b \cos (c+d x) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sqrt{\sin (c+d x)}}{\sqrt{1-\sin ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \sin ^2(c+d x)\right) \left(a^2+b^2 \left(\sin ^2(c+d x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \sin (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \sin (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right)}{(a+b \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right)}{8 (a-b) b (a+b) d \sin ^{\frac{3}{2}}(c+d x)}","-\frac{a e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{4 b^2 d \left(a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{a e^2 \left(a^2+2 b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{8 b^2 d \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \sin (c+d x)}}+\frac{a e^2 \left(a^2+2 b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{8 b^2 d \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \sin (c+d x)}}-\frac{a e \sqrt{e \sin (c+d x)}}{4 b d \left(a^2-b^2\right) (a+b \cos (c+d x))}-\frac{e^{3/2} \left(a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{3/2} d \left(b^2-a^2\right)^{7/4}}-\frac{e^{3/2} \left(a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{3/2} d \left(b^2-a^2\right)^{7/4}}+\frac{e \sqrt{e \sin (c+d x)}}{2 b d (a+b \cos (c+d x))^2}",1,"((1/(2*b*(a + b*Cos[c + d*x])^2) + a/(4*b*(-a^2 + b^2)*(a + b*Cos[c + d*x])))*Csc[c + d*x]*(e*Sin[c + d*x])^(3/2))/d - ((e*Sin[c + d*x])^(3/2)*((2*a*Cos[c + d*x]^2*(a + b*Sqrt[1 - Sin[c + d*x]^2])*((a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)) + (5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) - (4*b*Cos[c + d*x]*(a + b*Sqrt[1 - Sin[c + d*x]^2])*(((-1/8 + I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(-a^2 + b^2)^(3/4) + (5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(8*(a - b)*b*(a + b)*d*Sin[c + d*x]^(3/2))","C",0
84,1,837,529,14.3330081,"\int \frac{\sqrt{e \sin (c+d x)}}{(a+b \cos (c+d x))^3} \, dx","Integrate[Sqrt[e*Sin[c + d*x]]/(a + b*Cos[c + d*x])^3,x]","\frac{\sqrt{e \sin (c+d x)} \left(-\frac{5 a b \sin (c+d x)}{4 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{b \sin (c+d x)}{2 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}\right)}{d}+\frac{\sqrt{e \sin (c+d x)} \left(\frac{5 a \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sin ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \sin (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \sin (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \cos ^2(c+d x)}{12 \sqrt{b} \left(b^2-a^2\right) (a+b \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{2 \left(8 a^2+2 b^2\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sin ^{\frac{3}{2}}(c+d x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \sin (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \sin (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right)}{8 (a-b)^2 (a+b)^2 d \sqrt{\sin (c+d x)}}","-\frac{\sqrt{e} \left(3 a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 \sqrt{b} d \left(b^2-a^2\right)^{9/4}}+\frac{\sqrt{e} \left(3 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 \sqrt{b} d \left(b^2-a^2\right)^{9/4}}-\frac{5 a b (e \sin (c+d x))^{3/2}}{4 d e \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{b (e \sin (c+d x))^{3/2}}{2 d e \left(a^2-b^2\right) (a+b \cos (c+d x))^2}+\frac{5 a E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{4 d \left(a^2-b^2\right)^2 \sqrt{\sin (c+d x)}}+\frac{a e \left(3 a^2+2 b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{8 b d \left(a^2-b^2\right)^2 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \sin (c+d x)}}+\frac{a e \left(3 a^2+2 b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{8 b d \left(a^2-b^2\right)^2 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \sin (c+d x)}}",1,"(Sqrt[e*Sin[c + d*x]]*(-1/2*(b*Sin[c + d*x])/((a^2 - b^2)*(a + b*Cos[c + d*x])^2) - (5*a*b*Sin[c + d*x])/(4*(a^2 - b^2)^2*(a + b*Cos[c + d*x]))))/d + (Sqrt[e*Sin[c + d*x]]*((5*a*Cos[c + d*x]^2*(3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]) + 8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/(12*Sqrt[b]*(-a^2 + b^2)*(a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(8*a^2 + 2*b^2)*Cos[c + d*x]*(((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)) + (a*AppellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))/(3*(a^2 - b^2)))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(8*(a - b)^2*(a + b)^2*d*Sqrt[Sin[c + d*x]])","C",0
85,1,1226,535,11.3687291,"\int \frac{1}{(a+b \cos (c+d x))^3 \sqrt{e \sin (c+d x)}} \, dx","Integrate[1/((a + b*Cos[c + d*x])^3*Sqrt[e*Sin[c + d*x]]),x]","\frac{\left(-\frac{7 a b}{4 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{b}{2 \left(a^2-b^2\right) (a+b \cos (c+d x))^2}\right) \sin (c+d x)}{d \sqrt{e \sin (c+d x)}}+\frac{\left(\frac{2 \left(8 a^2+6 b^2\right) \cos (c+d x) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sqrt{\sin (c+d x)}}{\sqrt{1-\sin ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \sin ^2(c+d x)\right) \left(a^2+b^2 \left(\sin ^2(c+d x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \sin (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \sin (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right)}{(a+b \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}-\frac{14 a b \cos ^2(c+d x) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \sqrt{1-\sin ^2(c+d x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \sin ^2(c+d x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\sin ^2(c+d x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \sin (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \sin (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right)}{(a+b \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}\right) \sqrt{\sin (c+d x)}}{8 (a-b)^2 (a+b)^2 d \sqrt{e \sin (c+d x)}}","\frac{3 \sqrt{b} \left(5 a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d \sqrt{e} \left(b^2-a^2\right)^{11/4}}+\frac{3 \sqrt{b} \left(5 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d \sqrt{e} \left(b^2-a^2\right)^{11/4}}-\frac{7 a b \sqrt{e \sin (c+d x)}}{4 d e \left(a^2-b^2\right)^2 (a+b \cos (c+d x))}-\frac{b \sqrt{e \sin (c+d x)}}{2 d e \left(a^2-b^2\right) (a+b \cos (c+d x))^2}-\frac{7 a \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{4 d \left(a^2-b^2\right)^2 \sqrt{e \sin (c+d x)}}+\frac{3 a \left(5 a^2+2 b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{8 d \left(a^2-b^2\right)^2 \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \sin (c+d x)}}+\frac{3 a \left(5 a^2+2 b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{8 d \left(a^2-b^2\right)^2 \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \sin (c+d x)}}",1,"((-1/2*b/((a^2 - b^2)*(a + b*Cos[c + d*x])^2) - (7*a*b)/(4*(a^2 - b^2)^2*(a + b*Cos[c + d*x])))*Sin[c + d*x])/(d*Sqrt[e*Sin[c + d*x]]) + (Sqrt[Sin[c + d*x]]*((-14*a*b*Cos[c + d*x]^2*(a + b*Sqrt[1 - Sin[c + d*x]^2])*((a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)) + (5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(8*a^2 + 6*b^2)*Cos[c + d*x]*(a + b*Sqrt[1 - Sin[c + d*x]^2])*(((-1/8 + I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(-a^2 + b^2)^(3/4) + (5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(8*(a - b)^2*(a + b)^2*d*Sqrt[e*Sin[c + d*x]])","C",0
86,1,922,611,6.5361487,"\int \frac{1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2}} \, dx","Integrate[1/((a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(3/2)),x]","\frac{\sin ^2(c+d x) \left(\frac{13 a \sin (c+d x) b^3}{4 \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{\sin (c+d x) b^3}{2 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{2 \left(\cos (c+d x) a^3-3 b a^2+3 b^2 \cos (c+d x) a-b^3\right) \csc (c+d x)}{\left(a^2-b^2\right)^3}\right)}{d (e \sin (c+d x))^{3/2}}-\frac{\sin ^{\frac{3}{2}}(c+d x) \left(\frac{\left(8 b a^3+37 b^3 a\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sin ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \sin (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \sin (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \cos ^2(c+d x)}{12 b^{3/2} \left(b^2-a^2\right) (a+b \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{2 \left(8 a^4+72 b^2 a^2+10 b^4\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sin ^{\frac{3}{2}}(c+d x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \sin (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \sin (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right)}{8 (a-b)^3 (a+b)^3 d (e \sin (c+d x))^{3/2}}","-\frac{a \left(8 a^2+37 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{4 d e^2 \left(a^2-b^2\right)^3 \sqrt{\sin (c+d x)}}-\frac{9 a b}{4 d e \left(a^2-b^2\right)^2 \sqrt{e \sin (c+d x)} (a+b \cos (c+d x))}-\frac{b}{2 d e \left(a^2-b^2\right) \sqrt{e \sin (c+d x)} (a+b \cos (c+d x))^2}+\frac{5 b \left(7 a^2+2 b^2\right)-a \left(8 a^2+37 b^2\right) \cos (c+d x)}{4 d e \left(a^2-b^2\right)^3 \sqrt{e \sin (c+d x)}}-\frac{5 a b \left(7 a^2+2 b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{8 d e \left(a^2-b^2\right)^3 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \sin (c+d x)}}-\frac{5 a b \left(7 a^2+2 b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{8 d e \left(a^2-b^2\right)^3 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \sin (c+d x)}}-\frac{5 b^{3/2} \left(7 a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d e^{3/2} \left(b^2-a^2\right)^{13/4}}+\frac{5 b^{3/2} \left(7 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d e^{3/2} \left(b^2-a^2\right)^{13/4}}",1,"(Sin[c + d*x]^2*((-2*(-3*a^2*b - b^3 + a^3*Cos[c + d*x] + 3*a*b^2*Cos[c + d*x])*Csc[c + d*x])/(a^2 - b^2)^3 + (b^3*Sin[c + d*x])/(2*(a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) + (13*a*b^3*Sin[c + d*x])/(4*(a^2 - b^2)^3*(a + b*Cos[c + d*x]))))/(d*(e*Sin[c + d*x])^(3/2)) - (Sin[c + d*x]^(3/2)*(((8*a^3*b + 37*a*b^3)*Cos[c + d*x]^2*(3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]) + 8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/(12*b^(3/2)*(-a^2 + b^2)*(a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(8*a^4 + 72*a^2*b^2 + 10*b^4)*Cos[c + d*x]*(((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)) + (a*AppellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))/(3*(a^2 - b^2)))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(8*(a - b)^3*(a + b)^3*d*(e*Sin[c + d*x])^(3/2))","C",0
87,1,1308,629,14.4878073,"\int \frac{1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{5/2}} \, dx","Integrate[1/((a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(5/2)),x]","\frac{\left(\frac{15 a b^3}{4 \left(a^2-b^2\right)^3 (a+b \cos (c+d x))}+\frac{b^3}{2 \left(a^2-b^2\right)^2 (a+b \cos (c+d x))^2}-\frac{2 \left(\cos (c+d x) a^3-3 b a^2+3 b^2 \cos (c+d x) a-b^3\right) \csc ^2(c+d x)}{3 \left(a^2-b^2\right)^3}\right) \sin ^3(c+d x)}{d (e \sin (c+d x))^{5/2}}+\frac{\left(\frac{2 \left(8 b a^3+69 b^3 a\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \sqrt{1-\sin ^2(c+d x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \sin ^2(c+d x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\sin ^2(c+d x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \sin (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \sin (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \cos ^2(c+d x)}{(a+b \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{2 \left(8 a^4-120 b^2 a^2-42 b^4\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sqrt{\sin (c+d x)}}{\sqrt{1-\sin ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right)\right) \sin ^2(c+d x)\right) \left(a^2+b^2 \left(\sin ^2(c+d x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \sin (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \sin (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right) \sin ^{\frac{5}{2}}(c+d x)}{24 (a-b)^3 (a+b)^3 d (e \sin (c+d x))^{5/2}}","\frac{a \left(8 a^2+69 b^2\right) \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{12 d e^2 \left(a^2-b^2\right)^3 \sqrt{e \sin (c+d x)}}-\frac{7 a b^2 \left(9 a^2+2 b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{8 d e^2 \left(a^2-b^2\right)^3 \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \sin (c+d x)}}-\frac{7 a b^2 \left(9 a^2+2 b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{8 d e^2 \left(a^2-b^2\right)^3 \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \sin (c+d x)}}-\frac{11 a b}{4 d e \left(a^2-b^2\right)^2 (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))}-\frac{b}{2 d e \left(a^2-b^2\right) (e \sin (c+d x))^{3/2} (a+b \cos (c+d x))^2}+\frac{7 b \left(9 a^2+2 b^2\right)-a \left(8 a^2+69 b^2\right) \cos (c+d x)}{12 d e \left(a^2-b^2\right)^3 (e \sin (c+d x))^{3/2}}+\frac{7 b^{5/2} \left(9 a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d e^{5/2} \left(b^2-a^2\right)^{15/4}}+\frac{7 b^{5/2} \left(9 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d e^{5/2} \left(b^2-a^2\right)^{15/4}}",1,"((b^3/(2*(a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) + (15*a*b^3)/(4*(a^2 - b^2)^3*(a + b*Cos[c + d*x])) - (2*(-3*a^2*b - b^3 + a^3*Cos[c + d*x] + 3*a*b^2*Cos[c + d*x])*Csc[c + d*x]^2)/(3*(a^2 - b^2)^3))*Sin[c + d*x]^3)/(d*(e*Sin[c + d*x])^(5/2)) + (Sin[c + d*x]^(5/2)*((2*(8*a^3*b + 69*a*b^3)*Cos[c + d*x]^2*(a + b*Sqrt[1 - Sin[c + d*x]^2])*((a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)) + (5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(8*a^4 - 120*a^2*b^2 - 42*b^4)*Cos[c + d*x]*(a + b*Sqrt[1 - Sin[c + d*x]^2])*(((-1/8 + I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(-a^2 + b^2)^(3/4) + (5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)])*Sin[c + d*x]^2)*(a^2 + b^2*(-1 + Sin[c + d*x]^2)))))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(24*(a - b)^3*(a + b)^3*d*(e*Sin[c + d*x])^(5/2))","C",0
88,1,1014,700,6.7089548,"\int \frac{1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{7/2}} \, dx","Integrate[1/((a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(7/2)),x]","\frac{\sin ^4(c+d x) \left(-\frac{21 a \sin (c+d x) b^5}{4 \left(a^2-b^2\right)^4 (a+b \cos (c+d x))}-\frac{\sin (c+d x) b^5}{2 \left(a^2-b^2\right)^3 (a+b \cos (c+d x))^2}-\frac{2 \left(\cos (c+d x) a^3-3 b a^2+3 b^2 \cos (c+d x) a-b^3\right) \csc ^3(c+d x)}{5 \left(a^2-b^2\right)^3}-\frac{2 \left(3 \cos (c+d x) a^5-24 b^2 \cos (c+d x) a^3+50 b^3 a^2-39 b^4 \cos (c+d x) a+10 b^5\right) \csc (c+d x)}{5 \left(a^2-b^2\right)^4}\right)}{d (e \sin (c+d x))^{7/2}}-\frac{3 \sin ^{\frac{7}{2}}(c+d x) \left(\frac{\left(8 b a^5-64 b^3 a^3-139 b^5 a\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sin ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \sin (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \sin (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \cos ^2(c+d x)}{12 b^{3/2} \left(b^2-a^2\right) (a+b \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{2 \left(8 a^6-64 b^2 a^4-304 b^4 a^2-30 b^6\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right) \sin ^{\frac{3}{2}}(c+d x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \sin (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \sin (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \left(a+b \sqrt{1-\sin ^2(c+d x)}\right) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right)}{40 (a-b)^4 (a+b)^4 d (e \sin (c+d x))^{7/2}}","-\frac{13 a b}{4 d e \left(a^2-b^2\right)^2 (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}-\frac{b}{2 d e \left(a^2-b^2\right) (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))^2}+\frac{9 b \left(11 a^2+2 b^2\right)-a \left(8 a^2+109 b^2\right) \cos (c+d x)}{20 d e \left(a^2-b^2\right)^3 (e \sin (c+d x))^{5/2}}-\frac{9 b^{7/2} \left(11 a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d e^{7/2} \left(b^2-a^2\right)^{17/4}}+\frac{9 b^{7/2} \left(11 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d e^{7/2} \left(b^2-a^2\right)^{17/4}}+\frac{9 a b^3 \left(11 a^2+2 b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{8 d e^3 \left(a^2-b^2\right)^4 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \sin (c+d x)}}+\frac{9 a b^3 \left(11 a^2+2 b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{8 d e^3 \left(a^2-b^2\right)^4 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \sin (c+d x)}}-\frac{3 a \left(8 a^4-64 a^2 b^2-139 b^4\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{20 d e^4 \left(a^2-b^2\right)^4 \sqrt{\sin (c+d x)}}-\frac{3 \left(15 b^3 \left(11 a^2+2 b^2\right)+a \left(8 a^4-64 a^2 b^2-139 b^4\right) \cos (c+d x)\right)}{20 d e^3 \left(a^2-b^2\right)^4 \sqrt{e \sin (c+d x)}}",1,"(Sin[c + d*x]^4*((-2*(50*a^2*b^3 + 10*b^5 + 3*a^5*Cos[c + d*x] - 24*a^3*b^2*Cos[c + d*x] - 39*a*b^4*Cos[c + d*x])*Csc[c + d*x])/(5*(a^2 - b^2)^4) - (2*(-3*a^2*b - b^3 + a^3*Cos[c + d*x] + 3*a*b^2*Cos[c + d*x])*Csc[c + d*x]^3)/(5*(a^2 - b^2)^3) - (b^5*Sin[c + d*x])/(2*(a^2 - b^2)^3*(a + b*Cos[c + d*x])^2) - (21*a*b^5*Sin[c + d*x])/(4*(a^2 - b^2)^4*(a + b*Cos[c + d*x]))))/(d*(e*Sin[c + d*x])^(7/2)) - (3*Sin[c + d*x]^(7/2)*(((8*a^5*b - 64*a^3*b^3 - 139*a*b^5)*Cos[c + d*x]^2*(3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + b*Sin[c + d*x]]) + 8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/(12*b^(3/2)*(-a^2 + b^2)*(a + b*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(8*a^6 - 64*a^4*b^2 - 304*a^2*b^4 - 30*b^6)*Cos[c + d*x]*(((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*b*Sin[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)) + (a*AppellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (b^2*Sin[c + d*x]^2)/(-a^2 + b^2)]*Sin[c + d*x]^(3/2))/(3*(a^2 - b^2)))*(a + b*Sqrt[1 - Sin[c + d*x]^2]))/((a + b*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(40*(a - b)^4*(a + b)^4*d*(e*Sin[c + d*x])^(7/2))","C",0