1,1,31,0,0.0428443,"\int \frac{\sin ^4(x)}{a+a \cos (x)} \, dx","Int[Sin[x]^4/(a + a*Cos[x]),x]","\frac{x}{2 a}-\frac{\sin ^3(x)}{3 a}-\frac{\sin (x) \cos (x)}{2 a}","\frac{x}{2 a}-\frac{\sin ^3(x)}{3 a}-\frac{\sin (x) \cos (x)}{2 a}",1,"x/(2*a) - (Cos[x]*Sin[x])/(2*a) - Sin[x]^3/(3*a)","A",3,3,13,0.2308,1,"{2682, 2635, 8}"
2,1,19,0,0.0399363,"\int \frac{\sin ^3(x)}{a+a \cos (x)} \, dx","Int[Sin[x]^3/(a + a*Cos[x]),x]","\frac{\cos ^2(x)}{2 a}-\frac{\cos (x)}{a}","\frac{\cos ^2(x)}{2 a}-\frac{\cos (x)}{a}",1,"-(Cos[x]/a) + Cos[x]^2/(2*a)","A",2,1,13,0.07692,1,"{2667}"
3,1,13,0,0.0377543,"\int \frac{\sin ^2(x)}{a+a \cos (x)} \, dx","Int[Sin[x]^2/(a + a*Cos[x]),x]","\frac{x}{a}-\frac{\sin (x)}{a}","\frac{x}{a}-\frac{\sin (x)}{a}",1,"x/a - Sin[x]/a","A",2,2,13,0.1538,1,"{2682, 8}"
4,1,10,0,0.0235585,"\int \frac{\sin (x)}{a+a \cos (x)} \, dx","Int[Sin[x]/(a + a*Cos[x]),x]","-\frac{\log (\cos (x)+1)}{a}","-\frac{\log (\cos (x)+1)}{a}",1,"-(Log[1 + Cos[x]]/a)","A",2,2,11,0.1818,1,"{2667, 31}"
5,1,11,0,0.0103396,"\int \frac{1}{a+a \cos (x)} \, dx","Int[(a + a*Cos[x])^(-1),x]","\frac{\sin (x)}{a \cos (x)+a}","\frac{\sin (x)}{a \cos (x)+a}",1,"Sin[x]/(a + a*Cos[x])","A",1,1,8,0.1250,1,"{2648}"
6,1,23,0,0.0495032,"\int \frac{\csc (x)}{a+a \cos (x)} \, dx","Int[Csc[x]/(a + a*Cos[x]),x]","\frac{1}{2 (a \cos (x)+a)}-\frac{\tanh ^{-1}(\cos (x))}{2 a}","\frac{1}{2 (a \cos (x)+a)}-\frac{\tanh ^{-1}(\cos (x))}{2 a}",1,"-ArcTanh[Cos[x]]/(2*a) + 1/(2*(a + a*Cos[x]))","A",4,3,11,0.2727,1,"{2667, 44, 206}"
7,1,24,0,0.046257,"\int \frac{\csc ^2(x)}{a+a \cos (x)} \, dx","Int[Csc[x]^2/(a + a*Cos[x]),x]","\frac{\csc (x)}{3 (a \cos (x)+a)}-\frac{2 \cot (x)}{3 a}","\frac{\csc (x)}{3 (a \cos (x)+a)}-\frac{2 \cot (x)}{3 a}",1,"(-2*Cot[x])/(3*a) + Csc[x]/(3*(a + a*Cos[x]))","A",3,3,13,0.2308,1,"{2672, 3767, 8}"
8,1,49,0,0.0735655,"\int \frac{\csc ^3(x)}{a+a \cos (x)} \, dx","Int[Csc[x]^3/(a + a*Cos[x]),x]","\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}","\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,"(-3*ArcTanh[Cos[x]])/(8*a) - 1/(8*(a - a*Cos[x])) + a/(8*(a + a*Cos[x])^2) + 1/(4*(a + a*Cos[x]))","A",4,3,13,0.2308,1,"{2667, 44, 206}"
9,1,37,0,0.0484188,"\int \frac{\csc ^4(x)}{a+a \cos (x)} \, dx","Int[Csc[x]^4/(a + a*Cos[x]),x]","-\frac{4 \cot ^3(x)}{15 a}-\frac{4 \cot (x)}{5 a}+\frac{\csc ^3(x)}{5 (a \cos (x)+a)}","-\frac{4 \cot ^3(x)}{15 a}-\frac{4 \cot (x)}{5 a}+\frac{\csc ^3(x)}{5 (a \cos (x)+a)}",1,"(-4*Cot[x])/(5*a) - (4*Cot[x]^3)/(15*a) + Csc[x]^3/(5*(a + a*Cos[x]))","A",3,2,13,0.1538,1,"{2672, 3767}"
10,1,11,0,0.0210279,"\int \frac{\sin (2 x)}{1+\cos (2 x)} \, dx","Int[Sin[2*x]/(1 + Cos[2*x]),x]","-\frac{1}{2} \log (\cos (2 x)+1)","-\log (\cos (x))",1,"-Log[1 + Cos[2*x]]/2","B",2,2,13,0.1538,1,"{2667, 31}"
11,1,13,0,0.0216338,"\int \frac{\sin (2 x)}{1-\cos (2 x)} \, dx","Int[Sin[2*x]/(1 - Cos[2*x]),x]","\frac{1}{2} \log (1-\cos (2 x))","\log (\sin (x))",1,"Log[1 - Cos[2*x]]/2","B",2,2,15,0.1333,1,"{2667, 31}"
12,1,6,0,0.0185305,"\int \frac{\sin (x)}{(1+\cos (x))^2} \, dx","Int[Sin[x]/(1 + Cos[x])^2,x]","\frac{1}{\cos (x)+1}","\frac{1}{\cos (x)+1}",1,"(1 + Cos[x])^(-1)","A",2,2,9,0.2222,1,"{2667, 32}"
13,1,10,0,0.019053,"\int \frac{\sin (x)}{(1-\cos (x))^2} \, dx","Int[Sin[x]/(1 - Cos[x])^2,x]","-\frac{1}{1-\cos (x)}","-\frac{1}{1-\cos (x)}",1,"-(1 - Cos[x])^(-1)","A",2,2,11,0.1818,1,"{2667, 32}"
14,1,14,0,0.0311965,"\int \frac{\sin ^2(x)}{(1+\cos (x))^2} \, dx","Int[Sin[x]^2/(1 + Cos[x])^2,x]","\frac{2 \sin (x)}{\cos (x)+1}-x","\frac{2 \sin (x)}{\cos (x)+1}-x",1,"-x + (2*Sin[x])/(1 + Cos[x])","A",2,2,11,0.1818,1,"{2680, 8}"
15,1,16,0,0.030765,"\int \frac{\sin ^2(x)}{(1-\cos (x))^2} \, dx","Int[Sin[x]^2/(1 - Cos[x])^2,x]","-x-\frac{2 \sin (x)}{1-\cos (x)}","-x-\frac{2 \sin (x)}{1-\cos (x)}",1,"-x - (2*Sin[x])/(1 - Cos[x])","A",2,2,13,0.1538,1,"{2680, 8}"
16,1,10,0,0.0357753,"\int \frac{\sin ^3(x)}{(1+\cos (x))^2} \, dx","Int[Sin[x]^3/(1 + Cos[x])^2,x]","\cos (x)-2 \log (\cos (x)+1)","\cos (x)-2 \log (\cos (x)+1)",1,"Cos[x] - 2*Log[1 + Cos[x]]","A",3,2,11,0.1818,1,"{2667, 43}"
17,1,12,0,0.0362866,"\int \frac{\sin ^3(x)}{(1-\cos (x))^2} \, dx","Int[Sin[x]^3/(1 - Cos[x])^2,x]","\cos (x)+2 \log (1-\cos (x))","\cos (x)+2 \log (1-\cos (x))",1,"Cos[x] + 2*Log[1 - Cos[x]]","A",3,2,13,0.1538,1,"{2667, 43}"
18,1,10,0,0.0189541,"\int \frac{\sin (x)}{(1+\cos (x))^3} \, dx","Int[Sin[x]/(1 + Cos[x])^3,x]","\frac{1}{2 (\cos (x)+1)^2}","\frac{1}{2 (\cos (x)+1)^2}",1,"1/(2*(1 + Cos[x])^2)","A",2,2,9,0.2222,1,"{2667, 32}"
19,1,12,0,0.0188377,"\int \frac{\sin (x)}{(1-\cos (x))^3} \, dx","Int[Sin[x]/(1 - Cos[x])^3,x]","-\frac{1}{2 (1-\cos (x))^2}","-\frac{1}{2 (1-\cos (x))^2}",1,"-1/(2*(1 - Cos[x])^2)","A",2,2,11,0.1818,1,"{2667, 32}"
20,1,14,0,0.0308714,"\int \frac{\sin ^2(x)}{(1+\cos (x))^3} \, dx","Int[Sin[x]^2/(1 + Cos[x])^3,x]","\frac{\sin ^3(x)}{3 (\cos (x)+1)^3}","\frac{\sin ^3(x)}{3 (\cos (x)+1)^3}",1,"Sin[x]^3/(3*(1 + Cos[x])^3)","A",1,1,11,0.09091,1,"{2671}"
21,1,16,0,0.0308387,"\int \frac{\sin ^2(x)}{(1-\cos (x))^3} \, dx","Int[Sin[x]^2/(1 - Cos[x])^3,x]","-\frac{\sin ^3(x)}{3 (1-\cos (x))^3}","-\frac{\sin ^3(x)}{3 (1-\cos (x))^3}",1,"-Sin[x]^3/(3*(1 - Cos[x])^3)","A",1,1,13,0.07692,1,"{2671}"
22,1,14,0,0.0377091,"\int \frac{\sin ^3(x)}{(1+\cos (x))^3} \, dx","Int[Sin[x]^3/(1 + Cos[x])^3,x]","\frac{2}{\cos (x)+1}+\log (\cos (x)+1)","\frac{2}{\cos (x)+1}+\log (\cos (x)+1)",1,"2/(1 + Cos[x]) + Log[1 + Cos[x]]","A",3,2,11,0.1818,1,"{2667, 43}"
23,1,20,0,0.0372703,"\int \frac{\sin ^3(x)}{(1-\cos (x))^3} \, dx","Int[Sin[x]^3/(1 - Cos[x])^3,x]","-\frac{2}{1-\cos (x)}-\log (1-\cos (x))","-\frac{2}{1-\cos (x)}-\log (1-\cos (x))",1,"-2/(1 - Cos[x]) - Log[1 - Cos[x]]","A",3,2,13,0.1538,1,"{2667, 43}"
24,1,104,0,0.2562142,"\int \frac{\sin ^4(x)}{a+b \cos (x)} \, dx","Int[Sin[x]^4/(a + b*Cos[x]),x]","-\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}","-\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,"-(a*(2*a^2 - 3*b^2)*x)/(2*b^4) + (2*(a - b)^(3/2)*(a + b)^(3/2)*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/b^4 + ((2*(a^2 - b^2) - a*b*Cos[x])*Sin[x])/(2*b^3) - Sin[x]^3/(3*b)","A",5,5,13,0.3846,1,"{2695, 2865, 2735, 2659, 205}"
25,1,40,0,0.06125,"\int \frac{\sin ^3(x)}{a+b \cos (x)} \, dx","Int[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)}{2 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[x]^2/(2*b) + ((a^2 - b^2)*Log[a + b*Cos[x]])/b^3","A",3,2,13,0.1538,1,"{2668, 697}"
26,1,59,0,0.1088808,"\int \frac{\sin ^2(x)}{a+b \cos (x)} \, dx","Int[Sin[x]^2/(a + b*Cos[x]),x]","\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}","\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)/b^2 - (2*Sqrt[a - b]*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/b^2 - Sin[x]/b","A",4,4,13,0.3077,1,"{2695, 2735, 2659, 205}"
27,1,12,0,0.0252045,"\int \frac{\sin (x)}{a+b \cos (x)} \, dx","Int[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",2,2,11,0.1818,1,"{2668, 31}"
28,1,42,0,0.0256633,"\int \frac{1}{a+b \cos (x)} \, dx","Int[(a + b*Cos[x])^(-1),x]","\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}}","\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*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b])","A",2,2,8,0.2500,1,"{2659, 205}"
29,1,53,0,0.0722379,"\int \frac{\csc (x)}{a+b \cos (x)} \, dx","Int[Csc[x]/(a + b*Cos[x]),x]","\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)}","\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,"Log[1 - Cos[x]]/(2*(a + b)) - Log[1 + Cos[x]]/(2*(a - b)) + (b*Log[a + b*Cos[x]])/(a^2 - b^2)","A",6,4,11,0.3636,1,"{2668, 706, 31, 633}"
30,1,67,0,0.0949208,"\int \frac{\csc ^2(x)}{a+b \cos (x)} \, dx","Int[Csc[x]^2/(a + b*Cos[x]),x]","\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}}","\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*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)) + ((b - a*Cos[x])*Csc[x])/(a^2 - b^2)","A",4,4,13,0.3077,1,"{2696, 12, 2659, 205}"
31,1,92,0,0.1541307,"\int \frac{\csc ^3(x)}{a+b \cos (x)} \, dx","Int[Csc[x]^3/(a + b*Cos[x]),x]","-\frac{b^3 \log (a+b \cos (x))}{\left(a^2-b^2\right)^2}+\frac{\csc ^2(x) (b-a \cos (x))}{2 \left(a^2-b^2\right)}+\frac{(a+2 b) \log (1-\cos (x))}{4 (a+b)^2}-\frac{(a-2 b) \log (\cos (x)+1)}{4 (a-b)^2}","-\frac{b^3 \log (a+b \cos (x))}{\left(a^2-b^2\right)^2}+\frac{\csc ^2(x) (b-a \cos (x))}{2 \left(a^2-b^2\right)}+\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,"((b - a*Cos[x])*Csc[x]^2)/(2*(a^2 - b^2)) + ((a + 2*b)*Log[1 - Cos[x]])/(4*(a + b)^2) - ((a - 2*b)*Log[1 + Cos[x]])/(4*(a - b)^2) - (b^3*Log[a + b*Cos[x]])/(a^2 - b^2)^2","A",4,3,13,0.2308,1,"{2668, 741, 801}"
32,1,110,0,0.2708869,"\int \frac{\csc ^4(x)}{a+b \cos (x)} \, dx","Int[Csc[x]^4/(a + b*Cos[x]),x]","\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}}","\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*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)) - ((3*b^3 + a*(2*a^2 - 5*b^2)*Cos[x])*Csc[x])/(3*(a^2 - b^2)^2) + ((b - a*Cos[x])*Csc[x]^3)/(3*(a^2 - b^2))","A",5,5,13,0.3846,1,"{2696, 2866, 12, 2659, 205}"
33,1,129,0,0.0965506,"\int (a+b \cos (c+d x)) (e \sin (c+d x))^{7/2} \, dx","Int[(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(7/2),x]","-\frac{10 a e^3 \cos (c+d x) \sqrt{e \sin (c+d x)}}{21 d}+\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{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}","-\frac{10 a e^3 \cos (c+d x) \sqrt{e \sin (c+d x)}}{21 d}+\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{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,"(10*a*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(21*d*Sqrt[e*Sin[c + d*x]]) - (10*a*e^3*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(21*d) - (2*a*e*Cos[c + d*x]*(e*Sin[c + d*x])^(5/2))/(7*d) + (2*b*(e*Sin[c + d*x])^(9/2))/(9*d*e)","A",5,4,23,0.1739,1,"{2669, 2635, 2642, 2641}"
34,1,100,0,0.0697559,"\int (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2} \, dx","Int[(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2),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}","\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,"(6*a*e^2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*d*Sqrt[Sin[c + d*x]]) - (2*a*e*Cos[c + d*x]*(e*Sin[c + d*x])^(3/2))/(5*d) + (2*b*(e*Sin[c + d*x])^(7/2))/(7*d*e)","A",4,4,23,0.1739,1,"{2669, 2635, 2640, 2639}"
35,1,100,0,0.0713243,"\int (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2} \, dx","Int[(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2),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}","\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*a*e^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*d*Sqrt[e*Sin[c + d*x]]) - (2*a*e*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(3*d) + (2*b*(e*Sin[c + d*x])^(5/2))/(5*d*e)","A",4,4,23,0.1739,1,"{2669, 2635, 2642, 2641}"
36,1,68,0,0.0489842,"\int (a+b \cos (c+d x)) \sqrt{e \sin (c+d x)} \, dx","Int[(a + b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]],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}","\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*a*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(d*Sqrt[Sin[c + d*x]]) + (2*b*(e*Sin[c + d*x])^(3/2))/(3*d*e)","A",3,3,23,0.1304,1,"{2669, 2640, 2639}"
37,1,66,0,0.0509949,"\int \frac{a+b \cos (c+d x)}{\sqrt{e \sin (c+d x)}} \, dx","Int[(a + b*Cos[c + d*x])/Sqrt[e*Sin[c + d*x]],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}","\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[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(d*Sqrt[e*Sin[c + d*x]]) + (2*b*Sqrt[e*Sin[c + d*x]])/(d*e)","A",3,3,23,0.1304,1,"{2669, 2642, 2641}"
38,1,96,0,0.0731285,"\int \frac{a+b \cos (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx","Int[(a + b*Cos[c + d*x])/(e*Sin[c + d*x])^(3/2),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)}}","-\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)/(d*e*Sqrt[e*Sin[c + d*x]]) - (2*a*Cos[c + d*x])/(d*e*Sqrt[e*Sin[c + d*x]]) - (2*a*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(d*e^2*Sqrt[Sin[c + d*x]])","A",4,4,23,0.1739,1,"{2669, 2636, 2640, 2639}"
39,1,102,0,0.0707494,"\int \frac{a+b \cos (c+d x)}{(e \sin (c+d x))^{5/2}} \, dx","Int[(a + b*Cos[c + d*x])/(e*Sin[c + d*x])^(5/2),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)}{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}}","\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)/(3*d*e*(e*Sin[c + d*x])^(3/2)) - (2*a*Cos[c + d*x])/(3*d*e*(e*Sin[c + d*x])^(3/2)) + (2*a*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*d*e^2*Sqrt[e*Sin[c + d*x]])","A",4,4,23,0.1739,1,"{2669, 2636, 2642, 2641}"
40,1,131,0,0.0910235,"\int \frac{a+b \cos (c+d x)}{(e \sin (c+d x))^{7/2}} \, dx","Int[(a + b*Cos[c + d*x])/(e*Sin[c + d*x])^(7/2),x]","-\frac{6 a \cos (c+d x)}{5 d e^3 \sqrt{e \sin (c+d x)}}-\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{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}}","-\frac{6 a \cos (c+d x)}{5 d e^3 \sqrt{e \sin (c+d x)}}-\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{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,"(-2*b)/(5*d*e*(e*Sin[c + d*x])^(5/2)) - (2*a*Cos[c + d*x])/(5*d*e*(e*Sin[c + d*x])^(5/2)) - (6*a*Cos[c + d*x])/(5*d*e^3*Sqrt[e*Sin[c + d*x]]) - (6*a*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*d*e^4*Sqrt[Sin[c + d*x]])","A",5,4,23,0.1739,1,"{2669, 2636, 2640, 2639}"
41,1,193,0,0.1897185,"\int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{7/2} \, dx","Int[(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(7/2),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{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{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}","-\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{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{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,"(10*(11*a^2 + 2*b^2)*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(231*d*Sqrt[e*Sin[c + d*x]]) - (10*(11*a^2 + 2*b^2)*e^3*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(231*d) - (2*(11*a^2 + 2*b^2)*e*Cos[c + d*x]*(e*Sin[c + d*x])^(5/2))/(77*d) + (26*a*b*(e*Sin[c + d*x])^(9/2))/(99*d*e) + (2*b*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(9/2))/(11*d*e)","A",6,5,25,0.2000,1,"{2692, 2669, 2635, 2642, 2641}"
42,1,154,0,0.1609368,"\int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2} \, dx","Int[(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(5/2),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}","\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,"(2*(9*a^2 + 2*b^2)*e^2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(15*d*Sqrt[Sin[c + d*x]]) - (2*(9*a^2 + 2*b^2)*e*Cos[c + d*x]*(e*Sin[c + d*x])^(3/2))/(45*d) + (22*a*b*(e*Sin[c + d*x])^(7/2))/(63*d*e) + (2*b*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(7/2))/(9*d*e)","A",5,5,25,0.2000,1,"{2692, 2669, 2635, 2640, 2639}"
43,1,154,0,0.1680324,"\int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2} \, dx","Int[(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(3/2),x]","\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}","\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,"(2*(7*a^2 + 2*b^2)*e^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(21*d*Sqrt[e*Sin[c + d*x]]) - (2*(7*a^2 + 2*b^2)*e*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(21*d) + (18*a*b*(e*Sin[c + d*x])^(5/2))/(35*d*e) + (2*b*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2))/(7*d*e)","A",5,5,25,0.2000,1,"{2692, 2669, 2635, 2642, 2641}"
44,1,114,0,0.1310063,"\int (a+b \cos (c+d x))^2 \sqrt{e \sin (c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^2*Sqrt[e*Sin[c + d*x]],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}","\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*(5*a^2 + 2*b^2)*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*d*Sqrt[Sin[c + d*x]]) + (14*a*b*(e*Sin[c + d*x])^(3/2))/(15*d*e) + (2*b*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2))/(5*d*e)","A",4,4,25,0.1600,1,"{2692, 2669, 2640, 2639}"
45,1,114,0,0.1294921,"\int \frac{(a+b \cos (c+d x))^2}{\sqrt{e \sin (c+d x)}} \, dx","Int[(a + b*Cos[c + d*x])^2/Sqrt[e*Sin[c + d*x]],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}","\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[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*d*Sqrt[e*Sin[c + d*x]]) + (10*a*b*Sqrt[e*Sin[c + d*x]])/(3*d*e) + (2*b*(a + b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(3*d*e)","A",4,4,25,0.1600,1,"{2692, 2669, 2642, 2641}"
46,1,118,0,0.1380437,"\int \frac{(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx","Int[(a + b*Cos[c + d*x])^2/(e*Sin[c + d*x])^(3/2),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)}}","-\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,"(-2*(b + a*Cos[c + d*x])*(a + b*Cos[c + d*x]))/(d*e*Sqrt[e*Sin[c + d*x]]) - (2*(a^2 + 2*b^2)*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(d*e^2*Sqrt[Sin[c + d*x]]) - (2*a*b*(e*Sin[c + d*x])^(3/2))/(d*e^3)","A",4,4,25,0.1600,1,"{2691, 2669, 2640, 2639}"
47,1,124,0,0.139355,"\int \frac{(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{5/2}} \, dx","Int[(a + b*Cos[c + d*x])^2/(e*Sin[c + d*x])^(5/2),x]","\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}}","\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*(b + a*Cos[c + d*x])*(a + b*Cos[c + d*x]))/(3*d*e*(e*Sin[c + d*x])^(3/2)) + (2*(a^2 - 2*b^2)*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*d*e^2*Sqrt[e*Sin[c + d*x]]) - (2*a*b*Sqrt[e*Sin[c + d*x]])/(3*d*e^3)","A",4,4,25,0.1600,1,"{2691, 2669, 2642, 2641}"
48,1,165,0,0.1711971,"\int \frac{(a+b \cos (c+d x))^2}{(e \sin (c+d x))^{7/2}} \, dx","Int[(a + b*Cos[c + d*x])^2/(e*Sin[c + d*x])^(7/2),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 \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 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}}","-\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 \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 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,"(-2*(b + a*Cos[c + d*x])*(a + b*Cos[c + d*x]))/(5*d*e*(e*Sin[c + d*x])^(5/2)) - (2*a*b)/(5*d*e^3*Sqrt[e*Sin[c + d*x]]) - (2*(3*a^2 - 2*b^2)*Cos[c + d*x])/(5*d*e^3*Sqrt[e*Sin[c + d*x]]) - (2*(3*a^2 - 2*b^2)*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*d*e^4*Sqrt[Sin[c + d*x]])","A",5,5,25,0.2000,1,"{2691, 2669, 2636, 2640, 2639}"
49,1,242,0,0.3203939,"\int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{7/2} \, dx","Int[(a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(7/2),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{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{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}","-\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{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{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,"(10*a*(11*a^2 + 6*b^2)*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(231*d*Sqrt[e*Sin[c + d*x]]) - (10*a*(11*a^2 + 6*b^2)*e^3*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(231*d) - (2*a*(11*a^2 + 6*b^2)*e*Cos[c + d*x]*(e*Sin[c + d*x])^(5/2))/(77*d) + (2*b*(177*a^2 + 44*b^2)*(e*Sin[c + d*x])^(9/2))/(1287*d*e) + (34*a*b*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(9/2))/(143*d*e) + (2*b*(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(9/2))/(13*d*e)","A",7,6,25,0.2400,1,"{2692, 2862, 2669, 2635, 2642, 2641}"
50,1,202,0,0.2927216,"\int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{5/2} \, dx","Int[(a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(5/2),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}","\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,"(2*a*(3*a^2 + 2*b^2)*e^2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*d*Sqrt[Sin[c + d*x]]) - (2*a*(3*a^2 + 2*b^2)*e*Cos[c + d*x]*(e*Sin[c + d*x])^(3/2))/(15*d) + (2*b*(43*a^2 + 12*b^2)*(e*Sin[c + d*x])^(7/2))/(231*d*e) + (10*a*b*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(7/2))/(33*d*e) + (2*b*(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(7/2))/(11*d*e)","A",6,6,25,0.2400,1,"{2692, 2862, 2669, 2635, 2640, 2639}"
51,1,202,0,0.2978237,"\int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2} \, dx","Int[(a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(3/2),x]","\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}","\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,"(2*a*(7*a^2 + 6*b^2)*e^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(21*d*Sqrt[e*Sin[c + d*x]]) - (2*a*(7*a^2 + 6*b^2)*e*Cos[c + d*x]*Sqrt[e*Sin[c + d*x]])/(21*d) + (2*b*(89*a^2 + 28*b^2)*(e*Sin[c + d*x])^(5/2))/(315*d*e) + (26*a*b*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2))/(63*d*e) + (2*b*(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(5/2))/(9*d*e)","A",6,6,25,0.2400,1,"{2692, 2862, 2669, 2635, 2642, 2641}"
52,1,161,0,0.2498901,"\int (a+b \cos (c+d x))^3 \sqrt{e \sin (c+d x)} \, dx","Int[(a + b*Cos[c + d*x])^3*Sqrt[e*Sin[c + d*x]],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}","\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,"(2*a*(5*a^2 + 6*b^2)*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*d*Sqrt[Sin[c + d*x]]) + (2*b*(57*a^2 + 20*b^2)*(e*Sin[c + d*x])^(3/2))/(105*d*e) + (22*a*b*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2))/(35*d*e) + (2*b*(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(3/2))/(7*d*e)","A",5,5,25,0.2000,1,"{2692, 2862, 2669, 2640, 2639}"
53,1,157,0,0.2427021,"\int \frac{(a+b \cos (c+d x))^3}{\sqrt{e \sin (c+d x)}} \, dx","Int[(a + b*Cos[c + d*x])^3/Sqrt[e*Sin[c + d*x]],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}","\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,"(2*a*(a^2 + 2*b^2)*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(d*Sqrt[e*Sin[c + d*x]]) + (2*b*(11*a^2 + 4*b^2)*Sqrt[e*Sin[c + d*x]])/(5*d*e) + (6*a*b*(a + b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(5*d*e) + (2*b*(a + b*Cos[c + d*x])^2*Sqrt[e*Sin[c + d*x]])/(5*d*e)","A",5,5,25,0.2000,1,"{2692, 2862, 2669, 2642, 2641}"
54,1,165,0,0.2527627,"\int \frac{(a+b \cos (c+d x))^3}{(e \sin (c+d x))^{3/2}} \, dx","Int[(a + b*Cos[c + d*x])^3/(e*Sin[c + d*x])^(3/2),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)}}","-\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*(b + a*Cos[c + d*x])*(a + b*Cos[c + d*x])^2)/(d*e*Sqrt[e*Sin[c + d*x]]) - (2*a*(a^2 + 6*b^2)*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(d*e^2*Sqrt[Sin[c + d*x]]) - (2*b*(3*a^2 + 4*b^2)*(e*Sin[c + d*x])^(3/2))/(3*d*e^3) - (2*a*b*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2))/(d*e^3)","A",5,5,25,0.2000,1,"{2691, 2862, 2669, 2640, 2639}"
55,1,169,0,0.2596266,"\int \frac{(a+b \cos (c+d x))^3}{(e \sin (c+d x))^{5/2}} \, dx","Int[(a + b*Cos[c + d*x])^3/(e*Sin[c + d*x])^(5/2),x]","-\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}}","-\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,"(-2*(b + a*Cos[c + d*x])*(a + b*Cos[c + d*x])^2)/(3*d*e*(e*Sin[c + d*x])^(3/2)) + (2*a*(a^2 - 6*b^2)*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*d*e^2*Sqrt[e*Sin[c + d*x]]) - (2*b*(a^2 + 4*b^2)*Sqrt[e*Sin[c + d*x]])/(3*d*e^3) - (2*a*b*(a + b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(3*d*e^3)","A",5,5,25,0.2000,1,"{2691, 2862, 2669, 2642, 2641}"
56,1,192,0,0.2712214,"\int \frac{(a+b \cos (c+d x))^3}{(e \sin (c+d x))^{7/2}} \, dx","Int[(a + b*Cos[c + d*x])^3/(e*Sin[c + d*x])^(7/2),x]","-\frac{2 b \left(3 a^2-4 b^2\right) (e \sin (c+d x))^{3/2}}{5 d e^5}+\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{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 (a \cos (c+d x)+b) (a+b \cos (c+d x))^2}{5 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{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{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 (a \cos (c+d x)+b) (a+b \cos (c+d x))^2}{5 d e (e \sin (c+d x))^{5/2}}",1,"(-2*(b + a*Cos[c + d*x])*(a + b*Cos[c + d*x])^2)/(5*d*e*(e*Sin[c + d*x])^(5/2)) + (2*(a + b*Cos[c + d*x])*(a*b - (3*a^2 - 4*b^2)*Cos[c + d*x]))/(5*d*e^3*Sqrt[e*Sin[c + d*x]]) - (6*a*(a^2 - 2*b^2)*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*d*e^4*Sqrt[Sin[c + d*x]]) - (2*b*(3*a^2 - 4*b^2)*(e*Sin[c + d*x])^(3/2))/(5*d*e^5)","A",5,5,25,0.2000,1,"{2691, 2861, 2669, 2640, 2639}"
57,1,193,0,0.2735662,"\int \frac{(a+b \cos (c+d x))^3}{(e \sin (c+d x))^{9/2}} \, dx","Int[(a + b*Cos[c + d*x])^3/(e*Sin[c + d*x])^(9/2),x]","-\frac{2 b \left(5 a^2-4 b^2\right) \sqrt{e \sin (c+d x)}}{21 d e^5}-\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 \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 (a \cos (c+d x)+b) (a+b \cos (c+d x))^2}{7 d e (e \sin (c+d x))^{7/2}}","-\frac{2 b \left(5 a^2-4 b^2\right) \sqrt{e \sin (c+d x)}}{21 d e^5}-\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 \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 (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*(b + a*Cos[c + d*x])*(a + b*Cos[c + d*x])^2)/(7*d*e*(e*Sin[c + d*x])^(7/2)) - (2*(a + b*Cos[c + d*x])*(a*b + (5*a^2 - 4*b^2)*Cos[c + d*x]))/(21*d*e^3*(e*Sin[c + d*x])^(3/2)) + (2*a*(5*a^2 - 6*b^2)*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(21*d*e^4*Sqrt[e*Sin[c + d*x]]) - (2*b*(5*a^2 - 4*b^2)*Sqrt[e*Sin[c + d*x]])/(21*d*e^5)","A",5,5,25,0.2000,1,"{2691, 2861, 2669, 2642, 2641}"
58,1,544,0,1.8989289,"\int \frac{(e \sin (c+d x))^{11/2}}{a+b \cos (c+d x)} \, dx","Int[(e*Sin[c + d*x])^(11/2)/(a + b*Cos[c + d*x]),x]","\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{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(-49 a^2 b^2+21 a^4+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{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 (e \sin (c+d x))^{9/2}}{9 b d}","\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{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(-49 a^2 b^2+21 a^4+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{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 (e \sin (c+d x))^{9/2}}{9 b d}",1,"((-a^2 + b^2)^(9/4)*e^(11/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(11/2)*d) + ((-a^2 + b^2)^(9/4)*e^(11/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(11/2)*d) + (2*a*(21*a^4 - 49*a^2*b^2 + 33*b^4)*e^6*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(21*b^6*d*Sqrt[e*Sin[c + d*x]]) - (a*(a^2 - b^2)^3*e^6*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b^6*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) - (a*(a^2 - b^2)^3*e^6*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b^6*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) - (2*e^5*(21*(a^2 - b^2)^2 - a*b*(7*a^2 - 12*b^2)*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(21*b^5*d) + (2*e^3*(7*(a^2 - b^2) - 5*a*b*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2))/(35*b^3*d) - (2*e*(e*Sin[c + d*x])^(9/2))/(9*b*d)","A",15,12,25,0.4800,1,"{2695, 2865, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
59,1,461,0,1.3360817,"\int \frac{(e \sin (c+d x))^{9/2}}{a+b \cos (c+d x)} \, dx","Int[(e*Sin[c + d*x])^(9/2)/(a + b*Cos[c + d*x]),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{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 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{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 e (e \sin (c+d x))^{7/2}}{7 b d}","-\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{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 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{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 e (e \sin (c+d x))^{7/2}}{7 b d}",1,"-(((-a^2 + b^2)^(7/4)*e^(9/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(9/2)*d)) + ((-a^2 + b^2)^(7/4)*e^(9/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(9/2)*d) + (a*(a^2 - b^2)^2*e^5*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b^5*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (a*(a^2 - b^2)^2*e^5*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b^5*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) - (2*a*(5*a^2 - 8*b^2)*e^4*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*b^4*d*Sqrt[Sin[c + d*x]]) + (2*e^3*(5*(a^2 - b^2) - 3*a*b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2))/(15*b^3*d) - (2*e*(e*Sin[c + d*x])^(7/2))/(7*b*d)","A",14,12,25,0.4800,1,"{2695, 2865, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
60,1,474,0,1.3374917,"\int \frac{(e \sin (c+d x))^{7/2}}{a+b \cos (c+d x)} \, dx","Int[(e*Sin[c + d*x])^(7/2)/(a + b*Cos[c + d*x]),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 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 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 (e \sin (c+d x))^{5/2}}{5 b d}","\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 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 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 (e \sin (c+d x))^{5/2}}{5 b d}",1,"((-a^2 + b^2)^(5/4)*e^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(7/2)*d) + ((-a^2 + b^2)^(5/4)*e^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(7/2)*d) - (2*a*(3*a^2 - 4*b^2)*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*b^4*d*Sqrt[e*Sin[c + d*x]]) + (a*(a^2 - b^2)^2*e^4*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b^4*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (a*(a^2 - b^2)^2*e^4*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b^4*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (2*e^3*(3*(a^2 - b^2) - a*b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(3*b^3*d) - (2*e*(e*Sin[c + d*x])^(5/2))/(5*b*d)","A",14,12,25,0.4800,1,"{2695, 2865, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
61,1,399,0,0.8982097,"\int \frac{(e \sin (c+d x))^{5/2}}{a+b \cos (c+d x)} \, dx","Int[(e*Sin[c + d*x])^(5/2)/(a + b*Cos[c + d*x]),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}","-\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,"-(((-a^2 + b^2)^(3/4)*e^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(5/2)*d)) + ((-a^2 + b^2)^(3/4)*e^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(5/2)*d) - (a*(a^2 - b^2)*e^3*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b^3*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) - (a*(a^2 - b^2)*e^3*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b^3*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*a*e^2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(b^2*d*Sqrt[Sin[c + d*x]]) - (2*e*(e*Sin[c + d*x])^(3/2))/(3*b*d)","A",13,11,25,0.4400,1,"{2695, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
62,1,410,0,0.9041849,"\int \frac{(e \sin (c+d x))^{3/2}}{a+b \cos (c+d x)} \, dx","Int[(e*Sin[c + d*x])^(3/2)/(a + b*Cos[c + d*x]),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{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{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}","\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{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{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,"((-a^2 + b^2)^(1/4)*e^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(3/2)*d) + ((-a^2 + b^2)^(1/4)*e^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(3/2)*d) + (2*a*e^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b^2*d*Sqrt[e*Sin[c + d*x]]) - (a*(a^2 - b^2)*e^2*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) - (a*(a^2 - b^2)*e^2*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) - (2*e*Sqrt[e*Sin[c + d*x]])/(b*d)","A",13,11,25,0.4400,1,"{2695, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
63,1,302,0,0.6074551,"\int \frac{\sqrt{e \sin (c+d x)}}{a+b \cos (c+d x)} \, dx","Int[Sqrt[e*Sin[c + d*x]]/(a + b*Cos[c + d*x]),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)}}","-\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,"-((Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(Sqrt[b]*(-a^2 + b^2)^(1/4)*d)) + (Sqrt[e]*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(Sqrt[b]*(-a^2 + b^2)^(1/4)*d) + (a*e*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (a*e*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]])","A",9,7,25,0.2800,1,"{2701, 2807, 2805, 329, 298, 205, 208}"
64,1,307,0,0.5883082,"\int \frac{1}{(a+b \cos (c+d x)) \sqrt{e \sin (c+d x)}} \, dx","Int[1/((a + b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]]),x]","\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)}}","\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,"(Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(3/4)*d*Sqrt[e]) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(3/4)*d*Sqrt[e]) + (a*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (a*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]])","A",9,7,25,0.2800,1,"{2702, 2807, 2805, 329, 212, 208, 205}"
65,1,426,0,0.9644383,"\int \frac{1}{(a+b \cos (c+d x)) (e \sin (c+d x))^{3/2}} \, dx","Int[1/((a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2)),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}}-\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}}-\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)}}",1,"-((b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(5/4)*d*e^(3/2))) + (b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(5/4)*d*e^(3/2)) + (2*(b - a*Cos[c + d*x]))/((a^2 - b^2)*d*e*Sqrt[e*Sin[c + d*x]]) - (a*b*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - (a*b*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - (2*a*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)*d*e^2*Sqrt[Sin[c + d*x]])","A",13,11,25,0.4400,1,"{2696, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
66,1,447,0,1.0130611,"\int \frac{1}{(a+b \cos (c+d x)) (e \sin (c+d x))^{5/2}} \, dx","Int[1/((a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2)),x]","\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}}+\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}}+\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}}",1,"(b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(7/4)*d*e^(5/2)) + (b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(7/4)*d*e^(5/2)) + (2*(b - a*Cos[c + d*x]))/(3*(a^2 - b^2)*d*e*(e*Sin[c + d*x])^(3/2)) + (2*a*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*(a^2 - b^2)*d*e^2*Sqrt[e*Sin[c + d*x]]) - (a*b^2*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Sin[c + d*x]]) - (a*b^2*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Sin[c + d*x]])","A",13,11,25,0.4400,1,"{2696, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
67,1,501,0,1.3546713,"\int \frac{1}{(a+b \cos (c+d x)) (e \sin (c+d x))^{7/2}} \, dx","Int[1/((a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(7/2)),x]","-\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{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{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)}}+\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{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{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)}}+\frac{2 (b-a \cos (c+d x))}{5 d e \left(a^2-b^2\right) (e \sin (c+d x))^{5/2}}",1,"-((b^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(9/4)*d*e^(7/2))) + (b^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(9/4)*d*e^(7/2)) + (2*(b - a*Cos[c + d*x]))/(5*(a^2 - b^2)*d*e*(e*Sin[c + d*x])^(5/2)) - (2*(5*b^3 + a*(3*a^2 - 8*b^2)*Cos[c + d*x]))/(5*(a^2 - b^2)^2*d*e^3*Sqrt[e*Sin[c + d*x]]) + (a*b^3*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)^2*(b - Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Sin[c + d*x]]) + (a*b^3*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)^2*(b + Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Sin[c + d*x]]) - (2*a*(3*a^2 - 8*b^2)*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*(a^2 - b^2)^2*d*e^4*Sqrt[Sin[c + d*x]])","A",14,12,25,0.4800,1,"{2696, 2866, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
68,1,557,0,1.5566202,"\int \frac{(e \sin (c+d x))^{11/2}}{(a+b \cos (c+d x))^2} \, dx","Int[(e*Sin[c + d*x])^(11/2)/(a + b*Cos[c + d*x])^2,x]","\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{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(-28 a^2 b^2+21 a^4+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 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{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))}","\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{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(-28 a^2 b^2+21 a^4+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 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{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,"(9*a*(-a^2 + b^2)^(5/4)*e^(11/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(11/2)*d) + (9*a*(-a^2 + b^2)^(5/4)*e^(11/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(11/2)*d) - (3*(21*a^4 - 28*a^2*b^2 + 5*b^4)*e^6*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(7*b^6*d*Sqrt[e*Sin[c + d*x]]) + (9*a^2*(a^2 - b^2)^2*e^6*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*b^6*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (9*a^2*(a^2 - b^2)^2*e^6*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*b^6*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (3*e^5*(21*a*(a^2 - b^2) - b*(7*a^2 - 5*b^2)*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(7*b^5*d) - (9*e^3*(7*a - 5*b*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2))/(35*b^3*d) + (e*(e*Sin[c + d*x])^(9/2))/(b*d*(a + b*Cos[c + d*x]))","A",15,12,25,0.4800,1,"{2693, 2865, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
69,1,473,0,1.1572277,"\int \frac{(e \sin (c+d x))^{9/2}}{(a+b \cos (c+d x))^2} \, dx","Int[(e*Sin[c + d*x])^(9/2)/(a + b*Cos[c + d*x])^2,x]","-\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 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 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^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))}","-\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 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 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^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*a*(-a^2 + b^2)^(3/4)*e^(9/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(9/2)*d) + (7*a*(-a^2 + b^2)^(3/4)*e^(9/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(9/2)*d) - (7*a^2*(a^2 - b^2)*e^5*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*b^5*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) - (7*a^2*(a^2 - b^2)*e^5*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*b^5*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (7*(5*a^2 - 3*b^2)*e^4*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*b^4*d*Sqrt[Sin[c + d*x]]) - (7*e^3*(5*a - 3*b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2))/(15*b^3*d) + (e*(e*Sin[c + d*x])^(7/2))/(b*d*(a + b*Cos[c + d*x]))","A",14,12,25,0.4800,1,"{2693, 2865, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
70,1,487,0,1.15077,"\int \frac{(e \sin (c+d x))^{7/2}}{(a+b \cos (c+d x))^2} \, dx","Int[(e*Sin[c + d*x])^(7/2)/(a + b*Cos[c + d*x])^2,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))}","\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,"(5*a*(-a^2 + b^2)^(1/4)*e^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(7/2)*d) + (5*a*(-a^2 + b^2)^(1/4)*e^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(7/2)*d) + (5*(3*a^2 - b^2)*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*b^4*d*Sqrt[e*Sin[c + d*x]]) - (5*a^2*(a^2 - b^2)*e^4*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*b^4*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) - (5*a^2*(a^2 - b^2)*e^4*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*b^4*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) - (5*e^3*(3*a - b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(3*b^3*d) + (e*(e*Sin[c + d*x])^(5/2))/(b*d*(a + b*Cos[c + d*x]))","A",14,12,25,0.4800,1,"{2693, 2865, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
71,1,404,0,0.8424054,"\int \frac{(e \sin (c+d x))^{5/2}}{(a+b \cos (c+d x))^2} \, dx","Int[(e*Sin[c + d*x])^(5/2)/(a + b*Cos[c + d*x])^2,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)}}","-\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,"(-3*a*e^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(5/2)*(-a^2 + b^2)^(1/4)*d) + (3*a*e^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(5/2)*(-a^2 + b^2)^(1/4)*d) + (3*a^2*e^3*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*b^3*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (3*a^2*e^3*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*b^3*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) - (3*e^2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(b^2*d*Sqrt[Sin[c + d*x]]) + (e*(e*Sin[c + d*x])^(3/2))/(b*d*(a + b*Cos[c + d*x]))","A",13,11,25,0.4400,1,"{2693, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
72,1,418,0,0.9068066,"\int \frac{(e \sin (c+d x))^{3/2}}{(a+b \cos (c+d x))^2} \, dx","Int[(e*Sin[c + d*x])^(3/2)/(a + b*Cos[c + d*x])^2,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{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{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)}}","\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{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{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,"(a*e^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(3/2)*(-a^2 + b^2)^(3/4)*d) + (a*e^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(3/2)*(-a^2 + b^2)^(3/4)*d) - (e^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(b^2*d*Sqrt[e*Sin[c + d*x]]) + (a^2*e^2*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*b^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (a^2*e^2*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*b^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (e*Sqrt[e*Sin[c + d*x]])/(b*d*(a + b*Cos[c + d*x]))","A",13,11,25,0.4400,1,"{2693, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
73,1,438,0,0.9052617,"\int \frac{\sqrt{e \sin (c+d x)}}{(a+b \cos (c+d x))^2} \, dx","Int[Sqrt[e*Sin[c + d*x]]/(a + b*Cos[c + d*x])^2,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)}}","\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,"(a*Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*Sqrt[b]*(-a^2 + b^2)^(5/4)*d) - (a*Sqrt[e]*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*Sqrt[b]*(-a^2 + b^2)^(5/4)*d) + (a^2*e*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*b*(a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (a^2*e*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*b*(a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)*d*Sqrt[Sin[c + d*x]]) - (b*(e*Sin[c + d*x])^(3/2))/((a^2 - b^2)*d*e*(a + b*Cos[c + d*x]))","A",13,11,25,0.4400,1,"{2694, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
74,1,445,0,0.9328817,"\int \frac{1}{(a+b \cos (c+d x))^2 \sqrt{e \sin (c+d x)}} \, dx","Int[1/((a + b*Cos[c + d*x])^2*Sqrt[e*Sin[c + d*x]]),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)}}","-\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,"(-3*a*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(7/4)*d*Sqrt[e]) - (3*a*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(7/4)*d*Sqrt[e]) - (EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/((a^2 - b^2)*d*Sqrt[e*Sin[c + d*x]]) + (3*a^2*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*(a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (3*a^2*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*(a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) - (b*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)*d*e*(a + b*Cos[c + d*x]))","A",13,11,25,0.4400,1,"{2694, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
75,1,507,0,1.2711003,"\int \frac{1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx","Int[1/((a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(3/2)),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}}-\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}}-\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)}}",1,"(5*a*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(9/4)*d*e^(3/2)) - (5*a*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(9/4)*d*e^(3/2)) - b/((a^2 - b^2)*d*e*(a + b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]]) + (5*a*b - (2*a^2 + 3*b^2)*Cos[c + d*x])/((a^2 - b^2)^2*d*e*Sqrt[e*Sin[c + d*x]]) - (5*a^2*b*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*(a^2 - b^2)^2*(b - Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - (5*a^2*b*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*(a^2 - b^2)^2*(b + Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - ((2*a^2 + 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^2*d*e^2*Sqrt[Sin[c + d*x]])","A",14,12,25,0.4800,1,"{2694, 2866, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
76,1,530,0,1.3666167,"\int \frac{1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2}} \, dx","Int[1/((a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(5/2)),x]","-\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}}+\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}}+\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}}",1,"(-7*a*b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(11/4)*d*e^(5/2)) - (7*a*b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(11/4)*d*e^(5/2)) - b/((a^2 - b^2)*d*e*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2)) + (7*a*b - (2*a^2 + 5*b^2)*Cos[c + d*x])/(3*(a^2 - b^2)^2*d*e*(e*Sin[c + d*x])^(3/2)) + ((2*a^2 + 5*b^2)*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(3*(a^2 - b^2)^2*d*e^2*Sqrt[e*Sin[c + d*x]]) - (7*a^2*b^2*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*(a^2 - b^2)^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Sin[c + d*x]]) - (7*a^2*b^2*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*(a^2 - b^2)^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Sin[c + d*x]])","A",14,12,25,0.4800,1,"{2694, 2866, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
77,1,590,0,1.6800945,"\int \frac{1}{(a+b \cos (c+d x))^2 (e \sin (c+d x))^{7/2}} \, dx","Int[1/((a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(7/2)),x]","\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{3 \left(\left(-10 a^2 b^2+2 a^4-7 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)}}-\frac{3 \left(-10 a^2 b^2+2 a^4-7 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{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{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{3 \left(\left(-10 a^2 b^2+2 a^4-7 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)}}-\frac{3 \left(-10 a^2 b^2+2 a^4-7 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{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{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}}",1,"(9*a*b^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(13/4)*d*e^(7/2)) - (9*a*b^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(13/4)*d*e^(7/2)) - b/((a^2 - b^2)*d*e*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2)) + (9*a*b - (2*a^2 + 7*b^2)*Cos[c + d*x])/(5*(a^2 - b^2)^2*d*e*(e*Sin[c + d*x])^(5/2)) - (3*(15*a*b^3 + (2*a^4 - 10*a^2*b^2 - 7*b^4)*Cos[c + d*x]))/(5*(a^2 - b^2)^3*d*e^3*Sqrt[e*Sin[c + d*x]]) + (9*a^2*b^3*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*(a^2 - b^2)^3*(b - Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Sin[c + d*x]]) + (9*a^2*b^3*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(2*(a^2 - b^2)^3*(b + Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Sin[c + d*x]]) - (3*(2*a^4 - 10*a^2*b^2 - 7*b^4)*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(5*(a^2 - b^2)^3*d*e^4*Sqrt[Sin[c + d*x]])","A",15,12,25,0.4800,1,"{2694, 2866, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
78,1,590,0,1.4725828,"\int \frac{(e \sin (c+d x))^{13/2}}{(a+b \cos (c+d x))^3} \, dx","Int[(e*Sin[c + d*x])^(13/2)/(a + b*Cos[c + d*x])^3,x]","\frac{11 e^{13/2} \left(-11 a^2 b^2+9 a^4+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(-11 a^2 b^2+9 a^4+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 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 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 a e^7 \left(-11 a^2 b^2+9 a^4+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(-11 a^2 b^2+9 a^4+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}","\frac{11 e^{13/2} \left(-11 a^2 b^2+9 a^4+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(-11 a^2 b^2+9 a^4+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 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 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 a e^7 \left(-11 a^2 b^2+9 a^4+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(-11 a^2 b^2+9 a^4+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*(9*a^4 - 11*a^2*b^2 + 2*b^4)*e^(13/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(13/2)*(-a^2 + b^2)^(1/4)*d) - (11*(9*a^4 - 11*a^2*b^2 + 2*b^4)*e^(13/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(13/2)*(-a^2 + b^2)^(1/4)*d) - (11*a*(9*a^4 - 11*a^2*b^2 + 2*b^4)*e^7*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^7*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) - (11*a*(9*a^4 - 11*a^2*b^2 + 2*b^4)*e^7*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^7*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (11*a*(45*a^2 - 37*b^2)*e^6*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(20*b^6*d*Sqrt[Sin[c + d*x]]) - (11*e^5*(5*(9*a^2 - 2*b^2) - 27*a*b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2))/(60*b^5*d) + (11*e^3*(9*a + 2*b*Cos[c + d*x])*(e*Sin[c + d*x])^(7/2))/(28*b^3*d*(a + b*Cos[c + d*x])) + (e*(e*Sin[c + d*x])^(11/2))/(2*b*d*(a + b*Cos[c + d*x])^2)","A",15,13,25,0.5200,1,"{2693, 2863, 2865, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
79,1,604,0,1.581722,"\int \frac{(e \sin (c+d x))^{11/2}}{(a+b \cos (c+d x))^3} \, dx","Int[(e*Sin[c + d*x])^(11/2)/(a + b*Cos[c + d*x])^3,x]","-\frac{9 e^{11/2} \left(-9 a^2 b^2+7 a^4+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(-9 a^2 b^2+7 a^4+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{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{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{9 a e^6 \left(-9 a^2 b^2+7 a^4+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(-9 a^2 b^2+7 a^4+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}","-\frac{9 e^{11/2} \left(-9 a^2 b^2+7 a^4+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(-9 a^2 b^2+7 a^4+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{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{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{9 a e^6 \left(-9 a^2 b^2+7 a^4+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(-9 a^2 b^2+7 a^4+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,"(-9*(7*a^4 - 9*a^2*b^2 + 2*b^4)*e^(11/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(11/2)*(-a^2 + b^2)^(3/4)*d) - (9*(7*a^4 - 9*a^2*b^2 + 2*b^4)*e^(11/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(11/2)*(-a^2 + b^2)^(3/4)*d) + (3*a*(21*a^2 - 13*b^2)*e^6*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(4*b^6*d*Sqrt[e*Sin[c + d*x]]) - (9*a*(7*a^4 - 9*a^2*b^2 + 2*b^4)*e^6*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^6*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) - (9*a*(7*a^4 - 9*a^2*b^2 + 2*b^4)*e^6*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^6*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) - (3*e^5*(3*(7*a^2 - 2*b^2) - 7*a*b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(4*b^5*d) + (9*e^3*(7*a + 2*b*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2))/(20*b^3*d*(a + b*Cos[c + d*x])) + (e*(e*Sin[c + d*x])^(9/2))/(2*b*d*(a + b*Cos[c + d*x])^2)","A",15,13,25,0.5200,1,"{2693, 2863, 2865, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
80,1,498,0,1.1148215,"\int \frac{(e \sin (c+d x))^{9/2}}{(a+b \cos (c+d x))^3} \, dx","Int[(e*Sin[c + d*x])^(9/2)/(a + b*Cos[c + d*x])^3,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{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{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{e (e \sin (c+d x))^{7/2}}{2 b d (a+b \cos (c+d x))^2}","-\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{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{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{e (e \sin (c+d x))^{7/2}}{2 b d (a+b \cos (c+d x))^2}",1,"(-7*(5*a^2 - 2*b^2)*e^(9/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(9/2)*(-a^2 + b^2)^(1/4)*d) + (7*(5*a^2 - 2*b^2)*e^(9/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(9/2)*(-a^2 + b^2)^(1/4)*d) + (7*a*(5*a^2 - 2*b^2)*e^5*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^5*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (7*a*(5*a^2 - 2*b^2)*e^5*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^5*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) - (35*a*e^4*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(4*b^4*d*Sqrt[Sin[c + d*x]]) + (7*e^3*(5*a + 2*b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2))/(12*b^3*d*(a + b*Cos[c + d*x])) + (e*(e*Sin[c + d*x])^(7/2))/(2*b*d*(a + b*Cos[c + d*x])^2)","A",14,12,25,0.4800,1,"{2693, 2863, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
81,1,512,0,1.137759,"\int \frac{(e \sin (c+d x))^{7/2}}{(a+b \cos (c+d x))^3} \, dx","Int[(e*Sin[c + d*x])^(7/2)/(a + b*Cos[c + d*x])^3,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{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{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{e (e \sin (c+d x))^{5/2}}{2 b d (a+b \cos (c+d x))^2}","\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{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{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{e (e \sin (c+d x))^{5/2}}{2 b d (a+b \cos (c+d x))^2}",1,"(5*(3*a^2 - 2*b^2)*e^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(7/2)*(-a^2 + b^2)^(3/4)*d) + (5*(3*a^2 - 2*b^2)*e^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(7/2)*(-a^2 + b^2)^(3/4)*d) - (15*a*e^4*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(4*b^4*d*Sqrt[e*Sin[c + d*x]]) + (5*a*(3*a^2 - 2*b^2)*e^4*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^4*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (5*a*(3*a^2 - 2*b^2)*e^4*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^4*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (5*e^3*(3*a + 2*b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])/(4*b^3*d*(a + b*Cos[c + d*x])) + (e*(e*Sin[c + d*x])^(5/2))/(2*b*d*(a + b*Cos[c + d*x])^2)","A",14,12,25,0.4800,1,"{2693, 2863, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
82,1,520,0,1.1196506,"\int \frac{(e \sin (c+d x))^{5/2}}{(a+b \cos (c+d x))^3} \, dx","Int[(e*Sin[c + d*x])^(5/2)/(a + b*Cos[c + d*x])^3,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^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^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{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{e (e \sin (c+d x))^{3/2}}{2 b d (a+b \cos (c+d x))^2}","-\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^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^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{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{e (e \sin (c+d x))^{3/2}}{2 b d (a+b \cos (c+d x))^2}",1,"(-3*(a^2 - 2*b^2)*e^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(5/2)*(-a^2 + b^2)^(5/4)*d) + (3*(a^2 - 2*b^2)*e^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(5/2)*(-a^2 + b^2)^(5/4)*d) - (3*a*(a^2 - 2*b^2)*e^3*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^3*(a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) - (3*a*(a^2 - 2*b^2)*e^3*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^3*(a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (3*a*e^2*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(4*b^2*(a^2 - b^2)*d*Sqrt[Sin[c + d*x]]) + (e*(e*Sin[c + d*x])^(3/2))/(2*b*d*(a + b*Cos[c + d*x])^2) - (3*a*e*(e*Sin[c + d*x])^(3/2))/(4*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",14,12,25,0.4800,1,"{2693, 2864, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
83,1,534,0,1.2101887,"\int \frac{(e \sin (c+d x))^{3/2}}{(a+b \cos (c+d x))^3} \, dx","Int[(e*Sin[c + d*x])^(3/2)/(a + b*Cos[c + d*x])^3,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{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 \sqrt{e \sin (c+d x)}}{2 b d (a+b \cos (c+d x))^2}","-\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{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 \sqrt{e \sin (c+d x)}}{2 b d (a+b \cos (c+d x))^2}",1,"-((a^2 + 2*b^2)*e^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(3/2)*(-a^2 + b^2)^(7/4)*d) - ((a^2 + 2*b^2)*e^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(3/2)*(-a^2 + b^2)^(7/4)*d) - (a*e^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(4*b^2*(a^2 - b^2)*d*Sqrt[e*Sin[c + d*x]]) + (a*(a^2 + 2*b^2)*e^2*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^2*(a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (a*(a^2 + 2*b^2)*e^2*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b^2*(a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (e*Sqrt[e*Sin[c + d*x]])/(2*b*d*(a + b*Cos[c + d*x])^2) - (a*e*Sqrt[e*Sin[c + d*x]])/(4*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x]))","A",14,12,25,0.4800,1,"{2693, 2864, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
84,1,529,0,1.2116476,"\int \frac{\sqrt{e \sin (c+d x)}}{(a+b \cos (c+d x))^3} \, dx","Int[Sqrt[e*Sin[c + d*x]]/(a + b*Cos[c + d*x])^3,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)}}","-\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,"-((3*a^2 + 2*b^2)*Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*Sqrt[b]*(-a^2 + b^2)^(9/4)*d) + ((3*a^2 + 2*b^2)*Sqrt[e]*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*Sqrt[b]*(-a^2 + b^2)^(9/4)*d) + (a*(3*a^2 + 2*b^2)*e*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b*(a^2 - b^2)^2*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (a*(3*a^2 + 2*b^2)*e*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*b*(a^2 - b^2)^2*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Sin[c + d*x]]) + (5*a*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(4*(a^2 - b^2)^2*d*Sqrt[Sin[c + d*x]]) - (b*(e*Sin[c + d*x])^(3/2))/(2*(a^2 - b^2)*d*e*(a + b*Cos[c + d*x])^2) - (5*a*b*(e*Sin[c + d*x])^(3/2))/(4*(a^2 - b^2)^2*d*e*(a + b*Cos[c + d*x]))","A",14,12,25,0.4800,1,"{2694, 2864, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
85,1,535,0,1.2289434,"\int \frac{1}{(a+b \cos (c+d x))^3 \sqrt{e \sin (c+d x)}} \, dx","Int[1/((a + b*Cos[c + d*x])^3*Sqrt[e*Sin[c + d*x]]),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)}}","\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,"(3*Sqrt[b]*(5*a^2 + 2*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(11/4)*d*Sqrt[e]) + (3*Sqrt[b]*(5*a^2 + 2*b^2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(11/4)*d*Sqrt[e]) - (7*a*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(4*(a^2 - b^2)^2*d*Sqrt[e*Sin[c + d*x]]) + (3*a*(5*a^2 + 2*b^2)*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*(a^2 - b^2)^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) + (3*a*(5*a^2 + 2*b^2)*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*(a^2 - b^2)^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Sin[c + d*x]]) - (b*Sqrt[e*Sin[c + d*x]])/(2*(a^2 - b^2)*d*e*(a + b*Cos[c + d*x])^2) - (7*a*b*Sqrt[e*Sin[c + d*x]])/(4*(a^2 - b^2)^2*d*e*(a + b*Cos[c + d*x]))","A",14,12,25,0.4800,1,"{2694, 2864, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
86,1,611,0,1.6474033,"\int \frac{1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2}} \, dx","Int[1/((a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(3/2)),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}}-\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}}-\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)}}",1,"(-5*b^(3/2)*(7*a^2 + 2*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(13/4)*d*e^(3/2)) + (5*b^(3/2)*(7*a^2 + 2*b^2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(13/4)*d*e^(3/2)) - b/(2*(a^2 - b^2)*d*e*(a + b*Cos[c + d*x])^2*Sqrt[e*Sin[c + d*x]]) - (9*a*b)/(4*(a^2 - b^2)^2*d*e*(a + b*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]]) + (5*b*(7*a^2 + 2*b^2) - a*(8*a^2 + 37*b^2)*Cos[c + d*x])/(4*(a^2 - b^2)^3*d*e*Sqrt[e*Sin[c + d*x]]) - (5*a*b*(7*a^2 + 2*b^2)*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*(a^2 - b^2)^3*(b - Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - (5*a*b*(7*a^2 + 2*b^2)*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*(a^2 - b^2)^3*(b + Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Sin[c + d*x]]) - (a*(8*a^2 + 37*b^2)*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(4*(a^2 - b^2)^3*d*e^2*Sqrt[Sin[c + d*x]])","A",15,13,25,0.5200,1,"{2694, 2864, 2866, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
87,1,629,0,1.7749249,"\int \frac{1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{5/2}} \, dx","Int[1/((a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(5/2)),x]","\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}}+\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}}+\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}}",1,"(7*b^(5/2)*(9*a^2 + 2*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(15/4)*d*e^(5/2)) + (7*b^(5/2)*(9*a^2 + 2*b^2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(15/4)*d*e^(5/2)) - b/(2*(a^2 - b^2)*d*e*(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(3/2)) - (11*a*b)/(4*(a^2 - b^2)^2*d*e*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(3/2)) + (7*b*(9*a^2 + 2*b^2) - a*(8*a^2 + 69*b^2)*Cos[c + d*x])/(12*(a^2 - b^2)^3*d*e*(e*Sin[c + d*x])^(3/2)) + (a*(8*a^2 + 69*b^2)*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(12*(a^2 - b^2)^3*d*e^2*Sqrt[e*Sin[c + d*x]]) - (7*a*b^2*(9*a^2 + 2*b^2)*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*(a^2 - b^2)^3*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Sin[c + d*x]]) - (7*a*b^2*(9*a^2 + 2*b^2)*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*(a^2 - b^2)^3*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Sin[c + d*x]])","A",15,13,25,0.5200,1,"{2694, 2864, 2866, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
88,1,700,0,2.1182242,"\int \frac{1}{(a+b \cos (c+d x))^3 (e \sin (c+d x))^{7/2}} \, dx","Int[1/((a + b*Cos[c + d*x])^3*(e*Sin[c + d*x])^(7/2)),x]","-\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{3 \left(a \left(-64 a^2 b^2+8 a^4-139 b^4\right) \cos (c+d x)+15 b^3 \left(11 a^2+2 b^2\right)\right)}{20 d e^3 \left(a^2-b^2\right)^4 \sqrt{e \sin (c+d x)}}-\frac{3 a \left(-64 a^2 b^2+8 a^4-139 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{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{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{3 \left(a \left(-64 a^2 b^2+8 a^4-139 b^4\right) \cos (c+d x)+15 b^3 \left(11 a^2+2 b^2\right)\right)}{20 d e^3 \left(a^2-b^2\right)^4 \sqrt{e \sin (c+d x)}}-\frac{3 a \left(-64 a^2 b^2+8 a^4-139 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{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{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}}",1,"(-9*b^(7/2)*(11*a^2 + 2*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(17/4)*d*e^(7/2)) + (9*b^(7/2)*(11*a^2 + 2*b^2)*ArcTanh[(Sqrt[b]*Sqrt[e*Sin[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(17/4)*d*e^(7/2)) - b/(2*(a^2 - b^2)*d*e*(a + b*Cos[c + d*x])^2*(e*Sin[c + d*x])^(5/2)) - (13*a*b)/(4*(a^2 - b^2)^2*d*e*(a + b*Cos[c + d*x])*(e*Sin[c + d*x])^(5/2)) + (9*b*(11*a^2 + 2*b^2) - a*(8*a^2 + 109*b^2)*Cos[c + d*x])/(20*(a^2 - b^2)^3*d*e*(e*Sin[c + d*x])^(5/2)) - (3*(15*b^3*(11*a^2 + 2*b^2) + a*(8*a^4 - 64*a^2*b^2 - 139*b^4)*Cos[c + d*x]))/(20*(a^2 - b^2)^4*d*e^3*Sqrt[e*Sin[c + d*x]]) + (9*a*b^3*(11*a^2 + 2*b^2)*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*(a^2 - b^2)^4*(b - Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Sin[c + d*x]]) + (9*a*b^3*(11*a^2 + 2*b^2)*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(8*(a^2 - b^2)^4*(b + Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Sin[c + d*x]]) - (3*a*(8*a^4 - 64*a^2*b^2 - 139*b^4)*EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(20*(a^2 - b^2)^4*d*e^4*Sqrt[Sin[c + d*x]])","A",16,13,25,0.5200,1,"{2694, 2864, 2866, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"